ADAPTIVE IMAGE COMPRESSION WITH WAVELET PACKETS …

LINKÖPING STUDIES IN SCIENCE AND TECHNOLOGYDISSERTATION NO. 909

ADAPTIVE IMAGE COMPRESSIONWITH WAVELET PACKETS

AND EMPIRICAL MODEDECOMPOSITION

ANNA LINDERHED

DEPARTMENT OF ELECTRICAL ENGINEERING

LINKÖPING UNIVERSITY, SE - 581 83 LINKÖPING, SWEDEN

LINKÖPING 2004

Adaptive Image Compression with Wavelet Packets and Empirical Mode Decomposition

© 2004 Anna Linderhed

[email protected]

Deptartment of Electrical Engineering

Linköping University, SE-581 83 Linköping, Sweden

ISBN 91-85295-81-7 ISSN 0345-7524

Printed in Sweden by LTAB, Linköping 2004, no. 844.

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To My HusbandHåkan

and Our ChildrenUlrika, Karl, and Matilda

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AbstractThis thesis addresses the problem of using wavelet packets and empirical mode decom-position (EMD) for image compression. The wavelet packet basis selection algorithm isstudied through an extensive experimental survey of the generated decomposition trees.We formulate the "triplet problem" for image compression as follows: How is the decom-position tree related to the image content, filter set and cost function? Our aim is to findan optimal basis for compression of images. Results are presented using test images fromthe Brodatz texture set. We also present a method to analytically calculate the cost of split-ting a node, for a given signal model and filter, without actually performing the split.

A totally different approach to signal decomposition is the EMD. This is an adaptivedecomposition scheme with which any complicated signal is decomposed into its intrinsicmode functions (IMF). The concept of EMD is extended to two dimensions to make it use-ful for image processing. The EMD and the sifting process to generate the IMFs are de-scribed. Different known and newly found difficulties with implementation of the methodin two dimensions are highlighted and solutions are proposed. The method of variablesampling of the EMD, using overlapping blocks, is presented and the concept of em-piquency is introduced to describe spatial frequency since the traditional Fourier-basedfrequency concept is not applicable.

Several ways to use EMD for image compression are examined and presented. Thetwo-dimensional extension of the EMD is original as well as its application for imagecompression.

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PrefaceI started my stay in the Image Coding Group with my master thesis work on three-dimen-sional sound algorithms in an image coding project. This introduced me to the compres-sion side of the image processing problem. Before that I only knew about traditionalimage processing such as image analysis and restoration.

I knew that if I could find a very good representation of the image, the compressionwould be easy. This is the very core of image compression but I was not satisfied with theway the algorithms at the time treated all images the same way. I wanted an adaptive rep-resentation.

In 1993 I attended a lecture by Professor Coifman and fell in love with the idea ofwavelet packets and best basis. This was brand new, not only to me. The following yearsI spent with my babies, learning about image compression and wavelet theory, teachingsignal theory at the university and struggling with my main question that no one couldgive me the answer to: Which one of all these possible best bases is the best for compres-sion purposes? And why? Are wavelet packets the best way to represent the image forcompression? During this time several good wavelet image compression algorithms weredeveloped by others but the wavelet packets seemed to be of no good use.

By 2000 I nearly gave up my work towards a PhD thesis and left the Image CodingGroup for a position at the Swedish Defence Research Agency.

Then it happened again! I fell in love with an idea.

In April 2001 Dr. Norden E. Huang presented his idea of Empirical Mode Decompo-sition at a wavelet conference. This was what I had been trying to do with wavelet packets.A totally signal adaptive decomposition. I had to expand the method to two dimensionsfor the use on images and then find out how to use it for compression, if possible.

So what motivated the work in this thesis?

Love..........and Curiosity.......

Anna LinderhedSeptember 2004

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AcknowledgementsThe wavelet part of this thesis was funded by the Image Coding Group at Linköping Uni-versity while the EMD part of this thesis was conducted during my spare time, funded bymy husband, and also partly funded by the Swedish Defence Research Agency, FOI.

Many thanks to my supervisor Dr. Robert Forchheimer for letting me do this my way andfor many inspiring discussions. Thanks also to all my friends and previous colleagues atthe Image Coding and Information Theory groups at the Dept. of Electrical Engineeringat Linköping University, for good company and support. To all my friends and colleaguesat the Swedish Defence Research Agency, thank you for believing in me and pushing meto finish this thesis. Special thanks to Lena Klasén and Stefan Sjökvist for the company atlate hours and weekends doing our thesis writing together. Thank you also Dr. RobertForchheimer, Dr. Lena Klasén, and Håkan Linderhed for careful proof reading of the man-uscript. And also thanks to the anonymous reviewers of my papers for their valuable com-ments.

Most of all I thank my family, my dear husband Håkan and my children Ulrika, Karl andMatilda, for still loving me. Mummy’s book is finally finished!!!!

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I aningen finns sanningen

Björn Rosendal

Contents1. Introduction 1

1.1 Properties of images 21.1.1 Spatial frequency 21.1.2 Texture 31.1.3 Resonance 41.1.4 Stationarity 51.1.5 Spectral properties 5

1.2 The coding chain 61.2.1 Transform 71.2.2 Quantization and bit allocation 81.2.3 Entropy coding 9

1.3 Wavelet packet image coding 10

1.4 Empirical mode decomposition 10

1.5 Overview of the thesis and summary of publications 11

1.6 Scientific contributions 13

1.7 Possible applications 13

PART ONE: WAVELET PACKET IMAGE COMPRESSION

2. Wavelet and Wavelet Packet Methods for Image Compression 17

2.1 Introduction 172.1.1 Wavelet transform 172.1.2 Wavelet packets 19

2.2 Wavelet image coding algorithms 222.2.1 Wavelet coding with vector quantization 222.2.2 Embedded zerotree wavelet algorithm (EZW) 232.2.3 Set partitioning in hierarchical trees (SPIHT) 242.2.4 JPEG2000 252.2.5 Morphological representation of wavelet data (MRWD) 26

2.3 Wavelet packet coding algorithms 272.3.1 Best basis subband coding 272.3.2 Rate-distortion optimized wavelet packet coding 282.3.3 Optimal entropy constrained lattice vector quantization (ECLVQ) 282.3.4 Constrained wavelet packet coding (CWP) 292.3.5 WP-IMRWD 332.3.6 Compatible zerotree quantization of wavelet packets (CZQ-WP) 34

2.4 Segmentation-based image coding 352.4.1 Edge separating wavelet image coding 352.4.2 Joint use of segmentation and wavelet packet 35

2.5 Summary 38

3. The Wavelet Packet Triplet Problem 41

3.1 Decomposing the Brodatz textures 42

3.2 Results from the triplet database 423.2.1 Entropy estimations 423.2.2 Cost function effects 473.2.3 Measurements of cost function effects 483.2.4 Filter effects 543.2.5 Measurements of filter effects 563.2.6 Conclusion 58

3.3 Performance of the CWP coder 593.3.1 Conclusion 653.3.2 Note on the choice of filter 65

3.4 Calculated decision rule 663.4.1 System 663.4.2 Applying the cost function 683.4.3 Conclusion 72

3.5 Summary 72

4. Discussion - Wavelet packet coding 75

PART TWO: EMPIRICAL MODE DECOMPOSITION

5. Empirical Mode Decomposition 81

5.1 EMD for time signals 825.1.1 Time-frequency analysis with Hilbert-Huang Transform (HHT) 825.1.2 Sifting process for finding the IMF 855.1.3 Interpolation methods 925.1.4 Empiquency 985.1.5 EMD of noise 995.1.6 Conclusion 102

5.2 EMD of two-dimensional signals 1025.2.1 Sifting for the two-dimensional IMF 1035.2.2 Implementation for image EMD 1045.2.3 Significant extrema points 1065.2.4 Image EMD 1075.2.5 Conclusion 110

5.3 Summary 110

6. Compression of the EMD 111

6.1 Entropy coding of the EMD 1116.1.1 Results 112

6.2 Extrema point coding 1156.2.1 One-dimensional extrema point coding 1156.2.2 Two-dimensional extrema point coding 1196.2.3 Results 1206.2.4 Conclusion 124

6.3 Block-based single-component DCT coding 1246.3.1 Coder outline 1246.3.2 Result 1276.3.3 Two-coefficient block-based DCT 1306.3.4 Conclusion 130

6.4 DCT threshold coding 1306.4.1 Coder outline 1306.4.2 Implementation 1316.4.3 Result 1346.4.4 Conclusion 137

6.5 Summary 137

7. Variable Sampling of EMD 139

7.1 Variable sampling of overlapping blocks 1397.1.1 Sampling of overlapping blocks 1407.1.2 Interpolation 141

7.2 Implementation 1427.2.1 The variable sampling block 1427.2.2 The reconstruction block 142

7.3 Results 144

7.4 Summary 152

8. Entropy Coding of Variable Sampling Components 153

8.1 Coder outline 1538.1.1 Results 154

9. EMD Image Coding using Variable Sampling of Overlapping Blocks 163

9.1 Coding of the EMD using DCT of the variable sampled blocks (VSDCTEMD) 1639.1.1 Coder outline 1639.1.2 Results 165

9.2 Image coding using VSDCTEMD and DCT threshold coding 1739.2.1 Coder outline 1739.2.2 Results 173

9.3 Summary 178

10. Empiquency-Controlled Image Coding 179

10.1 Empiquency-controlled variable sampling of the original image 17910.1.1 Results 181

10.2 Empiquency-controlled entropy coded variable sampling (Entropy coded VS) 18310.2.1 Results 184

10.3 Empiquency-controlled image coding using variable sampling and DCT coding (VSDCT) 18710.3.1 Results 188

10.4 Summary 192

11. Summary and Discussion 195

11.1 EMD 195

11.2 Empiquency 197

11.3 Variable sampling 198

11.4 EMD image coding 199

11.5 Open problems 200

Appendix A 201

Appendix B 203

Appendix C 215

References 219

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Chapter 1

Introduction

This chapter gives a short introduction to the underlying theories that form the basis forthe forthcoming chapters. It also includes the problem formulation and the outline of thework discussed in the thesis. The scientific contribution is summarized in the last sectionwhere possible applications are also discussed.

The basis for this research is the problem of electronic communication of imageswhich includes storing of images and transmission over the Internet, telephone networkor television channel. The framework is the communication model by Shannon [77] asviewed in Figure 1.1. This model shows how a message is communicated over a disturbedchannel. The message is coded in the transmitter, the source coding and the channel cod-ing can be treated separately following [77]. The source coding block can be further de-composed into its subblocks according to the scheme in Figure 1.10. In the first part ofthis thesis the work is concentrated on the problem of efficient representation of the imagein order to make it easy for the source compression algorithms, while in the second partof this thesis the search for an efficient representation is complemented with differentcompression methods. We also restrict ourselves to work only with gray scale images.

Figure 1.1. The Shannon communication model.

Source

Transmitter

Channel Destination

NoiseSource

SourceCoder Coder

ChannelDecoder Decoder

Receiver{ {Source

Chapter 1 Introduction

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1.1 Properties of imagesTo be able to find an efficient representation of an image we have to understand the prop-erties of images. An image can often be modelled as areas of texture, single or combined,and separated by edges. Some theories treat the set of image signals as a random processwhere the image is one realization of the random process, while others look upon the im-age as a deterministic signal. In this section we look at different ways to describe an im-age and its properties. Throughout this thesis we use the word image to denote the digitalpicture. The image is regarded as a square integer array in which each element is denoteda pixel. We ignore the digitizing process and assume all images we use are representedwith 8 bits per pixel before our treatment starts. Pixel values thus range from 0 (black) to255 (white).

1.1.1 Spatial frequencySpatial frequency is a property that is often used for estimation of size and scale in images[30]. The images in Figure 1.2 show patterns of different spatial frequency along the hor-izontal direction.

Figure 1.2. Patterns with different spatial frequency, a) low, b) medium, and c) high.

In a two-dimensional signal like an image the spatial frequency can have different di-rections. A spatial frequency component is traditionally described by

(1.1)

which is a spatial pattern in which u and v are the spatial frequencies along the x- and y-axes [74]. The pattern has a spatial period of along a direction that has anangle with the x-axis.

Example 1.1. We analyse the spatial frequency in the horizontal direction of the imagein Figure 1.3 by visual inspection of one row of the image.

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ej2π ux vy+( )

u2 v2+( ) 1 2⁄–

u v⁄( )atan

1.1 Properties of images

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Figure 1.3. Lenna 512x512 with row 450 marked in black.

Figure 1.4. Row 450 of Lenna 512X512.

The fast varying values in Figure 1.4, part a, indicate high spatial frequency in the featherand the hair while the trend of slowly rising values in part b indicates the low spatial fre-quency in the shoulder. Edges are indicated by a dramatic change of value as in c, d, e andf in Figure 1.4.

1.1.2 Texture Textures are homogeneous patterns or spatial arrangements of pixels that regional inten-sity or colour alone does not sufficiently describe. They are composed of a large numberof more or less ordered similar patterns, giving rise to a perception of homogeneity [30].The three principal approaches to describe the texture of a region used in image process-ing are statistical properties, structural properties, and spectral properties [29]. A texturemay consist of the structured and/or random placement of elements, but also may be with-out fundamental sub-units. The properties that are important in the perception of visual

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texture are complex visual patterns composed of subpatterns that have characteristicbrightness, slopes, sizes, etc. The local subpattern properties give rise to the perceivedlightness, directionality, coarseness, etc., of the texture as a whole [74]. In this thesis apure texture image is characterized in that the whole image contains the same structureor a repeated pattern that the human eye/brain system interprets as a homogenous area ofthe same pattern all over the image.

1.1.3 ResonanceBy spatial resonance we will mean the dominating repeated pattern, a periodic texture,in the image. An example of a resonance image is the brickwall texture. The dominantrepeating structure is easily seen from the plot in Figure 1.6 of one row of the image inFigure 1.5.

Figure 1.5. The brickwall texture.

Figure 1.6. One row of the brickwall texture.

In Figure 1.4 we see one resonance in part a, the feather, and we see another resonancein part b, the shoulder.

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1.1.4 StationarityLet X be the collection of all possible images. Following Rosenfeld & Kak [74], such atwo-dimensional entity should be properly denoted random field corresponding to theone-dimensional random process. However, most of the time we will treat the images asone-dimensional vectors after a line scan in one of the directions. Alternatively we applyour algorithms in one direction at a time in a separable manner [30]. Thereforee we willuse the notation image random process in this thesis. Any ensemble mean taken over thewhole ensemble for one part of the image can be expected to be the same as the ensemblemean taken over the whole ensemble for another part of the image when the set includesall types of images. Also the autocorrelation function is not dependent on absolute loca-tion in the image thus and the process is wide sense stationary [22]. As a counter exampleof a set of images that is non-stationary in this sense we consider the set of face imagesused in video phone applications. Here all the images have a background and a face in thecentre of the picture, a nose in the middle of the face, and shoulders and neck attached tothe face. But in general we consider the images to be a weak sense stationary process andwe can also talk about the spectral representation of our process.

We will talk about deterministic non-stationarity when there are different resonancesin different parts of the image. In this sense a stationary image would be a pure textureimage, such as the images treated in Chapter 3.

1.1.5 Spectral propertiesAnother way of describing the spatial frequency content of the image is to use the Fouriertransform [9]. This is essentially the same as decomposing the image into its sinusoidalcomponents. The frequency content of the process is contained in the spectral density[22] of the signal, also called power spectrum [74,64]. This can be calculated by takingthe Fourier transform of the autocorrelation function [22] which describes the dependen-cies between pixels. The power spectrum can also be estimated by the periodogram [64]of the input signal.

When transforming the entire image we get no information on the location of the fre-quencies, and there is rarely a single clear resonance. When we transform only a smallhomogenous texture region of the image we get another result. This principle is used inthe JPEG coding [65] of an image where the transform is taken over small parts of theimage with the expectation that each part can be described by only a few frequency com-ponents.

Example 1.2. Compare the periodogram of the Lenna image in Figure 1.7 with the per-iodogram of the feather in her hat in Figure 1.8. The periodogram is taken over all the hor-izontal lines in the image together. The overall image is dominated by the low frequencycomponents in the image while the high frequency components stands out when analys-ing only a high frequency part of the image.


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Figure 1.7. Periodogram of Lenna.

Figure 1.8. A small part of the feather in Lenna’s hat and the periodogram of the same.

Example 1.3. The frequency content of the same part of the image can be different indifferent directions. Taking the periodogram of the brickwall texture in both the horizon-tal direction and the vertical direction shows the dominating low spatial frequency struc-ture in one direction and the high spatial frequency resonance in the other direction, seeFigure 1.9.

Figure 1.9. Horizontal and vertical periodogram of the brickwall texture.

1.2 The coding chainSource coding involves not only the transform itself but also quantization, followed by

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entropy coding. This is shown in Figure 1.10. If the signals to be encoded are realizationsof a Gaussian process the design is well known. The basis set of optimal linear transformsfor this signal is the Karhunen-Loeve (K-L) basis [40].

Figure 1.10. The coding chain.

1.2.1 TransformThe fast Fourier transform (FFT) algorithm [9] makes it possible to manipulate the imagein the frequency plane by transforming the image. The decorrelation property of a trans-form is very important for compression efficiency. Compression is achieved when quan-tization is applied to the transform components.

The transform can be seen as a way to change the statistics of the source, a change ofbasis is a change of point of view. The transform of a vector is represented by a vectormultiplication by a transform matrix .

(1.2)

After manipulations of , such as quantization, the reconstructed vector is achieved by

(1.3)

if the inverse exists.A good decorrelating transform will remove linear dependencies from the data, thus

producing a set of components such that, when individually quantized and entropy coded,the resulting symbol stream is reduced substantially, compared to applying the quantiza-tion directly on the image data.

The short time Fourier transform is the most widely used method for studying timevarying signals. It is well understood and for many signals and situations it gives a goodtime-frequency structure, however, for certain situations it is not the best method.

The discrete cosine transform (DCT) established itself before the wavelet revolutionand is one of the building blocks of the JPEG still image coding standard [65]. This trans-form is very close to the Karhunen-Loeve Transform (KLT) which produces uncorrelatedtransform coefficients of a Gaussian source. In the JPEG standard the two dimensionalDCT uses a set of 64 8x8 tap basis functions shown in Figure 1.11. The image is dividedinto 8x8 pixel sized blocks and the transform components for each block found by corre-lation with the basis functions. This representation of images relates to the short time Fou-rier transform in that it analyses the image in fixed sized windows. Because the image issegmented into 8x8 pixel blocks, a basis function of a given spatial frequency can only

transform

imagequantization entropy

coding

bit stream

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represent partial phase of a spatial oscillation within the block. The segmentation alsogives rise to visible discontinuities between adjacent blocks, this is called the “blockingartifacts”. For some applications in this thesis we use sizes other than the usual 8x8 block.

Figure 1.11. The set of 64 DCT basis functions used in the JPEG standard, from [65].

1.2.2 Quantization and bit allocationThe image transform used in a coder is usually invertible. That is, the image can be re-constructed from its transform coefficients. The quantization and thresholding stagehowever introduces errors, but also reduces the bitrate needed to represent the image. Inthis thesis uniform quantization is used except for some sections where SPIHT (seesection 2.2.3) is used.

We use the mean square error (MSE)

(1.4)

as distortion measure. The signal-to-noise ratio for any signal is

(1.5)

For images represented with 8 bits per pixel (bpp) we use the peak-signal-to-noise ra-tio measure (PSNR).

MSE 1N---- xi xi–( )

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i 1=

N

∑=

SNRdB 10 σ2 m2+MSE-------------------

log=

1.2 The coding chain

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(1.6)

1.2.3 Entropy codingIn most of the wavelet work in this thesis the bitrate is estimated by the entropy of thequantized components. However, in other parts of the thesis we will perform “real” cod-ing using one of the following methods.

Huffman coding The Huffman coder [18] assigns codewords to each component in relation to the proba-bility of the value it holds. More probable values gets shorter codewords while rare valuesget long codewords. This way the components can be represented more efficiently.

Runlength codingAfter quantization of the transform components many of them have the value zero. Thelength of zero runs can be coded separately. The simplest way to code the runlengths isto send a sequence of zeroes followed by a 1 (unary coding). 1 is coded as 01, 2 is codedas 001, 9 is coded as 0000000001 etc. Sorting the runlengths and coding the most frequentones with the shortest codewords can give some improvement.

Elias code Using Elias code [25] gives us an efficient way to assign codewords to the runlength val-ues x. This code represents the integer x with its binary representation and a prefix tellingthe length of the binary representation of x. There are different ways to generate the pre-fix.

The simplest one is to use unary coding of the length of the binary representation giv-ing codewords of length .

Using Elias code to code the length of the binary representation as well gives a codethat maps the integer x to a codeword consisting of bits followed by the bi-nary value of x with the leading 1 deleted. The resulting codeword has length

. This gives shorter codewords for large values ofx.

Coding of amplitudeFor the coding of amplitude levels we will commonly use fixlength code of varying sizes.

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1.3 Wavelet packet image codingImage properties, filter and cost function make up a closely related triplet in selecting awavelet packet basis. Changing any of these three affects the compression performance.Analysing the triplet problem is to understand why and how the wavelet packet basis isaffected by the choice of attributes image, decomposition filter and cost function. In thisthesis we analyse the triplet problem in a compression application. We use the expressionbest basis for the basis found by the best basis search algorithm of Coifman & Wicker-hauser [17]. This algorithm uses a bottom up search strategy and is stated to find a globaloptimal basis. The expression wavelet packet basis is used for any non-redundant basisextracted from the full wavelet packet tree. In this thesis we are concerned mainly withthe compression ratio achieved by the wavelet packet transform. We use texture imagesin this work because we are specifically interested in efficient representation of the reso-nance in the image.

Previous work used the fixed wavelet basis while discussing suitable filters, or thewavelet packet basis with discussion of cost functions but mostly without considering fil-ters and image properties.

A triplet database consisting of coding results from 2295 different triplets has beencreated using the best basis algorithm. This data is analysed and discussed with respectto different images, cost functions, and filters.

An important question is whether, and when, a wavelet packet basis is better than thefixed wavelet basis. We will attempt to answer this question by combining the best filterand cost function with a flexible wavelet packet coder described in Chapter 2.

We present a method to analytically calculate the expected wavelet packet basis, witha top-down approach, for certain triplets using signal models to represent the image.

1.4 Empirical mode decompositionA totally different approach to signal decomposition was first presented by Huang et al.[36]. Empirical mode decomposition (EMD) is an adaptive decomposition with whichany complicated signal can be decomposed into its intrinsic mode functions (IMF). TheFourier spectral analysis as well as wavelet methods require the system to be linear andthe data must be strictly periodic or stationary. EMD is an analysis method that in manyaspects gives a better understanding of the physics behind the signals. Because of its abil-ity to describe short time changes in frequencies that can not be resolved by Fourier spec-tral analysis it can be used for non-linear and non-stationary time series analysis.

In the second part of the thesis we will investigate how to apply EMD to images andimage coding. Previous work often mentions the lack of a mathematical formalism to de-scribe the EMD; the concept is truly empirical. In this work we keep the empirical ap-proach as we extend the method to two-dimensional signals, such as images. We presentan EMD method that can decompose the image into a number of IMFs and a residue withno, or only a few extrema points. This method makes it possible to use the EMD for im-age processing.

Next we examine the use of EMD for image compression purposes. The set of IMFsand the residue image is a very redundant way to represent an image. Our first goal is torepresent the image by its EMD with the same number of samples as the original image.

1.5 Overview of the thesis and summary of publications

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Our second goal is to find an EMD-based image coder that can achieve reasonable com-pression. The concept of empiquency, short for empirical mode frequency, is introducedto describe the signal oscillations, since the traditional frequency concept is not applica-ble in this work.

1.5 Overview of the thesis and summary of publicationsThis thesis consists of two main parts. The first part, Chapters 2 to 4, deals with waveletpacket coding of images, an area where much research has already been done. However,in this part we focus specifically on how the choice of filters and cost functions influencesthe basis selection in relation to the images. The second part, Chapters 5 to 11, deals withthe idea of using empirical mode decomposition for signal compression. This is new, asis applying EMD to images. Parts of the thesis work are included in the following reports:

• J. Karlholm, M. Ulvklo, S. Nyberg, A. Lauberts, A. Linderhed, “A survey of meth-ods for detection of extended ground targets in EO/IR imagery”. FOI 2003, 144pp. Scientific report FOI-R--0892--SE, Linköping

in which the author of this thesis contributed with a discussion on image compression ofsensor data.

• J. Karlholm, M. Ulvklo, J. Nygårds, M. Karlsson, S. Nyberg, M. Bengtsson, L.Klasén, A. Linderhed, M. Elmqvist, “Target detection and tracking processingchain: a survey of methods with special reference to EO/IR sequences”. FOA2000, User report FOA-R--00-01767-408,616--SE, Linköping

in which the author of this thesis contributed with the wavelet packet representation oftexture.

Chapter 2 deals with image coding, wavelet and wavelet packet methods, and presentsa survey of such image coding methods. The focus is on image compression methods thathave had major impact on the development of wavelet and wavelet packet image com-pression.

In Chapter 3 the wavelet packet triplet problem is analysed to understand why andhow the wavelet packet basis is affected by the image, decomposition filter and cost func-tion.

This question arose early when different best bases occurred for the same image whenusing different filters or cost functions. The problem is analysed by the author in:

• A. Linderhed, “Cost function and filter influence on wavelet packet decomposi-tions of texture images”. Proceedings of SSAB symposium on image analysis,SSAB’00, Halmstad, 7-8 March 2000, pp.5-8.

The exhaustive search of the combination of 17 images, 9 cost functions and 15 differentwavelet filters was presented in

• A. Linderhed, “Wavelet packet decompositions of texture images- Cost functionand filter influence” Proceedings of the Internal Workshop for Information Theoryand Image Coding 2001, Report LiTH-ISY-R-2345, Linköping University, 2001,pp. 49-73.


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In this thesis we draw some conclusions from this work that explain the different bas-es and guides to a choice of filter and cost function. With this in mind, one of the waveletpacket coders from Chapter 2 is tested on different images. The test verifies the hypoth-esis that for pure texture images a wavelet packet basis adapted to the actual image isbetter than a general wavelet basis.

The chapter continues with an analytic calculation of the expected wavelet packet ba-sis based on two simple signal models, filters and cost functions. Part of this work waspresented in:

• A. Linderhed, “Wavelet packet decompositions of texture images: analysis of costfunctions, filter influences, and image models”. SPIE Proceedings Vol. 4738,Wavelet and Independent Component Analysis Applications IX, Orlando, FL2002, April 2002, pp.9-20.

In this thesis the work is extended to show that the best basis can be found, for a se-lection of triplets, with a top down approach using calculations based on image statistics.

Chapter 4 presents a discussion of the first part of the thesis. With Chapter 5 the second part of this thesis starts. This chapter focuses on empirical

mode decomposition, and starts with a description of the original work and its purpose aswe understand it. The EMD and the sifting process for the IMF are described. The EMDis extended to two-dimensional signals. This was first presented in:

• A. Linderhed, “2D empirical mode decompositions in the spirit of image compres-sion”, SPIE Proceedings Vol. 4738, Wavelet and Independent Component Analy-sis Applications IX, Orlando, FL 2002, April 2002, pp.1-8.

In this chapter different known and newly found difficulties with implementation ofthe EMD method in two dimensions are highlighted and solutions are proposed. We alsointroduce the concept of empiquency, published in [53], to describe the spatial frequencyin the image since traditional the Fourier-based frequency is not applicable in this work.

In Chapter 6 we present different attempts to compress the EMD. The results fromthis work are only published in this thesis.

The EMD is a redundant representation of an image. To overcome this a method ofvariable sampling of the EMD, using overlapping blocks, is proposed in Chapter 7. Thisis published in:

• A. Linderhed, “Variable Sampling of the Empirical Mode Decomposition of Two-Dimensional Signals”, International Journal of Wavelets, Multiresolution andInformation Processing, special issue on Sampling Problems Related to WaveletTheory and Time-Frequency Analysis, forthcoming.

In Chapters 8 to 10 we present different image coders using this approach. A variantof these is presented in:

• A. Linderhed, “Image compression based on empirical mode decomposition”.Proceedings of SSBA symposium on image analysis, SSBA’04, Halmstad, 11-12March 2004, p. 110-113.

The rest of the image coding methods in this chapter are only published in this thesisso far, although several papers are in preparation.

Chapter 11 presents a discussion of the EMD work and how the result relates to thework of others. This chapter also present some open problems in this area.

1.6 Scientific contributions

13

1.6 Scientific contributions The contributions of this thesis are:• Survey of image coding methods using wavelet and wavelet packet techniques,

• Analysis of the triplet problem,

• Discussion of filters suitable for wavelet packet decomposition from an experi-mental and theoretical view,

• Analytic calculation wavelet packet basis desicion rule,

• Extension of EMD to two dimensions,

• Concept of empiquency,

• Different methods to compress the EMD,

• Variable sampling of EMD,

• Image compression algorithms based on variable sampling in combination withentropy coding and a DCT coder.

1.7 Possible applicationsThe work presented in this thesis is strictly concerned with image coding applications.During the time frame of the PhD studies the author has also been involved in anothertype of work that may benefit from the results described here.

Humanitarian demining is an important task to support the population of a region af-fected by a conflict in returning to the normal use of the country, e.g. for agriculture orthe construction of infrastructure facilities. In this context the EU funded ARC projectaimed at supporting Mine Actions by providing a tool for the fast, accurate and cost-effi-cient mapping of a mine suspected area and for minefield area reduction. The project ispresented in [87] and [42]. The major contribution by the author to this project is the ideaand implementation of temporal signal processing, detecting modelled object signaturesin a diurnal IR-image sequence. The temporal signal processing in [79,80,81,49] stemsfrom the author of this thesis. Including EMD in the analysis may provide yet another ap-proach in the search for better mine detection methods. The EMD method has alreadyshown excellent classification performance in the analysis of laser vibrometry signals[61].


14

PART ONE

WAVELET PACKET IMAGE COMPRESSION

17

Chapter 2

Wavelet and Wavelet Packet Methods for Image Compression

The first part of this thesis starts with an introduction of the wavelet and wavelet packetconcept and continues with a survey of different image coding methods using these meth-ods that have evolved over the years. We will go through the history and literature andpresent these techniques that lead us on to the problem we have focused on in this thesis.The compression results given in this chapter are only the results reported in the reviewedpapers.

2.1 IntroductionHierarchical methods (pyramidal, multi-resolution) for representing images have severaladvantages over plain sampled images or the flat representation. The first to use hierar-chical representation of images for the purpose of compression were Adelson & Burt [1].In subband representation [94] of an image the subsampled approximation of the originalimage is produced in the same way as in the pyramid coding scheme, but instead of thedifference image used in [1], the subband coding scheme produces the detail image by afiltering of the original image with a high pass filter. In this section we will introduce twovariations of the subband coding scheme, namely the wavelet transform and the waveletpacket transform.

2.1.1 Wavelet transformThe wavelet transform of an image is also a multiresolution description. The possibility

Chapter 2 Wavelet and Wavelet Packet Methods for Image Compression

18

of using the wavelet theory in image compression was outlined in Mallat [54] where themultiresolution approach used in the subband coding schemes and the wavelet theory wasmerged. The wavelet theory introduced new nomenclature into the subband technique be-cause it wed together different disciplines such as mathematics, seismology, physics andsignal processing into the same field. The difference compared to subband coding is thatthe wavelet theory has equipped us with a tool for proper filter construction [20]. Thewavelet transform uses scaled and translated versions of a prototype wavelet as basisfunctions to represent a signal. For details of the wavelet theory, the reader is referred tothe text books [19,56,90,93].

The high-pass filtered data set is the wavelet transform detail components at that levelof scale of the transform. The low-pass filtered data set is the approximation componentsat that level of scale. Thanks to the subsampling, the four sets of components have fourtimes fewer elements than the original data set. The approximation components can now

re 2.1. Component correspondence between subbands on different levels, starting with a pixel in the original image.

Figure 2.2. Components correspondence between subbands on different levels starting with one component in the lowest subband (right).

2.1 Introduction

19

be used as the sampled data input for another pair of wavelet filters, identical to the firstpair, generating another set of detail and approximation components at the next lower lev-el of scale. The wavelet theory is an impressive mathematical tool to produce proofs thatthis really produces a representation of the image, proofs that did not exist in the time ofsubband theory. Today, wavelet coding has become the substitute name for essentiallyany kind of subband coding of images. This thesis includes a survey of wavelet and wave-let packet image coding methods.

Another way of looking at the representation is by looking at the subband correspond-ence of components. Due to the subsampling the size of the data matrix is the same for alllevels but higher levels contain more subbands. One component in a higher subband rep-resents the spatial frequency in an area of several pixels in the original image, having thesame size and location as the component would cover if the subband is upsampled to theoriginal image size. This is illustrated in Figure 2.2. A particular pixel in the original im-age is represented in every subband, indicating the amount of spatial frequency of the sub-band at the location of the pixel as shown in Figure 2.1.

2.1.2 Wavelet packetsSince texture typically is a composition of mid and high spatial frequency components

Figure 2.3. The subband stack. Subbands of different scales, the chosen subbands in gray form the resulting orthogonal basis shown at the bottom, to the left the wavelet basis and to the right a

wavelet packet basis.


20

the wavelet decomposition does not capture its typical structure. These structures are bet-ter found by the wavelet packet algorithms. The wavelet packet technique was introducedin [17] as a natural extension of the wavelet techniques and was immediately investigatedin various areas of signal analysis, [70,45,66] among others.

The wavelet packet basis is constructed adaptively based on some cost function andchoice of decomposition filter applied to an image. The decomposition of a signal can beviewed as a tree, where the left branch represents the low-pass horizontal/low-pass ver-tical filtering, the right branch represents the high-pass horizontal/high-pass vertical fil-tering and the middle branches represents the low-pass horizontal/high-pass verticalfiltering and the high-pass horizontal/low-pass vertical filtering, respectively.

The wavelet packet tree is constructed in two steps. First, the image is filtered andsubsampled into four new images representing the spatial frequency subbands. Each ofthe subbands are filtered and subsampled into four new images, a process that is repeatedto a certain level. By keeping the components in every subband at every level the waveletpacket tree has the advantage of attaining the complete hierarchy of segmentation in fre-quency and is a redundant expansion of the image. This is illustrated by the subband stackin Figure 2.3. In the second step we can choose our best basis to represent the texture bycutting off branches in the tree controlled by the cost function applied on one node andon its children nodes. These subbands are to be seen as a collection of bases for this par-ticular image and filter choice. Choosing a best basis is a way to represent the image inthe most effective way.

The fully decomposed wavelet packet (WP) tree is shown in Figure 2.4. The waveletpacket transform generates an orthogonal basis if a proper wavelet is chosen.

For basis selection we apply a cost function on the unsplit subband A, M(A), inFigure 2.5, and the same cost function on the subbands B, C, D and E resulting from theone level subband decomposition.

Split the subband if M(A) >M(B) + M(C) + M(D) + M(E), otherwise keep the sub-band A as it is.

Figure 2.5. One step subband split.

Figure 2.4. Fully decomposed 3-level wavelet packet tree.

A

B C

D E

split

2.1 Introduction

21

In Figure 2.6 are shown two pruned trees, the left representing the wavelet basis andthe right representing a possible best basis. The leaves of the tree represent the basis.Below are shown some texture images from the Brodatz collection [10] and their best bas-es, using a certain decomposition filter and cost function, specified in section 3.1. Wave-let packet decomposition of an image and basis selection results in different bases whenusing different filters or different cost functions. The best basis found for a specific im-age, using a particular filter and cost function, is best for this triplet of image, filter andcost function. Changing at least one of the three may result in another best basis. InFigure 2.7 the same filter and cost function are used on three different images.

Figure 2.7. Texture images and their best wavelet packet trees.

Figure 2.6. Tree representation of the bases in Figure 2.3.

(0,0)

(1,0) (1,1) (1,2) (1,3

(2,0) (2,1) (2,2)(2,3) (2,4) (2,5) (2,6)(2,7) (2,8) (2,9)(2,10)(2,11

(3,0) (3,1)(3,2)(3,3) (3,4) (3,5)(3,6)(3,7) (3,16) (3,17) (3,18) (3,19) (3,20) (3,21)(3,22)(3,23) (3,32) (3,33)(3,34)(3,35)

(4,0)(4,1)(4,2)(4,3)(4,16)(4,17)(4,18)(4,19) (4,64)(4,65)(4,66)(4,67) (4,72) (4,73)(4,74)(4,75)(4,80)(4,81)(4,82)(4,83) (4,128) (4,129)(4,130)(4,131)(4,132)(4,133)(4,134)(4,135)

(5,0)(5,1)(5,2)(5,3) (5,72)(5,73)(5,74)(5,75) (5,256)(5,257)(5,258)(5,259)(5,288)(5,289)(5,290)(5,291)(5,296)(5,297)(5,298)(5,299)(5,320)(5,321)(5,322)(5,323)(5,328)(5,329)(5,330)(5,331) (5,512)(5,513)(5,514)(5,515)(5,516)(5,517)(5,518)(5,519)(5,528)(5,529)(5,530)(5,531)

(6,0)(6,1)(6,2)(6,3)(6,8)(6,9)(6,10)(6,11) (6,288)(6,289)(6,290)(6,291)(6,300)(6,301)(6,302)(6,303)(6,1024)(6,1025)(6,1026)(6,1027)(6,1028)(6,1029)(6,1030)(6,1031)(6,1032)(6,1033)(6,1034)(6,1035)(6,1152)(6,1153)(6,1154)(6,1155)(6,1156)(6,1157)(6,1158)(6,1159)(6,1160)(6,1161)(6,1162)(6,1163)(6,1164)(6,1165)(6,1166)(6,1167)(6,1184)(6,1185)(6,1186)(6,1187)(6,1192)(6,1193)(6,1194)(6,1195)(6,1288)(6,1289)(6,1290)(6,1291)(6,1316)(6,1317)(6,1318)(6,1319)(6,1324)(6,1325)(6,1326)(6,1327)(6,2048)(6,2049)(6,2050)(6,2051)(6,2052)(6,2053)(6,2054)(6,2055)(6,2064)(6,2065)(6,2066)(6,2067)(6,2068)(6,2069)(6,2070)(6,2071)(6,2116)(6,2117)(6,2118)(6,2119)(6,2120)(6,2121)(6,2122)(6,2123)

(0,0)

(1,0) (1,1) (1,2) (1,3

(2,0) (2,1) (2,2) (2,3) (2,8) (2,9) (2,10) (2,11

(3,0) (3,1) (3,2) (3,3) (3,8) (3,9) (3,10) (3,11) (3,32) (3,33) (3,34) (3,35) (3,40) (3,41) (3,42) (3,43

(4,0)(4,1)(4,2)(4,3)(4,8)(4,9)(4,10)(4,11)(4,32)(4,33)(4,34)(4,35)(4,40)(4,41)(4,42)(4,43)(4,128)(4,129)(4,130)(4,131)(4,136)(4,137)(4,138)(4,139)(4,160)(4,161)(4,162)(4,163)(4,168)(4,169)(4,170)(4,171

(5,0)(5,1)(5,2)(5,3)(5,8)(5,9)(5,10)(5,11)(5,32)(5,33)(5,34)(5,35)(5,40)(5,41)(5,42)(5,43)(5,128)(5,129)(5,130)(5,131)(5,136)(5,137)(5,138)(5,139)(5,160)(5,161)(5,162)(5,163)(5,168)(5,169)(5,170)(5,171)(5,512)(5,513)(5,514)(5,515)(5,520)(5,521)(5,522)(5,523)(5,544)(5,545)(5,546)(5,547)(5,552)(5,553)(5,554)(5,555)(5,640)(5,641)(5,642)(5,643)(5,648)(5,649)(5,650)(5,651)(5,672)(5,673)(5,674)(5,675)(5,680)(5,681)(5,682)(5,683)

(6,0)(6,1)(6,2)(6,3)(6,8)(6,9)(6,10)(6,11)(6,32)(6,33)(6,34)(6,35)(6,40)(6,41)(6,42)(6,43)(6,128)(6,129)(6,130)(6,131)(6,136)(6,137)(6,138)(6,139)(6,160)(6,161)(6,162)(6,163)(6,168)(6,169)(6,170)(6,171)(6,512)(6,513)(6,514)(6,515)(6,520)(6,521)(6,522)(6,523)(6,544)(6,545)(6,546)(6,547)(6,552)(6,553)(6,554)(6,555)(6,556)(6,557)(6,558)(6,559)(6,640)(6,641)(6,642)(6,643)(6,648)(6,649)(6,650)(6,651)(6,672)(6,673)(6,674)(6,675)(6,680)(6,681)(6,682)(6,683)(6,2048)(6,2049)(6,2050)(6,2051)(6,2056)(6,2057)(6,2058)(6,2059)(6,2080)(6,2081)(6,2082)(6,2083)(6,2088)(6,2089)(6,2090)(6,2091)(6,2176)(6,2177)(6,2178)(6,2179)(6,2184)(6,2185)(6,2186)(6,2187)(6,2208)(6,2209)(6,2210)(6,2211)(6,2216)(6,2217)(6,2218)(6,2219)(6,2560)(6,2561)(6,2562)(6,2563)(6,2568)(6,2569)(6,2570)(6,2571)(6,2592)(6,2593)(6,2594)(6,2595)(6,2600)(6,2601)(6,2602)(6,2603)(6,2688)(6,2689)(6,2690)(6,2691)(6,2696)(6,2697)(6,2698)(6,2699)(6,2720)(6,2721)(6,2722)(6,2723)(6,2728)(6,2729)(6,2730)(6,2731)

(0,0)

(1,0) (1,1)(1,2)(1,3

(2,0) (2,1)(2,2)(2,3)

(3,0) (3,1)(3,2) (3,3)

(4,0) (4,1) (4,2) (4,3) (4,12) (4,13) (4,14) (4,15)

(5,0) (5,1)(5,2)(5,3)(5,4)(5,5)(5,6)(5,7)(5,12)(5,13)(5,14)(5,15) (5,52)(5,53)(5,54)(5,55)(5,56)(5,57)(5,58)(5,59)(5,60)(5,61)(5,62)(5,63)

(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)(6,7) (6,24)(6,25)(6,26)(6,27) (6,52)(6,53)(6,54)(6,55)(6,60)(6,61)(6,62)(6,63) (6,216)(6,217)(6,218)(6,219)(6,224)(6,225)(6,226)(6,227)(6,232)(6,233)(6,234)(6,235)(6,244)(6,245)(6,246)(6,247)


22

2.2 Wavelet image coding algorithmsWavelet transform coders are mostly used at rates below 1 bpp and are able to recoverimages at nearly perfect visual quality. Already from the beginning of the wavelet theory,the method was considered for image compression [3,55], due to the close connection tosubband coding [94].

2.2.1 Wavelet coding with vector quantization The coding algorithm of [3] uses the wavelet decomposition together with a multiresolu-tionvector quantization scheme and a noise shaping bit allocation procedure as explainedbelow. The image is decomposed into its wavelet basis using a set of biorthogonal filters[19].

The vector quantization procedure was already accepted in the subband communityas a powerful tool for image compression. The principle involves encoding a sequence ofsamples (a vector) rather than encoding each sample individually. Encoding is performedby approximating the sequence to be encoded by a vector belonging to a codebook. Amultiresolution codebook is created and optimized using a classifying algorithm with amean square error criterion; it contains vectors for each resolution level and direction. InAntonini et al. [3] the training set is a set of vectors belonging to different images includ-ing the image to be coded.

The bit allocation is based on the fact that the human eye is not equally sensitive toall spatial frequencies. The bit allocation, which is a function of the image, is transmittedas side information. For the coding of the 256x256 Lenna image the bit allocation schemeis shown in Figure 2.8. In Table 2.1 the coding performance of the Lenna image with dif-ferent biorthogonal filters is shown.

Figure 2.8. The bit allocation of the wavelet coder.

8 bppscalar

quantization

2 bpp

VQ

2 bpp

VQ

VQ

VQ

VQ0.5 bpp

0.5 bpp

0 bpp

0.5 bpp

2.2 Wavelet image coding algorithms

23

Table 2.1. Coding performance of the Lenna image with different biorthogonal filters.

2.2.2 Embedded zerotree wavelet algorithm (EZW)For transform coders operating below 1 bpp with optimized scalar quantization and bitallocation the performance depends mostly on the encoding efficiency of the position ofthe zero components. Also an important part of the low bitrate image compression is thecoding of the amplitude of those components that will be transmitted as nonzero values.

The zerotree was defined in order to improve the compression of the significance mapin which only the components which are significant according to a specified threshold arerepresented. The idea of coding the significance map as a zerotree was introduced byLewis & Knowles in [46] and beautifully developed by Shapiro in [78] to become an em-bedded image coder. The zerotree is based on the hypothesis that if a wavelet componentat a coarse scale is insignificant with respect to a given threshold , then all wavelet com-ponents of the same orientation in the same spatial location at finer scales, the “children”,are likely to be insignificant with respect to .

A wavelet component is said to be insignificant with respect to a threshold if. Such a component is quantized to zero. The others are significant components.

Coding the positions of the significant coefficients is the same as storing a binary signif-icance map defined by

(2.1)

Furthermore, a component is said to be an element of a zerotree for threshold ifitself and all of its descendants are insignificant with respect to . An element of a zero-tree for threshold is a zerotree root if it is not the descendant of a previously found ze-rotree root for threshold .

The EZW algorithm [78] is a simple but effective compression algorithm that has theproperty of generating the bits in the bitstream in order of importance, yielding a fully em-bedded code. With an embedded bit stream the reception of code bits can be stopped atany point and the image can still be reconstructed. The algorithm is based on four keyconcepts, 1) a discrete wavelet transform, 2) zerotree coding of the components, whichprovides a compact multiresolutionrepresentation of significance maps, 3) entropy codedsuccessive-approximation quantization, and 4) arithmetic coding.

Two separate lists of coordinates of the wavelet components are maintained. Thedominant list contains the coordinates of the components that have not yet been found tobe significant. The subordinate list contains the magnitude of the significant components.The scanning of the components during a dominant pass is performed in such a way that

PSNR

Bitrate 9-3 filter 5-7 filter 9-7 filter0.8 bpp 31.82dB 31.46 dB 32.10 dB

T

Ta T

a T<

b

b 0 if a T≤1 if a T>

=

a TT

TT


24

no child is scanned before its parent. During the scan, each component that is not markedinsignificant by another zerotree is coded with one of four symbols; 1) zerotree root, 2)isolated zero, 3) positive significant, and 4) negative significant. Each time a componentis coded significant its magnitude is appended to the subordinate list. The string of sym-bols is then encoded using an arithmetric coder. The performance of the EZW coder, us-ing a 6 level QMF pyramid decomposition, for two different test images is shown inTable 2.2.

Table 2.2. The performance of the EZW coder, using a 6 level QMF pyramid decomposition, for two different test images.

2.2.3 Set partitioning in hierarchical trees (SPIHT)SPIHT was introduced in [75] as a refinement of the EZW algorithm. In SPIHT the waysubsequent components are partitioned and the way the significance information is con-veyed are fundamentally different than from EZW. SPIHT uses the same hypothesis asEZW that if a parent component is below a certain threshold then it is likely that all itsdescendants are below the threshold too. If this prediction is successful then SPIHT rep-resents the parent and all its descendants with a single symbol called a zerotree, as insection 2.2.2.

SPIHT works in two passes, the ordering pass and the refinement pass. One of themain features is that the ordering is not explicitly transmitted. Another important fact isthat it is not necessary to sort all the components. A component is said to be sig-nificant with respect to a threshold if for a given n. The sorting algorithmdivides the set of components into partitioning subsets and performs a magnitude test

(2.2)

If the subset is insignificant, all components in are insignificant. Otherwise thesubset is significant and is partitioned again into new subsets , and the procedure isrepeated. The partitioning uses the hierarchy of the wavelet decomposition.

In the refinement pass the quantization of components is refined. SPIHT maintains

PSNR

Bitrate Lenna Barbara1.0 bpp 39.55 dB 35.14 dB0.5 bpp 36.28 dB 30.53 dB0.25 bpp 33.17 dB 26.77 dB0.125 bpp 30.23 dB 24.03 dB0.0625 bpp 27.54 dB 23.10 dB0.03125 bpp 25.38 dB 21.94 dB0.015625 bpp 23.63 dB 20.75 dB

a i j,( )a i j,( ) 2n>

Tm

maxi j Tm∈,

a i j,( ){ } 2n≥

Tm TmTm l,

2.2 Wavelet image coding algorithms

25

three lists of coordinates of components in the decomposition. These are the List of In-significant Pixels (LIP), the List of Significant Pixels (LSP) and the List of InsignificantSets (LIS). The LIP contains coordinates of components that are insignificant at the cur-rent threshold. The LSP contains the coordinates of components that are significant at thesame threshold. The LIS contains coordinates of the roots of the spatial parent-childrentrees.

The efficiency of SPIHT is heavily dependent on finding zerotrees in the LIS, yet pro-ducing an embedded bit-stream code. The performance of the SPIHT coder with arithmet-ric coding of the bit stream is shown in Table 2.3.

Table 2.3. Performance of the SPIHT coder with arithmetric coding of the bit stream, applied to the Lenna image.

2.2.4 JPEG2000JPEG2000 is a recently (2000) developed coding standard for still images. The use ofwavelet compression is one of its many features. Here we will only review the use of thewavelet coding and leave the rest of the standard, which is well described in [13] for theinterested reader. The standard is working on image tiles of size 128x128 or 256x256, asif they were individual images, and the wavelet decomposition is applied to each tile asshown in Figure 2.9.

Figure 2.9. Wavelet decomposition of the tiles in JPEG2000.

The wavelet transform uses the 9-7 tap biorthogonal filter and Swelden’s lifting tech-nique [83]. The result of coding the Woman image using JPEG2000 with all its featuresis shown in Table 2.4.

bitrate PSNR0.5 bpp 37.2 dB0.25 bpp 34.1 dB0.15 bpp 31.9 dB


26

Table 2.4. Performance of the JPEG2000 coder applied to the Woman image.

2.2.5 Morphological representation of wavelet data (MRWD)The goal of Servetto et al. [76] in their morphological representation of wavelet data isto find an improved statistical description of image subbands, capable of explicitly takinginto account the space-frequency localization properties of the wavelet.

This alternative approach to the zerotree is based on a region growing technique,where the significance map is encoded using local probability models depending on theregion being encoded. If a given component is known to be classified as significant, thencomponents in a small neighbourhood, with high probability, will be significant too (like-wise for the insignificant ones). In a sense, the morphological approach proposed in [76]is a concept dual to that of zerotrees, in that it exploits directly the clustering of signifi-cant components. Also, arbitrarily shaped regions of nonzero components are defined atno extra cost.

The coder is a pure waveform coder. The morphological sets considered are not in-tended to capture any image domain objects or to exploit any property of the human vis-ual system, but instead are tools used to assign probabilities to sets of waveletcomponents.

The probability describing intraband dependencies is determined, for each subband,by starting with a map of components all labelled insignificant, followed by computinga partition of components labelled significant or insignificant as follows.

Select a component at a random location within the current subband. If this selectedcomponent is not significant, or if it is significant but it has already been labelled signif-icant, select another component at a random location until an unlabelled significant com-ponent is found. Then apply one step of morphological dilation at this location, and labelall the dilated components significant. The morphological dilation is described in [76] asfollows. Consider a set to which the dilation operation will be applied, and let represent some structuring element. Let denote the morphological dilation operator.The dilated set is defined to be the union of all points falling under the supportof the structuring element , when centred at each point in . For each new significantcomponents found by the dilation, recursively repeat this step, until no new significantcomponents are found. Stop selecting components at random when all the significantones have been labelled significant. Those without a label are labelled insignificant.

Define two histograms, corresponding to the frequency of occurrence of the compo-nents labelled significant and insignificant respectively by this partitioning method, andestimate the entropy of the composite model.

The encoder then scans subbands in raster scan order until a significant component is

bitrate PSNR1 bpp 41 dB0.5 bpp 38 dB0.25 bpp 35 dB

S B⊕

S B⊕B S

2.3 Wavelet packet coding algorithms

27

detected. Then the encoder signals this event to the decoder with a special symbol, as sideinformation. One step of dilation is performed: the encoder sends to the decoder and la-bels significant those coefficients in a neighbourhood of the current significant one. Thesenew ones are examined: if new significant components are found, the process is appliedrecursively on each of these new values; otherwise, the process stops and raster scanningof insignificant components resumes, until the whole subband is exhausted. As a result,clusters of significant components are efficiently captured by morphological dilation: theonly side information required to encode a new cluster is the symbol used to signal thetransition from an insignificant to a significant region.

The resulting symbol stream is compressed using arithmetic coding based on the sep-arate probability tables from above. The performance on the 512x512 Lenna image, us-ing a five-level wavelet decomposition based on the spline 9-7 biorthogonal filters, witha uniform quantizer is shown in Table 2.5.

Table 2.5. The performance of MRWD on the 512x512 Lenna image.

2.3 Wavelet packet coding algorithmsThe idea behind the wavelet packet coder is that the adaptive structure will lead to bettercoding results. The wavelet packet tree is any sub-set of a fully decomposed tree of a cer-tain depth. To find the basis (sub-tree) a cost function is introduced. The tree is thensearched for the basis that minimizes the selected cost function. If the filter and cost func-tion are appropriately chosen the decomposition produced by the wavelet packet treeshould, theoretically, always be better than that produced by the wavelet basis since thisbasis is included as a special case of the admissible wavelet packet structures.

2.3.1 Best basis subband codingAlthough the wavelet packet and best basis concept were revealed to the public in 1992[17] work had been done on images using this technique before that under the name ofbest basis subband coding [92]. It is based on the work of [16], at the time unpublished,extending it to two dimensions.

The best basis scheme avoids an exhaustive search over all bases. The algorithm startsby growing a full tree to the desired depth. Then, starting from the leaves, for every nodethe cost of keeping the node and pruning it is compared. The node is kept if the cost ofkeeping it is lower then that of pruning it. The process is continued at every level until the

Bitrate PSNR1 bpp 40.33 dB0.75 bpp 38.98 dB0.5 bpp 37.17 dB0.25 bpp 34.25 dB0.125 bpp 31.09 dB0.0325 25.82 dB


28

root of the tree is reached. In [17] QMF filters were used and the cost function denoted energy entropy (Equation

3.12). Compression is achieved by using a threshold ε on the components from the bestbasis and simply discharging the components below this threshold. Alternatively, for afixed compression rate, the components are sorted and the appropriate number of com-ponents are used. Note that no quantization is applied to the components.

2.3.2 Rate-distortion optimized wavelet packet codingThe problem of optimal bit allocation in a wavelet packet framework was addressed inRamchandran & Vetterli [69,70]. They used a rate-distortion criterion to find the best de-composition for a given signal. The idea is that each node in the fully decomposed tree isassigned the Lagrangian cost

(2.3)

Then the basis is chosen by forming the subtree which has the minimum sum of cost forits leaves. The decision to prune is based on the Lagrangian cost function for a fixed valueof . The value of is found by sweeping through the sequence of values and for eachvalue of the least cost quantization for each node is selected as the one that minimizesEquation 2.3. The total rate is compared to the budget rate , if then is adjusted and the procedure starts over. The sequence of is achieved by us-ing Newton’s method. The basis selected should minimize the distortion given a bit budg-et, which is the essence of image coding.

In [70] the image is decomposed using the Db4 filter and trees were grown to a depthof 4 for a 512x512 image. First order entropy is used as the bitrate measure as scalarquantization is assumed, MSE is used as a distortion criterion. This codes the Barbara im-age to PSNR 36.4 dB at 0.92 bpp. The main achievement with this work was that itshowed that the wavelet packet best basis approach performs better on high frequencytextures than traditional coders.

2.3.3 Optimal entropy constrained lattice vector quantization (ECLVQ)One of the advantages of wavelet packets for image compression is that it is adapted tothe spatial frequency characteristic of the image. A wavelet packet basis can be chosento maximize the energy compaction in each subband. In [43] wavelet packet representa-tion is applied for entropy constrained image compression using lattice vector quantiza-tion. In resolving the mutual dependencies of the wavelet basis and source coding a twophase algorithm is used. First the optimal entropy constrained lattice vector quantizationis developed for a given wavelet packet structure, then a greedy heuristic search algo-rithm is adopted to identify a locally optimal wavelet packet basis. The greedy algorithmdoes not guarantee an optimal basis; still, it performs as well as the other coders describedhere. Performance of the coder on the 512x512 Lenna image is presented in Table 2.6.

L Di λRi+=

λ λλ

Ri∑ RT Ri∑ RT>

λ λ


29

Table 2.6. Performance of the ECLVQ coder on the 512X512 Lenna image.

2.3.4 Constrained wavelet packet coding (CWP)In this section a more thorough description of a SPIHT-WP-based coder is presented fromKhalil et al. [41]. Especially the modifications necessary for using SPIHT with the adap-tive tree structure are explained. The coder is used in the experiments in section 3.3 as acomparison with the more theoretical results.

With the wavelet-transform-based coder the octave band decomposition structure wasfixed. The strength of the SPIHT algorithm for coding wavelet components is obvious. Itwould be desirable to combine the basis selection from the wavelet packet transform withSPIHT.

The parent-children relationship for SPIHT assumes that the decomposition structureis an octave-band decomposition. This means that in order to get SPIHT working with themore general wavelet packet structure (WP-structure) some modifications must be made.

Figure 2.10. Parent-children relationship for a WP-structure.

In Figure 2.10 the region Parent is spatially co-located with the regions marked Chil-dren. In the wavelet transform case each child occupies a size corresponding to ¼ of thatoccupied by the parent. In the WP-structure in Figure 2.10 each child-region is at thesame scale as the parent and thus occupies a region of the same size as the parent. Thereason for choosing Parent as parent is that Parent is in a subband that has more low-passcharacter than the ones denoted Children, and low-pass subbands better predict the energyof high-pass subbands than the other way around.

Bitrate PSNR0.5 bpp 36.97 dB0.25 bpp 33.83 dB0.15 bpp 31.82 dB


30

Spatial reorderingTo be able to use SPIHT efficiently on the WP-structure we have to take into accountthese new definitions of parent-children relationships, where children may reside in dif-ferent subbands of the same scale. The modifications should be made so that inter-bandsimilarities can be exploited. In Figure 2.11 this is exemplified for the same structure asin Figure 2.10.

The reordering permits the use of the SPIHT-algorithm without any changes to theSPIHT-algorithm. The simple 1:4 (two-dimensional case) parent-children mapping of awavelet structure does not always hold in a general WP-structure. Instead the WP-de-composed data is reordered with some constraints explained below. In Figure 2.11 thesimple 1:4 (two-dimensional case) parent-children mapping holds; in a general WP-structure this may not be the case. In Figure 2.12

Figure 2.11. The upper figure is the WP-structure with the desired parent-children relationships. In the lower figure the data has been reordered so that the simple parent-children relationship hold.

Parent have 7 children of which four represent a region 4 times the size of the parent. Ifthe children represent a larger region than the parent then parts of the signal that do notaffect the parent can affect the children. This leads to reduced possibilities of finding ze-rotrees

Following [41], the problem of in-coherent parent-children mappings is solved by us-

Parent

Children


31

ing a limited selection of admissible bases for the WP-decomposition. The limitations areimplemented with two constraints on the algorithm for wavelet packet basis search.

The problem presented in Figure 2.12 arise from the fact that a subband that shouldbe a child subband is decomposed one level more than its parent-subband. The constraintfor avoiding these situations is formulated as:

“Rule B: When pruning parents, their spatially co-located child subbands may need tobe pruned also in order to keep the tree structure valid.”

Another mismatch in the original 1:4 parent-children mapping occurs if a child sub-band is not decomposed enough.

Figure 2.12. A WP-structure that does not have a one to four parent-children mapping.

For the one to four parent-children mapping to work a child subband must be decom-posed to the same level as its parent subband or at most one level less. To avoid the prob-lem of child bands not being decomposed enough the constraint added is:

“Rule A: Any four child bands will be considered as candidates for pruning only iftheir current decomposition level is the same as that of the spatially co-located parentband”

The reordering of a number of child subbands is done so that the resulting structureresembles a subband that is decomposed one level less than the reordered bands. That is,components representing the same spatial region in the child subbands are grouped to-gether. This leads to larger regions representing the same spatial region of the original sig-nal, which should serve as a motivation for reordered subbands to resemble a next highersubband.

Using the decomposition tree we would like to prune nodes in such a way that the treeafter pruning is an octave-band decomposition tree of the same depth as the original tree.

The reordering is started from the bottom of the decomposition tree reordering indi-vidual components as seen in Figure 2.13. The next level up the reordering has to be madein units of 2x2 components (two-dimensional case) shown in Figure 2.14. This process iscontinued until the tree shown in Figure 2.15 is achieved.

When the reordering is complete the reordered structure can be fed to the SPIHT codersince now the parent-children relationships are adapted to SPIHT. On the decoding sidethe bit stream from SPIHT can be decoded producing a reconstruction of the reordered.


32

Figure 2.13. The original decomposition structure.

Figure 2.14. The first level of reordering as pruning of the WP tree (the dotted lines mark the spatial location of the subbands that are reordered).

Figure 2.15. Lastly the final stage of reordering, which makes the pruned tree into an octave-band tree.


33

structure. Before this structure can be inversely transformed an inverse reordering has tobe made to transform the structure into its original form. This inverse reordering can alsobe explained by utilizing the decomposition tree. This time starting from the root eachnode in an octave-band tree is split (inverse reordered) until the split tree has the sameshape as the original tree.

This coder compresses the Barbara image to 32.61 dB at 0.5 bpp.

2.3.5 WP-IMRWDIn [96] an improved version of the coder in section 2.2.5 is applied to a wavelet packetbasis. The improvements are the introduction of some new features: 1) Exploitation of theknowledge of component magnitude in the parent band, 2) Reordering of the residual partof the significant map according to the distance to the previously visited point. 3) Em-ployment of hierarchic numerative entropy coder.

As in section 2.2.5 the problem of children nodes having more than one parent mustbe solved. This is done by applying inverse wavelet packet transform to the parent blockuntil the full resolution parent block is retained. Then forward wavelet packet transformis applied to the retained parent block by using the decomposition basis of the child. Sincethe layout of the two blocks is now the same, the parent-child relation can be established.The procedure is pictured in Figure 2.16. The performance of the coder is presented inTable 2.7.

Parent block

IWPT

WPT

Descendant block

Figure 2.16. Parent-child relation establishing procedure.


34

Table 2.7. The performance of the WP-IMRWD coder.

2.3.6 Compatible zerotree quantization of wavelet packets (CZQ-WP)The issue of wavelet packet zerotrees was also addressed partially by Xiong et al. in thespace-frequency quantization (SFQ) algorithm presented in [95], which employs a rate-distortion (R-D) optimization framework to select the best basis and to assign an optimalquantizer to each of the wavelet packet subbands. In their work, however, the subbanddecomposition was restricted to avoid the parenting conflict.

In [68] a general zerotree structure for an arbitrary wavelet packet geometry in an im-age coding framework is presented. A Markov chain-based cost estimate of encoding theimage is used. This adaptive wavelet zerotree image coder has a relatively low computa-tional complexity, performs comparably to the state-of-the-art image coders, and is capa-ble of progressively encoding images.

In order to resolve the parenting conflict, Rajpoot et al. [68] suggest that the childnodes are moved up in the tree so that they are linked directly to the root node of the pri-mary compatible zerotree associated with the family of subbands. This generates a com-patible zerotree structure without restricting the basis selection.

The algorithm uses a set of rules to construct the overall compatible zerotree structurefor an arbitrary wavelet packet basis. These are reviewed here. The only assumptionmade is that the lowest frequency subband of the selected basis is always at the coarsestresolution.

Rule (a) If a node P at a coarser resolution is followed only by a node C at the nextfiner resolution (as in a wavelet transform), the node P is declared as a parent of C.

Rule (b) If a node P is followed by four nodes C1 , C2 , C3 , and C4 (at the same res-olution), then P is declared to be the parent of all these four nodes.

Rule (c) If four subbands P1, P2, P3, and P4 at a coarser resolution are followed byfour subbands C1 , C2 , C3 , and C4 at the next finer resolution, then node Pi is declaredto be the parent of node Ci (for i=1,2,3,4).

Rule (d) If a node P is at a finer resolution than four of its children, say C1 , C2, C3 ,and C4 , then P is disregarded as being the parent of all these nodes and all of them aremoved in the tree under a node at the same or a coarser resolution.

In order to ensure that the selected basis is zerotree friendly, [68] use a cost functionthat can estimate the entropy of quantized coefficients belonging to the compatible zero-trees. The cost can be estimated by computing the entropy of a discrete random variable

PSNR

Bitrate Barbara Lenna1bpp 38.03 dB 40.95dB0.5bpp 33.05 dB 37.54 dB0.25bpp 28.52 dB 34.51 dB0.125 bpp 25.56 dB 31.36 dB

2.4 Segmentation-based image coding

35

whose value is drawn from the set of codewords used to encode the significance map. TheCZQ-WP coder performs as presented in Table 2.8 with the factorized 9-7 biorthogonalfilter.

Table 2.8. The performance of the CZQ-WP coder.

2.4 Segmentation-based image coding During the past years there have been many different approaches to achieve better imagecompression than the JPEG standard. The wavelet transform is now a part of theJPEG2000 standard but the wavelet packet techniques have not yet reached their ultimateapplication and performance. One important drawback of the methods described so far inthis chapter is that they all operate on the entire image. One approach is to make the as-sumption that the wavelet packet best basis search finds the resonance in the image andthereforee should be applied to pure textures only. This is done by segmenting the imageand finding the best basis for each segment.

2.4.1 Edge separating wavelet image codingWavelet-based image coders are good at representing smooth regions and isolated pointsingularities. However, they are less adaptive at representing perceptually important edgesingularities, and coding performance suffers significantly as a result. In [28] Froment &Mallat proposed a two-stage image coder framework based on modelling images as edgecartoons plus textures. The coder works by first inferring and efficiently coding the edgeinformation from the image using a multiscale wavelet decomposition. Second, the resid-ual, “edgeless” texture image is coded using a standard wavelet coder. This preliminarycoder improves significantly over standard wavelet coding techniques in terms of visualquality, according to [28].

2.4.2 Joint use of segmentation and wavelet packetThe best way to find an optimal basis representation for an image is to search jointly forthe best segmentation and wavelet packet tree. This is the idea behind the following meth-ods.

PSNR

Bitrate Barbara Lenna1bpp 36.15 dB 40.10dB0.5bpp 31.60 dB 37.55 dB0.25bpp 28.12 dB 34.56 dB0.1 bpp 24.27 dB 29.95 dB


36

Double tree (DT)The wavelet packet tree adapts to frequency content, the spatial quadtree adapts to spatialcontent. The spatial quad tree does not have the same capacity as a WP tree for compact-ing the energy of the image. The double tree was first suggested by Herley et al. [32,33]to address the problem of non-stationarity without sacrificing the energy compaction, bywhich is meant the deterministic non-stationarity described in section 1.1.4. The expan-sion finds the optimal wavelet packet trees over spatial segments of the image. This al-gorithm is extended to the use on images in [34]. The algorithm calculates the bestwavelet packet tree for the entire image and stores the cost. Then the image is segmentedinto a first level quad tree and separate wavelet packet trees are calculated for each of thefour segments and the cost is stored. Then the image is segmented into a second-levelquad tree and separate wavelet packet trees are calculated for each of the sixteen seg-ments and the cost is stored. This is iterated until sufficient depth of the quad tree isreached. The overall basis search identifies the most efficient dyadic segmentation andcorresponding wavelet packet expansions for the segments.

The double tree performance with a 7-9 tap linear phase filter and uniform quantiza-tion is shown in Table 2.9 [34].

Table 2.9. Performance of the double tree coder.

The dual double-tree (DDT)The dual double-tree in Smith & Chang [82] finds the optimal spatial segmentations ofthe wavelet packet nodes. The dual double tree is produced by first growing a singlewavelet packet tree followed by the full segmentation of all subbands. The dual doubletree performance with a 7-9 tap linear phase filter and uniform quantization is shown inTable 2.10 [34].

Table 2.10. Performance of the dual double tree coder.

The joint space and frequency best basisBoth the double tree and the dual double tree provide an asymmetric treatment of space

PSNR

Bitrate Barbara Lenna1bpp 36.55 dB 39.32 dB0.5bpp 31.73 dB 36.26 dB0.25bpp 28.06 dB 33.24 dB

PSNR

Bitrate Barbara Lenna1bpp 36.71dB 39.72 dB0.5bpp 32.03 dB 36.62 dB0.25bpp 28.27 dB 33.49 dB

2.4 Segmentation-based image coding

37

and frequency. As a result, neither tree attains the full joint decomposition in space andfrequency.

An algorithm that considers both frequency and spatial split at each node is presentedin [82]. It combines the double-tree and dual double-tree to attain the best joint space andfrequency decomposition for image compression.

Figure 2.17. The best joint space and frequency decomposition uses both frequency and spatial split at each node. Solid lines describe frequency split F, dashed line describe spatial split S.

In Figure 2.17 the two step joint space and frequency decomposition is shown. Solidlines describe frequency split F, dashed lines describe spatial split S. If we approximatethe output from FS to be equal to the output from SF these data need not be producedtwice. The accuracy of this approximation is filter dependent. The decomposing can beviewed as the graph in Figure 2.18.

Figure 2.18. Joint spatial frequency decomposition graph.

F

F S

S

F S

FF FS SF SS

F

F

F F

F

F

S

S S

S S S


38

The left branch FFF provides the fully decomposed wavelet packet tree while theright branch SSS provides the fully decomposed spatial quadtree. The middle nodes pro-vide different combinations of space and frequency decompositions where DT is one andDDT is another. This gives a larger set of possible bases to choose from when selectingthe best basis. An example for a one-dimensional (time-varying) signal is shown inFigure 2.19 [82].

Figure 2.19. One example of a time frequency tiling possible with the basis set in Figure 2.18 but not with WP, DT or DDT.

Performance of the joint space and frequency decomposition of the Barbara image isshown in Table 2.11, when using a R-D optimized cost function for best basis selection[82].

Table 2.11. Performance of the joint space and frequency decomposition of the Barbara image using different filters.

2.5 SummaryIn this chapter we have presented some image coding algorithms using wavelet andwavelet packet methods that have contributed to the development of the field. The wave-let compression methods have found their way into use in standards while the waveletpacket algorithms still struggle with problems.

PSNR

BitrateHaar filter

QMF filter

2.0bpp 43.8 dB 43.8dB1.0bpp 37.7 dB 37.5 dB0.5bpp 33.0 dB 22.7 dB

time

frequency

2.5 Summary

39

The algorithms presented in this chapter are typical in the sense that they focus on thequantization, bit allocation, and entropy coding steps of the coding chain shown inFigure 1.10. The problem of choosing the basis is often not treated apart from the basicscheme of each method. In the wavelet case one can argue that the basis is fixed but theresult still changes with the choice of filters. For the wavelet packet case the bases are to-tally different depending on the image, the choice of filter, and the choice of cost function.Also the choice of basis selection method, bottom-up or a greedy top-down search affectsthe result. This is not treated in the earlier works.

The wavelet packet methods demonstrate the difficulties of using the principle to geta good representation of the image for compression purpose. The main problem identifiedis the use of wavelet packets on the entire image while an optimal basis is only found forpure textures. Different filters are also shown to influence the results. But the waveletpacket methods also demonstrate the potential to find a good representation, although theproblems are not yet solved.

In the next chapter we will investigate the wavelet packet trees in more detail and howthe different bases depend on image structure and on the choice of filter as well as costfunction.


40

41

Chapter 3

The Wavelet Packet Triplet Problem

When working with image coding algorithms using the wavelet packet transform differentbases appear as the best basis for the same image when using different cost functions orfilters. For different images, the best compression is achieved with different bases due tothe different image statistics. This is almost intuitively understood. The triplet of image,filter, and cost function is closely related in the analysis of the compression ability of thewavelet packet transform. The influence of filter and cost function in connection with dif-ferent image statistics is analysed in this chapter.

We start the presentation of the triplet problem by analysing a database of coding re-sults using different filters and cost functions on a subset of the Brodatz texture images.The data base contains coding results from 2295 different triplets, the combination of 17images, 15 filters, and 9 cost functions. The database is analysed and discussed with re-spect to images, cost function effects, and filter effects. Due to the limited space in thisthesis only a part of the result is presented in plots. From this we get some guidance howto choose filter and cost function in the implementation of one of the wavelet packet cod-ers from Chapter 2. The same filter and cost function is also used in the next part of thechapter where we present a method to analytically calculate the expected wavelet packetbasis, with a top-down approach. Here we use a signal model to represent the image. Wethen apply this method to different images and compare the top-down calculated baseswith the bottom-up best bases for the same images.

We use the expression best basis for the basis found by the best basis search algorithmof Coifman & Wickerhauser [17]. This algorithm uses a bottom up search strategy and isstated to find a global optimal basis. The expression wavelet packet basis is used for basesfound to be optimal by any search algorithm. In this part of the thesis we are mainly con-cerned with the compression ratio achieved by the wavelet packet transform, and restrictourselves to estimating the entropy of the components as a measure of possible compres-sion performance except in one section where we test the performance of the SPIHT coderand the CWP coder.

Chapter 3 The Wavelet Packet Triplet Problem

42

3.1 Decomposing the Brodatz texturesThe reason for using texture images in this work is that we are specifically interested inefficient representation of the resonance in the image. We have used the Brodatz collec-tion of texture images for most of our experiments [10]. The test images available for ourexperiments are presented in Appendix A, and are a set of 17 digitized images out of thecollection in [10].

The images have different resonance due to their different textures. The cost functionapplies to the output from the filtering process. This leads to the questions: Does the filtermatter? What is the result if another cost function is used? The resulting best bases whenapplying the Haar filter and Db4 filter with the energy entropy cost function, specified inEquation 3.12, and the log energy cost function, specified in Equation 3.14, respectivelyto the images in Figure 3.1 are shown in Figure 3.2 - 3.7.

Clearly we can see that the change of filter or cost function results in different bases.The different bases representing the best basis for the triplet image, cost function and fil-ter, raise several questions. Why is there no typical tree for a particular image or why isthere no strong trend for a certain type of tree in combination with a fixed filter? Whichcombination is best for compression of the image?

Figure 3.1. The textures analysed in Figure 3.2 - 3.7, beach, brickwall, gravel.

3.2 Results from the triplet databaseIn this section the compression performance of the wavelet packet transform is evaluatedthrough extensive testing of all combinations of 17 images, 9 cost functions and 15 dif-ferent wavelet filters. The motivation is to compare the filter theory and cost function dis-cussion with experimental results and find explanation as to why the common adhoc useof wavelet packets does not always work as expected.

3.2.1 Entropy estimationsAn important measure in image coding applications is the cost for coding the transformcomponents. It may be even more important than the cost for choosing and representingthe basis. We simulate the coding cost by measuring the entropy of the components bythe theoretically calculated entropy presented in Equation 3.1.

3.2 Results from the triplet database

43

The image is represented by a sequence of pixels, , created bya scan of the two-dimensional image array, where is the pixel at location i, is the number of pixels in the image, as a message generated by a discrete source togetherwith a set of probabilities, . Then the image is a finite dimensional random vectorvariable where is the pixel random variable at location i. Inlimes, , the random vector is a random process, and the random process is the in-finite dimensional generalization of a vector random variable [22]. One image is a reali-sation of the random process. We assume ergodicity and thereforee we can use onerealisation of our process (one image) to compute statistics [74].

Let pxk be the probability that the pixel x takes a value k out of an alphabet defined bythe set of amplitude values, then the entropy for this pixel x is

(3.1)

and we know that we can represent the pixel with H bits/ pixel. We want to know the en-tropy of the whole image or of the part of the image representing the subband. When weconsider two pixels at the same time we use the joint probability for the pixels xiand xj, can be represented by the relative frequency of the pair (i,j), all i,j.

only if xi and xj are independent. The entropy for the pair of pix-els is the joint entropy and we have

(3.2)

with equality if and only if the pixels xi and xj are independent! For the whole image we can write the joint entropy as

(3.3)

and if the pixels values are drawn from the same distribution then

(3.4)

with equality if they are independent.The entropy rate of the random vector variable representing our image is

bits / pixel. (3.5)

The collection of random variables in a vector is independent if and only if the joint(for the whole image) probability density function (pdf) can be writ-ten as

(3.6)

x x1 x2 … xN N×, , ,( )=xi n N N×=

p xi( )X X1 X2 … Xn, , ,( )= Xi

n ∞→

H x( ) pxk pxklogk 1=

N∑–=

p i j,( )p i j,( )

p i j,( ) p i( ) p j( )( )=H xi xj,( )

H xi xj,( ) H xi( ) H xj( )+≤

H image( ) H xi( )i 1=

n∑≤

H image( ) nH xi( )≤

X

H image( )n------------------------ H xi( )≤

p x1 x2 … xN N×, , ,( )

p x1 x2 … xN N×, , ,( ) p x1( )p x2( )…p xN N×( )=


44

Figure 3.2. The beach texture, its best basis tree using Haar filter and the energy entropy cost function to the left and Haar filter and the log energy cost function to the right.

Figure 3.3. The brickwall texture, its best basis tree using Haar filter and the energy entropy cost function to the left and Haar filter and the log energy cost function to the right.

Figure 3.4. The gravel texture, its best basis tree using Haar filter and the energy entropy cost function to the left and Haar filter and the log energy cost function to the right.

(0,0)

(1,0) (1,1)(1,2)(1,3

(2,0) (2,1)(2,2)(2,3)

(3,0) (3,1)(3,2) (3,3)

(4,0) (4,1) (4,2) (4,3) (4,12) (4,13) (4,14) (4,15)

(5,0) (5,1)(5,2)(5,3)(5,4)(5,5)(5,6)(5,7)(5,12)(5,13)(5,14)(5,15) (5,52)(5,53)(5,54)(5,55)(5,56)(5,57)(5,58)(5,59)(5,60)(5,61)(5,62)(5,63)

(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)(6,7) (6,24)(6,25)(6,26)(6,27) (6,52)(6,53)(6,54)(6,55)(6,60)(6,61)(6,62)(6,63) (6,216)(6,217)(6,218)(6,219)(6,224)(6,225)(6,226)(6,227)(6,232)(6,233)(6,234)(6,235)(6,244)(6,245)(6,246)(6,247)

(0,0)

(1,0) (1,1) (1,2) (1,3)

(2,0) (2,1)(2,2)(2,3)(2,4)(2,5)(2,6)(2,7)(2,12)(2,13)(2,14)(2,15

(3,0) (3,1)(3,2)(3,3)

(4,0) (4,1) (4,2) (4,3)

(5,0) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (5,7)(5,12)(5,13)(5,14)(5,15)

(6,0)(6,1)(6,2)(6,3) (6,8)(6,9)(6,10)(6,11)(6,12)(6,13)(6,14)(6,15)(6,16)(6,17)(6,18)(6,19)(6,24)(6,25)(6,26)(6,27)

(0,0)

(1,0) (1,1) (1,2) (1,3

(2,0) (2,1) (2,2) (2,3) (2,8) (2,9) (2,10) (2,11

(3,0) (3,1) (3,2) (3,3) (3,8) (3,9) (3,10) (3,11) (3,32) (3,33) (3,34) (3,35) (3,40) (3,41) (3,42) (3,43

(4,0)(4,1)(4,2)(4,3)(4,8)(4,9)(4,10)(4,11)(4,32)(4,33)(4,34)(4,35)(4,40)(4,41)(4,42)(4,43)(4,128)(4,129)(4,130)(4,131)(4,136)(4,137)(4,138)(4,139)(4,160)(4,161)(4,162)(4,163)(4,168)(4,169)(4,170)(4,171

(5,0)(5,1)(5,2)(5,3)(5,8)(5,9)(5,10)(5,11)(5,32)(5,33)(5,34)(5,35)(5,40)(5,41)(5,42)(5,43)(5,128)(5,129)(5,130)(5,131)(5,136)(5,137)(5,138)(5,139)(5,160)(5,161)(5,162)(5,163)(5,168)(5,169)(5,170)(5,171)(5,512)(5,513)(5,514)(5,515)(5,520)(5,521)(5,522)(5,523)(5,544)(5,545)(5,546)(5,547)(5,552)(5,553)(5,554)(5,555)(5,640)(5,641)(5,642)(5,643)(5,648)(5,649)(5,650)(5,651)(5,672)(5,673)(5,674)(5,675)(5,680)(5,681)(5,682)(5,683)

(6,0)(6,1)(6,2)(6,3)(6,8)(6,9)(6,10)(6,11)(6,32)(6,33)(6,34)(6,35)(6,40)(6,41)(6,42)(6,43)(6,128)(6,129)(6,130)(6,131)(6,136)(6,137)(6,138)(6,139)(6,160)(6,161)(6,162)(6,163)(6,168)(6,169)(6,170)(6,171)(6,512)(6,513)(6,514)(6,515)(6,520)(6,521)(6,522)(6,523)(6,544)(6,545)(6,546)(6,547)(6,552)(6,553)(6,554)(6,555)(6,556)(6,557)(6,558)(6,559)(6,640)(6,641)(6,642)(6,643)(6,648)(6,649)(6,650)(6,651)(6,672)(6,673)(6,674)(6,675)(6,680)(6,681)(6,682)(6,683)(6,2048)(6,2049)(6,2050)(6,2051)(6,2056)(6,2057)(6,2058)(6,2059)(6,2080)(6,2081)(6,2082)(6,2083)(6,2088)(6,2089)(6,2090)(6,2091)(6,2176)(6,2177)(6,2178)(6,2179)(6,2184)(6,2185)(6,2186)(6,2187)(6,2208)(6,2209)(6,2210)(6,2211)(6,2216)(6,2217)(6,2218)(6,2219)(6,2560)(6,2561)(6,2562)(6,2563)(6,2568)(6,2569)(6,2570)(6,2571)(6,2592)(6,2593)(6,2594)(6,2595)(6,2600)(6,2601)(6,2602)(6,2603)(6,2688)(6,2689)(6,2690)(6,2691)(6,2696)(6,2697)(6,2698)(6,2699)(6,2720)(6,2721)(6,2722)(6,2723)(6,2728)(6,2729)(6,2730)(6,2731)

(0,0)

(1,0) (1,1) (1,2) (1,3)

(2,0) (2,1) (2,2) (2,3)(2,4)(2,5)(2,6)(2,7) (2,8) (2,9)(2,10)(2,11)(2,12)(2,13(2,14(2,1

(3,0) (3,1) (3,2) (3,3)(3,4)(3,5)(3,6)(3,7) (3,8) (3,9)(3,10)(3,11) (3,32) (3,33) (3,34) (3,35)

(4,0)(4,1)(4,2)(4,3)(4,8)(4,9)(4,10)(4,11) (4,32)(4,33) (4,34)(4,35)(4,40)(4,41)(4,42)(4,43) (4,128) (4,129) (4,130)(4,131)(4,136)(4,137)(4,138)(4,139)

(5,0)(5,1)(5,2)(5,3)(5,8)(5,9)(5,10)(5,11)(5,32)(5,33)(5,34)(5,35)(5,40)(5,41)(5,42)(5,43)(5,128)(5,129)(5,130)(5,131)(5,132)(5,133)(5,134)(5,135)(5,136)(5,137)(5,138)(5,139)(5,160)(5,161)(5,162)(5,163) (5,512)(5,513)(5,514)(5,515)(5,516)(5,517)(5,518)(5,519)(5,520)(5,521)(5,522)(5,523)(5,552)(5,553)(5,554)(5,555)(5,556)(5,557)(5,558)(5,559)

(6,0)(6,1)(6,2)(6,3)(6,8)(6,9)(6,10)(6,11)(6,32)(6,33)(6,34)(6,35)(6,40)(6,41)(6,42)(6,43)(6,128)(6,129)(6,130)(6,131)(6,136)(6,137)(6,138)(6,139)(6,160)(6,161)(6,162)(6,163)(6,168)(6,169)(6,170)(6,171)(6,520)(6,521)(6,522)(6,523)(6,532)(6,533)(6,534)(6,535)(6,544)(6,545)(6,546)(6,547)(6,552)(6,553)(6,554)(6,555)(6,556)(6,557)(6,558)(6,559)(6,640)(6,641)(6,642)(6,643)(6,648)(6,649)(6,650)(6,651)(6,2048)(6,2049)(6,2050)(6,2051)(6,2056)(6,2057)(6,2058)(6,2059)(6,2064)(6,2065)(6,2066)(6,2067)(6,2072)(6,2073)(6,2074)(6,2075)(6,2076)(6,2077)(6,2078)(6,2079)(6,2080)(6,2081)(6,2082)(6,2083)(6,2088)(6,2089)(6,2090)(6,2091)(6,2092)(6,2093)(6,2094)(6,2095)(6,2208)(6,2209)(6,2210)(6,2211)(6,2216)(6,2217)(6,2218)(6,2219)(6,2232)(6,2233)(6,2234)(6,2235)

(0,0)

(1,0) (1,1) (1,2) (1,3

(2,0) (2,1) (2,2)(2,3) (2,4) (2,5) (2,6)(2,7) (2,8) (2,9)(2,10)(2,11

(3,0) (3,1)(3,2)(3,3) (3,4) (3,5)(3,6)(3,7) (3,16) (3,17) (3,18) (3,19) (3,20) (3,21)(3,22)(3,23) (3,32) (3,33)(3,34)(3,35)

(4,0)(4,1)(4,2)(4,3)(4,16)(4,17)(4,18)(4,19) (4,64)(4,65)(4,66)(4,67) (4,72) (4,73)(4,74)(4,75)(4,80)(4,81)(4,82)(4,83) (4,128) (4,129)(4,130)(4,131)(4,132)(4,133)(4,134)(4,135)

(5,0)(5,1)(5,2)(5,3) (5,72)(5,73)(5,74)(5,75) (5,256)(5,257)(5,258)(5,259)(5,288)(5,289)(5,290)(5,291)(5,296)(5,297)(5,298)(5,299)(5,320)(5,321)(5,322)(5,323)(5,328)(5,329)(5,330)(5,331) (5,512)(5,513)(5,514)(5,515)(5,516)(5,517)(5,518)(5,519)(5,528)(5,529)(5,530)(5,531)

(6,0)(6,1)(6,2)(6,3)(6,8)(6,9)(6,10)(6,11) (6,288)(6,289)(6,290)(6,291)(6,300)(6,301)(6,302)(6,303)(6,1024)(6,1025)(6,1026)(6,1027)(6,1028)(6,1029)(6,1030)(6,1031)(6,1032)(6,1033)(6,1034)(6,1035)(6,1152)(6,1153)(6,1154)(6,1155)(6,1156)(6,1157)(6,1158)(6,1159)(6,1160)(6,1161)(6,1162)(6,1163)(6,1164)(6,1165)(6,1166)(6,1167)(6,1184)(6,1185)(6,1186)(6,1187)(6,1192)(6,1193)(6,1194)(6,1195)(6,1288)(6,1289)(6,1290)(6,1291)(6,1316)(6,1317)(6,1318)(6,1319)(6,1324)(6,1325)(6,1326)(6,1327)(6,2048)(6,2049)(6,2050)(6,2051)(6,2052)(6,2053)(6,2054)(6,2055)(6,2064)(6,2065)(6,2066)(6,2067)(6,2068)(6,2069)(6,2070)(6,2071)(6,2116)(6,2117)(6,2118)(6,2119)(6,2120)(6,2121)(6,2122)(6,2123)

(0,0)

(1,0) (1,1) (1,2) (1,3)

(2,0) (2,1) (2,2)(2,3)(2,4)(2,5)(2,6)(2,7)(2,8)(2,9)(2,10)(2,11)(2,12)(2,13)(2,14)(2,15)

(3,0) (3,1)(3,2)(3,3)(3,4)(3,5)(3,6)(3,7)

(4,0) (4,1)(4,2)(4,3)

(5,0) (5,1)(5,2)(5,3)

(6,0)(6,1)(6,2)(6,3)


45

Figure 3.5. The beach image, its best basis tree using the Haar filter and the energy entropy cost function to the left and the DB4 filter with the energy entropy cost function to the right.

Figure 3.6. The brickwall image, its best basis tree using the Haar filter and the energy entropy cost function to the left and the DB4 filter with the energy entropy cost function to the right.

Figure 3.7. The gravel image, its best basis tree using the Haar filter and the energy entropy cost function to the left and the DB4 filter with the energy entropy cost function to the right.

(0,0)

(1,0) (1,1) (1,2) (1,3

(2,0) (2,1) (2,2)(2,3) (2,4) (2,5)(2,6)(2,7) (2,8) (2,9)(2,10)(2,11

(3,0) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6)(3,7)(3,8)(3,9)(3,10)(3,11)(3,12)(3,13)(3,14)(3,15) (3,16) (3,17)(3,18)(3,19) (3,32)(3,33)(3,34)(3,35)

(4,0) (4,1)(4,2)(4,3) (4,4) (4,5)(4,6)(4,7)(4,8)(4,9)(4,10)(4,11)(4,12)(4,13)(4,14)(4,15)(4,16) (4,17)(4,18)(4,19)(4,20) (4,21)(4,22)(4,23) (4,60)(4,61)(4,62)(4,63)(4,64)(4,65)(4,66)(4,67)(4,128)(4,129)(4,130)(4,131)

(5,0)(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)(5,7)(5,16)(5,17)(5,18)(5,19)(5,20)(5,21)(5,22)(5,23)(5,32)(5,33)(5,34)(5,35)(5,40)(5,41)(5,42)(5,43)(5,64)(5,65)(5,66)(5,67)(5,68)(5,69)(5,70)(5,71)(5,80)(5,81)(5,82)(5,83)(5,84)(5,85)(5,86)(5,87) (5,256)(5,257)(5,258)(5,259)(5,260)(5,261)(5,262)(5,263) (5,520)(5,521)(5,522)(5,523)

(6,0)(6,1)(6,2)(6,3)(6,20)(6,21)(6,22)(6,23)(6,64)(6,65)(6,66)(6,67)(6,68)(6,69)(6,70)(6,71)(6,76)(6,77)(6,78)(6,79)(6,84)(6,85)(6,86)(6,87)(6,128)(6,129)(6,130)(6,131)(6,136)(6,137)(6,138)(6,139)(6,168)(6,169)(6,170)(6,171)(6,260)(6,261)(6,262)(6,263)(6,264)(6,265)(6,266)(6,267)(6,276)(6,277)(6,278)(6,279)(6,324)(6,325)(6,326)(6,327)(6,340)(6,341)(6,342)(6,343)(6,348)(6,349)(6,350)(6,351) (6,1044)(6,1045)(6,1046)(6,1047)(6,1052)(6,1053)(6,1054)(6,1055) (6,2088)(6,2089)(6,2090)(6,2091)

(0,0)

(1,0) (1,1)(1,2)(1,3

(2,0) (2,1)(2,2)(2,3)

(3,0) (3,1)(3,2) (3,3)

(4,0) (4,1) (4,2) (4,3) (4,12) (4,13) (4,14) (4,15)

(5,0) (5,1)(5,2)(5,3)(5,4)(5,5)(5,6)(5,7)(5,12)(5,13)(5,14)(5,15) (5,52)(5,53)(5,54)(5,55)(5,56)(5,57)(5,58)(5,59)(5,60)(5,61)(5,62)(5,63)

(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)(6,7) (6,24)(6,25)(6,26)(6,27) (6,52)(6,53)(6,54)(6,55)(6,60)(6,61)(6,62)(6,63) (6,216)(6,217)(6,218)(6,219)(6,224)(6,225)(6,226)(6,227)(6,232)(6,233)(6,234)(6,235)(6,244)(6,245)(6,246)(6,247)

(0,0)

(1,0) (1,1) (1,2) (1,3

(2,0) (2,1) (2,2) (2,3)(2,4) (2,5) (2,6)(2,7) (2,8) (2,9) (2,10) (2,11

(3,0) (3,1) (3,2) (3,3)(3,4) (3,5)(3,6)(3,7)(3,8)(3,9)(3,10)(3,11)(3,16)(3,17)(3,18)(3,19)(3,20)(3,21)(3,22)(3,23)(3,32)(3,33)(3,34)(3,35)(3,40)(3,41)(3,42)(3,43

(4,0)(4,1)(4,2)(4,3)(4,4)(4,5)(4,6)(4,7)(4,8)(4,9)(4,10)(4,11)(4,16)(4,17)(4,18)(4,19)(4,20)(4,21)(4,22)(4,23)(4,32)(4,33)(4,34)(4,35)(4,40)(4,41)(4,42)(4,43)(4,64)(4,65)(4,66)(4,67)(4,80)(4,81)(4,82)(4,83)(4,84)(4,85)(4,86)(4,87)(4,128)(4,129)(4,130)(4,131)(4,136)(4,137)(4,138)(4,139)(4,160)(4,161)(4,162)(4,163)(4,168)(4,169)(4,170)(4,171

(5,0)(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)(5,7)(5,8)(5,9)(5,10)(5,11)(5,16)(5,17)(5,18)(5,19)(5,20)(5,21)(5,22)(5,23)(5,32)(5,33)(5,34)(5,35)(5,40)(5,41)(5,42)(5,43)(5,64)(5,65)(5,66)(5,67)(5,68)(5,69)(5,70)(5,71)(5,80)(5,81)(5,82)(5,83)(5,84)(5,85)(5,86)(5,87)(5,128)(5,129)(5,130)(5,131)(5,136)(5,137)(5,138)(5,139)(5,160)(5,161)(5,162)(5,163)(5,168)(5,169)(5,170)(5,171)(5,256)(5,257)(5,258)(5,259)(5,260)(5,261)(5,262)(5,263)(5,264)(5,265)(5,266)(5,267)(5,320)(5,321)(5,322)(5,323)(5,324)(5,325)(5,326)(5,327)(5,336)(5,337)(5,338)(5,339)(5,512)(5,513)(5,514)(5,515)(5,520)(5,521)(5,522)(5,523)(5,552)(5,553)(5,554)(5,555)(5,640)(5,641)(5,642)(5,643)(5,648)(5,649)(5,650)(5,651)(5,672)(5,673)(5,674)(5,675)(5,680)(5,681)(5,682)(5,683

(6,0)(6,1)(6,2)(6,3)(6,8)(6,9)(6,10)(6,11)(6,16)(6,17)(6,18)(6,19)(6,20)(6,21)(6,22)(6,23)(6,32)(6,33)(6,34)(6,35)(6,40)(6,41)(6,42)(6,43)(6,64)(6,65)(6,66)(6,67)(6,68)(6,69)(6,70)(6,71)(6,84)(6,85)(6,86)(6,87)(6,128)(6,129)(6,130)(6,131)(6,136)(6,137)(6,138)(6,139)(6,160)(6,161)(6,162)(6,163)(6,168)(6,169)(6,170)(6,171)(6,276)(6,277)(6,278)(6,279)(6,324)(6,325)(6,326)(6,327)(6,344)(6,345)(6,346)(6,347)(6,520)(6,521)(6,522)(6,523)(6,552)(6,553)(6,554)(6,555)(6,640)(6,641)(6,642)(6,643)(6,648)(6,649)(6,650)(6,651)(6,672)(6,673)(6,674)(6,675)(6,680)(6,681)(6,682)(6,683)(6,1044)(6,1045)(6,1046)(6,1047)(6,1284)(6,1285)(6,1286)(6,1287)(6,1344)(6,1345)(6,1346)(6,1347)(6,1348)(6,1349)(6,1350)(6,1351)(6,2048)(6,2049)(6,2050)(6,2051)(6,2056)(6,2057)(6,2058)(6,2059)(6,2088)(6,2089)(6,2090)(6,2091)(6,2560)(6,2561)(6,2562)(6,2563)(6,2568)(6,2569)(6,2570)(6,2571)(6,2592)(6,2593)(6,2594)(6,2595)(6,2688)(6,2689)(6,2690)(6,2691)(6,2720)(6,2721)(6,2722)(6,2723)(6,2728)(6,2729)(6,2730(6,2731

(0,0)

(1,0) (1,1) (1,2) (1,3

(2,0) (2,1) (2,2) (2,3) (2,8) (2,9) (2,10) (2,11

(3,0) (3,1) (3,2) (3,3) (3,8) (3,9) (3,10) (3,11) (3,32) (3,33) (3,34) (3,35) (3,40) (3,41) (3,42) (3,43

(4,0)(4,1)(4,2)(4,3)(4,8)(4,9)(4,10)(4,11)(4,32)(4,33)(4,34)(4,35)(4,40)(4,41)(4,42)(4,43)(4,128)(4,129)(4,130)(4,131)(4,136)(4,137)(4,138)(4,139)(4,160)(4,161)(4,162)(4,163)(4,168)(4,169)(4,170)(4,171

(5,0)(5,1)(5,2)(5,3)(5,8)(5,9)(5,10)(5,11)(5,32)(5,33)(5,34)(5,35)(5,40)(5,41)(5,42)(5,43)(5,128)(5,129)(5,130)(5,131)(5,136)(5,137)(5,138)(5,139)(5,160)(5,161)(5,162)(5,163)(5,168)(5,169)(5,170)(5,171)(5,512)(5,513)(5,514)(5,515)(5,520)(5,521)(5,522)(5,523)(5,544)(5,545)(5,546)(5,547)(5,552)(5,553)(5,554)(5,555)(5,640)(5,641)(5,642)(5,643)(5,648)(5,649)(5,650)(5,651)(5,672)(5,673)(5,674)(5,675)(5,680)(5,681)(5,682)(5,683)

(6,0)(6,1)(6,2)(6,3)(6,8)(6,9)(6,10)(6,11)(6,32)(6,33)(6,34)(6,35)(6,40)(6,41)(6,42)(6,43)(6,128)(6,129)(6,130)(6,131)(6,136)(6,137)(6,138)(6,139)(6,160)(6,161)(6,162)(6,163)(6,168)(6,169)(6,170)(6,171)(6,512)(6,513)(6,514)(6,515)(6,520)(6,521)(6,522)(6,523)(6,544)(6,545)(6,546)(6,547)(6,552)(6,553)(6,554)(6,555)(6,556)(6,557)(6,558)(6,559)(6,640)(6,641)(6,642)(6,643)(6,648)(6,649)(6,650)(6,651)(6,672)(6,673)(6,674)(6,675)(6,680)(6,681)(6,682)(6,683)(6,2048)(6,2049)(6,2050)(6,2051)(6,2056)(6,2057)(6,2058)(6,2059)(6,2080)(6,2081)(6,2082)(6,2083)(6,2088)(6,2089)(6,2090)(6,2091)(6,2176)(6,2177)(6,2178)(6,2179)(6,2184)(6,2185)(6,2186)(6,2187)(6,2208)(6,2209)(6,2210)(6,2211)(6,2216)(6,2217)(6,2218)(6,2219)(6,2560)(6,2561)(6,2562)(6,2563)(6,2568)(6,2569)(6,2570)(6,2571)(6,2592)(6,2593)(6,2594)(6,2595)(6,2600)(6,2601)(6,2602)(6,2603)(6,2688)(6,2689)(6,2690)(6,2691)(6,2696)(6,2697)(6,2698)(6,2699)(6,2720)(6,2721)(6,2722)(6,2723)(6,2728)(6,2729)(6,2730)(6,2731)

(0,0)

(1,0) (1,1) (1,2) (1,3

(2,0) (2,1) (2,2)(2,3) (2,4) (2,5) (2,6)(2,7) (2,8) (2,9) (2,10) (2,11

(3,0) (3,1) (3,2) (3,3) (3,4) (3,5)(3,6)(3,7)(3,8)(3,9)(3,10)(3,11)(3,12)(3,13)(3,14)(3,15)(3,16)(3,17)(3,18)(3,19) (3,20) (3,21)(3,22)(3,23) (3,32)(3,33)(3,34)(3,35)(3,40)(3,41)(3,42)(3,43

(4,0)(4,1)(4,2)(4,3)(4,4) (4,5)(4,6)(4,7)(4,8)(4,9)(4,10)(4,11)(4,16)(4,17)(4,18)(4,19) (4,64)(4,65)(4,66)(4,67)(4,80)(4,81)(4,82)(4,83)(4,84)(4,85)(4,86)(4,87)(4,128)(4,129)(4,130)(4,131)(4,160)(4,161)(4,162)(4,163)(4,168)(4,169)(4,170)(4,171

(5,0)(5,1)(5,2)(5,3)(5,16)(5,17)(5,18)(5,19)(5,20)(5,21)(5,22)(5,23)(5,32)(5,33)(5,34)(5,35)(5,40)(5,41)(5,42)(5,43)(5,64)(5,65)(5,66)(5,67)(5,68)(5,69)(5,70)(5,71)(5,76)(5,77)(5,78)(5,79) (5,256)(5,257)(5,258)(5,259)(5,260)(5,261)(5,262)(5,263)(5,320)(5,321)(5,322)(5,323)(5,324)(5,325)(5,326)(5,327)(5,332)(5,333)(5,334)(5,335)(5,336)(5,337)(5,338)(5,339)(5,512)(5,513)(5,514)(5,515)(5,520)(5,521)(5,522)(5,523)(5,640)(5,641)(5,642)(5,643)(5,648)(5,649)(5,650)(5,651)(5,672)(5,673)(5,674)(5,675)

(6,0)(6,1)(6,2)(6,3)(6,64)(6,65)(6,66)(6,67)(6,68)(6,69)(6,70)(6,71)(6,84)(6,85)(6,86)(6,87)(6,128)(6,129)(6,130)(6,131)(6,136)(6,137)(6,138)(6,139)(6,168)(6,169)(6,170)(6,171)(6,256)(6,257)(6,258)(6,259)(6,260)(6,261)(6,262)(6,263)(6,276)(6,277)(6,278)(6,279) (6,1024)(6,1025)(6,1026)(6,1027)(6,1044)(6,1045)(6,1046)(6,1047)(6,1052)(6,1053)(6,1054)(6,1055)(6,1284)(6,1285)(6,1286)(6,1287)(6,1292)(6,1293)(6,1294)(6,1295)(6,1296)(6,1297)(6,1298)(6,1299)(6,1344)(6,1345)(6,1346)(6,1347)(6,1352)(6,1353)(6,1354)(6,1355)(6,2048)(6,2049)(6,2050)(6,2051)(6,2088)(6,2089)(6,2090)(6,2091)(6,2568)(6,2569)(6,2570)(6,2571)(6,2592)(6,2593)(6,2594)(6,2595)(6,2688)(6,2689)(6,2690)(6,2691)

(0,0)

(1,0) (1,1) (1,2) (1,3

(2,0) (2,1) (2,2)(2,3) (2,4) (2,5) (2,6)(2,7) (2,8) (2,9)(2,10)(2,11

(3,0) (3,1)(3,2)(3,3) (3,4) (3,5)(3,6)(3,7) (3,16) (3,17) (3,18) (3,19) (3,20) (3,21)(3,22)(3,23) (3,32) (3,33)(3,34)(3,35)

(4,0)(4,1)(4,2)(4,3)(4,16)(4,17)(4,18)(4,19) (4,64)(4,65)(4,66)(4,67) (4,72) (4,73)(4,74)(4,75)(4,80)(4,81)(4,82)(4,83) (4,128) (4,129)(4,130)(4,131)(4,132)(4,133)(4,134)(4,135)

(5,0)(5,1)(5,2)(5,3) (5,72)(5,73)(5,74)(5,75) (5,256)(5,257)(5,258)(5,259)(5,288)(5,289)(5,290)(5,291)(5,296)(5,297)(5,298)(5,299)(5,320)(5,321)(5,322)(5,323)(5,328)(5,329)(5,330)(5,331) (5,512)(5,513)(5,514)(5,515)(5,516)(5,517)(5,518)(5,519)(5,528)(5,529)(5,530)(5,531)

(6,0)(6,1)(6,2)(6,3)(6,8)(6,9)(6,10)(6,11) (6,288)(6,289)(6,290)(6,291)(6,300)(6,301)(6,302)(6,303)(6,1024)(6,1025)(6,1026)(6,1027)(6,1028)(6,1029)(6,1030)(6,1031)(6,1032)(6,1033)(6,1034)(6,1035)(6,1152)(6,1153)(6,1154)(6,1155)(6,1156)(6,1157)(6,1158)(6,1159)(6,1160)(6,1161)(6,1162)(6,1163)(6,1164)(6,1165)(6,1166)(6,1167)(6,1184)(6,1185)(6,1186)(6,1187)(6,1192)(6,1193)(6,1194)(6,1195)(6,1288)(6,1289)(6,1290)(6,1291)(6,1316)(6,1317)(6,1318)(6,1319)(6,1324)(6,1325)(6,1326)(6,1327)(6,2048)(6,2049)(6,2050)(6,2051)(6,2052)(6,2053)(6,2054)(6,2055)(6,2064)(6,2065)(6,2066)(6,2067)(6,2068)(6,2069)(6,2070)(6,2071)(6,2116)(6,2117)(6,2118)(6,2119)(6,2120)(6,2121)(6,2122)(6,2123)


46

If each random variable has the same distribution we have an iid (independent, iden-tically distributed) random vector .

The probability density function (pdf) of a pixel describes how probable a pixel valueis. The relative frequency of the pixel value measures the probability of the pixel valueindependent of its neighbours. This memory-less assumption, together with the ergodic-ity assumption, is the common approximation made when estimating the pdf with thegrayscale histogram of the image.

Common models for the pdf of a memory-less source, used in image processing, arethe Laplacian density

, mx=0, (3.7)

and the Gaussian density

(3.8)

The generalized Gaussian density

(3.9)

(3.10)

can be used as a model of the pdf [54,23]. α models the variance and β is inverse propor-tional to the decreasing rate of the peak. This model can represent both the Laplacian(β=1), and the Gaussian (β =2) distribution as special cases.

For a memory-less source we can estimate the pdf with the grayscale histogram of theimage and the entropy rate of the source is given by Equation 3.1.

The entropy of the pixel value is the entropy true to the intentions of Shannon andcommon in the telecommunication community [77]. The calculated entropy of the pixelvalue gives us a hint of the compression rate achieved for a memory-less source. In orderto estimate the coding performance, the entropy of the pixel value has been used in ourexperiments knowing that we can have better compression with source coding algorithmsthat take care of the memory in the signal. The cost of representing the basis tree structureis not taken into consideration here. This is because we are interested in the properties ofthe transform alone and the cost of coding the decomposition tree is minor compared withthe cost of coding the values and positions of the transform components. Fine quantiza-tion of the components is used as these are typically represented with 8 bpp. The trans-form itself generates lossless compression with perfect reconstruction filters. No efficient

X

p x( ) 1σx 2-------------e

2 xσx

-------------–=

p x( ) 1σx 2π-----------------e

x mx–( )2

2σx2----------------------–

=

p x( ) Ke x α⁄( )β–=

K β

2αΓ 1β---

--------------------= Γ t( ) e u– ut 1– ud0

∞

∫=


47

bit allocating procedures are here applied on the component matrix in order to achievebetter compression.

3.2.2 Cost function effectsHere we describe the cost functions used in the triplet database. The behaviour of the costfunctions is very different depending on the range of the value of the components. Thenorm |x| and x2 have the same basic behaviour in the interval 0..1 as in 1..255 but costfunctions based on log(x) show different characteristics in the two different intervals. Al-though the curve form of the functions log(x2) and log(x) looks the same in the two inter-vals, working in one interval gives large negative values for inputs close to the lower endof the interval and values close to zero for the upper end of the interval while the otherinterval gives values close to zero for inputs close to the lower end of the interval andlarge values for the upper end of the interval

The functions x2log(x2), and xlog(x) show an even larger difference in the two inter-vals, see Figure 3.8 and Figure 3.9. In the interval 0..1 these functions are close to zero atboth ends of the interval with a maximum negative value in the middle of the intervalwhile in the interval 1..255 the functions are dominated by their x and x2 components andcan almost be approximated by them.

Figure 3.8. For the interval 0..1, x2log(x2), xlog(x), log(x2), log(x), x, x2.

Figure 3.9. For the interval 1..255, x2log(x2), xlog(x), log(x2), log(x), x, x2.

In the following, the inputs are in the interval 1..255. This is why the energy entropyshows almost the same results as the quadratic norm in most of the plots. Cost functionsbased on the form x2log(x2) and xlog(x) were originally designed for use on the probabil-ity distribution, on the interval 0..1 [77] and have the originally intended properties onlyin images normalized to the interval 0...1.

The best basis algorithm is used in the construction of the triplet database. The algo-rithm is dependent on the assumption that the cost function is additive. An additive costmeasure is defined by

(3.11)

where is the cost for the signal component i.The different cost functions we have used are presented below.

M C xi( )i

∑=

C xi( )


48

Entropy of the pixel energy (3.12)

This is the cost function used in the wavelet packet community under the misleadingname Shannon entropy. In this thesis we call this cost function the Energy Entropy. Thiscost function is applied to the subbands components in the interval 1..255, thus it func-tions almost like the quadratic norm.

We estimate the true entropy of a subband by calculating

Entropy of a subband (3.13)

with being the normalized histogram of the same subband. This is a fair approxi-mation of the entropy of the subband as long as the subband is not too small. It should benoted that this measure, if used as a cost function, is not additive.

Logarithmic cost (3.14)

The logarithm of the pixel energy is used as a cost function applied on the image inthe interval 1..255.

The Lp -norm (3.15)

The L1-norm and the L1.99-norm have been used as the cost function most likely to matchthe Laplacian and the Gaussian probability distributions. The L1 - norm, representing theabsolute mean of the grayscale values, and the L2 - norm are popular measures. The en-ergy measure L2 - norm, the quadratic norm, representing the variance of the grayscalevalues if the mean = 0, is not useful to make tree-pruning decisions when the decompo-sition is orthonormal. We use the L1.99 - norm as a measure close to the energy measureand we use some norms lying in between to see if any of them possibly match some gen-eralized Gaussian distribution.

3.2.3 Measurements of cost function effectsWe first present results on how cost functions affect the decompositions. We compare theentropy estimations for best bases chosen with different cost functions, while the othertwo variables are held constant.

Consider the texture images in Figure 3.10, brickwall and cslfleath.

Csh xk2 xk

2logk∑–=

Chest p x( ) p x( )logx 1=

N

∑–=

p x( )

Clog xk2log

k∑=

Cnp xkp

k∑=


49

Figure 3.10. The images brickwall (left), cslfleath (right).

In Figure 3.11, and Figure 3.14 the images are analysed with the Haar filter and theirassociated best bases are chosen with the different cost functions described insection 3.2.2. The cost of coding the components, estimated by Equation 3.1, from thebest basis is marked by o and the cost of coding the components from the correspondingwavelet basis is marked by x. It is expected that the x:s form a horizontal line as the meas-ure is taken on the same wavelet tree using the same decomposition filter. Analysis ofmore images is collected in Appendix B.

For both images, the wavelet packet basis performs better than the wavelet basis.Figure 3.12 and Figure 3.13 show the bases chosen with the Haar filter and the cost func-tions defined in Equation 3.13 and Equation 3.14 for the image cslfleath in Figure 3.10.These figures show basis trees with very different structure, and performance worse thanthe other cost functions, but still better than for the wavelet basis. However, note the scal-ing of the y-axis, in bits/pixels. The difference is very small! Since we assume losslesscoding without coarse quantization of the transform components the visual performance(measured in PSNR) is not affected, hence the performance of different bases is almostin-distinguishable.

The next example image (brickwall) produces similar results with the analysis filterHaar. Figure 3.15, Figure 3.16 and Figure 3.17 show bases with different performance.The basis in Figure 3.15 looks similar to the wavelet basis. However, this is not the basiswith performance most similar to the wavelet basis; that goes to the basis in Figure 3.16,with performance in position 3 in Figure 3.14.

While the bases chosen with the Haar filter and different cost functions all performvery close to the wavelet basis performance, we see another result when choosing anotheranalysis filter. When the images are analysed with the bior1,5 filter we see that instead ofhaving the difference in performance of different cost functions in the range of 0,01 bit/pixel we now have the difference in 2 or 3 bit/pixel between best and worse performance,Figure 3.18.

The plots in Figure 3.18, Figure 3.19, and Figure 3.20 representing the analysis withthe bior1.5 filter and the db4 filter clearly demonstrate the importance of using a filter anda cost function that work well together. For some combinations of filter, cost functionsand image, the best basis selection algorithm does not work well at all. Either the fullydecomposed tree is chosen or the algorithm chooses not to decompose the image at all,giving the entropy of the image itself. The fully decomposed image has reduced entropybut the best result is achieved with a carefully chosen best basis that represents the reso-nance in the image.


50

Figure 3.11. The cslfleath image decomposed with the Haar filter and where the cost function is varied. The cost of coding the components of the best basis is marked by o and the cost of coding

the components of the corresponding wavelet basis is marked by x.

Figure 3.12. The best basis for the triplet filter Haar, image cslfleath, and cost function hest, defined in Equation 3.13. The estimated entropy is shown in position 2 in Figure 3.11.

Figure 3.13. The best basis for the triplet filter Haar, image cslfleath, and cost function log, defined in Equation 3.14. The estimated entropy is shown in position 3 in Figure 3.11.

1 2 3 4 5 6 7 8 92.74

2.75

2.76

2.77

2.78

2.79

2.8

2.81

sh hest log n1 n12 n14 n16 n18 n199

haarcslfleath

(0,0)

(1,0) (1,1) (1,2) (1,3

(2,0) (2,1) (2,2) (2,3) (2,8) (2,9) (2,10) (2,11)

(3,0)(3,1)(3,2)(3,3)(3,8)(3,9)(3,10)(3,11)(3,12)(3,13)(3,14)(3,15)(3,32)(3,33)(3,34)(3,35)(3,36)(3,37)(3,38)(3,39)(3,40)(3,41)(3,42)(3,43)(3,44)(3,45)(3,46)(3,47)

(4,0)(4,1)(4,2)(4,3)(4,8)(4,9)(4,10)(4,11)(4,12)(4,13)(4,14)(4,15)(4,32)(4,33)(4,34)(4,35)(4,36)(4,37)(4,38)(4,39)(4,40)(4,41)(4,42)(4,43)(4,44)(4,45)(4,46)(4,47)(4,48)(4,49)(4,50)(4,51)(4,52)(4,53)(4,54)(4,55)(4,56)(4,57)(4,58)(4,59)(4,60)(4,61)(4,62)(4,63)(4,128)(4,129)(4,130)(4,131)(4,132)(4,133)(4,134)(4,135)(4,136)(4,137)(4,138)(4,139)(4,140)(4,141)(4,142)(4,143)(4,144)(4,145)(4,146)(4,147)(4,148)(4,149)(4,150)(4,151)(4,152)(4,153)(4,154)(4,155)(4,156)(4,157)(4,158)(4,159)(4,160)(4,161)(4,162)(4,163)(4,164)(4,165)(4,166)(4,167)(4,168)(4,169)(4,170)(4,171)(4,172)(4,173)(4,174)(4,175)(4,176)(4,177)(4,178)(4,179)(4,180)(4,181)(4,182)(4,183)(4,184)(4,185)(4,186)(4,187)(4,188)(4,189)(4,190(4,19

(5,0)(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)(5,7)(5,8)(5,9)(5,10)(5,11)(5,12)(5,13)(5,14)(5,15)(5,32)(5,33)(5,34)(5,35)(5,36)(5,37)(5,38)(5,39)(5,40)(5,41)(5,42)(5,43)(5,44)(5,45)(5,46)(5,47)(5,48)(5,49)(5,50)(5,51)(5,52)(5,53)(5,54)(5,55)(5,56)(5,57)(5,58)(5,59)(5,60)(5,61)(5,62)(5,63)(5,128)(5,129)(5,130)(5,131)(5,132)(5,133)(5,134)(5,135)(5,136)(5,137)(5,138)(5,139)(5,140)(5,141)(5,142)(5,143)(5,144)(5,145)(5,146)(5,147)(5,148)(5,149)(5,150)(5,151)(5,152)(5,153)(5,154)(5,155)(5,156)(5,157)(5,158)(5,159)(5,160)(5,161)(5,162)(5,163)(5,164)(5,165)(5,166)(5,167)(5,168)(5,169)(5,170)(5,171)(5,172)(5,173)(5,174)(5,175)(5,176)(5,177)(5,178)(5,179)(5,180)(5,181)(5,182)(5,183)(5,184)(5,185)(5,186)(5,187)(5,188)(5,189)(5,190)(5,191)(5,192)(5,193)(5,194)(5,195)(5,196)(5,197)(5,198)(5,199)(5,200)(5,201)(5,202)(5,203)(5,204)(5,205)(5,206)(5,207)(5,208)(5,209)(5,210)(5,211)(5,212)(5,213)(5,214)(5,215)(5,216)(5,217)(5,218)(5,219)(5,220)(5,221)(5,222)(5,223)(5,224)(5,225)(5,226)(5,227)(5,228)(5,229)(5,230)(5,231)(5,232)(5,233)(5,234)(5,235)(5,236)(5,237)(5,238)(5,239)(5,240)(5,241)(5,242)(5,243)(5,244)(5,245)(5,246)(5,247)(5,252)(5,253)(5,254)(5,255)(5,512)(5,513)(5,514)(5,515)(5,516)(5,517)(5,518)(5,519)(5,520)(5,521)(5,522)(5,523)(5,524)(5,525)(5,526)(5,527)(5,528)(5,529)(5,530)(5,531)(5,532)(5,533)(5,534)(5,535)(5,536)(5,537)(5,538)(5,539)(5,540)(5,541)(5,542)(5,543)(5,544)(5,545)(5,546)(5,547)(5,548)(5,549)(5,550)(5,551)(5,552)(5,553)(5,554)(5,555)(5,556)(5,557)(5,558)(5,559)(5,560)(5,561)(5,562)(5,563)(5,564)(5,565)(5,566)(5,567)(5,568)(5,569)(5,570)(5,571)(5,572)(5,573)(5,574)(5,575)(5,576)(5,577)(5,578)(5,579)(5,580)(5,581)(5,582)(5,583)(5,584)(5,585)(5,586)(5,587)(5,588)(5,589)(5,590)(5,591)(5,592)(5,593)(5,594)(5,595)(5,596)(5,597)(5,598)(5,599)(5,600)(5,601)(5,602)(5,603)(5,604)(5,605)(5,606)(5,607)(5,608)(5,609)(5,610)(5,611)(5,612)(5,613)(5,614)(5,615)(5,616)(5,617)(5,618)(5,619)(5,620)(5,621)(5,622)(5,623)(5,624)(5,625)(5,626)(5,627)(5,628)(5,629)(5,630)(5,631)(5,632)(5,633)(5,634)(5,635)(5,636)(5,637)(5,638)(5,639)(5,640)(5,641)(5,642)(5,643)(5,644)(5,645)(5,646)(5,647)(5,648)(5,649)(5,650)(5,651)(5,652)(5,653)(5,654)(5,655)(5,656)(5,657)(5,658)(5,659)(5,660)(5,661)(5,662)(5,663)(5,664)(5,665)(5,666)(5,667)(5,668)(5,669)(5,670)(5,671)(5,672)(5,673)(5,674)(5,675)(5,676)(5,677)(5,678)(5,679)(5,680)(5,681)(5,682)(5,683)(5,684)(5,685)(5,686)(5,687)(5,688)(5,689)(5,690)(5,691)(5,692)(5,693)(5,694)(5,695)(5,696)(5,697)(5,698)(5,699)(5,700)(5,701)(5,702)(5,703)(5,704)(5,705)(5,706)(5,707)(5,708)(5,709)(5,710)(5,711)(5,712)(5,713)(5,714)(5,715)(5,716)(5,717)(5,718)(5,719)(5,720)(5,721)(5,722)(5,723)(5,724)(5,725)(5,726)(5,727)(5,728)(5,729)(5,730)(5,731)(5,732)(5,733)(5,734)(5,735)(5,736)(5,737)(5,738)(5,739)(5,740)(5,741)(5,742)(5,743)(5,744)(5,745)(5,746)(5,747)(5,748)(5,749)(5,750)(5,751)(5,752)(5,753)(5,754)(5,755)(5,756)(5,757)(5,758)(5,759)(5,760(5,761(5,762(5,763(5,764(5,76(5,76(5,76

(0,0)

(1,0) (1,1)(1,2)(1,3

(2,0) (2,1) (2,2) (2,3)

(3,0) (3,1) (3,2) (3,3)(3,4)(3,5)(3,6) (3,7) (3,8) (3,9) (3,10) (3,11)

(4,0)(4,1)(4,2)(4,3)(4,4)(4,5)(4,6)(4,7)(4,8)(4,9)(4,10)(4,11)(4,12)(4,13)(4,14)(4,15)(4,24)(4,25)(4,26)(4,27)(4,28)(4,29)(4,30)(4,31)(4,32)(4,33)(4,34)(4,35)(4,36)(4,37)(4,38)(4,39)(4,40)(4,41)(4,42)(4,43)(4,44)(4,45)(4,46)(4,47)

(5,0)(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)(5,7)(5,8)(5,9)(5,10)(5,11) (5,32)(5,33)(5,34)(5,35)(5,36)(5,37)(5,38)(5,39)(5,40)(5,41)(5,42)(5,43)(5,44)(5,45)(5,46)(5,47)(5,52)(5,53)(5,54)(5,55)(5,100)(5,101)(5,102)(5,103)(5,116)(5,117)(5,118)(5,119)(5,120)(5,121)(5,122)(5,123)(5,136)(5,137)(5,138)(5,139)(5,140)(5,141)(5,142)(5,143)(5,144)(5,145)(5,146)(5,147)(5,148)(5,149)(5,150)(5,151)(5,164)(5,165)(5,166)(5,167)(5,168)(5,169)(5,170)(5,171)(5,172)(5,173)(5,174)(5,175)(5,176)(5,177)(5,178)(5,179)(5,180)(5,181)(5,182)(5,183)(5,188)(5,189)(5,190)(5,191)


51

Figure 3.14. The brickwall image decomposed with the Haar filter and where the cost function is varied. The cost of coding the components of the best basis is marked by o and the cost of coding

the components of the corresponding wavelet basis is marked by x.

Figure 3.15. The best basis for the triplet filter Haar, image brickwall, and cost function hest, defined in Equation 3.13. The estimated entropy is shown in position 2 in Figure 3.14.

Figure 3.16. The best basis for the triplet filter Haar, image brickwall, and cost function log, defined in Equation 3.14. The estimated entropy is shown in position 3 in Figure 3.14.

1 2 3 4 5 6 7 8 91.32

1.33

1.34

1.35

1.36

1.37

1.38

1.39

1.4

1.41


haarbrickwall

(0,0)

(1,0) (1,1) (1,2) (1,3)

(2,0) (2,1) (2,2) (2,3)

(3,0) (3,1) (3,2) (3,3) (3,8) (3,9) (3,10)(3,11)

(4,0) (4,1) (4,2) (4,3)

(5,0) (5,1) (5,2) (5,3)

(0,0)

(1,0) (1,1) (1,2) (1,3)

(2,0) (2,1) (2,2) (2,3)(2,4)(2,5)(2,6)(2,7) (2,8) (2,9)(2,10)(2,11)(2,12)(2,13)(2,14)(2,15

(3,0) (3,1) (3,2) (3,3)(3,4)(3,5)(3,6)(3,7) (3,8) (3,9)(3,10)(3,11) (3,32) (3,33)(3,34)(3,35)

(4,0)(4,1)(4,2)(4,3) (4,8)(4,9)(4,10)(4,11) (4,32)(4,33)(4,34)(4,35)(4,40)(4,41)(4,42)(4,43) (4,128)(4,129)(4,130)(4,131)(4,136)(4,137)(4,138)(4,139)

(5,0)(5,1)(5,2)(5,3)(5,8)(5,9)(5,10)(5,11)(5,32)(5,33)(5,34)(5,35) (5,128)(5,129)(5,130)(5,131) (5,512)(5,513)(5,514)(5,515)


52

Figure 3.17. The best basis for the triplet filter Haar, image brickwall, and cost function sh, defined in Equation 3.12. The estimated entropy is shown in position 1 in Figure 3.14.

Figure 3.18. The cslfleath image decomposed with the filter bior1.5 and where the cost function is varied. The cost of coding the components of the optimal wavelet packet basis is marked by o and

the cost of coding the components of the corresponding wavelet basis is marked by x.

(0,0)

(1,0) (1,1) (1,2) (1,3

(2,0) (2,1) (2,2) (2,3) (2,8) (2,9) (2,10) (2,11)

(3,0) (3,1) (3,2) (3,3) (3,8) (3,9) (3,10) (3,11) (3,32) (3,33) (3,34) (3,35) (3,40) (3,41) (3,42) (3,43)

(4,0)(4,1)(4,2)(4,3)(4,8)(4,9)(4,10)(4,11)(4,32)(4,33)(4,34)(4,35)(4,40)(4,41)(4,42)(4,43) (4,128)(4,129)(4,130)(4,131)(4,136)(4,137)(4,138)(4,139)(4,160)(4,161)(4,162)(4,163)(4,168)(4,169)(4,170)(4,171)

(5,0)(5,1)(5,2)(5,3)(5,8)(5,9)(5,10)(5,11)(5,32)(5,33)(5,34)(5,35)(5,40)(5,41)(5,42)(5,43)(5,128)(5,129)(5,130)(5,131)(5,136)(5,137)(5,138)(5,139)(5,160)(5,161)(5,162)(5,163)(5,168)(5,169)(5,170)(5,171)(5,512)(5,513)(5,514)(5,515)(5,520)(5,521)(5,522)(5,523)(5,544)(5,545)(5,546)(5,547)(5,552)(5,553)(5,554)(5,555)(5,640)(5,641)(5,642)(5,643)(5,648)(5,649)(5,650)(5,651)(5,672)(5,673)(5,674)(5,675)(5,680)(5,681)(5,682)(5,683)

1 2 3 4 5 6 7 8 92.5

3

3.5

4

4.5

5

5.5

6

6.5

7

7.5


bior1.5cslfleath


53

Figure 3.19. The brickwall image decomposed with the db4 filter and where the cost function is varied. The cost of coding the components of the optimal wavelet packet basis is marked by o and


Figure 3.20. The cslfleath image decomposed with the db4 filter and where the cost function is varied. The cost of coding the components of the optimal wavelet packet basis is marked by o and


1 2 3 4 5 6 7 8 91.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2


db4brickwall

1 2 3 4 5 6 7 8 92.5

3

3.5

4

4.5

5

5.5

6

6.5


db4cslfleath


54

3.2.4 Filter effectsIn this section we review some filter properties important to image analysis and compres-sion and present the filters used in the triplet database. This is done in preparation for theanalysis of the triplet database from a filter point of view presented in section 3.2.5. Theresults presented in this section are proved in the text books [19,56,90,93] if no other ref-erence is given.

Orthogonal filters implement a unitary transform between the original and its sub-bands. One of the important features of unitary transforms is conservation of energy. Thisis essential in section 3.4. For compression applications we want the wavelet and waveletpacket transform to produce only a few non-zero components.

Definition. Vanishing moments [56]. has p vanishing moments if

(3.16)

Suppose we want to decompose a function into wavelets. We compute all thewavelet components . If is L-1 times continuously differentiable and has L vanishing moments, we will only have large values of the wavelet components forsingularities of or its derivatives. This is why the edges are clearly visible in the trans-form images. This also means that the signal properties dictate the number of vanishingmoments required for the wavelet to be able to decompose the signal in a manner suitablefor compression.

Because of the cascading effects of the wavelets in a wavelet or wavelet packet de-composition we get accumulated errors from the filtering process. The decompositioncan be described by the equivalent filter obtained by the cascading of the filtersH0 and H1.

(3.17)

where dr is the depth of the cascade of path r and indicates which of thefilters H0 or H1 is used.

Linear phase is desired since such filters can be cascaded in a pyramidal filter struc-ture without the need for phase compensation. Symmetric filters have linear phase. Ex-cept for the Haar basis, all real orthonormal wavelet bases with compact support areasymmetric. Using compactly supported orthonormal wavelets (other than Haar) symme-try and exact reconstruction are incompatible requirements if the same filter is used forreconstruction and decomposition. Biorthogonal wavelet bases uses two dual (symmet-ric) wavelet filters. Linear filters are wanted in order to avoid phase distortion aroundedges.

There is a link between the regularity of and the multiplicity of the zero at of , where denoted the Fourier transform of .

ψ

ykψ t( ) td∞–

∞

∫ 0= 0 k p<≤

F x( )F ψj k,,⟨ ⟩ F ψ

F

Hr z( )

Hr z( ) HPrz2k

( )k 1=

dr 1–

∏=

Pr k( ) 0 1,{ }∈

ψ ξ 0=ψ ψ ψ


55

Definition. Lipschitz regularity [56]. A function f is bounded and uniformly Lipschitz over if

(3.18)

These zeros are important because they “kill” aliasing components when cascadingthe filters. Regularity is concerned with differentiability. In the Fourier domain the exist-ence of derivatives is related to the decay of the Fourier spectra. For fixed support widthof i.e. for fixed length of the filters in the associated subband coding scheme, thechoice with maximum vanishing moments for is different from the maximum regu-larity choice of . Antonini et al. [3] found that for wavelet coding, the regularity of thereconstruction wavelet is important in a wavelet compression scheme. For the samenumber of vanishing moments for the scheme with most regular performs the best.Increasing regularity of , even at the expense of the number of vanishing moments of

performs better. For comparable regularity of the scheme with the largest vanishingmoment for is likely to perform best.

The situation is slightly different in a WP coding scheme but it still remains that if has k moments which are zero then the coefficients will represent the fea-tures of f which are k times differentiable with a high compression potential, many coef-ficients will be close to zero.

The smoothness of a wavelet is regarded as an important property. Since images aremostly smooth except for occasional edges it is argued [3] that an exact reconstructionsubband coding schema for image analysis should correspond to an orthonormal basiswith a reasonably smooth mother wavelet. However, the frequency domain criteria suchas constraints on high frequency components, are less important to image compressionapplications than time domain criteria such as ringing.

The compact support of a filter kernel is vital to the time frequency resolution. It istherefore important to have filters that are well localized in space (in time for time sig-nals). The fact that the wavelet is not smooth, such as db4, does not exclude the represen-tation of smooth functions. The right linear combination of these unsmooth wavelets isvery good for representing smooth functions as the irregularities cancel out. Compact andunsmooth functions are efficient in representing functions with edges and other disconti-nuities but smooth functions are not compact.

During the years of the development of the wavelet theory new filters have appeared,fullfilling the demands from the wavelet theory, and have given new life to older signalprocessing algorithms. Below the filters used in the triplet database are presented.

Haar wavelet is a compactly supported wavelet, the oldest and the simplest wavelet.The scaling function = 1 on [0 1] and 0 otherwise. The wavelet function = 1 on [00.5[ , = -1 on [0.5 1] and 0 otherwise. The Haar wavelet is orthogonal and symmetricbut not continuous. The number of vanishing moments for is 1.

In the group of orthogonal and compactly supported wavelets, the Daubechies wave-let, symlets and coiflets reside. Their common properties are that exists and the decom-position is orthogonal, and are compactly supported and has a given number ofvanishing moments. The regularity is poor.

Daubechies wavelets are compactly supported wavelets with extremal phase and a

α R

f ω( ) 1 ω α+( ) ωd∞–

∞

∫ ∞<

φ ψ ,ψ

hnψ

ψ ψψ

ψ ψψ

ψψm n, f >,<

φ ψψ

ψ

φψ φ ψ


56

high number of vanishing moments for a given support width. Associated scaling filteris minimum-phase filter. For order N the support width is 2N-1, filter length is 2N, reg-ularity about 0.2 N for large N. The wavelets are asymmetric and the number of vanishingmoments for is N.

Symlets are compactly supported wavelets with least asymmetry and a high numberof vanishing moments for a given support width. The associated scaling filter is near lin-ear-phase filter. For order N the support width is 2N-1, filter length is 2N. The filters arenearly symmetric and the number of vanishing moments for is N.

Coiflets are compactly supported nearly symmetric wavelets with high number ofvanishing moments for both and for a given support width. For order N the supportwidth is 6N-1 and filter length is 6N. The number of vanishing moments for is 2N andthe number of vanishing moments for is 2N-1.

The biorthogonal wavelets are compactly supported wavelet pairs constructed fromB-splines such that symmetry with FIR filters is achieved and desirable properties for de-composition and reconstruction are achieved. functions exist and the decomposition isbiorthogonal, and both for decomposition and reconstruction are compactly sup-ported, and for decomposition have vanishing moments. and for reconstruc-tion have known regularity. For order Nr, Nd (r for reconstruction, d for decomposition)the support width is 2Nr+1 for reconstruction and 2Nd+1 for decomposition. Regularityfor the symmetric reconstruction is Nr-1 and the number of vanishing moments for decomposition is Nr.

For the compactly supported biorthogonal spline wavelets symmetry and exact recon-struction are possible with FIR filters while in the orthogonal case this is impossible ex-cept for the Haar wavelet.

3.2.5 Measurements of filter effectsWe test the performance of the best basis algorithm with a set of the first generationwavelet filters. A listing of filter coefficients can be found in Appendix C.

The filters tested here are Haar, db2, db4, db6, db8, bior1.5, bior2.8, bior3.5, bior6.8,coif1, coif2, coif5, sym2, sym4, sym8, with respective vanishing moments, 1, 2, 4, 6, 8,5, 8, 5, 8, 2, 4, 10, 2, 4, 8. The results for different combinations are shown in AppendixB and in figures Figure 3.21 and Figure 3.22.

Only the decomposition part is considered here, the theoretically important factor isvanishing moments for the decomposition wavelet. The regularity of the reconstructionwavelet affects the image quality but not the compression ratio. When looking at the im-ages in Figure 3.21, Figure 3.22, and in Appendix B, the vanishing moments do not seemto be the important factor. In fact the performance of the WP coder gets worse with in-creasing vanishing moments. The theory holds, in practice, only for the wavelet coderwith symlet filters.

One explanation is that filtering with the wavelet filter gives large values for imagesingularities. Most of the filtering in the WP decomposition is done on the detail images.We need a wavelet with more vanishing moments than the numbers of continuous deriv-atives of the signal for a good compression potential. However the detail bands are notcontinuously differentiable. Thus we need not have more than one vanishing moment forthe wavelet on the detail bands.

The cost functions in Equation 3.12, in Equation 3.13 and in Equation 3.14 give dif-

ψ

ψ

φ ψψ

φ

φψ φ

φ ψ ψ φ

ψ ψ


57

ferent performance for the biorthogonal filters. However, the results are worse than thewavelet tree by 0,5 to 5 bpp while the cost function in Equation 3.15, with p=1, followsthe wavelet performance more closely. The Haar filter always gives better performancefor the WP tree than for the wavelet tree but not necessarily the best performance of thefilters. For the wavelet coder the filters db6, bior2.8, coif5 and sym8 perform well with

0 5 10 152

2.5

3

3.5norm1cslfleath

haar db2 db4 db6 db8 bior1.5 bior2.8 bior3.5 bior6.8 coif1 coif2 coif5 sym2 sym4 sym8

Figure 3.21. The cslfleath image decomposed with varying filters and the cost function n1. The cost of coding the components of the optimal wavelet packet basis is marked by o and the cost of

coding the components of the corresponding wavelet basis is marked by x.

0 5 10 152

3

4

5

6

7

8hestcslfleath


Figure 3.22. The cslfleath image decomposed with varying filters and the cost function hest. The cost of coding the components of the best basis is marked by o and the cost of coding the

components of the corresponding wavelet basis is marked by x.


58

the brickwall image while the bior3.5 is outstanding for the other images. The symmetry of the decomposing wavelet is more important in a wavelet packet de-

composition than in a wavelet decomposition. In the wavelet decomposition only the lin-ear phase of the scaling function matters since the repetition of decomposition is doneonly on the approximation branch. Daubechies compactly supported wavelets have ex-tremal phase and the highest number of vanishing moments for a given support width.Only the associated scaling filters are minimum-phase filters. This makes them not assuitable for WP decomposition. Symlets are least asymmetric and compactly supportedhaving the highest number of vanishing moments for a given support width with associ-ated scaling filters near linear-phase making them a little bit more suitable. Both Daub-echies compactly supported wavelets and Symlet wavelets have poor regularity. Coifletsare orthonormal and nearly symmetric wavelets with vanishing moments for both and

. They are much more symmetric than the compactly supported wavelets but at theprice of wider support.

3.2.6 ConclusionAfter analysing the result of combining 17 images, 9 cost functions and 15 differentwavelet filters in all possible ways we come to the conclusion that in the context of loss-less image compression we are free to choose cost function by a criterion other than theexplicit cost of coding the components. The compression does not vary very much withthe choice of cost function. Almost none of the tested cost functions for choosing a basisgenerally finds a basis better than the wavelet tree for all filters.

The L1-norm and the L1.99-norm have been used as the cost function most likely tomatch the Laplacian and the Gaussian probability distributions. We have used somenorms lying in between to see if any of them possibly match some generalized Gaussiandistribution. The main part of the subbands has a distribution very like the Laplacian.There is very little difference in the performance with different values of the p parameterin the Lp-norm. The best choice of cost function is the L1-norm.

The bases chosen with the Haar filter and different cost functions all perform veryclose to the wavelet basis performance. When the images are analysed with the bior1.5filter, a difference of up to 3 bit/pixel between different cost functions is produced.

The fact that cascading of filters is done both on the approximation band and the detailbands causes trouble in a wavelet packet decomposition with wavelet filters. The sym-metry of the decomposing wavelet is more important in a wavelet packet decompositionthan in a wavelet decomposition where only the linear phase of the scaling function mat-ters since the repetition of decomposition is done only on the approximation branch. Mostcompactly supported wavelets have extremal phase and a high number of vanishing mo-ments for a given support width. Only the associated scaling filters are minimum-phasefilters. This makes them not suitable for wavelet packet decomposition due to the cascad-ing effects and, thus accumulated phase errors in the detail bands. Most of the filtering inthe wavelet packet decomposition is done on the detail images which are not continuous-ly differentiable. We need a wavelet with more vanishing moments than the number ofcontinuous derivatives of the signal for a good compression potential. This means that weneed only one vanishing moment for the wavelet on the detail bands. The Haar waveletfulfills this demand.

The conclusion is that the choice of filter is important when we want to minimize the

ψφ

3.3 Performance of the CWP coder

59

coding cost. The Haar filter gives the wavelet packet basis better performance in combi-nation with almost all of the cost functions tested here. The Haar wavelet is the only real,compactly supported, symmetric, linear phase wavelet that gives perfect reconstruction.Indeed it is the only filter in this set suitable for wavelet packet transform for compres-sion.

3.3 Performance of the CWP coderThe CWP coder, described in section 2.3.4, is a wavelet packet coder that is based on theSPIHT coder, presented in section 2.2.3, for the bit allocation and quantization step. CWPuses a constrained WP tree because the SPIHT coder is designed for the wavelet tree andthe reordering algorithm cannot handle all possible WP trees. A suboptimal basis is cho-sen as close to the best basis as possible.

We use the CWP coder to show the effects of using an optimal basis compared to theuse of a suboptimal basis and also compared to the performance of the correspondingwavelet coder. The coder is implemented with the use of the Haar filter and the n1 costfunction for the WP decomposition as this combination has given better coding perform-ance than the corresponding wavelet coder with the same filter for all images tested insection 3.2. In addition to the analysis in section 3.2, we here take the coding of the basisitself into account. The final step of the SPIHT coder is not implemented. This step con-sists of entropy coding of the bitstream. Thus we will get higher bitrates than publishedresults from using the SPIHT coder on the same images. Since we use the same codingwhen comparing the different bases, this absence of the last coding step is expected to beof minor importance.


60

LennaThe above SPIHT wavelet coder and the CWP coder are both used to code the Lenna im-age (512x512 pixels). Both coders use the Haar wavelet, decomposing to level 6. TheCWP uses n1 as cost function. Figure 3.23 shows the result. The suboptimal tree of theCWP coder is seen in Figure 3.24. There are virtually no differences between the CWPcoder and the SPIHT coder. This is expected as the Lenna image contains mixed textures.Thus the WP algorithm has no chance to find a common resonance in the image.

Figure 3.23. PSNR as a function of bpp for the Lenna images, * is wavelet coder and o is wavelet packet coder.

Figure 3.24. Left: the constrained tree used in the SPIHT coder, right: the unconstrained wavelet packet tree representing the best basis chosen for the actual triplet.

0 0.5 1 1.515

20

25

30

35

40

(0,0)

(1,0) (1,1) (1,2)(1,3)

(2,0) (2,1) (2,2)(2,3) (2,4) (2,5)(2,6)(2,7)

(3,0) (3,1)(3,2)(3,3)(3,4)(3,5)(3,6)(3,7) (3,16)(3,17)(3,18)(3,19)

(4,0) (4,1)(4,2)(4,3)

(5,0) (5,1)(5,2)(5,3)

(6,0)(6,1)(6,2)(6,3)

(0,0)

(1,0) (1,1) (1,2)(1,3

(2,0) (2,1) (2,2)(2,3) (2,4) (2,5)(2,6)(2,7)

(3,0) (3,1)(3,2)(3,3) (3,4) (3,5)(3,6)(3,7) (3,16) (3,17)(3,18)(3,19)

(4,0) (4,1) (4,2)(4,3) (4,16) (4,17)(4,18)(4,19) (4,64) (4,65)(4,66)(4,67)

(5,0) (5,1)(5,2)(5,3)(5,4)(5,5)(5,6)(5,7) (5,64) (5,65)(5,66)(5,67) (5,256) (5,257)(5,258)(5,259)

(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)(6,7) (6,16)(6,17)(6,18)(6,19) (6,256)(6,257)(6,258)(6,259)(6,260)(6,261)(6,262)(6,263) (6,1024)(6,1025)(6,1026)(6,1027)(6,1028)(6,1029)(6,1030)(6,1031)


61

BarkFigure 3.25 shows the corresponding results for the Bark image. Only marginal differencebetween the CWP coder and the SPIHT coder can be seen. Also for the bark image a con-strained, suboptimal tree is used, similar to the wavelet tree in Figure 3.26.

Figure 3.25. PSNR as a function of bpp for the bark images, * is wavelet coder and o is wavelet packet code

Figure 3.26. Left: the constrained tree used in the SPIHT coder, right: the unconstrained wavelet packet tree representing the best basis chosen for the actual triplet.

0 0.5 1 1.5 2 2.514

16

18

20

22

24

26

28

30

32

(0,0)

(1,0) (1,1) (1,2)(1,3)

(2,0) (2,1) (2,2)(2,3) (2,4) (2,5)(2,6)(2,7)

(3,0) (3,1)(3,2)(3,3)(3,4)(3,5)(3,6)(3,7) (3,16)(3,17)(3,18)(3,19)

(4,0) (4,1)(4,2)(4,3)

(5,0) (5,1)(5,2)(5,3)

(6,0)(6,1)(6,2)(6,3)

(0,0)

(1,0) (1,1) (1,2) (1,3)

(2,0) (2,1) (2,2) (2,3) (2,8) (2,9)(2,10)(2,11)

(3,0) (3,1) (3,2) (3,3) (3,8) (3,9)(3,10)(3,11) (3,32) (3,33)(3,34)(3,35)

(4,0) (4,1)(4,2)(4,3) (4,8) (4,9)(4,10)(4,11) (4,32) (4,33)(4,34)(4,35) (4,128) (4,129)(4,130)(4,131)

(5,0)(5,1)(5,2)(5,3) (5,32)(5,33)(5,34)(5,35) (5,128)(5,129)(5,130)(5,131)(5,132)(5,133)(5,134)(5,135) (5,512)(5,513)(5,514)(5,515)(5,516)(5,517)(5,518)(5,519)


62

GrassThe results in Figure 3.28 show the degradation in PSNR when coding the grass image.

Figure 3.27. To the left: grass coded with SPIHT at 0.7593 bpp, psnr=19.9694. To the right: grass coded with constrained CWP at 0.7499 bpp, psnr=20.1663.

Although the constrained tree for the grass image is not exactly the same as the bestbasis tree the constrained tree is more similar to the best tree than to the wavelet tree andcaptures most of the resonance (Figure 3.29). The visual structure is kept at lower bitratesfor the WP coder than for the wavelet coder, Figure 3.27.

Figure 3.28. PSNR as a function of bpp for the grass images, * is wavelet coder and o is wavelet packet coder.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.812

14

16

18

20

22

24

26


63

Figure 3.29. Left: the constrained tree used in the CWP coder, right: the unconstrained wavelet packet tree representing the best basis chosen for the actual triplet.

BrickwallFigure 3.34 shows the corresponding results for the brickwall image.

For the brickwall image the CWP coder uses the best basis (Figure 3.30). A clear dif-ference is seen both visually and in the PSNR, Figure 3.34. The best basis captures theresonance which is better represented by the CWP coder. From the reconstructed imagesin Figure 3.31 to Figure 3.33 we see the different degradation in visual performance forthe SPIHT coder and for the CWP coder at low bitrates. The typical structure for the tex-ture is kept longer for the WP coder.

Figure 3.30. left: the constrained tree used in the CWP coder, right: the unconstrained wavelet packet tree representing the best basis chosen for the actual triplet.

(0,0)

(1,0) (1,1) (1,2) (1,3)

(2,0) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (2,7) (2,12) (2,13) (2,14) (2,15)

(3,0) (3,1)(3,2)(3,3)(3,4)(3,5)(3,6)(3,7)(3,12)(3,13)(3,14)(3,15)(3,16)(3,17)(3,18)(3,19)(3,20)(3,21)(3,22)(3,23)(3,24)(3,25)(3,26)(3,27)(3,28)(3,29)(3,30)(3,31)(3,48)(3,49)(3,50)(3,51)(3,52)(3,53)(3,54)(3,55)(3,56)(3,57)(3,58)(3,59)(3,60)(3,61)(3,62)(3,63

(4,0)(4,1)(4,2)(4,3)(4,4)(4,5)(4,6)(4,7)(4,12)(4,13)(4,14)(4,15)(4,16)(4,17)(4,18)(4,19)(4,20)(4,21)(4,22)(4,23)(4,24)(4,25)(4,26)(4,27)(4,28)(4,29)(4,30)(4,31)(4,48)(4,49)(4,50)(4,51)(4,52)(4,53)(4,54)(4,55)(4,56)(4,57)(4,58)(4,59)(4,60)(4,61)(4,62)(4,63)(4,64)(4,65)(4,66)(4,67)(4,68)(4,69)(4,70)(4,71)(4,72)(4,73)(4,74)(4,75)(4,80)(4,81)(4,82)(4,83)(4,84)(4,85)(4,86)(4,87)(4,88)(4,89)(4,90)(4,91)(4,92)(4,93)(4,94)(4,95)(4,96)(4,97)(4,98)(4,99)(4,100)(4,101)(4,102)(4,103)(4,104)(4,105)(4,106)(4,107)(4,108)(4,109)(4,110)(4,111)(4,112)(4,113)(4,114)(4,115)(4,116)(4,117)(4,118)(4,119)(4,120)(4,121)(4,122)(4,123)(4,124)(4,125)(4,126)(4,127)(4,192)(4,193)(4,194)(4,195)(4,196)(4,197)(4,198)(4,199)(4,200)(4,201)(4,202)(4,203)(4,208)(4,209)(4,210)(4,211)(4,212)(4,213)(4,214)(4,215)(4,216)(4,217)(4,218)(4,219)(4,220)(4,221)(4,222)(4,223)(4,224)(4,225)(4,226)(4,227)(4,232)(4,233)(4,234)(4,235)(4,236)(4,237)(4,238)(4,239)(4,240)(4,241)(4,242)(4,243)(4,244)(4,245)(4,246)(4,247)(4,248)(4,249)(4,250(4,251(4,252(4,25(4,25(4,25

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(2,0) (2,1) (2,2)(2,3) (2,4) (2,5) (2,6)(2,7)

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(0,0)

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(4,0) (4,1)(4,2)(4,3)(4,4) (4,5)(4,6)(4,7)(4,16) (4,17)(4,18)(4,19)(4,20)(4,21)(4,22)(4,23)(4,64)(4,65)(4,66)(4,67)(4,68)(4,69)(4,70)(4,71)(4,80)(4,81)(4,82)(4,83)(4,84)(4,85)(4,86)(4,87)

(5,0)(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)(5,7)(5,16)(5,17)(5,18)(5,19)(5,20)(5,21)(5,22)(5,23)(5,64)(5,65)(5,66)(5,67)(5,68)(5,69)(5,70)(5,71)(5,80)(5,81)(5,82)(5,83)(5,84)(5,85)(5,86)(5,87)(5,256)(5,257)(5,258)(5,259)(5,260)(5,261)(5,262)(5,263)(5,272)(5,273)(5,274)(5,275)(5,276)(5,277)(5,278)(5,279)(5,320)(5,321)(5,322)(5,323)(5,324)(5,325)(5,326)(5,327)(5,328)(5,329)(5,330)(5,331)(5,336)(5,337)(5,338)(5,339)(5,340)(5,341)(5,342)(5,343)

(6,0)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)(6,7)(6,16)(6,17)(6,18)(6,19)(6,20)(6,21)(6,22)(6,23)(6,64)(6,65)(6,66)(6,67)(6,68)(6,69)(6,70)(6,71)(6,80)(6,81)(6,82)(6,83)(6,84)(6,85)(6,86)(6,87)(6,256)(6,257)(6,258)(6,259)(6,260)(6,261)(6,262)(6,263)(6,272)(6,273)(6,274)(6,275)(6,276)(6,277)(6,278)(6,279)(6,284)(6,285)(6,286)(6,287)(6,320)(6,321)(6,322)(6,323)(6,324)(6,325)(6,326)(6,327)(6,336)(6,337)(6,338)(6,339)(6,340)(6,341)(6,342)(6,343)(6,1024)(6,1025)(6,1026)(6,1027)(6,1028)(6,1029)(6,1030)(6,1031)(6,1040)(6,1041)(6,1042)(6,1043)(6,1044)(6,1045)(6,1046)(6,1047)(6,1088)(6,1089)(6,1090)(6,1091)(6,1092)(6,1093)(6,1094)(6,1095)(6,1104)(6,1105)(6,1106)(6,1107)(6,1108)(6,1109)(6,1110)(6,1111)(6,1280)(6,1281)(6,1282)(6,1283)(6,1284)(6,1285)(6,1286)(6,1287)(6,1296)(6,1297)(6,1298)(6,1299)(6,1300)(6,1301)(6,1302)(6,1303)(6,1316)(6,1317)(6,1318)(6,1319)(6,1324)(6,1325)(6,1326)(6,1327)(6,1344)(6,1345)(6,1346)(6,1347)(6,1348)(6,1349)(6,1350)(6,1351)(6,1360)(6,1361)(6,1362)(6,1363)(6,1364)(6,1365)(6,1366)(6,1367)


64

Figure 3.31. To the left: brickwall coded with SPIHT at 0.1621 bpp, PSNR=23.9492. To the right: brickwall coded with constrained CWP at 0.1443 bpp, PSNR=24.3551.

Figure 3.32. To the left: brickwall coded with SPIHT at 0.0631 bpp, PSNR=22.0631. To the right: brickwall coded with constrained CWP at 0.0624 bpp, PSNR=22.7652

Figure 3.33. To the left: brickwall coded with SPIHT at 0.0215 bpp, PSNR=19.9064. To the right: brickwall coded with constrained CWP at 0.0274 bpp, PSNR=20.9682.


65

Figure 3.34. PSNR as a function of bpp for the brickwall images, * is wavelet coder and + is wavelet packet coder.

3.3.1 ConclusionWe see that the visual performance as well as PSNR is better when we use an optimal ba-sis rather than a suboptimal basis or a wavelet basis. When the CWP coder is required touse a suboptimal basis the performance is only slightly better than the wavelet coder. AllCWP coded images have better quality than the SPIHT coded images. Also, visually theCWP coder gives a better result.

3.3.2 Note on the choice of filterThe Haar filter yields poor reconstruction when used at low bitrates. The PSNR valuesand the visual performance would definitely be much better with the use of another filter.Why then do we use this filter? The careful reader has already noted some of the reasonsfrom the previous text in this thesis. From section 3.2.4 we note that the Haar filter hasthe same behaviour for all the images, even if it is not the best filter. From section 3.2.5we learned that the Haar wavelet is the only real, symmetric, linear phase wavelet for per-fect reconstruction out of our set of filters. With this filter we do not need to care aboutthe phase distortion, which can be large for some other filters when cascading on the de-tail coefficients, as the filters are designed only for cascading on the approximation coef-ficients. Another reason is that we take advantage of the reconstruction distortion to seethe visual differences between the coders.

0 0.5 1 1.5 2 2.5 315

20

25

30

35

40


66

3.4 Calculated decision rule In this section we will deduce the decision rule analytically for simple image models, fil-ter and cost function. We do this analysis in one dimension. The extension to two dimen-sions is straightforward following the separable filtering. These rules are developed for aGaussian source, Laplacian source, and generalized Gaussian source.

3.4.1 SystemWe will view the image as a wide-sense stationary random process, X, with the autocor-relation function . This function, given by Eq. 3.19), describes the time dependencein the process. It is an ensemble mean taken over all realizations of the process (all im-ages) and is defined by the expected value of the random process X at two different times

(3.19)

The deterministic autocorrelation of a sequence is a measure of the time de-pendencies within the sequence of one image

(3.20)

where denotes the inner product. In an ergodic [74] model this is expected to es-timate the autocorrelation function. The estimate of the statistical autocorrelation func-tion for a weak sense stationary sequence can also be computed by the inverse Fouriertransform of the periodogram.

Figure 3.35. Cost measurements before and after splitting.

The situation we want to analyse is described in Figure 3.35. We want to compare thecost of the result from highpass/lowpass filtering and downsampling a signalto the cost of the input signal. The subband is to be split if

rx k[ ]

rx k[ ] E XtXt k+( )=

p m[ ] h n[ ]

p m[ ] h n[ ] h n m+[ ],⟨ ⟩=

,⟨ ⟩

2

2

H1

H2

C

C

y1 z1 Mz1

y2 z2 Mz2

X C Mx

Mz1Mz2

+Mx

3.4 Calculated decision rule

67

(3.21)

With the Haar filter we have

(3.22)

(3.23)

(3.24)

(3.25)

By definition

(3.26)

we have the autocorrelation functions and by

(3.27)

and

(3.28)

giving the variances as

(3.29)

(3.30)

We choose as our first source model the correlations between neighbours described by the

Mz1Mz2

+ Mx<

y1 n[ ] 12

------- x n[ ] x n 1–[ ]+( )=

y2 n[ ] 12

------- x n[ ] x– n 1–[ ]( )=

z1 n[ ] y1 2n[ ] 12

------- x 2n[ ] x 2n 1–[ ]+( )= =

z2 n[ ] y2 2n[ ] 12

------- x 2n[ ] x– 2n 1–[ ]( )= =

rx i[ ] E X k[ ]X k i–[ ]{ }=

rz1k[ ] rz2

k[ ]

rz1i[ ] E z1 k[ ]z1 k i–[ ]{ } 1

2--- 2rx 2i[ ] rx 2i 1+[ ] rx 2i 1–[ ]+ +( )= =

rz2i[ ] E z2 k[ ]z2 k i–[ ]{ } 1

2--- 2rx 2i[ ] rx– 2i 1+[ ] rx– 2i 1–[ ]( )= =

σz1rz1

0[ ] mz1

2– rx 0[ ] rx 1[ ] 2mx2–+= =

σz2rz2

0[ ] mz2– rx 0[ ] rx– 1[ ]= =


68

autocorrelation function

, (3.31)

and secondly we use a Markov source model with autocorrelation function

. (3.32)

With these signal models we have the variances for both signal models:

(3.33)

(3.34)

This result is reasonable since the Haar filter only considers two neighbouring pixels.The Haar filter makes no difference between the two models.

3.4.2 Applying the cost functionApplying the cost function to the input signal x we get a cost measure of X

We are interested in the measure of the whole signal or subband, thus the cost func-tion output , assuming that the cost function is additive, is given by

(3.35)

Apart from a normalization constant, depending on the signal size, this is the mean ofthe signal under the influence of the cost function. This can be expressed by the expected

rx k[ ]

σx2ρ mx

2 k, 1=+

σx2 mx

2 k, 0=+

mx2 k, 1>

=

rx k[ ]σx

2ρ k mx2 k, 0>+

σx2 mx

2 k, 0=+=

σz1

2 σx2 1 ρ+( )=

σz2

2 σx2 1 ρ–( )=

C g x( )= C x( )

CX C(x)

MX

MX C xi( )i

∑=


69

value of a random process X representing our signal.

(3.36)

where p(x) is the probability density function.One cost function that lends itself easily to analytic computation is the norm . For-

tunately, it has proven to work well together with the Haar filter as shown in section 3.2.The cost associated with the input signal is given by

(3.37)

for a Gaussian distributed process, and

(3.38)

for a Laplacian distributed process.Following [54], we have an expression for the same integral with the generalized

Gaussian source (see Equation 3.9).

(3.39)

where α and β are extracted from the signal, for details see [23]. After filtering and subsampling we get the following expressions for the expected

costs for a Gaussian source.

(3.40)

and

E g X( ){ } g x( )p x( ) xd∫=

x

MG E X{ }G x 1σx 2π-----------------e

x mx–( )2

2σx2----------------------–

xd∞–

∞

∫2π---σx mx+= = =

ML E X{ }L x 1σx 2π-----------------e

x mx–

2σx2-----------------–

xd∞–

∞

∫σx

2-------e

2 mx

σx----------------–

mx+= = =

MGG x Ke x α⁄( )β– xd∞–

∞

∫ 2Kα2

β------Γ 2

β--- = =

MGz1

2π---σx 1 ρ+( ) 2mx+=


70

(3.41)

Splitting is done when

(3.42)

which leads us to the following decision rule for both the source models and a Gaussiandistribution

(3.43)

For mean values , for ranging from 0 to 1 and the variance ranging from0 to 100 the decision is to always split.

For the Laplacian distribution the decision rule for both source models is described by

(3.44)

and

(3.45)

The split decision rule is written

(3.46)

leading to

MGz2

2π---σx 1 ρ–( )=

N2----MGz1

N2----MGz2

+ NMG<

2π---

σx2----- 1 ρ+ 1 ρ– 2–+( ) 2 1–( )mx 0<+

mx 5.3–< ρ σx2

MLz1

σx 1 ρ+( )

2---------------------------e

2 mx

σx 1 ρ+( )---------------------------–

2mx+=

MLz2

σx 1 ρ–( )

2---------------------------e

2 mx

σx 1 ρ–( )---------------------------–

=

N2----MLz1

N2----MLz2

+ NML<


71

(3.47)

In Figure 3.36 we present plots of Equation 3.47 for ranging from 0 to 1 and thevariance ranging from 0 to 100, for different mean values . The black area de-scribes for which values of and the band will be split. A mean value of 0 indicatesthat a split is made.

Figure 3.36. The decision rule plotted for a Laplacian process with different mean values, for between 0 and 1 and between 0 and 100. The black area describes for which values of and

the band will be split.

We have produced decision rules for the split/no-split decision to be applied to everysubband. These rules uses the autocorrelation function of the mother band to calculate theeffect of splitting the band without actually performing the split. This can be used bothtop down in a greedy search and bottom up like the best basis search. For a typical imagewhere the mean is positive, the first level will be split. The approximation band will al-ways have a positive mean value and is thereforee more likely to split than the detailbands. This is probably the main reason why the wavelet decomposition comes so closeto the optimal solution.

The brickwall image is one of the images with an autocorrelation function that doesnot let itself be modelled by the image models used here. Grass, on the other hand, has anautocorrelation function that better fits the models used in the calculations. With the as-sumption that the approximation band always will be split, we will use the calculated de-cision rules in a top down fashion on the grass image. The autocorrelation function for

σx

2 2---------- 1 ρ+ e

2 mx

σx 1 ρ+( )---------------------------–

1 ρ– e

2 mx

σx 1 ρ–( )---------------------------–

2e

2 mx

σx----------------–

–+

2 1–( ) mx 0<+

ρσx

2 mxρ σ

|m|=0.01

σ

ρ

2 4 6 8 10

0.2

0.4

0.6

0.8

1|m|=0.1

σ

ρ

2 4 6 8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

|m|=1

σ

ρ

2 4 6 8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1|m|=3

σ

ρ

2 4 6 8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

ρσ ρ

σ


72

each subband, in both horizontal and vertical direction, is plotted together with the cor-responding values of mean, variance and correlation coefficient in Figure 3.37. Thesevalues are fed into the decision rules to achieve the second-level split. The calculated de-cision rule for an image with a Gaussian pdf suggests to split every subband while thedecision rule for an image with a Laplacian pdf suggests to keep every subband. The truthis probably somewhere in between and the decision rule for an image with a generalizedGaussian pdf could be used.

Figure 3.37. The first level decomposition of the Grass texture, with plots of the ACF and the values.

3.4.3 ConclusionThe calculated decision rule does not work with an image with an autocorrelation func-tion that cannot be modelled by the image models used in the calculations, but could finda basis for an image with a reasonable autocorrelation function. The Gaussian and Lapla-cian pdf models used here are to the two extremes when modelling the amplitude distri-bution of the image. Using the deneralized Gaussian can lead to a useful method fortaking the decision of split in a top-down manner, but this is left to future work.

3.5 SummaryThis chapter deals with the problem of finding the basis that is best for representing theimage in a compression application. The focus is on the transform in contradiction to themethods presented in Chapter 2. We start with an unbiased investigation of what happenswith the compression ratio when different images are wavelet packet analysed usingcombinations of different filters and cost function. The compression ratio is estimated bythe entropy of the coefficient matrix. This is motivated by the assumption of a memory-less source generating the image. This assumption is often used in the literature when try-

m=0.0235 σ2=0.0131

m=0.0252 m=0.0098σ2=0.0196 σ2=0.0034

ρ=0.4296

ρ=0.3343ρ=0.2790

ρ=0.2885 ρ=0.2802ρ=0.3413

3.5 Summary

73

ing to adapt methods to the image statistics only by taking the probability density functioninto account. The results are discussed with respect to the properties of the filters and costfunctions used. One filter and one cost function are chosen and put into an implementa-tion of a wavelet packet coder presented in Chapter 2. This is in order to view the effectof using a basis which is optimal with respect to image, filter and cost function.

Next we analytically calculate the decision rule based on a selection of filter and costfunction. The image is modelled by the spatial dependencies described by the autocorre-lation function. The probability density function is set to be Gaussian or Laplacian butcould be adapted to the image by the use of the generalized Gaussian.

We find that the choice of filter is more important to the compression performancethan the choice of cost function. Due to the cascaded filtering process applied on the detailbands, the phase effects disqualify most wavelet filters for the wavelet packet use. Theneeded number of vanishing moments of the filters depends on the differentiability of theimage. For the detail bands we do not need more than one vanishing moment. The Haarfilter is in fact the only filter out of our filter set that can be used, although it has poorreconstruction performance. We also note that a wavelet packet basis is better than thewavelet basis only when the optimal basis is used. Any sub-optimal basis, even close tothe optimal, gives performance close to that of a wavelet basis using the same filter.


74

75

Chapter 4

Discussion - Wavelet packet coding

The wavelet packet part of this thesis starts with a survey of different wavelet and waveletpacket image coding methods that evolved over the years. Adelson and others provide astart through pyramid image coding [1,2,12]. Even though the pyramid was created by alow-pass filtering process by successive convolutions of the images with Gaussian-likekernels, the difference between the resulting images was the equivalent to convolving theimage with an appropriately scaled Laplacian weighting function [1,2]. In [12] the theoryis expanded to including band-splitting transforms using quadrature mirror filters (QMF)and Haar basis. The QMF approach was also treated by Woods & O'Neil [94], in whichthe expansion to the multi-dimensional subband coding of [89] to image subband codingappeared. With Mallat [54,55] the connection between the Laplacian pyramid image cod-ing, subband coding and wavelet theory [20,21] was established.

Many ideas evolved for signal analysis and image processing applications such as im-age coding. The wavelet packet transform [16,17] which was a natural extension of thewavelet analysis was applied to images in [92]. Herley et al.'s work [32,33] on tilings ofthe time frequency plane to adapt the wavelet packet basis to time varying structures wasapplied to images in [34], and further developed to joint dual double trees in [82]. The lat-ter method can tile the time frequency plane in ways that the wavelet packet method can-not and thus can reach a more efficient decomposition of the signal. Ramchandran et al.extended this [71]. While these tiling are still rectangular, another approach was taken byBaraniuk & Jones [5,6] with the introduction of the chirplet transform which gives non-rectangular tilings of the time frequency plane.

Recently the multiwavelet and multiwavelet packets were introduced in image com-pression applications [57]. These methods use more than one scaling function and waveletfunction to reach better results.

Apparently there are many variations on the theme of finding the best representation.In this part we concentrated on the wavelet packet transform and dissect the problems ofthe method that affects the result and which cannot be solved by the extensions described

Chapter 4 Discussion - Wavelet packet coding

76

above. The wavelet compression methods have developed into use in standards while the

wavelet packet algorithms still struggle with problems. Understanding the Triplet Prob-lem, treated in Chapter 3, is essential for the development of the wavelet packet methods.Almost everyone using wavelet packet methods refers to the fact that the wavelet packettransform is better than the wavelet transform on images with high spatial frequency pat-terns, such as the Barbara image (in Figure 5.38). This is true to some degree. It was earlynoted [15] that the wavelet packet transform can only represent the resonance in the im-age. A high spatial frequency pattern is a resonance and the Barbara image is full of that,with different orientation. Each orientation of this pattern generates a different waveletpacket basis when the algorithm is applied on the pure pattern. The Barbara image is, asmost natural images, a mixture of different spatial frequency patterns, or textures, andthus, the wavelet packet transform fails to find the very best representation for the imageby one basis alone. However, this particular image contains textures similar enough tomake the wavelet packet transform outperform the wavelet transform and the image is apopular choice for the testing of wavelet packet image compression algorithms.

The problem how to decompose different parts of an image differently was treated by[34] and [82] as discussed above. In the wavelet part of this thesis the work ignore thisproblem and is concentrated on pure texture images. The problem of proper choice of fil-ter and cost function still remains and this is treated in this thesis. The reviewed algo-rithms presented in Chapter 2 are typical in the sense that they focus on the quantization,bit allocation, and entropy coding steps of the coding chain shown in Figure 1.10. Theproblem of choosing the basis is often not treated. In the wavelet case one can argue thatthe basis is fixed but the coding result change with the choice of filters. Most waveletpacket image coding algorithms use wavelet filters without any motivation other thanthat it is a good filter for wavelet image coding. In this thesis we have shown experimen-tally the effects of using wavelet filters in both wavelet and wavelet packet coding of tex-ture images. The wavelet filters are designed for iterative use only on the low-pass partof the signal. Only the low-pass filter corresponding to the scaling function has propertiesappropriate for cascading the filters. When using the high-pass filter corresponding to thewavelet in a cascading manner, errors are easily introduced in the image. The filters treat-ed in this thesis are the first generation wavelet filters. Using these are sufficient to showthe differences between the wavelet transform and the wavelet packet transform regard-ing the needs for proper filters. We find that the choice of filter is more important to thecompression performance than the choice of cost function. Due to the cascaded filteringprocess applied on the detail bands, the phase effects disqualify most of the filters forwavelet packet use. For the detail bands we do not need more than one vanishing mo-ment. The Haar filter is in fact the only filter out of our test set that can be used, althoughit has poor reconstruction performance.

The focus of the first part of this thesis has been to understand the behaviour of thewavelet packet transform rather than finding a new good filter set. However, an embryoof basic criteria for such a filter set is outlined in that both low-pass and high-pass filtershould be linear phase filters and the decomposing filter set can be Haar-like while thereconstruction filter set needs to be smoother.

For the wavelet packet case the bases become totally different depending on the im-age, the choice of filter, and the choice of cost function. When using a proper filter set,like Haar, we found that the choice of cost function is of minor importance for the com-pression result. The choice of cost function in a wavelet packet application is treated in

77

[84,85] for speech compression where they examine additive and non-additive cost func-tions in both top-down and bottom-up search approaches and present a near-best basis se-lection algorithm. The work is very interesting due to its extensive examination of costfunctions in different basis search algorithms and their conclusion that the choice of costfunction has significant impact on the result in both time-frequency analysis and com-pression. However, they used filters which were optimal for the wavelet transform, andthus the result of their work is not valid for our evaluation of the wavelet packet trans-form.

The best basis method of [17] is a bottom-up search algorithm often stated to find theglobal optimal basis. This method decomposes the image down to some level and appliesthe cost function on the nodes from the bottom up to decide whether or not to cut the treeat this node. However, we have found in this thesis that the best basis method finds dif-ferent bases when the image is decomposed to different levels.

In this thesis a method of analytic calculation of the expected basis has been present-ed. The method uses the analytic expression of the cost function, the filter parameters andthe image statistics by means of two different models for the pixel interdependencies de-scribed by the autocorrelation function. This is an approach that differs from using the im-age or subband histogram only as in [11] for example. The memory free assumption isused in most wavelet coding algorithm, including the best basis algorithm, when the his-togram is used for approximation of the probability function. When the image is modelledby its spatial dependencies described by the autocorrelation function we are taking thememory of the source into account. This is to take the wavelet packet coding algorithmsone step further towards a useful algorithm.

The examination of wavelet packet methods in this thesis demonstrates the difficultiesto utilize these methods for improved image coding. The potential seems to be there, how-ever, it has to be admitted that the problem is not yet solved.

Chapter 4 Discussion - Wavelet packet coding

78

PART TWO

EMPIRICAL MODEDECOMPOSITION

81

Chapter 5

Empirical Mode Decomposition

The wavelet packet algorithms treated in previous chapters search for the resonance in thesignal. One of the purposes for the thorough examination of the algorithms was to find outif the wavelet packet decomposition could serve as the totally adaptive decomposition forany signal. The limitations due to filters and cost functions in combination with differentsignals made this difficult. The totally different approach of the Empirical Mode Decom-position (EMD) was revealed in [36]. EMD is an adaptive decomposition with which anycomplicated signal can be decomposed into its Intrinsic Mode Functions (IMF). EMD isan analysis method that in many aspects gives a better understanding of the physics be-hind the signals. Because of its ability to describe short time changes in frequencies thatcan not be resolved by Fourier spectral analysis it can be used for nonlinear and non-sta-tionary time series analysis. For the IMF we can have a well-defined instantaneous fre-quency. The original purpose for the EMD was to find a decomposition which made itpossible to use the instantaneous frequency for time-frequency analysis of non-stationarysignals. In this method we found the totally adaptive signal decomposition we had beenlooking for and we expanded the concept of EMD to two dimensions for image analysisand compression purposes [51].

This chapter is devoted to understanding EMD in both one and two dimensions. Weexplain the algorithms and put the focus on different limitations initialized by differentimplementation methods.

This chapter begins with a review of the EMD applied to time signals. Previous workoften mentions the lack of a mathematical formalism to describe the EMD. The conceptis truly empirical. In this work we keep the empirical approach as we expand the methodto two-dimensional signals, such as images. Some implementations of the EMD in twodimensions generate a residue with many extrema points. We will give an example of suchan implementation and explain its drawbacks. In this chapter we also present an improvedmethod that can decompose the image into a number of IMFs and a residue with none, orwith only a few extrema points. This method makes it possible to use the EMD for image

Chapter 5 Empirical Mode Decomposition

82

processing. We introduce the concept of empiquency, short for empirical mode frequen-cy, to describe the signal oscillations as the traditional frequency concept is not applicablein this context. We also discuss the selection of significant extrema points as a tool fornoise reduction.

5.1 EMD for time signalsThe EMD is an adaptive decomposition with which any complicated signal can be de-composed into a redundant set of signals denoted IMF and a residue. Adding all the IMFstogether with the residue reconstructs the original signal without information loss or dis-tortion.

An IMF is characterized by some specific properties. One is that the number of zerocrossings and the number of extrema points are equal or differs only by one. Anotherproperty of the IMF is that the envelopes defined by the local maxima and minima, re-spectively, are locally symmetric around the envelope mean. The original purpose for theEMD was to find a decomposition which made it possible to use the instantaneous fre-quency, defined as the derivative of the phase of an analytic signal, in the time frequencyanalysis of the signal [36]. Traditional Fourier analysis does not allow a signal's spectrumto change over time. The EMD, however, provides a decomposition method that analysesthe signal locally and separates the component holding locally the highest empiquency(see section 5.1.4), from the rest into a separate IMF. Within this IMF both high and lowempiquencies can coexist at different times. The key word is locally.

In this thesis the Hilbert-Huang Transform is presented in an example for the intro-duction of the EMD of time signals, and as the origin of EMD. We then leave the HHTand develop the EMD concept further in the time domain and image domain for compres-sion purpose.

5.1.1 Time-frequency analysis with Hilbert-Huang Transform (HHT)Many methods exist which analyse signals simultaneously in the time- and frequency-domain, wavelets [56] and short-time Fourier transform [14], among others. These meth-ods are based on the expansion of the signal into a set of basis functions where the basisfunctions are defined by the method. The concept of the EMD is to expand the signal intoa set of functions defined by the signal itself, the IMFs. Signal adaptive decompositionby means of Principal Component Analysis (PCA) [29] also expands the signal into a ba-sis defined by the signal itself. The PCA differs from EMD in that it is based on the signalstatistics, while EMD is deterministic and is based on local properties. The EMD processallows time-frequency analysis of transient signals for which Fourier based methods havebeen unsuccessful. Whenever we use the Fourier transform to represent frequencies weare limited by the uncertainty principle. For infinite signal length we can get exact infor-mation about the frequencies in the signal, but when we restrict ourselves to analyse asignal of finite length there is a bound on the precision of the frequencies that we can de-tect. The instantaneous frequency represents the frequency of the signal at one time, with-out any information about the signal at other times. There exist different definitions for

5.1 EMD for time signals

83

instantaneous frequency; a common one is to use the derivative of the phase of the signalas described in [36]. Another problem with using the instantaneous frequency is that itonly provides one value at each time. A signal usually consists of many intrinsic frequen-cies and this is where the EMD is used, to decompose the signal into its IMF, where onlyone frequency component is present at each time so that a well-defined instantaneous fre-quency can be conputed.

A relatively new technique has been developed to perform general analysis of highlytransient time-domain signals, called the Hilbert-Huang Transform (HHT) [36]. It hasshown great utility in time-frequency analysis of dispersive, nonlinear, or non-stationarysignals and systems. The transform uses the EMD, with which the signal is decomposedinto a redundant set of IMFs and a residue. By applying the Hilbert transform to each IMFwe get a set of analytical signals representing the input signal. The HHT calculates theinstantaneous frequency of each transformed IMF and presents the result as a time-fre-quency analysis in a Hilbert spectrum plot. Time-frequency analysis by means of HHT,or EMD separately, has been done in several areas, such as analysis of ocean waves, seis-mology, and one-dimensional analysis of SAR images [36,88,37]. In a study of laser vi-brometry data for target classification, the HHT outperforms wavelet analysis and thepower spectral density method [61].

Example 5.1. We create a composite chirp by adding together a 2-second long linearlyincreasing chirp ranging from DC to 300 Hz with a "convex" quadratic chirp of duration1 second which starts at 25 Hz and increases to 100 Hz, and a "concave" quadratic chirpof duration 1 second which starts at 100 Hz and decreases to 25Hz. The resulting signalis sampled at 1 kHz and is shown in Figure 5.1

Figure 5.2 shows the time frequency plot of the composite chirp signal by the win-dowed Fourier transform to the left and the corresponding plot of a wavelet packet bestbasis decomposition of the same signal to the right.

Figure 5.1. Time plot of the composite chirp test signal.

0 0.5 1 1.5 2−3

−2

−1

0

1

2

3


84

Figure 5.2. Time-frequency plot of the composite chirp signal by the windowed Fourier transform to the left and the corresponding plot of the wavelet packet best basis decomposition of the same

signal to the right.

Figure 5.3. The Hilbert spectrum of the composite chirp signal.

The Hilbert spectrum of the test signal is shown in Figure 5.3. It is composed byHilbert spectra of each IMF of the composite chirp signal shown in Figure 5.15.

Both the short-time Fourier transform and the wavelet packet best basis representa-tions suffer from the uncertainty principle. In the left plot in Figure 5.2 the signal re-sponse is smeared along the frequency axis as a result of the narrow time window andhence a broad frequency window used to analyse the signal. On the right side ofFigure 5.2 the wavelet packet best basis representation is plotted. Here all the small boxesare of the same area and the grayscale indicates the correlation between the signal and theanalysis filter associated with the box: white indicates high correlation and black indi-cates low correlation. The form and position of the box is dictated by the best basis searchalgorithm originally described in [17].

0

50

100

150

200

250

300

350

0 0,5 1 1,5 2

Time (s)

Freq

uenc

y (H

z)


85

The Hilbert spectrum is plotted against the time axis in Figure 5.3, and it shows notonly the inter frequencies generated by the construction of the signal, but also the intrafrequencies which come from the change of curve form by the frequency change and theaddition of signals. This is an advantage with the EMD compared to Fourier methods asit offers more features for the analysis of the signal. Artifacts are also introduced by theuse of sampled signals.

5.1.2 Sifting process for finding the IMFThe sifting process to find the IMFs is introduced in [36]. We describe and picture it by avery simple test signal denoted by . It is constructed by the addition of two sine curvesand a slowly varying trend, and is visualized by the green curve in Figure 5.4 (a).

To find the first IMF, start with the image itself as input signal . Thefirst index is the IMF number, l=1..L, and the second index is the iteration number,k=1..K, in the sifting process. To find the next IMF, use the residue corresponding to thepreviously found IMF as input signal .

The sifting process to find the IMFs of a signal x(t), comprises the following steps:(1) Find the positions and amplitudes of all local maxima, and find the positions and

amplitudes of all local minima in the input signal . These are marked by red andblue dots respectively in Figure 5.4(a)

(2) Create the upper envelope by spline interpolation of the local maxima and the low-er envelope by spline interpolation of the local minima, denoted and .These are shown as the red and blue curves in Figure 5.4(b). Cubic spline is used in thisexample.

(3) For each time instant t, calculate the mean of the upper envelope and the lower en-velope.

(5.1)

This signal is referred to as the envelope mean and is shown as the black linein Figure 5.4(b), (c), and (d).

(4) Subtract the envelope mean signal from the input signal (black in Figure 5.4(b)from green in Figure 5.4(b)), yielding the results illustrated by green in Figure 5.4(c).

(5.2)

This is one iteration of the sifting process. The next step is to check if the signal from step (4) is an IMF or not. In the original work [36] the sifting process stops

when the difference between two consecutive siftings is smaller than a selected thresholdε. In this thesis the process stops when the envelope mean signal is close enough to zero,as suggested in [51].

(5.3)

x t( )

in11 t( ) x t( )=

in21 t( ) r1 t( )=

inlk t( )

emax t( ) emin t( )

emlk t( )emax t( ) emin t( )+

2-----------------------------------------=

emlk t( )

hlk t( ) inlk t( ) emlk t( )–=

hlk t( )

emlk t( ) ε< t∀


86

The reason for this choice is that forcing the envelope mean to zero will guarantee thesymmetry of the envelope and the correct relation between the number of zero crossingsand number of extremes that define the IMF. A slightly more complicated version of thisstop criteria is presented in [73], along with a discussion of typical threshold values.

(5) Check if the mean signal is close enough to zero, based upon the stop criterion. Ifnot, iterate by repeating the process from step (1) with the resulting signal from step 4 asthe input signal a sufficient number of times.

(5.4)

Figure 5.4. a: The test signal with the local maxima marked by red dots and the local minima marked by blue dots. b: The test signal with upper envelope in red, lower envelope in blue and the

envelope mean in black. c: The result of one iteration in the IMF sifting process with the upper and lower envelope in red and blue and their mean in black. d: The resulting first IMF with the

upper and lower envelope and their mean. e and f: Iteration for the second IMF.

inl k 1+( ) t( ) hlk t( )=


87

When the stop criterion is met, k=K, the IMF is defined as the last result from (4).

(5.5)

After the IMF is found, (here illustrated by the green curve in Figure 5.4(d)),define the residue as the result of subtracting this IMF from the input signal.

(5.6)

The residue is illustrated by the green curve in Figure 5.4(e).(6) The next IMF is found by starting over from step 1, now with the residue as the

input signal.

(5.7)

The second IMF is illustrated in Figure 5.4(f).Steps (1) to (6) can be repeated for all the subsequent . The EMD is completed when

the residue, ideally, does not contain any extrema points. This means that it is either a con-stant or a monotonic function. The signal can be expressed as the sum of IMFs and thelast residue

(5.8)

The first IMF captures the highest empiquency in the signal as a function of time. Therange of empiquencies contained in the IMFs varies over time but the first IMF alwayscontains the locally highest one in the signal.

Example 5.2. We show the sifting process with a slightly more complicated signal de-fined by Equation 5.9 and plotted in Figure 5.5.

cl t( ) hlK t( )=

cl t( )rl t( )

rl t( ) inl1 t( ) cl t( )–=

in l 1+( )1 t( ) rl t( )=

rj

x t( ) rL t( ) cj t( )j 1=

L

∑+=

Figure 5.5. Plot of the test signal in Equation 5.9.

0 2 4 6 8 0 2 4 6 8−3

−2

−1

0

1

2

3


88

(5.9)

The signal has different frequency components in different parts. In the first twothirds of the signal the component of the highest frequency is , while in the lastthird of the signal the component of the highest frequency is . This is also theresult of the sifting for the first IMF shown in Figure 5.6. The start of the sifting with firstupper and lower envelope is shown in Figure 5.6 (a) and result from subsequent iterationsis presented in Figure 5.6 (b-f).

The first IMF is shown in Figure 5.6 (f) and is a signal very similar to the signal givenby Equation 5.10.

(5.10)

The sifting for the second IMF is presented in Figure 5.7 in the same manner as thesifting for the first IMF is presented in Figure 5.6. The result should, in theory, give usthe signal given by Equation 5.11 if we were concerned with only the inter frequencycomponents in the signal. Due to the limitations of the implementation, further discussedin section 5.1.3, and the fact that different signal components influence each other andproduce intra frequency effects in the signal, we have another result, shown in Figure 5.7.

(5.11)

Figure 5.6. Iteration for the first IMF.

x t( )πt( ) 2πt( ) 6πt( )sin+sin+sin

πt( ) 6πt( )sin+sinπt( ) 6πt( ) 12πt( )sin+sin+sin

=0 t 3≤<3 t 6≤<6 t 9≤<

6πt( )sin12πt( )sin

IMF16πt( )sin12πt( )sin

0 t 6≤<6 t 9≤<

=

IMF22πt( )sinπt( )sin

6πt( )sin

=0 t 3≤<3 t 6≤<

6 t 9 ≤<

a b c

d e f


89

Figure 5.7. Iteration for the second IMF.

The lowest frequency component used in the composition of the test signal appears inthe sifting for the third IMF, Figure 5.8, the next IMFs holds rest signals, Figure 5.9 toFigure 5.12, and the residue is a monotonic signal shown in Figure 5.13.

Figure 5.8. Iteration for the third IMF.

Figure 5.9. Iteration for the fourth IMF.

a b c

d e f


90

Figure 5.10. Iteration for the fifth IMF.

Figure 5.11. Iteration for the sixth IMF.

Figure 5.12. Iteration for the seventh IMF.

Figure 5.13. Iteration for the eighth IMF and the residue.


91

Example 5.3. When we use the composite chirp test signal from Example 5.1 insection 5.1.1, the result of the sifting for the first IMF is shown in Figure 5.14.

The composite chirp is shown in Figure 5.14 (a) and the result of one iteration in the IMFsifting process is shown in Figure 5.14 (b) in yellow with the upper envelope in red andthe lower envelope in blue; their mean is shown in black. The resulting first IMF with theupper and lower envelope and their mean is shown in Figure 5.14 (c). In the latter casethe envelopes are symmetric around their mean which is close to zero.

In Figure 5.15 the IMFs of the composite chirp signal are plotted together with theirrespective HHT. This shows the frequency separation capability of the EMD. The highestfrequencies of the signal are represented by the first IMF while the next highest frequen-cies are represented by the second IMF etc. The EMD does not separate the signal into itsoriginal components.

0 0.5 1 1.5 2

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

0 0.5 1 1.5 2−4

−3

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−1

0

1

2

3

4

0 0.5 1 1.5 2−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

Figure 5.14. a) The composite chirp, b) the result of one iteration in the IMF sifting process with the upper and lower envelope and their mean plotted in the same figure, c) the resulting first IMF

with the upper and lower envelope and their mean.

a

b

c


92

5.1.3 Interpolation methodsThe interpolation to produce envelopes from the extrema points can be done in differentways. The work done in [36] states that spline interpolation should be used. We here usepiecewise cubic spline interpolation and piecewise cubic Hermite interpolation. Thesemethods produce slightly different EMD results. The two interpolation methods are bothsplines but of different degrees.

On each subinterval , let P(x) be the interpolant to the given valuesand certain slopes at the two end points. Between any two neighbouring data sites x(k)and x(k+1), x(t) is a polynomial. Neighbouring polynomials match in value, first, andsecond derivative, across their common data site.

Figure 5.15. The first IMFs and the residue for the composite chirp (left) and their respective HHT (right).

x t( ), k t k 1+≤ ≤


93

For piecewise cubic Hermite interpolation [44], the interpolator is denoted Pp(x). Thefirst derivative, Pp'(x), is continuous, but Pp''(x) is not necessarily continuous. The slopesat the x(k) are chosen in such a way that Pp(x) is shape preserving and respects monoto-nicity. This means that, on intervals where the data are monotonic, so is Pp(x), and atpoints where the data have a local extremum, so does Pp(x): overshoots does not exist.The function Ps(x) supplied by the piecewise cubic spline interpolation is constructedsuch that the slopes at the x(k) are chosen to make Ps''(x) continuous also. This makesPs(x) smoother and more accurate if the data are values of a smooth function. The effectsof using linear interpolation, cubic Hermite interpolation, and cubic spline interpolationon regularly spaced data are shown in Figure 5.16. Note that in Figure 5.16 (b) the datapoints are no longer the extrema points of the interpolated signal. The interpolation hasproduced overshoots in order to achieve the continuous second derivative. These effectsintroduce new extrema points that may not have been present in the original signal ormove the extrema point to another location.

The EMD involves the interpolation of irregularly spaced data. It is preferred thatthese stay as extrema points. Linear interpolation, cubic Hermite interpolation, and cubicspline interpolation on irregularly spaced data are shown in Figure 5.17. Only linear in-terpolation and cubic Hermite interpolation keep the extrema points in place. These ef-fects are further treated in the examples.

Using the cubic spline interpolation method is good when decomposing signals, butat the same time it introduces new higher frequencies at false positions in longer periodsof very low frequencies present in a surrounding of higher frequencies. This is shown inthe Example 5.4 where the test signal from Example 5.2 is used, only modified such thatthe middle part is set to zero, Equation 5.12.

Figure 5.17. l = linear interpolation, s = piecewise cubic spline interpolation, p = piecewise cubic Hermite interpolation of scattered data points.

0 2 4 6 8 10−1

−0.8

−0.6

−0.4

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0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10−1

−0.8

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1

0 2 4 6 8 10−1

−0.8

−0.6

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0

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0.6

0.8

1

a b c

Figure 5.16. a) linear interpolation, b) piecewise cubic spline interpolation, c) piecewise cubic Hermite interpolation of regular data points.


94

Example 5.4. The test signal is plotted in Figure 5.18. The result of using the cubicspline interpolation method is shown in Figure 5.19 in the sifting for the first IMF andFigure 5.20 in the sifting for the second IMF. The cubic Hermite interpolation does notintroduce any new frequencies for the same case, shown in Figure 5.21 in the sifting forthe first IMF and Figure 5.22 in the sifting for the second IMF, but to the cost of slowerconvergence of the sifting process and a larger amount of IMFs found for the same valueon the stop criteria. When the cubic spline interpolation method is used the sifting intro-duces new frequency components in the zero zone, but the cubic spline interpolationmethod is recommended for most "normal" signals.

(5.12)

Figure 5.19. Iteration for first cubic IMF.

x t( )πt( ) 2πt( ) 6πt( )sin+sin+sin

0πt( ) 6πt( ) 12πt( )sin+sin+sin

=0 t 3≤<3 t 6≤<6 t 9≤<

Figure 5.18. Test signal.


95

Figure 5.20. Iteration for second cubic IMF.

Figure 5.21. Iteration for first Hermitian IMF.

Figure 5.22. Iteration for second Hermitian IMF.


96

Example 5.5. To further examine the cubic spline interpolation and the cubic Hermiteinterpolation, we use a test signal composed of two different sinuoids. We apply both in-terpolation methods and look at the frequency separation capability of the EMD. The testsignal

(5.13)

is plotted in Figure 5.23. The signal is simple and pure, both methods separate the firsttwo IMF as signals that look similar to the input signal components. In Figure 5.24 (a)the signal component of the highest frequency is plotted, in Figure 5.24 (b) the first cubicHermite interpolation IMF is plotted and in Figure 5.24 (c) their difference is plotted. Thesame is done for the second cubic Hermite interpolation IMF in Figure 5.25(b), which iscompared to the signal component of the next highest frequency, plotted inFigure 5.25(a). Their difference is plotted in Figure 5.25 (c). The error is approximately5% of the signal amplitude in the centre of the signal where the edge effects do not affectthe signal.

Figure 5.24. a) sin(6πt), b) first Hermitian IMF, c) Their difference.

2πt( ) 6πt( )sin+sin

Figure 5.23. Test Signal

0 2 4 6 8 0 2 4 6 8−1.5

−1

−0.5

0

0.5

1

1.5

0 2 4 6 8 0 2 4 6 8−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 0 2 4 6 8−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

a b c


97

Figure 5.25. a) sin(2πt), b) second Hermitian IMF, c) Their difference.

When we do the same for the cubic spline method we have a better performance. InFigure 5.26 (a) the signal component of the highest frequency is plotted, in Figure 5.26(b) the first cubic spline interpolation IMF is plotted and in Figure 5.26 (c) their differ-ence is plotted. The same is done for the second cubic spline interpolation IMF inFigure 5.27 (b), which is compared to the signal component of the next highest frequency,plotted in Figure 5.27 (a). Their difference is plotted in Figure 5.27 (c). The error is ap-proximately zero in the central region.

Figure 5.26. a) sin(6πt), b) first cubic IMF, c) Their difference.

Figure 5.27. a) sin(2πt), b) second cubic IMF, c) Their difference

The cubic spline interpolation has worse behaviour at the edges than cubic Hermiteinterpolation due to the treatment of edge points. No effort is made on this subject in thiswork, some solutions to the edge problem for time signals can be found in [36].

0 2 4 6 8 0 2 4 6 8−1

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a b c

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a b c

0 2 4 6 8 0 2 4 6 8−2

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−0.8

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1

a b c


98

5.1.4 EmpiquencyThe EMD is a truly empirical method, not based on the Fourier frequency approach but,as we will see, related to the locations of extrema points and zero crossings. Based on thiswe use the concept of empiquency [53], short for empirical mode frequency, instead of atraditional Fourier-based frequency measure to describe the signal oscillations. Let d bethe distance between two neighbouring extrema points. In [36] the concept of time scaleis presented. It can be seen as the mean of all d in the signal. In this context the empiquen-cy is the local time scale. The measure of empiquency is defined as “One half the recip-rocal distance between two consecutive extrema points”:

(5.14)

The empiquency value is assigned to every position between the respective ex-trema points. Each extrema point influences more than one empiquency definition; we letit take the highest value of the two empiquency values it defines.

The empiquency is accompanied by an amplitude describing the signal at the actualposition. At each extrema point the empiquency amplitude Ae is the absolute value of theextrema point. At all other positions, the empiquency amplitude takes the value of the ab-solute mean of the extrema points that define the empiquency for the actual position.

Thus,

(5.15)

where a and b are values of neighbouring extrema points. The very special properties of the IMFs are that these are locally zero mean and ide-

ally do not have more than one extrema point between two neighbouring zero crossings.Because of this relationship between zero-crossings and extrema points there is also a re-lation between the empiquency and the concept of sequency. Sequency is defined as “theaverage number of zero crossings per second divided by 2” [31]. For a sine or cosine thefrequency and the sequency is the same, but the concept of sequency has a meaning forother signals such as Walsh functions or IMFs as well.

Example 5.6. Maximum empiquency within a signal segment is found by examiningthe space between the extrema points. Figure 5.28 and Figure 5.29 show two discretetime signals, both with the properties of an IMF but with different maximum empiquen-cy, (0.5, in a normalized scale, for the signal in Figure 5.28 and 0.167 for the signal inFigure 5.29). Later on we will assume that a signal like the one in Figure 5.29 can be re-constructed with low distortion from a subsampled version, while a signal example likethe one in Figure 5.28 needs every sample to be represented without distortion due to itshigh maximum empiquency.

fe1

2d------=

fe

Ae p )( )a if p is an extrema point

a b+2------------------ if p is not an extrema points

=


99

Figure 5.28. A discrete time signal with its three extrema points marked x for local maxima and o for local minima. Maximum empiquency 0.5.

Figure 5.29. A discrete time signal with its three extrema points marked x for local maxima and o for local minima. Maximum empiquency 0.167.

5.1.5 EMD of noiseThe behaviour of the EMD as a filter bank is highlighted in [26] through the analysis ofnoise. We review this work with an example using the cubic spline interpolation. The testsignal used is white Gaussian noise with a variance of 1. The result from the EMD isshown in Figure 5.30 (a-y). The images in the left column show the time signal IMF andthe middle column shows the fast Fourier transform (FFT) of the corresponding IMF. InFigure 5.30(b), (d), (f), (h), the spectrum density plot shows the whole result and thus thenegative frequency components appear in the right half of the plot. The rest of the fre-quency plots show only positive frequency components in a zoomed view. The test signalis shown in Figure 5.30 (a) and its power spectra by means of the FFT of the signal isshown in Figure 5.30 (b). The first IMF show a “high-pass filtered” noise signal inFigure 5.30 (c) and Figure 5.30 (d). The rest of the IMFs show “band-bass filtered” noisesignals with their centre frequency fc decreasing in an octave band manner, just like a fil-ter bank. This experiment shows that the EMD sorts the frequency content of a noise sig-nal in the same way as it sorts any other signal. We should have this in mind when treatingnoisy signals. It is sometimes assumed that the noise of a signal is totally contained in thefirst IMF, an assumption that obviously, from this example, is not true.

time

xx

o

time

xx

o


100

.

a b

c d

e f

g h

i j

k l

zero mean Gaussiannoise,variance1

first IMF,variance 0.5750centre frequencyfc=5000Hz

Second IMFvariance 0.2117 centre frequencyfc=2500Hz

Third IMFvariance 0.1225centre frequencyfc=1250 Hz

Fourth IMFvariance 0.0742centre frequencyfc=500Hz

Fifth IMFvariance 0.0545centre frequencyfc=250Hz


101

Sixth IMFvar(c6)=0.0230fc=160Hz

Seventh IMFvar(c7)=0.0181fc=80Hz

Eighth IMFvar(c8)=0.0158fc=40Hz

Nineth IMFvar(c9)=0.0189fc=20Hz

Tenth IMFvar(c10)=0.0016fc=15Hz

Tenth residuevar(res10)=0.0233fc=10Hz

Figure 5.30. EMD of gaussian nose. Time plot of the IMF to the left, the spectral density from FFT of the IMF in the middle and specifications for the IMF to the right.

m n

o p

q r

s t

u v

x y


102

5.1.6 ConclusionThe concept of empiquency is new and is useful to describe time signals and, as will beseen later in this thesis, also when EMD is applied to images. HHT is suitable for time-frequency analysis. The performance depends on the interpolation method and its imple-mentation. Cubic spline interpolation is probably the best interpolation method for EMDof time signals despite its drawbacks. This is an area where more work is needed, but thisis not the focus of this thesis. The number of IMFs achieved with the EMD depends alsoon the choice of stop criterion value. If it is too large the number of IMFs is not largeenough to make the empiquency separation complete. If it is too small the empiquencyseparation completes but computation time is longer than necessary.

5.2 EMD of two-dimensional signalsHHT, the EMD-based time-frequency analysis, presented in section 5.1.1, is only one ofmany applications made possible by the use of EMD. In the two-dimensional case, whereadaptive spatial frequency separation presents a challenge, the EMD enables many newapproaches for image processing applications.

The results and ideas in time domain applications using the EMD technique apply totwo-dimensional signals, such as images, as well. Similarly to the one-dimensionalEMD, the first IMF extracts the locally highest spatial empiquencies in the image whilethe second IMF holds the locally next highest spatial empiquencies, etc. The EMD in twodimensions provides a tool for image processing by its special ability to locally separatespatial frequencies that build texture. Assume that a texture consists of a superposition ofa number of spatial frequency patterns. At each spatial position the texture is a result ofthe superposition of different spatial frequencies. The EMD sorts the spatial frequencycomponents into a set of IMFs where the highest spatial frequency component of eachspatial position is in the first IMF and the next highest spatial frequency component ofeach spatial position is in the second IMF, etc.

The use of the Hilbert transform for the creation of analytic signals in two dimensionsis made possible by the introduction of a direction of reference in the Fourier domain[30], but will not be further treated in this thesis. In the two-dimensional EMD, the sym-metry criterion on the IMF envelopes is relaxed. The stop criterion is based on the con-dition that the IMF envelope mean must be close enough to zero. This guarantees that wewill find the IMF without actually checking for symmetric envelopes.

Other work on EMD in two dimensions typically generates a residue with many ex-trema points. In this chapter we propose an improved method to decompose an image intoa number of IMFs and a residue image with a minimum number of extrema points.

The extension of the EMD to two dimensions relies on proper spline interpolation intwo dimensions. The edge constraints are important, as the errors introduced theretraverse into the image. The IMFs can be seen as spatial frequency subbands, with vari-ous centre frequency and bandwidth along the signal. Within each IMF both high and lowempiquencies can coexist at different locations. The key word still is locally.

5.2 EMD of two-dimensional signals

103

5.2.1 Sifting for the two-dimensional IMFThe sifting process for the IMF is here extended to two dimensional signals.

To find the first IMF, start with the image itself as input signal. The first index is the IMF number, l=1..L, and the second index

is the iteration number, k=1..K, in the sifting process. m and n represent the two spatialdimensions. To find the next IMF, use the residue corresponding to the previously foundIMF as input signal .

The sifting process to find the IMFs of a signal x(m,n), comprises the following steps:(1) Find the positions and amplitudes of all local maxima, and find the positions and

amplitudes of all local minima in the input signal .(2) Create the upper envelope by spline interpolation of the local maxima and the low-

er envelope by spline interpolation of the local minima. Denote the envelopes and . respectively.

(3) For each position (m,n), calculate the mean of the upper envelope and the lowerenvelope.

(5.16)

The signal is referred to as the envelope mean.(4) Subtract the envelope mean signal from the input signal

(5.17)

This is one iteration of the sifting process. The next step is to check if the signal from step 4 is an IMF or not. The process stops when the envelope mean signal

is close enough to zero.

(5.18)

In the same way as for time signals, forcing the envelope mean to zero will give us thewanted symmetry of the envelope and the correct relation between the number of zerocrossings and the number of extremes that define the IMF. This way we will find the IMFwithout actually having to check for symmetric envelopes.

(5) Check if the mean signal is close enough to zero, based upon the stop criterion. Ifnot, repeat the process from step 1 with the resulting signal from step 4 as the input signala sufficient number of times.

(5.19)

When the stop criterion is met, k=K, the IMF is defined as the last result of (4).

(5.20)

After the IMF is found, define the residue as

in11 m n,( ) x m n,( )=

in21 m n,( ) r1 m n,( )=

inlk m n,( )

emax m n,( )emin m n,( )

emlk m n,( )emax m n,( ) emin m n,( )+

2----------------------------------------------------------=

emlk m n,( )

hlk m n,( ) inlk m n,( ) emlk m n,( )–=

hlk m n,( )

emlk m n,( ) ε< m n,( )∀

inl k 1+( ) m n,( ) hlk m n,( )=

cl m n,( ) hlK m n,( )=

cl m n,( ) rl m n,( )


104

(5.21)

(6) The next IMF is found by starting over from step 1, now with the residue as theinput signal.

(5.22)

Steps (1) to (6) can be repeated for all the subsequent . The EMD is completedwhen the residue, ideally, does not contain any extrema points. This means that it is eithera constant or a monotonic function. The signal can be expressed as the sum of IMFs andthe last residue

(5.23)

5.2.2 Implementation for image EMDThe EMD of images relies on proper spline interpolation in two dimensions. Some effortsto implement the EMD in two dimensions have been published [51,62] but the methodshave not proven to be able to fully decompose the two-dimensional signal down to a res-idue with a low number of extrema points. We will give an example of such an imple-mentation and explain its drawbacks. The residue signal should have only a few extremapoints. Huang et al. [36] state that there should be no extrema points in the one-dimen-sional case and ideally we would like to have a similar criterion for the two-dimensionalcase.

It is mainly steps (1) and (2) in the sifting process that cause problems in the imple-mentation of the EMD in two dimensions. We here present our solution to these prob-lems. In step (1) we have the problem of selecting the extrema points. In 2D, there aremany possibilities to define extrema, each one yielding a different decomposition. Sincewe are only concerned with discrete two-dimensional signals we simply extract the ex-trema points by comparing the candidate data point with its nearest 8-connected neigh-bours. This does not allow for saddle points to be considered as extrema points. In thecase where saddle points are considered to be extrema points in the algorithm, these areboth maxima and minima at the same point. This nails the saddle point to the zero meanlevel in the first sifting round. Clearly, more sophisticated methods to extract extremapoints could handle the situation with saddle points to be considered as extrema points aswell, but the extrema points defined by an 8-connected neighbourhood serves the purposefor EMD at this stage, with further improvement being possible. In the following, saddlepoints are thus not considered to be extrema points.

The border constraints are even more important in two dimensions than they are inthe one-dimensional EMD case. One of the main objections for using spline interpolationin the EMD for two-dimensional signals such as images is that the borders cause toomany problems. The set of extrema points is very sparse and since the interpolation meth-ods only interpolate between points, the borders need special care. We propose the trick

rl m n,( ) inl1 m n,( ) cl m n,( )–=

in l 1+( )1 m n,( ) rl m n,( )=

rj

x m n,( ) rL m n,( ) cj m n,( )j 1=

L

∑+=


105

of adding extra data points at the borders to the set of extrema points. These extra pointsare placed at the corners of the image and equally spaced around the border. Withoutthese extra points, the areas not covered by the interpolation traverse into the image in thesifting process.

In the second step in the sifting process the problem is to fit a surface to the two-di-mensional scattered data points representing the extrema points. The interpolated surfacemust go through each data point. Overshot must be avoided and the second derivativemust be continuous everywhere for the signal to be smooth enough for two-dimensionalEMD. The interpolation methods considered here are triangle-based cubic spline interpo-lation and thin-plate smoothing spline interpolation. The first is an example of an inter-polation method using a piece-wise approach to interpolate the surface. It produces piece-wise smooth surfaces and is based on Delaunay triangulation [67] of the data. The piece-wise approach causes more problems in the two-dimensional case than with one-dimen-sional signals. Even though the interpolator produces continuous second derivatives, suchas the cubic spline in [51] or the radial basis functions used in [62], the borders of theneighbouring pieces cause problems. It is possible that the piece-wise approach will workif we use another set of extrema points where the saddle points are included. This is notexamined in this thesis but is left to future work.

We suggest that the thin-plate smoothing spline interpolation is used for the imple-mentation of the two-dimensional EMD. This method gives a surface with continuoussecond derivative everywhere. The thin-plate smoothing spline algorithm [8] calculatesthe function f that minimizes the integral bending norm If, in Equation 5.24, for givenscattered data in the plane. The integral is taken over the entire image, and involves thesecond derivatives of f.

(5.24)

This means that the determination of the smoothing spline involves the solution of a linearsystem with as many unknowns as there are data points. This method turns out to success-fully decompose an image into its IMFs and a smooth residue with no or only a few ex-trema points. We use the thin-plate smoothing spline interpolation for the two-dimensional EMD throughout this thesis.

Example 5.7. The scattered extrema points from the fourth IMF in Figure 5.35(d) areshown in Figure 5.31(a). Local maxima are plotted as light pixels and local minima areplotted as dark pixels. The different surfaces achieved when interpolating the same scat-tered data points using thin-plate interpolation and triangle-based cubic interpolation areshown in Figure 5.31(b,c). The artifacts when using a piece wise approach on very sparsedata are caused by the sharp edges and the un-smooth borders of the different areas cre-ated by the triangulation. This can be seen in Figure 5.31(c). These sharp borders intro-duce new extrema points in the iteration for the next IMF and prevent the decompositionfrom finding and separating any lower spatial frequencies. This also forces the EMDmethod into producing IMFs containing features that were not in the image originally.

Ifx2

2

∂∂ f

2

2 x y∂

2

∂∂ f

2

y2

2

∂∂ f

2

+ +

x y∂∂R2∫∫=


106

Figure 5.31. The different resulting surfaces when interpolating the scattered data points (a), using thin plate interpolation (b), and triangle-based cubic interpolation (c).

5.2.3 Significant extrema pointsIn the first and second IMFs the number of extrema points usually is very large. Not allof these are essential for the signal analysis We can reduce the number of extrema pointsby selecting the significant extrema points of the IMF. This is done by setting the extremapoints with low amplitude to zero, as described by Equation 5.25.

(5.25)

Figure 5.32(a) shows the extrema points of the first IMF at their position in the imagewhere the rest of the pixels are set to zero. Figure 5.32(b) shows the corresponding his-togram of the image in Figure 5.32(a).

Figure 5.32. a) The extrema points of the first IMF at their position in the image, the rest of the pixels zeroed. b) histogram of the image in a.

If we let the coefficients with absolute amplitude lower than a suitable threshold, inthis case 10, be set to zero we get the significant extrema points with respect to the thresh-old, shown in Figure 5.33.

a b c

b m n,( )0 if b m n,( ) T≤

b m n,( ) if b m n,( ) T>

=

a b


107

Figure 5.33. a) The image in Figure 5.32(a) with the smallest coefficients set to zero, b) histogram of the image in (a).

Keeping only the significant extrema points reduces the number of samples needed torepresent the IMF, with only a minor effect in the reconstructed signal. However, errorswill be introduced which will show up in the SNR measure.

This operation also serves as a tool for noise reduction. The insignificant extremapoints are defined to be noise. It is sometimes assumed that the first IMF contains all thenoise in the signal and that the noise can be removed by just skipping the first IMF. Thisis not true. Noise usually contains all empiquencies, both higher and lower than the max-imum empiquency of the signal. The effects of EMD of noise was treated in section 5.1.5.Image processing is often performed on digitized photos. This means that any image isnoisy, due to the physics of image capturing. High quality photos, such as the Lenna im-age, have a low noise level, not visible to the human eye, with a signal-to-noise ratio high-er than 40 dB. For signals with moderate signal-to-noise ratio, we can still assume thatthe noise is of lower amplitude than the significant extrema points. We can then reducethe noise by setting the extrema points with low amplitude to zero, as described by Equa-tion 5.25.

5.2.4 Image EMDIn this section we use three examples to illustrate the application of our method for imageEMD calculations. The first example decomposes the 128x128 pixel Lenna image shownin Figure 5.34. The second example uses a 64x64 pixel detail part of the 512x512 pixelLenna image and the third example treats a 64x64 detail part of the Barbara image inFigure 5.38.

Example 5.8. As an example of an image EMD we look at the well-known Lenna im-age, shown in Figure 5.34. The image's four IMFs and their corresponding residues areshown in Figure 5.35. This example clearly demonstrates the “band-pass filtering” effectsof the EMD. The first IMF in a) and the first residue in e) can be seen as the result of ahigh-pass and low-pass filtering process. Adding these together reconstructs the original.For creation of the second IMF the first residue in Figure 5.35(e) is treated and the resultshows as the second IMF in b) and the second residue in Figure 5.35(f). Using these toreconstruct the original we add a), b), and f) according to Equation 5.23. Continuing thedecomposition gives four IMFs and a final residue with only a few extrema points.

a b


108

Figure 5.34. The Lenna image at 128x128 pixel size.

Figure 5.35. a) First IMF, b) second IMF, c) third IMF, d) fourth IMF, e) first residue, f) second residue, g) third residue, h) fourth residue.

Example 5.9. Figure 5.36 shows a part of the textured hat of the Lenna image. Thistexture is only visible in the high resolution 512x512 pixel version of the image. This ex-ample shows the EMD of a highly detailed texture image in Figure 5.37, using a mediumlevel of the stop criterion. The first IMF captures the texture component with high spatialfrequency of the hat and in the same IMF the high frequency noise in the background.The texture components of lower spatial frequency remain in the first residue, as do theshape and curvature of the hat. Next step is to separate the first residue into the secondIMF and second residue. Here the second IMF still holds some high spatial frequenciesand the second residue holds the shape and curvature of the hat. While decomposing thisimage down to a residue with only a few extrema points, we see the potential of EMD toevolve into a texture representation tool. This application is not treated in this thesis butleft to future work.

a b c d

e f g h


109

Figure 5.36. Part of the hat of Lenna image, 512x512.

Figure 5.37. a) First IMF, b) second IMF, c) third IMF, d) fourth IMF, e) first residue, f) second residue, g) third residue, h) fourth residue.

Example 5.10. Figure 5.38 shows the Barbara image, with the lower right corner cutas test image. This image contains many parts with high spatial frequency and is oftenused to demonstrate the performance of image coders and texture analysis methods. Weuse the lower right corner of the image for this example. The EMD is shown inFigure 5.39, with, from left to right: the first IMF, the second IMF, the third IMF and thethird residue. This is an example of the use of too large a value of the stop criterion. Witha smaller value the first two IMFs would have been separated into some more IMFs andmore information of the image would have been revealed.

a b c d

e f g h


110

Figure 5.38. Left: Barbara. Right: lower right corner zoomed.

Figure 5.39. From left to right, the three IMF s of the image to the right in Figure 5.38, and the residue

It should be noted that all these IMFs are approximately zero mean, while the DC lev-el of the signal is contained in the residue. These IMFs have the very special property ofhaving one extrema point between two zero-crossings in almost any direction, exceptwhere there are saddle points. This will be used in the next chapter to get a more efficientrepresentation of the EMD.

5.2.5 ConclusionThe EMD concept is successfully extended into two dimensions. The main problem withthe implementation is the detection of the extrema points and the interpolation of the scat-tered extrema points. Performance of the EMD depends on the interpolation method andits implementation.

5.3 SummaryIn this chapter the Empirical Mode Decomposition (EMD) was introduced. The use ofEMD in the Hilbert-Huang Transform (HHT) was viewed as an example of EMD fortime signals. Different interpolation methods were discussed for time signals. The con-cept of empiquency is introduced for further use in subsequent chapters.

The EMD was extended to two dimensions and we proposed an improved method todecompose a full image into a number of IMFs and a residue image with a minimumnumber of extrema points. The selection of significant extrema points was discussed asa tool for noise reduction.

111

Chapter 6

Compression of the EMD

In Chapter 5 the EMD for images was developed. In this and the following chapters wewill investigate how to use it for compression. In this chapter a number of methods tocompress the EMD are tested. The set of IMFs and the residue image is a very redundantway to represent an image. Our first goal is to represent the image by its EMD with thesame number of bits as the original image. Our second goal is to invent an image coderthat achieves reasonable compression. Throughout this chapter the test image used is thepart of Lenna 512 shown in Figure 5.36 called “detail one”.

First we use a memory-less coding strategy, an entropy coder, on each IMF and theresidue to test the compression potential of the EMD.

The next attempt to use the EMD for compression is the Extrema Point Coder [51].This method is examined here both for time signals and images. The EMD sorts the em-piquencies in such a way that there is only one empiquency present at each position foreach IMF. Although the empiquency changes with position in the IMF the hypothesis forthe next compression method is that within a reasonable small image block the empiquen-cy should be approximately the same. This way the block-based DCT could represent theimage with only one or two of the DCT coefficients for each block.

Disregarding the blocking, the DCT of the IMFs shows an interesting structure. Apartfrom the first IMF they have all significant components in the low-frequency region of thecoefficient matrix. Zeroing out everything else gives an efficient representation of theIMF and the compression result is good. But there still remains to find a way to compressthe first IMF.

6.1 Entropy coding of the EMDThe scheme for entropy coding of the EMD is presented in Figure 6.1. The image is de-composed into its IMFs and a residue. These are entropy coded by a Huffman coder ap-plied to the quantized IMFs and residue.

Chapter 6 Compression of the EMD

112

6.1.1 Results

The set of IMFs and residue is compressed using fine quantization and Huffman codingresults in a bitrate of 5.49 bpp at PSNR of 58.95 dB for the first IMF, 4.96bpp at PSNRof 58.97 dB for the second IMF, 5.47 bpp at PSNR of 59.04 dB for the third IMF, 6.79bpp at PSNR of 58.89 dB for the fourth IMF, and 7.52 bpp at PSNR of 58.88 dB for theresidue. Compression result using quantization levels ranging from 256 levels down to 2levels are presented in Figure 6.2. An example of quantized IMFs and residue is shownin Figure 6.3 together with information on quantization, rate and distortion. We presentthe quantized IMFs and residue together with the original. The addition of the chosen re-constructions creates the reconstructed image in Figure 6.4 with 15.61 bpp and PSNR33.9 dB.

EMDDecomposition

First IMF

Second IMF

Nth IMF

Entropy Coder

MUX

DEMUX

Channel

First IMFreconstructed

reconstructed

reconstructedNth IMF

Second IMF

Signal

Signal

Reconstructed

Figure 6.1. The EMD Entropy coding scheme.

Residue reconstructed

Residue

+

Entropy Coder

Entropy Coder

Entropy Coder

Decoder

Decoder

Decoder

Decoder

6.1 Entropy coding of the EMD

113

Figure 6.2. Entropy Coding results for each IMF and Residue, a) first IMF, b) second IMF, c) third IMF, d) fourth IMF, e) residue.

a b

c d

e


114

IMF140.54dB2.60 bpp32 quantization levels

IMF241.29 dB2.07 bpp32 quantization levels



Residue40.80 dB4.57 bpp32 quantization levels

Figure 6.3. Left: original, the image on top and IMFs and the residue below. Middle: The quantized IMFs and residue. Right: Information on quantization levels, rate, and distortion for

the images.

6.2 Extrema point coding

115

Figure 6.4. Reconstruction using 15.61 bpp results in PSNR 33.9 dB.

6.2 Extrema point coding Our next idea to compress the EMD will be termed the extrema point coder.

The idea is to decompose the signal into its IMFs and transmit only the extrema pointsfor each IMF. Then we reconstruct the IMF in the decoder with spline interpolation. Thiswas first presented in [51]. The scheme is shown in Figure 6.5.

6.2.1 One-dimensional extrema point codingTo get a feeling for the behaviour of the IMFs in a coding application, we use the com-posite chirp test signal from example 5.1 in section 5.1. The IMFs are computed usingcubic spline interpolation. The bitrate is estimated through entropy calculations by Eq.3.1) in which the signal is quantized to the nearest integer. With the representation usedhere this means quantization to 256 levels. The entropy of a signal serves as an indicatorof performance of a memory-less and lossless compression algorithm. The entropy is cal-culated for the IMFs themselves, their extrema vector, and their quantized extrema vector,as listed in Figure 6.6. The sum of entropy of the IMFs is much higher than the entropyof the signal itself. Yet, choosing to use only the extrema points gives the opportunity tocode very sparse signals. The IMF extrema vectors are plotted in Figure 6.7. As can beseen, both in the plots and in the listing in Figure 6.6, the number of extrema points is lowcompared to the length of the original signal to code. The fewer extrema points to codethe better. The cost of coding these sparse signals is estimated by the entropy of the ex-trema vector in Figure 6.6 and even more compression can be achieved with further quan-tization. in The example quantization to 128 levels is used. The entropy of the test signalis 7.08 bits/sample, while the sum of the entropies of the IMFs equals 28.52 bits/sample.Using the extrema points of the IMFs for representing the signal gives a summed entropyof 5.90 bits/sample, while using quantized extrema points leaves us with a summed en-tropy of 3.42 bits/sample.


116

EMDDecomposition

First IMF

Second IMF

Nth IMF

Extract extrema

Extract extrema

Extract extrema

Source Coding

Source Decoding

Channel

Interpolation

Interpolation

InterpolationFirst IMFreconstructed

reconstructed


Second IMF

Decoder

Coder

Signal

Signal

Reconstructed

Figure 6.5. The extrema point coding scheme.

InterpolationResidue

Extract extrema

reconstructed

Residue

+

Figure 6.6. Listing of entropies of the IMFs and their extrema vectors.

Signal # sample entropy # extrema entropy for extremaentropy for quantized extrema

Composite chirp 2001 7.08

IMF 1 2001 7.19 637 2.72 2.03

IMF 2 2001 6.73 345 1.56 0.99

IMF3 2001 5.13 189 0.68 0.23

IMF4 2001 3.37 101 0.46 0.11

IMF5 2001 2.69 54 0.28 0.04

IMF6 2001 3.41 33 0.20 0.02

SUM of IMF entropy

28.52 5.90 3.42


117

Reconstruction of the IMF is done by cubic spline interpolation of the extrema points re-ceived. The signal can then be reconstructed by the addition of the reconstructed IMFsand the residue. The interpolation introduces errors which are plotted in Figure 6.8. Thisis because the IMF itself is not a spline function but approximated by a spline interpola-tion of the extrema points when reconstructed. When no quantization is done on the ex-trema vectors, the cubic spline interpolation causes the least reconstruction error ofseveral tested interpolation methods. This is in line with the examination of different in-terpolation methods in section 5.1.3.

a b

c d

e f

Figure 6.7. The extrema points of the IMFs respectively: a) IMF1, b) IMF2, c) IMF3, d) IMF4, e) IMF5 f) IMF6.


118

CommentsThe lossless compression algorithm ZIP compresses the whole set of IMFs into approx-imately the same number of bits as when applied to the original signal. This indicates thatthere could be a way to overcome the redundancy in the set, and use this for compression.The spline interpolation of the extrema points introduces large errors in the reconstructedsignal. This is particularly true for the first IMF which holds the most unsmooth part ofthe signal.

a b

c d

e f

Figure 6.8. The reconstruction error for the IMFs respectively, a) IMF1, b) IMF2, c) IMF3, d) IMF4, e) IMF5 f) IMF6.


119

6.2.2 Two-dimensional extrema point codingWith the result of extrema point coding of time signals in mind, we want to use the sameconcept on images.

Coder outlineThe two-dimensional extrema point coding is done by letting the extrema points of eachIMF and residue represent the image. The coding follows the same scheme as for one-dimensional signals shown in Figure 6.5. The image is decomposed into a number ofIMFs and a residue. For each IMF and the residue the extrema points are extracted andcoded. Reconstruction is done with spline interpolation, either triangle-based cubic orthin-plate. The first IMF is not particularly smooth and the number of extrema points ofthe first IMF is very large. The smoothing thin-plate interpolation cannot reconstruct thefirst IMF in a proper manner. Instead we use the triangle-based cubic spline interpolationfor the reconstruction of the first IMF.

The positions of the extrema points are coded using runlength coding and their valuesare quantized and fixlength coded. The exact positioning of the extrema point is extreme-ly important for the result. In the extrema point coding scheme we use Elias coding forthe runlength coding of the extrema point positions and fixlength coding for the quantizedamplitudes of the extrema points. The fixed length is in relation to the number of quanti-zation levels.

Example 6.1. Reconstructing the image with and without the cubic interpolating re-construction of the first IMF gives very different visual performance as shown inFigure (a). Here the thin-plate spline interpolation is used for reconstruction of all IMFsand the residue. In Figure (b) the thin-plate spline interpolation is used for reconstructionof all IMFs and the residue, except for the first IMF, where the triangle-based cubic splineinterpolation is used.

a) The test image “Detail one”, b) Reconstruction by thin plate interpolation of allIMFs and residue, PSNR 23.8dB. c), Reconstruction by thin plate interpolation of allIMFs except for the reconstruction of the first IMF where cubic spline is used, PSNR21.0dB.

It is obvious that the visual quality is better in the latter case, although the PSNRmeasure tells us the opposite.

a b c


120

Example 6.2. The significant extrema points of the first IMF of the image inFigure (a), quantized with enlarged zero zone, are shown in Figure 6.9. The length histo-gram of zero runs in a line scan of the significant extrema point image is plotted in thehistogram in the same figure.

Using Elias code gives us an effective way to assign codewords to the runlength val-ues, giving 2982 bits to code the positions in this particular case. The fixed length codefor coding of amplitude values uses here a 4-bit code to code 16 levels of amplitude. Inthis example the 767 nonzero positions can be coded with 767*4=3068 bits.

6.2.3 Results

The extrema points of the IMFs and the residue from the EMD of the test image inFigure 5.36 are shown in Figure 6.10.

The set of IMFs and residue is compressed using the extrema points shown inFigure 6.10. Five different sizes for the extended zero zone are used and the number ofquantization levels ranges from 256 levels down to 2 levels.

The curves in Figure 6.11 show that each IMF and the residue can be represented withan acceptable distortion with a rate below one bit per pixel, except for the first IMF andmaybe the second. In Figure 6.12 we have chosen to present one reconstruction of eachIMF and residue. We present the reconstructed image together with the original. The ad-dition of the chosen reconstructions creates the reconstructed image with 6.75 bpp at psnrequal to 19.79 dB.

Figure 6.9. a) Extrema points of the first IMF, grey is zero. b) Length histogram of zero runs


121

a b

c d

eFigure 6.10. From the EMD of Figure 5.36: a) extrema points of IMF 1, b) extrema points of IMF

2, c) extrema points of IMF3, d) extrema points of IMF 4, extrema points of the residue.


122

Figure 6.11. Extrema Point Coding results for each IMF and residue, a) first IMF, b) second IMF, c) third IMF, d) fourth IMF, e) residue.

a b

c d

e


123

Sum of IMFsand residue19.79 dB6.75 bpp

IMF130.27dB3.16 bpp1227extrema points

IMF235.21 dB1.60 bpp461 extrema points



Residue27.34 dB0.375 bpp6 extrema points

Figure 6.12. Left: original, the image on top and IMFs and the residue below. Middle: the reconstruction of the object to the left by interpolation of the extrema points. Right: information on rate and distortion for the reconstructed images. The number of extrema points includes extra

helping points at the image border.


124

Figure 6.13. Reconstruction using 8 bpp results in PSNR 22.78 dB.

6.2.4 ConclusionThe extrema point coding approach is an attractive way to represent the image but doesnot seem to be suitable for compression. Normally in other coding methods it is in thequantization process that the major reduction of data and quality loss is found. In thismethod the major distortion comes from the reconstruction by interpolation. However,we have almost reached our first goal of representing the image without too much distor-tion at its original bitrate, Figure 6.13.

6.3 Block-based single-component DCT codingWe saw in Chapter 6.2 that using the extrema points to represent the image results in alarge number of coefficients to code. This number has to be reduced before we can hopeto achieve good compression. In this chapter we use the DCT, presented in section 1.2.1,for this purpose.

6.3.1 Coder outlineThe sifting process for finding the IMFs sorts out one empiquency for each position ineach IMF. This should trig a transform coder to produce a very compact code, as there isonly one empiquency present in the IMF at each position. Most IMFs would only haveresponse in the low frequency band of the DCT.

6.3 Block-based single-component DCT coding

125

The idea is to divide each IMF into blocks of a specific size and use the DCT on eachblock. We use only one coefficient for each block. The motivation for this is the assump-tion that the IMF holds locally only one spatial frequency. If the DCT can represent thisspatial frequency with only one coefficient, this would yield a good compression of theIMF. Decoding of the IMF is done with inverse DCT.

The coder is implemented in accordance with the scheme in Figure 6.14, which isbased on the scheme in Figure 6.5.

The implementation of the single-component DCT coding block is illustrated inFigure 6.15. First the IMF is divided into blocks of a specific size. Each block is trans-formed using DCT. The largest magnitude is detected and the rest of the coefficients arezeroed. The last step in this block is to patch the DCT blocks together into one large ma-trix, the coefficient image.

EMDDecomposition

First IMF

Second IMF

Nth IMF

Source Coding

Source Decoding

Channel

+

Reconstruction


reconstructed


Second IMF

Decoder

Coder

Image

ImageReconstructed

Figure 6.14. Single-component DCT coding scheme.


Residue

Block-basedsingle-component DCT

Reconstruction

Reconstruction

Reconstruction





126

Figure 6.15. Detail of the block-based single-component DCT coding block.

Huffman code is used to represent the amplitudes of the DCT coefficient which arequantized to 256 levels. The positions are line scanned and runlength coded using Eliascode.

The details of the reconstruction block are shown in Figure 6.16. The first step in thereconstruction is to divide the coefficient image into blocks holding only one non-zerovalue. Next we reconstruct the image block with inverse DCT, and patch the imageblocks together to an image.

Figure 6.16. Detail of the reconstruction block.

Example 6.3. We show the effects of using two different block sizes for the first IMFin Figure 6.17 and Figure 6.18. In Figure 6.17 the first IMF is coded using blocksize 2x2.We see in (a) that the single component of each block has different positions within theblock. This results in the trigging of different basis functions in the reconstruction inFigure 6.17(b). In Figure 6.18 the first IMF is coded using blocksize 16x16. The differentpositions are shown in Figure 6.18(a) and the reconstruction is shown in Figure 6.18(b).With this blocksize it is even clearer that different positioning of the single componentgives different reconstruction results. The same basis function is used for the wholeblock, clearly seen in Figure 6.18(b). The high spatial frequency in the IMF is capturedusing both blocksizes, but in the first case the localization is more precise.

Block

DCT on each block

Choose highest DCT component

Patch one-coefficient blocks

Block-based single-component DCT

IMF to

to coefficient image

Block

IDCT on each block

Patch blocksImage to

Reconstruction


127

Figure 6.17. First IMF, blocksize 2x2, a) DCT-components, b) reconstructed IMF,

Figure 6.18. First IMF, blocksize 16x16, a) DCT-components, b) reconstructed IMF

6.3.2 ResultThe different blocksizes tested on each IMF and the residue are 2x2, 4x4, 8x8, 16x16, and32x32 pixels. In Figure 6.19 we can see the result of reconstructing the image by the ad-dition of the reconstructed IMFs and residue when the single-component block-basedDCT is used. Each dot represents a combination of blocksize from 2x2 to 32x32 for thefour IMFs and the residue needed to reconstruct the image. Excluding the option forblocksize 2x2 eliminates the higher bitrates. Excluding the option for blocksize 32x32 aswell eliminates the worst images in terms of PSNR. In this test we skipped coarse quan-tization of the amplitudes as this procedure only further reduces the performance in termsof PSNR. The method does not perform better than 20 dB due to blocking artifacts andthe lack of precision in frequency response from the single-component DCT representa-tion.

a b

a b


128

Figure 6.19. Each dot represents a combination of blocksizes from 2x2 to 32x32 for the four IMFs and the residue needed to reconstruct the image.

In Figure 6.20 some reconstructions are shown. The vector represents the blocksizesused for each IMF (first IMF, second IMF.....residue), where 5 is 32x32blocks, 4 is 16x16blocks, 3 is 8x8 blocks, 2 is 4x4 blocks, 1 is 2x2 blocks. The best performance in termsof PSNR is shown in Figure 6.21, when we use many 2x2 blocks, but then we have higherbitrates instead.

Figure 6.20. a) (3 4 3 3 3) PSNR =17.7624 0.9810 bpp, b) (3 4 5 3 3) PSNR =17.5502 0.7686 bpp, c) (5 4 5 2 2) PSNR =19.0502 1.6602 bpp

Figure 6.21. a) (1 3 2 1 1) PSNR =19.8284 9.3147 bpp.

bpp

dB

a b c


129

Figure 6.22. Result to left, in the middle a reconstructed example using the parameters specified to the right.

IMF129.07dB1.48 bppsize of block: 4x4128 quantization levels

IMF233.41dB0.79 bppsize of block: 4x416 quantization levels

IMF333.86 dB0.86 bppsize of block: 4x416 quantization levels

IMF432.53 dB0.87 bppsize of block: 4x416 quantization levels

Residue32.66 dB0.84 bppsize of block: 4x416 quantization levels


130

6.3.3 Two-coefficient block-based DCTEven though the IMF could be locally zero mean, the blocking tiles it into non-zero-meanparts. We need to include the DC component of each block to get a representation goodenough for a compression algorithm. Our method is the same as the one-coefficientblock-based DCT but with the DC component added to get better performance. In thisscheme the DCT components kept are also quantized using different numbers of quanti-zation levels.

ResultFigure 6.22 shows compression results from the two-component block-based DCT. Forbitrates below 1 bpp, the distortion is acceptable in terms of PSNR value for most of theIMFs except for the first IMF. But the visual performance shows severe blocking arti-facts.

6.3.4 ConclusionDue to the blocking artifacts, both the one-component block-based DCT and the two-component block-based DCT fail to be serious candidates for a compression algorithm.However, they illustrate several properties of the IMFs.

6.4 DCT threshold coding The sifting process for finding the IMFs sorts out higher empiquencies in the first IMF.The tendency is that the IMFs other than the first are low frequency images. This can beseen in the DCT of the whole IMFs in Figure 6.23, in which the image’s lowest frequen-cies in both directions are represented by the components in the upper left corner of theimage and the image’s highest frequencies in both directions are represented by the com-ponents in the lower right corner of the image. The upper right corner represents the highfrequencies in the horizontal direction and low frequencies in the vertical direction, whilethe lower left corner represents the high frequencies in the vertical direction and low fre-quencies in the horizontal direction. In Figure 6.23(a) the components are spread all overthe matrix, with a highlight in the middle representing the dominating frequencies of thefirst IMF. In Figure 6.23(b) the largest components are located along the upper edge,meaning that the second IMF is mostly low frequency in the vertical direction and haslittle higher frequency content in the horizontal direction. For the rest of the IMFs and theresidue, the frequency components concentrate in the upper left corner, where they rep-resent low spatial frequency.

6.4.1 Coder outlineThe idea is to keep only the significant components located in the upper left corner. Thesize of the triangle of components is determined by a sliding limit line parallel to the di-

6.4 DCT threshold coding

131

agonal from the lower left corner to the upper right corner. The limit line shall be as closeto the upper left corner as possible when still no components larger than the threshold areto the right of the line. All the components in the upper left corner to the left of the lineare undisturbed, only uniformly quantized.

6.4.2 ImplementationThe implementation scheme is shown in Figure 6.24. The image is decomposed into itsIMFs and the residue. Each IMF and the residue is fed into the coding block. After recon-struction the IMFs are added together with the residue to reconstruct the image. The de-tails of the coding block and the reconstruction block are further described below.

a b

c d

eFigure 6.23. a) 64x64 DCT transforms of the 64x64-sized IMFs (a-d) and the residue (e) of the

EMD shown in the left column in Figure 6.29.


132

The coding blockThe details of the coding block are shown in Figure 6.26. Each IMF and the residue aretransformed by the DCT. The resulting DCT component matrix is searched for the sizeof the upper left corner by scanning the component matrix in a zigzag pattern startingwith the DC level coefficient in the upper left corner and stopping at the lower right cor-ner. The scan pattern is illustrated in Figure 6.25. The scan is truncated at the position ofthe first component larger than the threshold found when scanned from the end. The re-sulting DCT components in the scan are uniformly quantized. The component values inthe truncated scan are Huffman coded.

EMD

First IMF

Second IMF

Nth IMF

MUX

DEMUX

Channel

+

Reconstruction


reconstructed


Second IMF

Decoder

Coder

Image

ImageReconstructed

Figure 6.24. The DCT Threshold coding scheme.


Residue

Coding

Coding

Coding

Coding

Reconstruction

Reconstruction

Reconstruction


133

Figure 6.25. Zigzag scan pattern.

Figure 6.26. Detail of the coding block.

The reconstruction blockThe details of the reconstruction block are shown in Figure 6.27. When the truncated andquantized scan is reconstructed from the source decoding step, the original length of thescan is established by adding zeros at the end. The next step is to resize the data into amatrix format. Finally the IMF is reconstructed by the inverse DCT.

imagezigzag scanDCT component

QuantizeTruncate scan

Coding

DCT of

matrix


134


6.4.3 ResultDCT Threshold Coding results for each IMF and residue are shown in Figure 6.28. Thefull result is presented to the left and to the right is a blow up of the result below one bitper pixel. For the first IMF no blow up is presented. Each graph consists of seven differ-ent curves, representing the seven different quantizations used in the test. The stars oneach curve present the result for the different truncation lengths. In this test we use anumber of predefined truncation lengths instead of a threshold-controlled truncation.

In Figure 6.29, we present one reconstruction of each IMF and the residue, togetherwith the original and the parameters. The reconstructed image is in Figure 6.30 (a) at28.26 dB and the sum of bitrate of. 2.26 bpp. In Figure 6.30 (b) the compression ratio isset to 1. The image is reconstructed at 34.41 dB using a sum of bitrate of 7.97 bpp wheremost of the bits, 4.2 bpp, 2.8 bpp, are spent on IMF1 and IMF2 respectively.

full size

resize tomatrix

Zero pad

Reconstruction

IDCT scan to


135

Figure 6.28. DCT threshold coding results for each IMF and residue, a) first IMF, b) second IMF, c) zoomed, d) third IMF, e) zoomed, f) fourth IMF, g) zoomed, h) residue, i) zoomed.

a

b c

d e

f g

h i


136

Figure 6.29. Original to left, reconstructed in the middle, using the parameters specified to the right.

IMF135.47dB1.94 bpplength of vector:204864 quantization levels

IMF235.28 dB0.28 bpplength of vector: 32832 quantization levels

IMF335.11 dB0.13 bpplength of vector: 16416 quantization levels

IMF435.09 dB0.027 bpplength of vector:824 quantization levels

Residue36.13 dB0.008 bpplength of vector: 294 quantization levels

6.5 Summary

137

Figure 6.30. a) Reconstructed image, PSNR =28.46 dB, sum of bitrate = 2.39 bpp, b) Reconstructed image, PSNR =40.51 dB, sum of bitrate = 8.00 bpp.

When implementing a more traditional threshold coding of the DCT of the EMD, keepingonly the significant components, the result is results for each IMF and residue are approx-imately the same as for the method presented here.

6.4.4 ConclusionThreshold coding of the DCT of the IMF and residue gives a compact representation ofthese images except for the first and perhaps the second IMF which would need bettermethods to be efficiently compressed. Although the two coding methods tested result indifferent components kept, the coding result is not very different. The traditional thresh-old coder is slightly better. We have reached our first goal of representing the image withthe EMD, using no more bits than the original, and without too much distortion. Using amore efficient way to code the first IMF we expect to achieve better image compression.

6.5 SummaryIn this chapter different methods to compress the EMD are tested. First the set of IMFsand residue is entropy coded with both fine quantization and coarser.

The next idea to compress the EMD is the extrema point coder. The idea is to decom-pose the signal into its IMFs and transmit only maxima and mean values for each IMF.Then we reconstruct the IMF in the decoder with spline interpolation. The method is eval-uated both for time signals and for images. The method is good for compression of thesmooth IMFs, but fails to compress the first IMF due to its high-empiquency content. Theextrema point coding approach is an attractive way to represent the image but is ill suitedfor compression. It is in the quantization process that the major reduction of data andquality loss normally appears. In this method the major distortion comes from the recon-

a b


138

struction by interpolation. Using the extrema points to represent the image results in alarge number of coefficients to code. This number has to be reduced to have a good com-pression algorithm.

Another idea is to divide each IMF into blocks of a specific size and use the DCT oneach block. In our first attempt we use only one coefficient for each block. The motiva-tion is the assumption that the IMF holds locally only one spatial frequency. If the DCTcan represent this frequency with only one coefficient this would yield good compressionof the IMF. Even though the IMF should be locally zero mean, the blocking tiles the IMFinto non-zero mean parts. We need to include the DC component of each block to get arepresentation good enough for a compression algorithm. Due to the blocking artifacts,both the one-coefficient block-based DCT and the two-coefficient block-based DCT failto be candidates for a compression algorithm. The two-component blockbased DCT rep-resentation gives us good results in terms of PSNR values and bitrates, but the blockingartifacts are annoying.

The tendency is that the IMFs other than the first are low frequency images. This canbe seen in the DCT of the whole image. The idea in our third and fourth scheme is to keeponly the significant components. This is done by truncation of a zigzag scan of the DCTmatrix or only thresholding the components. Threshold coding gives a compact represen-tation of the IMF and residue except for the first and maybe the second IMF. With thismethod we can represent the image with EMD, using no more bits than the original, andwithout too much distortion.

139

Chapter 7

Variable Sampling of EMD

In Chapter 6 our first goal was met even without finding a good method for the compres-sion of the first IMF. To meet our second goal we must find a method for the first IMF.With the variable sampling presented in this chapter we go back to the thinking that leadus to the one-coefficient block-based DCT in section 6.3. A method of using overlapping7x7 blocks is introduced to overcome blocking artifacts and to further reduce the numberof parameters required to represent the image. The very special properties of the IMF thatthese are locally zero mean and ideally do not have more than one extrema point betweentwo neighbouring zero-crossings enables us to use the empiquency to control the sam-pling rate of each block.

7.1 Variable sampling of overlapping blocksThe special property of the IMF that the empiquency varies can be used for variable sam-pling. Areas with many extrema points have high empiquency, while areas with a few orno extrema points have low empiquency. The IMFs are smoother than the image itself;only the first IMF holds the nonsmooth parts of the image. This means that it should bepossible to subsample the IMFs. Due to the different empiquencies in the different partsof the IMF, the subsampling can be different in different parts of the IMF.

It is known [59] that a band-limited signal can be uniquely determined from its non-uniform samples, provided that the average sampling rate exceeds the Nyquist rate. Theextrema points define the maximum empiquency in the IMF. Maximum empiquency isfound by examining the space between the extrema points. In the first IMF there are areaswhere two neighbouring pixels both are extrema points, thus the maximum empiquencyis 0.5. Letting this define our Nyquist rate, we expect that it is not possible to subsample

Chapter 7 Variable Sampling of EMD

140

this IMF without distortion.Our suggestion is to treat the IMF blockwise. This way the sampling rate for each

block can be defined according to its empiquency content. The high empiquency blocksthat cannot be subsampled according to our Nyquist rate are not modified. The remainingones are subsampled.

7.1.1 Sampling of overlapping blocksIn the implementation of the blocking process we choose to use overlapping blocks ofsize 7x7 pixels. This is to minimize the artifacts from the blocking and to further reducethe samples used to represent the IMF. The corners of the block are always represented,regardless of the chosen sampling rate.

The overlapping pixels in two neighbouring blocks will be the same, but used twice,as shown in Figure 7.1. This will ensure that the concatenated blocks have the same val-ues at the edge pixels. The overlapping pixels will only belong to one of the blocks whenpatching and counting the total number of samples.

Figure 7.1. Variable sampling with overlapping 7x7 blocks.

The sampling pattern within a block consists of every pixel, every second pixel, everythird pixel, and every sixth pixel in both directions, to represent 1/1, 1/4, 1/9, and 1/36 ofthe pixels, respectively, as shown in Figure 7.1.

To find maximum empiquency we use a separable approach, analysing the extremapoints by row and column separately and choosing the maximum empiquency.

x xxx xxx

x xxx xxx

x xxx xxx

x xxx xxx

x xxx xxx

x xxx xxx

x xxx xxx

x x xx

x x xx

x x xx

x x xx

x x

x x x

x

x x x

x x

x x

7.1 Variable sampling of overlapping blocks

141

7.1.2 InterpolationFor a signal that is band limited to and sampled according to the sam-pling theorem, the signal can be reconstructed without distortion from its sampled version

through the reconstruction formula

(7.1)

A sampling function used for reconstruction satisfies

(7.2)

The sampling function that should be used according to the sampling theorem is thebasis function sinc(t).

(7.3)

This function has infinite duration.The sampling theory relies on the use of ideal filters. The use of finite signals is in

contradiction to the band limitation approach. An alternative approach is to use splines.These are piecewise polynomials with pieces that are smoothly connected together at thesample points.

The cubic B-spline interpolator is given by

(7.4)

Considering the fact that we are treating finite signals, the use of the spline interpola-tor to reconstruct the image from its samples is superior to using a windowed sinc func-

f t( )1

2T------ 12T------,–

f nT( )

f t( ) f nT( )hT t nT–( )n ∞–=

∞

∑=

hT t( )

hT t( )1 if t 0=

0 if t nT=

=

hT t( ) πt T⁄( )sinπt T⁄( )

-------------------------=

b 3( )

23---

tT---

2–

tT---

3

2---------+ 0 tT---

1<≤

2 tT---

– 3

1 tT---

2<≤

0 2 tT---

≤

=


142

tion as interpolator [86]. The uniformly sampled points of the block are thus connectedby a surface created by the use of an interpolating cubic spline extended to two dimen-sions.

7.2 ImplementationThe method of variable sampling of overlapping blocks is implemented according to thescheme in Figure 7.2. Each IMF and the residue from the EMD of the image are proc-essed in the variable sampling block.

7.2.1 The variable sampling blockThe details of the variable sampling block are shown in Figure 7.3. The extrema pointimage holds only the significant extrema points with respect to some size of the zerozone. Both the image itself and the extrema point image are decomposed into overlappingsubblocks. The blocks of the extrema point image are used for the determination of themaximum empiquency in the block. This steers the sampling rate of the correspondingimage block. In the resulting bitstream each block is marked with a two-bit header thatholds information about the sampling rate.

7.2.2 The reconstruction blockThe details of the reconstruction block are shown in Figure 7.4. The two-bit header of thestream tells us not only the sampling rate but also the length of the stream for the block.With this information the coefficients can be placed in their proper place in the 7x7pixelblock. The stream only provides samples for the 6x6 block; the missing samples arefound in the neighbouring blocks.

Reconstruction of the IMF uses the 7x7 block for block reconstruction with interpo-lation. The nonoverlapping 6x6 part of the reconstructed block is used when patching theblocks together to reconstruct the image.

7.2 Implementation

143

Figure 7.3. Details of the variable sampling block.

EMDDecomposition

First IMF

Second IMF

Nth IMF

Concatenatestreams

separate stream+


reconstructed


Second IMF

Decoder

Coder

Image

ImageReconstructed

Figure 7.2. Variable sampling scheme.


Residue

Reconstruction

Reconstruction

Reconstruction

Reconstruction

variablesampling

variablesampling

variablesampling

variablesampling

Overlapping

Find MaxEmpiquencyin Block

Subsample

Variable Sampling

IMF to

Blockand add headerto coefficientstream

points

Find extrema

OverlappingImage to

Block

Extrema Point


144

Figure 7.4. Details of the reconstruction block.

7.3 ResultsThe method of variable sampling is tested on the Lenna image of size 128 by 128 pixels,shown in Figure 7.13(a), using the EMD, shown in Figure 7.9 to the left. Other imagesare two different detail parts of the 512x512 Lenna image, one of size 64x64 pixelsshown in Figure 7.11(a), using the EMD, shown in Figure 7.5 to the left, and the otherone of size 128x128 pixels shown in Figure 7.12(a), using the EMD, shown in Figure 7.7to the left. Each IMF is subsampled using the method of variable sampling with overlap-ping 7x7 blocks, as described in section 7.2. Reconstructing the subsampled blocks isdone with cubic interpolation over a regular set of samples. The PSNR measure is com-puted from the original and the reconstructed image. The average sampling rate measureis a count of the number of samples used in relation to the total number of pixels in theimage.

The results of the variable subsampling of the IMFs and the residue are presented inFigure 7.5 to Figure 7.10. To the left are the original IMFs, in the middle the reconstruct-ed IMFs and to the right the parameters in terms of sampling rate and distortion. Fromthis we can see that the first IMF needs more than a quarter of the pixels, every secondsample in both directions, to be reconstructed at acceptable distortion. For the second andthird IMF we see a behaviour which is more pleasing in that it has a signal-to-noise ratioof more than 36 dB, combined with a low sampling rate. For the fourth IMF and the res-idue we have reached the limit of subsampling which is one sample per block.

The image is reconstructed by the sum of reconstructed IMFs and residue. InFigure 7.11 to Figure 7.13 the original image is shown together with the reconstructedimage. The sum of samples to represent all of the IMFs and the residue gives a value ofthe average sampling rate needed to represent the image.

Patch part of Interpolate

Reconstruction

blocksblockreconstruct sampled blockfrom stream

7.3 Results

145

IMF 1 35.97 dB average samplingrate 0.658.

IMF 2 36.08 dBaverage samplingrate 0.2193.



residue 4 40.58 dBaverage samplingrate 0.0317.

Figure 7.5. Detail one. The result of the variable subsampling of the set of IMFs and residue with zero zone set to 1. Left: original, Middle: reconstructed, Right: sampling rate and

distortion.


146







distortion.

7.3 Results

147

IMF 133.16 dB average samplingrate 0.8038.




Figure 7.7. Detail two. The result of the variable subsampling of the set of IMFs and residue with zero zone set to 1. Left: original, Middle: reconstructed, Right: sampling rate and

distortion.


148






distortion.

7.3 Results

149




IMF 4 62.69dBaverage samplingrate 0.1094.


Figure 7.9. Lenna 128. The result of the variable subsampling of the set of IMFs and residue with zero zone set to 1. Left: original, Middle: reconstructed, Right: sampling rate and

distortion.


150






Figure 7.10. Lenna 128. The result of the variable subsampling of the set of IMFs and residue with zero zone set to 20. Left: original, Middle: reconstructed, Right: sampling rate and

distortion.

7.3 Results

151

Figure 7.11. Detail one. a) Original image, b) Reconstructed image by the summation of reconstructed IMFs and residue, 30.45 dB, average sampling rate 1.0637

Figure 7.12. Detail two. a) Original image, b) Reconstructed image by the summation of reconstructed IMFs and residue, 30.9934 dB, average sampling rate 0.9193.

Figure 7.13. Lenna 128. a) Original image, b) Reconstructed image by the sum of reconstructed IMFs and residue, 31.8 dB using an average sampling rate of 0.86.

a b

a b

a b


152

7.4 SummaryWe have proposed a method for variable sampling of the two-dimensional EMD. This isdone block wise using the non-uniformly located extrema points of the IMF to controlthe uniform sampling rate of the block. The very special properties of the IMF can beused for variable sampling of the IMFs and the residue in order to reduce the number ofparameters to represent the image. The empiquency varies in the IMF and thus can beused to control the sampling rate adaptive over the image. Using overlapping blocks re-duces blocking artifacts.

A method of using overlapping 7x7 blocks is introduced to overcome blocking arti-facts and to further reduce the number of parameters to represent the image. The resultspresented here show that subsampling offers a way to keep the total numbers of samplesgenerated by EMD approximately equal to the number of pixels of the original image.

153

Chapter 8

Entropy Coding of Variable Sampling Components

The variable sampling in Chapter 7 presents a promising method to treat the EMD. Thesampling process leaves us with a number of samples for each block. In this chapter wewill concentrate on entropy coding the samples from variable sampling.

8.1 Coder outlineThis approach uses almost the same scheme as the variable sampling. The difference isthat after the subsampling the samples are entropy coded using Huffman coding or fix-length coding of the samples. The coder is shown in Figure 8.1.

The details of the variable sampling block are shown in Figure 7.3. The IMF issearched for extrema points and both the extrema point image and the IMF itself are di-vided into overlapping blocks. The maximum empiquency found in the extrema point im-age block determines the sampling rate for the subsampling of the corresponding IMFblock. The samples are represented by one sample alone or 6x6, 3x3, or 2x2 blocks ofsamples. The whole set of samples is entropy coded and the components are quantized be-fore the two-bit block header is added to each block.

The details of the reconstruction block are shown in Figure 7.4. The two-bit header in-dicates the sampling rate. With this information the components achieved from the streamcan be placed in their proper place in the 7x7 pixel block. The stream only contains sam-ples for the 6x6 block, the missing samples are found in the reconstructed neighbouringblocks. Since the missing samples are located in the rightmost column and the lowest rowof the block, the reconstruction starts with the block in the lower right corner workingrow-wise through the blocks. For the blocks with no neighbours holding missing samples

Chapter 8 Entropy Coding of Variable Sampling Components

154

dummy samples are used. The 7x7 blocks of samples is interpolated and the nonoverlap-ping 6x6 part of the reconstructed block is used to generate the output image.

8.1.1 ResultsThe coder is tested on the EMD of three different images, Detail one, Detail two, andLenna128, using zero zone 1, 10, and 20. The quantization varies from 256 levels downto 4 levels. Figure 8.2, Figure 8.4, and Figure 8.6 show the coding results for the EMDof the images, respectively. In Figure 8.3, Figure 8.5, and Figure 8.7, example recon-structions of the IMFs and residue using fine quantization are presented along with theoriginal image and the parameters used. The coding by the Huffman method gives betterperformance than the fixlength coding method for the first IMF while for the rest of theIMFs and the residue the fixlength coding method gives better performance than theHuffman method. Figure 8.8, Figure 8.9, and Figure 8.9 show the images reconstructedusing fine quantization.

VariableSampling

EMDDecomposition

First IMF

Second IMF

Nth IMF

Sourcecoding

Source decoding+


reconstructed


Second IMF

Decoder

Coder

Image

Image

Reconstructed

Figure 8.1. Variable sampling entropy coding scheme.


Residue

VariableSampling

Reconstruction

Reconstruction

Reconstruction

Reconstruction

VariableSampling

VariableSampling

8.1 Coder outline

155

Figure 8.2. Entropy coding result of Detail one. Zero zone = 1, 10, 20 quantization varies from 256 levels down to 4 levels.

Fixlength code

Huffman code


156

Figure 8.3. The result of the variable subsampling entropy coding of the set of IMF and residue for Detail one using fine quantization. Original to left, reconstructed in the middle, the

parameters specified to the right.

IMF131.93 dB4.04 bppzero zone 10




Residue40.50 dB0.31 bppzero zone 10

8.1 Coder outline

157

Figure 8.4. Entropy coding result of Detail two. Zero zone=1, 10, 20 quantization varies from 256 levels down to 4 levels.

Fixlength code

Huffman code


158

IMF 133.15 dB 6.58 bpp.zero zone 10

IMF 2 38.05 dB1.99 bpp.zero zone 10


Residue 55.80 dB0.96 bpp.zero zone 10

Figure 8.5. The result of the variable subsampling entropy coding of the set of IMF and residue for Detail two with different values on the zero zone and fine quantization. Left:

original, Middle: reconstructed, Right: bitrate and distortion.

8.1 Coder outline

159

Figure 8.6. Entry coding result of Lenna128. Zero zone=1, 10, 20 quantization varies from 256 levels down to 4 levels.

Fixlength code

Huffman code


160

IMF 133.03 dB 4.69 bpp.zero zone 10




Residue58.16 dB0.28 bpp.zero zone 10

Figure 8.7. The result of the variable subsampling entropy coding of the set of IMF and residue for Lenna128 with zero zone set to 10 and fine quantization. Left: original, Middle:

reconstructed, Right: sampling rate and distortion.

8.1 Coder outline

161

Figure 8.8. a) Detail one original, b) Detail one reconstructed to 28.49 dB using 7.47 bpp. Fine quantization.

Figure 8.9. a) Detail two original, b). Detail two image reconstructed to 31.41 dB using 6.55 bpp. Fine quantization.

Figure 8.10. a) Lenna128 original, b) Lenna128 image reconstructed to 30.15 dB using 6.67 bpp. Fine quantization.

ba

ba

a b


162

163

Chapter 9

EMD Image Coding using Variable Sampling of Overlapping Blocks

The entropy coding of the variable sampling presented in Chapter 8 shows the compres-sion potential of the EMD. In this chapter we will concentrate on using the variable sam-pling method for compression. We start with coding the samples of the IMF and residuewith the DCT. We then try coding the image by using the DCT of the variable sampledblocks for the first IMF and coding the first residue using the threshold DCT method pre-sented in Chapter 6.

9.1 Coding of the EMD using DCT of the variable sampled blocks (VSDCTEMD)The structure of the variable sampling described in Chapter 7 is inherited in this codingapproach. The sampling process leaves us with a number of samples for each block. Thesecan be squeezed into blocks of smaller size which can be DCT coded. The DCT compo-nents are then quantized and thresholded, leaving us with even fewer components to rep-resent the block.

9.1.1 Coder outlineThis approach uses almost the same scheme as the variable sampling. The difference is

Chapter 9 EMD Image Coding using Variable Sampling of Overlapping Blocks

164

that after the subsampling the samples are coded using DCT. The coder is shown inFigure 9.1

The details of the coding block are shown in Figure 9.2. the imf is searched for ex-trema points and both the extrema point image and the IMF itself are divided into over-lapping blocks. The maximum empiquency found in the extrema point image blockdetermines the sampling rate for the subsampling of the corresponding IMF block. Thesamples are represented by one sample alone or 6x6, 3x3, or 2x2 blocks of samples.These are DCT coded and the components are quantized and thresholded before the two-bit block header is added. The resulting component stream is Huffman coded or fixlengthcoded.

The details of the reconstruction block are shown in Figure 9.3. The two-bit headerindicates the sampling rate. With this information the components achieved from the in-verse DCT transform of the stream can be placed in their proper place in the 7x7 pixelblock. The stream only contains samples for the 6x6 block, the missing samples are foundin the reconstructed neighbouring blocks. Since the missing samples are located in therightmost column and the lowest row of the block, the reconstruction starts with the blockin the lower right corner working row-wise through the blocks. For the blocks with noneighbours holding missing samples dummy samples are used. The 7x7 blocks of sam-ples are interpolated and the nonoverlapping 6x6 part of the reconstructed block is usedto generate the output image.

EMDDecomposition

First IMF

Second IMF

Nth IMF

Source coding

Source+


reconstructed


Second IMF

Decoder

Coder

Image

ImageReconstructed

Figure 9.1. Variable sampling DCT coding scheme (VSDCTEMD).


Residue

Reconstruction

Reconstruction

Reconstruction

Reconstruction

Coding

Coding

Coding

Coding

decoding

9.1 Coding of the EMD using DCT of the variable sampled blocks (VSDCTEMD)

165

Figure 9.2. Details of the coding block.


9.1.2 ResultsThe VSDCTEMD coder is tested on three different images, Detail one, Detail two, andLenna128, using zero zones 1, 10, and 20. The quantization varies from 256 levels downto 8 levels. The DCT components are thresholded leaving only those having a value largerthan or equal to the threshold. The threshold varies from 0.1% to 100% of the maximumvalue. Figure 9.4, Figure 9.6, and Figure 9.8, show the coding results for the EMD of theimages, respectively, and in Figure 9.5, Figure 9.7, and Figure 9.9, example reconstruc-tions of the IMFs and residue are presented along with the original image and the param-eters used. The coding method gives good results for all IMFs and the residue. Even forthe difficult first IMF the result is over 30 dB for a bitrate of 1bpp. Example reconstruc-tions are presented in Figure 9.10, Figure 9.11 and Figure 9.12, along with the originalimage.

Overlapping


Subsample

IMF to

Block

points

Find extrema OverlappingImage to

Block

Extrema Pointaddheader

Threshold and quantize

DCTof

squeezedsamples

Patch part of Interpolate blocksblock

Read header and IDCT

Positioningsamples and


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Figure 9.4. VSDCTEMD Coding results of Detail one. Zero zone=1, 10, 20 quantization varies from 256 levels down to 8 levels.

Fixlength codeHuffman code


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Figure 9.5. The result of VSDCTEMD coding of Detail one. Original to left, reconstructed in the middle, using the parameters specified to the right.





Residue39.07 dB0.22 bppzero zone 1


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Figure 9.6. VSDCTEMD Coding result of Detail two. Zero zone=1, 10, 20 quantization varies from 256 levels down to 8 levels



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IMF 13309 dB 2.73 bpp.zero zone 10


IMF 3 44.86 dB0.22bpp.zero zone 20

Residue 48.75 dB0.22 bpp.zero zone 20

Figure 9.7. The result of VSDCTEMD coding of Detail two with different values on the zero zone. Left: original, Middle: reconstructed, Right: sampling rate and distortion.


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Figure 9.8. VSDCTEMD Coding result of Lenna128. Zero zone=1, 10, 20 quantization varies from 256 levels down to 8 levels.



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IMF 32.70 dB 3.12bpp.zero zone 1




Residue49.28 dB0.20 bpp.zero zone 1

Figure 9.9. The result of VSDCTEMD coding of Lenna128. Left: original, Middle: reconstructed, Right: sampling rate and distortion.


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Figure 9.10. a) Detail one original, b) Detail one image reconstructed to 28.27 dB using 4.79 bpp.

Figure 9.11. a) Detail two original, b). Detail two image reconstructed to 30.76 dB using 3.56 bpp.

Figure 9.12. a) Lenna128 original, b) Lenna128 image reconstructed to 29.33 dB using 4.33 bpp.

a b

a b

ba

9.2 Image coding using VSDCTEMD and DCT threshold coding

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9.2 Image coding using VSDCTEMD and DCT threshold coding In this modified coder we will only decompose into one IMF and one residue. We will usethe VSDCTEMD from section 9.1 on the first IMF and DCT threshold coding describedin section 6.4 on the first residue.

9.2.1 Coder outlineThis coder uses only the first IMF and the corresponding first residue to code the image.The coder is shown in Figure 9.13.

The first IMF is coded with the VSDCTEMD. The details of this block are shown inFigure 9.2. The first residue is coded using DCT threshold coding. The details of this cod-er are shown in Figure 6.26. The reconstruction is done according to the reconstructionblocks of each coder, shown in Figure 9.3 and Figure 6.27 respectively.

9.2.2 ResultsThe coder is tested on the images Detail one, Detail two, and Lenna128. The result of thecoding of the first IMF using VSDCTEMD and DCT threshold coding on the first residueis shown in Figure 9.14, Figure 9.16 and Figure 9.18 for the images respectively. All pos-

DCTThreshold coding

EMDFirst IMF

MUX

DEMUX

Channel

Reconstruction

Decoder

Coder

Image

ImageReconstructed

Figure 9.13. Combined VSDCTEMD and DCT threshold coding scheme.

VSDCTEMD

First residue

Reconstruction

DCT threshold

VSDCTEMDFirst IMF

First residue

+ coding


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sible pairs of these two sets of results for each image are presented in Figure 9.15,Figure 9.17 and Figure 9.19 as the result for this coder. Two different reconstruction ex-amples for each image are presented in Figure 9.20, Figure 9.21 and Figure 9.22.

Figure 9.14. Result of coding Detail one by VSDCTEMD in first IMF and DCT threshold coding on first residue.

Figure 9.15. Result of combining the coding of Detail one by VSDCTEMD in first IMF and DCT threshold coding on first residue.


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Figure 9.16. Result of coding Detail two by VSDCTEMD in first IMF and DCT threshold coding on first residue.

Figure 9.17. Result of combining the coding of Detail two by VSDCTEMD in first IMF and DCT threshold coding on first residue.


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Figure 9.18. Result of coding Lenna128 by VSDCTEMD in first IMF and DCT threshold coding on first residue.

Figure 9.19. Result of coding Lenna128 by VSDCTEMD in first IMF and DCT threshold coding on first residue.


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Figure 9.20. Result of coding Detail one by VSDCTEMD in first IMF and DCT threshold coding on first residue.a) 1.00 bpp 30.09 dB, b) 0.79 bpp 29.23 dB

Figure 9.21. Result of coding Detail two by VSDCTEMD in first IMF and DCT threshold coding on first residue.a) 1.2788 bpp 30.0208 dB, b) 0.69 bpp 28.88 dB

Figure 9.22. Result of coding Lenna128 by VSDCTEMD in first IMF and DCT threshold coding on first residue.a) 1.35 bpp 30.52 dB, b) 0.89 bpp 28.86 dB

a b

a b

a b


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9.3 SummaryIn this chapter we have used the variable sampling method for compression. We startedwith coding the samples of the IMF and residue with the DCT followed by an entropycoder. We then tried the approach of using only one IMF and coding the first IMF by us-ing the DCT of the variable sampled blocks and coding the first residue using the thresh-old DCT method presented in Chapter 6.

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Chapter 10

Empiquency-Controlled Image Coding

In this chapter we present alternative ways to code an image using the blockbased variablesampling. We assume that the same method that gives good results coding the first IMFalso can be used on the image itself. We use the empiquency of the first IMF to controlthe choice of coding method and sampling rate used in the different blocks of the image.In the first scheme, we use the empiquency of the first IMF to control the variable sam-pling of the image itself. In our second approach we apply quantization and entropy cod-ing on the variable sampling components. Finally in our third scheme we use theempiquency-controlled variable sampling together with DCT coding of the samples.

10.1 Empiquency-controlled variable sampling of the original imageThis approach is based on the variable sampling technique described in Chapter 7. Thedifference is that the EMD process now stops when the first IMF has been found. The em-piquency of the first IMF is used to control the subsampling of the image.

Coder outlineThe coder is shown in Figure 10.1. The details of the VS block are presented inFigure 10.2. The first IMF is searched for extrema points and both the image itself and themaximum empiquency image is decomposed into overlapping 7x7 pixel blocks, as de-scribed in section 7.1.1. The value of the maximum empiquency found in the extremapoint image block determines the sampling rate for the subsampling of the correspondingimage block. The samples are quantized and Huffman coded. The details of the recon-

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struction block are shown in Figure 10.3. The two-bit header indicates the sampling rate.With this information the components achieved from the stream can be placed in theirproper place in the 7x7 pixel block. The stream only contain samples for the 6x6 block,the missing samples are found in the neighbouring blocks. The 7x7 blocks of samples areinterpolated and the nonoverlapping 6x6 part of the reconstructed block is used whenpatching the blocks to reconstruct the image.

EMD First IMF

MUX

DEMUX

Channel

Reconstruction

Decoder

Coder

Image

ImageReconstructed

Figure 10.1. Empiquency controlled variable sampling scheme.

VS

10.1 Empiquency-controlled variable sampling of the original image

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Figure 10.2. Details of the VS block.


10.1.1 ResultsThe empiquency-controlled variable subsampling of the image is tested on the imagesDetail one, Detail two, and Lenna128. The resulst of subsampling using zero zone 10 areshown in Figure 10.4, Figure 10.5 and Figure 10.6 for the images, respectively. Only thevariable subsampling of the image renders compression to some degree. The image qual-ity shown here is the best we can expect from the next two coders in this section as theyonly differ in that they apply compression algorithms to the samples.

Overlapping


Subsample

Variable Sampling

Image to

Blockand add headerto coefficient

stream

points

Find extrema

OverlappingImage to

Block

Extrema Point

Image

First IMF


Reconstruction

blocksblock




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Figure 10.4. Variable sampling of Detail one, 34.60 dB at average sampling rate 0.71.

Figure 10.5. Variable sampling of Detail two, 32.93 dB at average sampling rate 0.86.

Figure 10.6. Variable sampling of Lenna128, 32.85 dB at average sampling rate 0.71.

10.2 Empiquency-controlled entropy coded variable sampling (Entropy coded VS)

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10.2 Empiquency-controlled entropy coded variable sampling (Entropy coded VS)The structure of the variable sampling in section 10.1 is inherited in this coding approach.The sampling process leaves us with a set of samples for each block which are quantizedand entropy coded using Huffman code or fixlength code.

Coder outlineThe coder is shown in Figure 10.7

The first IMF is searched for extrema points and as in previous schemes using variablesampling, only the significant extrema points are kept for the definition of empiquency.The details of the VS block are shown in Figure 10.2. Both the image itself and the ex-trema point image are decomposed into overlapping 7x7pixel blocks as described insection 7.1.1. The value of the maximum empiquency in the extrema point image blockdecides the sampling rate for the subsampling of the corresponding image block. Thesamples are represented by one sample alone or 6x6, 3x3, or 2x2 blocks of samples. Atwo-bit header is added to the samples of each block to indicate the sampling rate of thisspecific block.

The details of the reconstruction block are shown in Figure 10.3. The two-bit header

EMD First IMF

Source Coding

Source Decoding

Channel

Reconstruction

Decoder

Coder

Image

ImageReconstructed

Figure 10.7. Coding scheme.

VS


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indicates the sampling rate. With this information the components can be placed in theirproper place in the 7x7 pixel block. The stream only contains samples for the 6x6 block,the missing samples are found in the reconstructed neighbouring blocks. The 7x7 blocksof samples are interpolated and the nonoverlapping 6x6 part of the reconstructed block isused to reconstruct the image.

10.2.1 ResultsThe coder is tested on three different images, Detail one, Detail two, and Lenna128, usingzero zones 1, 10, and 20. The quantization varies from 256 levels to 4 levels. Figure 10.8,Figure 10.9, and Figure 10.10, show the coding result for the images respectively, and inFigure 10.11, Figure 10.12, and Figure 10.13, two example reconstructions for each im-age are presented.

Figure 10.8. Entropy coding result of Detail one. Zero zone=1, 10, 20; quantization varies from 256 levels down to 4 levels

10.2 Empiquency-controlled entropy coded variable sampling (Entropy coded VS)

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Figure 10.9. Entropy coding result of Detail two. Zero zone=1, 10, 20; quantization varies from 256 levels down to 4 levels

Figure 10.10. Entropy coding result of Lenna128. Zero zone=1, 10, 20; quantization varies from 256 levels down to 4 levels


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Figure 10.11. Entropy coded VS of Detail one: a) 4.00 bpp 34.38 dB, b) 1.15 bpp 26.90 dB

Figure 10.12. Entropy coded VS of Detail two: a) 4.07 bpp 32.34 dB, b) 1.80 bpp 28.41 dB

Figure 10.13. Entropy coded VS of Lenna128: a) 3.31 bpp 32.56 dB, b) 1.90 bpp 28.96dB.

a b

a b

a b

10.3 Empiquency-controlled image coding using variable sampling and DCT coding

187

10.3 Empiquency-controlled image coding using variable sampling and DCT coding (VSDCT)The structure of the variable sampling in section 10.1 is inherited in this coding approach.The sampling process leaves us with a set of samples for each block which are squeezedinto blocks of smaller size and DCT coded. The DCT components are then quantized andthresholded.

Coder outlineThe VSDCT coder is shown in Figure 10.14

The first IMF is searched for extrema points and as in previous schemes using variablesampling, only the significant extrema points are kept for the definition of empiquency.The details of the VS+DCT block are shown in Figure 10.15. Both the image itself andthe extrema point image are decomposed into overlapping 7x7 pixel blocks as describedin section 7.1.1. The value of the maximum empiquency in the extrema point image blockdecides the sampling rate for the subsampling of the corresponding image block. Thesamples are represented by one sample alone or 6x6, 3x3, or 2x2 blocks of samples. Theseare DCT coded and the components are quantized and thresholded before the two-bitblock header is added.

EMD First IMF

Source Coding

Source Decoding

Channel

Reconstruction

Decoder

Coder

Image

ImageReconstructed

Figure 10.14. VSDCT coding scheme.

VS+DCT


188

The details of the reconstruction block are shown in Figure 10.16. The two-bit headerindicates the sampling rate. With this information the components achieved from the in-verse DCT transform of the stream can be placed in their proper place in the 7x7 pixelblock. The stream only contains samples for the 6x6 block, the missing samples are foundin the reconstructed neighbouring blocks. The 7x7 blocks of samples are interpolated andthe nonoverlapping 6x6 part of the reconstructed block is used to reconstruct the image.

Figure 10.15. Detail of the VS+DCT coding block.


10.3.1 ResultsThe VSDCT coder is tested on three different images: Detail one, Detail two, andLenna128, using zero zone 1, 10, and 20. The quantization varies from 256 levels to 8levels. The DCT components are thresholded. The threshold varies from 0.1% to 100%of the maximum value. Figure 10.17, Figure 10.18, and Figure 10.19, show the codingresult for the images respectively, and in Figure 10.20, Figure 10.21, and Figure 10.22,two example reconstructions for each image are presented. We see that this coder givesgood compression results for bitrates around 1 bpp.

Overlapping


Subsample

Image to

Block

points

Find extrema OverlappingImage to

Block

Extrema Point

DCTof

addheader

squeezed

Threshold and quantize

samples

Image

IMF


Reconstruction

blocksblock




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Figure 10.17. The result of using VSDCT coder on the Detail one image.

Figure 10.18. The result of using the VSDCT coder on the Detail two image.


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Figure 10.19. The result of using the VSDCT coder on the Lenna128 image.


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Figure 10.20. VSDCT of Detail one: a) 1.17 bpp 33.83 dB, b) 0.37 bpp 31.04 dB

Figure 10.21. VSDCT of Detail two: a) 1.33 bpp 33.28 dB, b) 0.43 bpp 30.17 dB

Figure 10.22. VSDCT of Lenna128: a) 1.35 bpp 28.93 dB, b) 0.66 bpp 21.17 dB.

a b

a b

a b


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10.4 SummaryIn this chapter we have presented new methods to code an image using the blockbasedvariable sampling. The EMD is only used to find the first IMF. The empiquency of thefirst IMF is used to control the choice of coding method and sampling rate used in thedifferent image blocks. First the empiquency of the first IMF was used to control the var-iable sampling of the image itself. This gives us the limit of the performance we can ex-pect from the other coders tested. The variable subsampling of the image rendercompression to some degree. This is the basis for using the empiquency-controlled vari-able sampling together with two different entropy coders and the empiquency-controlledvariable sampling together with DCT coding of the samples. The key results from eachcoder are summarized in Figure 10.23, Figure 10.24, and Figure 10.25.

Figure 10.23. Summary of results from the different coders applied to the Detail one image.

10.4 Summary

193

Figure 10.24. Summary of results from the different coders applied to the Detail two image.

Figure 10.25. Summary of results from the different coders applied to the Lenna128 image.


194

195

Chapter 11.

Summary and Discussion

In this chapter we summarize the work in the second part of this thesis, trying to put it ina wider perspective, discussing the results and relating the work to the work of others.

11.1 EMD The second part of this thesis discusses the Empirical Mode Decomposition (EMD),which is an adaptive decomposition with which any complicated signal can be decom-posed into its Intrinsic Mode Functions (IMF), corresponding to the physical reality thatgenerated the signal [36]. The purpose of the work on EMD on time signals in this thesisis to introduce the EMD concept.

An IMF is characterized by some specific properties. One is that the number of zerocrossings and the number of extrema points are equal or differ only by one. Another prop-erty of the IMF is that the envelopes defined by the local maxima and minima, respective-ly, are locally symmetric around the envelope mean. The original purpose for the EMDwas to find a decomposition which made it possible to use the instantaneous frequency,defined as the derivative of the phase of an analytic signal, in the time frequency analysisof the signal. Traditional Fourier analysis does not allow a signal's spectrum to changeover time. The EMD, however, provides a decomposition method that analyses the signallocally. The use of EMD in the Hilbert-Huang Transform (HHT) [36] is viewed as an ex-ample of EMD for time signals. Different interpolation methods are discussed for timesignals as an introduction to the same problem for images.

Previous work often mention the lack of a mathematical formalism to describe theEMD, the concept is truly empirical. In this thesis we keep the empirical approach as weextend the EMD concept into two dimensions.

The sifting process of [36] is here extended and given a more strict formulation in twodimensions.

The behaviour of the EMD as a filter bank is highlighted in Flandrin et al. [26] throughthe analysis of noise. The first IMF resemble a “high-pass filtered” noise signal. The rest

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196

of the IMFs resembles “band-bass filtered” noise signals with their centre frequency de-creasing in an octave band manner, just like a filter bank. The experiment shows that theEMD sorts the energy of the signal similarly to a subband coder. The same can be seenwhen EMD is applied to images. The tendency is that the IMFs other than the first arelow frequency images. By examining of the DCT of the whole image we see that for thefirst IMF the energy is spread all over the matrix, with a highlight in the centre part rep-resenting the dominating spatial frequencies. For the subsequent IMFs and the residue theenergy concentrates in the upper left corner, where it represents low frequency.

Some implementations of the EMD in two dimensions generate a residue with manyextrema points [51,62,63]. These methods have not proven capable of fully decomposingthe two-dimensional signal down to a residue with a low number of extrema points. Inthis thesis we present an improved method that can decompose the image into a numberof IMFs and a residue with none, or with only a few extrema points. This method makesit possible to use the EMD for image processing.

For two-dimensional signals there are several possibilities to define extrema, eachone yielding a different decomposition. In this thesis the extrema points are simply ex-tracted by comparing the candidate data point with its nearest 8-connected neighbours.Clearly, more sophisticated methods could be used, but the extrema points defined by an8-connected neighbourhood serve the purpose for EMD at this stage, with further im-provement being possible. In Nunes et al. [63] the extrema points are selected using mor-phological reconstruction. Neither of these methods allow for saddle points to beconsidered as extrema points. Nunes et al. proposed the use of a watershed operation tofind these extrema points in the future [63]. In this thesis we note that in the case wheresaddle points are considered to be extrema points, these are both maxima and minima atthe same point. This nails the saddle point to the zero mean level in the first sifting round.For this reason, saddle points are not considered to be extrema points. Future work mayfind methods for extrema point extraction that could also handle saddle points.

In the second step in the sifting process the problem is to fit a surface to the two-di-mensional scattered data points representing the extrema points. The interpolated surfacemust go through each data point. Overshot shall be avoided and the second derivativemust be continuous everywhere for the signal to be smooth enough for two-dimensionalEMD. The interpolation methods considered here are triangle-based cubic spline interpo-lation and thin-plate smoothing spline interpolation. The first is an example of an inter-polation method using a piecewise approach to interpolate the surface. It produces piece-wise smooth surfaces and is based on Delaunay triangulation [67] of the data. The piece-wise approach causes more problems in the two-dimensional case than with one-dimen-sional signals. Even though the interpolators have continuous second derivatives, such asthe cubic spline in [51] or the radial basis functions used in [63], the borders of the neigh-bouring pieces cause problems. It is possible that the piece-wise approach will work ifwe use another set of extrema points where the saddle points are included. This is not ex-amined in this thesis but is left to future work.

The border constraints are even more important in two dimensions than they are inthe one-dimensional EMD case and are the main objection to using spline interpolationin the EMD for two-dimensional signals. The set of extrema points is very sparse andsince the interpolation methods only interpolate between these points, the borders needspecial care. In this thesis we introduce the trick of adding extra data points at the bordersto the set of extrema points. These extra points are placed at the corners of the image andequally spaced around the border. Without these extra points, the areas not covered by

11.2 Empiquency

197

the interpolation traverse into the image in the sifting process.In this thesis the thin-plate smoothing spline interpolation [8] is suggested for two-di-

mensional EMD analysis. This algorithm calculates the surface over the entire image andgives a surface with continuous second derivatives everywhere. The determination of thesmoothing spline involves the solution of a linear system with as many unknowns as thereare data points. This method turns out to successfully decompose an image into its IMFsand a smooth residue with no or only a few extrema points.

In [36] the sifting process stops when the difference between two consecutive siftingsis smaller than a selected threshold. In this thesis the process stops when the envelopemean signal is close enough to zero as suggested in Linderhed [51]. The reason for thischoice is that forcing the envelope mean to zero will guarantee the symmetry of the en-velope and the correct relation between the number of zero crossings and number of ex-tremes that define the IMF. A slightly more complicated version of this stop criterion ispresented in Rilling et al. [73] along with a discussion of typical values on the threshold.The number of IMFs achieved with the EMD depends on the choice of value of the stopcriterion. If it is too large, the number of IMFs is lower. If it is too small, the number ofIMFs is large enough but computation time is longer than necessary.

11.2 EmpiquencyThe EMD is a truly empirical method, not based on the Fourier frequency approach butrelated to the locations of extrema points and zero-crossings. Using the EMD for signalanalysis requires a mind free from the traditional Fourier-based frequency concept. Basedon this we introduce the concept of empiquency [53], short for empirical mode frequency,instead of a traditional Fourier-based frequency measure to describe the signal oscilla-tions. The measure of empiquency is defined as “One half the reciprocal distance betweentwo consecutive extrema points”. With this definition the values relate to normalized fre-quency used with discrete Fourier transform. We define d to be the distance between twoneighbouring extrema points. In [36] the concept of time scale is used to describe the sig-nal oscillations. In relation to empiquency this can be seen as the mean of all d in the sig-nal. In this context the empiquency is the local time scale.

The IMF have the very special property of being locally zero mean and ideally havingnot more than one extrema point between two neighbouring zero-crossings. Because ofthis relationship between zero-crossings and extrema points there is also a relation be-tween the empiquency, time scale, and the concept of sequency [31]. Sequency is definedas, “the average number of zero crossings per second divided by 2”. For a sine or cosinethe frequency and the sequency are the same, but the concept of sequency has a meaningfor other signals such as Walsh functions or IMF as well. There is a possibility that themathematical theory developed for sequency could be used for the development of math-ematical methods for EMD. This idea is not further treated in this thesis but left to futurework.

The empiquency value is assigned to every position between the two extrema pointsused to compute it. Each extrema point influences more than one set of empiquency def-initions. We let it take the highest value of the empiquency values it defines. The em-piquency is accompanied by an amplitude describing the signal at the actual position. Ateach extrema point the empiquency amplitude Ae is the absolute value of the extrema


198

point. At any other position the empiquency amplitude takes the value of the absolutemean of the extrema points that define the empiquency for the actual position. The em-piquency concept will be further developed towards a texture analysis tool. In the imagecoding applications in this thesis the empiquency is only used as an indication of thehighest spatial frequency present in the block to be coded.

In this thesis we have also discussed the selection of significant extrema points. Thishas turned out to be essential for image compression with EMD similar to the definitionof significant coefficients in zerotree coding [75]. The selection of significant extremapoints can also be used as a tool for noise reduction.

11.3 Variable samplingIt is known [59,35] that a band-limited signal can be uniquely determined from its non-uniform samples, provided that the average sampling rate exceeds the Nyquist rate. Fromthis we assume that the IMF can be subsampled with special requirements on the sam-pling rate and that this rate is given by the maximum empiquency.

The special property of the IMF, namely that the empiquency varies, is used for var-iable sampling. The IMFs are smoother than the image itself; only the first IMF holds thenonsmooth parts of the image. This means that it should be possible to subsample the IM-Fs. Due to the different empiquencies in the different parts of the IMF, the subsamplingshould vary. The extrema points define the maximum empiquency in the IMF. Maximumempiquency is found by examining the space between these points. In the first IMF thereare often areas where two neighbouring pixels both are extrema points, thus the maxi-mum empiquency is 0.5. Letting this define our Nyquist bandwidth, we expect that it isnot possible to subsample this IMF without distortion. Our suggestion is to treat the IMFblock wise. This way the sampling rate for each block can be defined according to its em-piquency content. The high empiquency blocks which cannot be subsampled accordingto our criterion are not modified. The remaining ones are subsampled.

Another proposal for variable sampling of an image is presented in [7], where an im-age coder based on non-uniform sampling of the image is presented. The sampling ratewas determined by the complexity of the region. The use of Delaunay triangulation to-gether with irregular sampling of an image is presented in Rila [72] in an image codingapproach with promising results.

In this thesis we use overlapping blocks of size 7x7 pixels to minimize the artifactsfrom the blocking and to further reduce the samples used to represent the IMF. The cor-ners of the block are always represented. The overlapping pixels in two neighbouringblocks will be the same, but used twice. This will ensure that the concatenated blockshave the same values at the edge pixels. The overlapping pixels will only belong to oneof the blocks when patching together the reconstructed image.

In our EMD image coding proposals the image content steers the choice of methodfor each part of the image. This is the opposite of dynamic coding [27,24] where the se-lection is guided by the coding result. Dynamic coding deals with coding of images inwhich multiple coding schemes are used in different regions of an image.

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11.4 EMD image coding The image compression efforts in this work can be divided into two groups. The firstgroup of methods use the full EMD, the image is decomposed down to a residue with onlya few extrema points. The second group of methods use an EMD where only the first IMFis computed.

In the first group of methods we have: Entropy coding of the EMDExtrema Point CoderBlock-based DCTThreshold coding of the full size DCTVSDCTEMDThe first attempt to use the EMD for compression was the Extrema Point Coder [51].

This method is examined in this thesis both for time signals and images. The method ofrepresenting the IMF with its extrema points gives good visual performance but needsmany extrema points for the representation of the first IMF and the coding of positions ofthese generate a high bitrate.

The block-based DCT is an attempt to use the property of the IMF that only one em-piquency is present at each position. Using only one or two of the DCT components foreach block in the representation gives bitrate and distortion measures that are acceptablebut the visual performance suffers from severe blocking artifacts.

The Threshold DCT method works on the whole IMF and takes advantage of the en-ergy compaction ability of the EMD. The energy of the IMFs other than the first IMF isconcentrated in the low frequency regions and thus the rest of the frequency componentscan be left out of the representation of the EMD. Even for the first IMF this method givesacceptable distortion, both in terms of PSNR measure and visual performance, with bi-trates below 2bpp. For the rest of the IMFs we have good performance at bitrates way be-low 1 bpp.

The variable sampling presents a promising method to treat the EMD. The structureof the variable sampling is inherited in the VSDCTEMD coding approach. The samplingprocess applied on each IMF and the residue leaves us with a reduced number of samplesfor each block. These can be squeezed into blocks of smaller sizes which can be DCT cod-ed. The DCT components are then quantized and thresholded, leaving us with even fewercomponents to represent the block. The DCT coding of the block samples provides a sig-nificant compression of the variable sampled IMFs.

For image coding purposes there is no need to decompose the image into a full EMD.The second group of image compression methods use the empiquency of the first IMF tocontrol the choice of coding method used in the different blocks of the image. The meth-ods that we have developed are:

Entropy coded VSVSDCTEMD+Threshold DCTVSDCT

These new methods to code an image use the block-based variable sampling controlledby the first IMF. With the VS method we start with using the empiquency of the first IMFto control the variable sampling of the image. The image quality shown here is the bestwe can expect compared to the remaining coders in this section as they only differ in thatthey further compress the samples.

Threshold coding gives a very compact representation of the residue. Together with


200

the VSDCTEMD on the first IMF the compression result is very good at bitrates around1 bpp.

The structure of the variable sampling is inherited in the VSDCT coding approach.The first IMF controls the sampling rate of the image block. The set of samples for eachblock left by the sampling process are squeezed into blocks of smaller size and DCT cod-ed. The DCT components are then quantized and thresholded. For the VSDCT the com-pression result is also very good at bitrates around 1 bpp. This coder performs slightlybetter than the use of VSDCTEMD together with the Threshold DCT

The coders presented in this thesis are only early examples of image coding algo-rithms using the EMD. There is still much room for improvement and new ideas. A sim-ilar image compression method, but still totally different, is the multi-layered imagerepresentation of Meyer et al. [60], where the detail image is produced with a waveletpacket method, compressed and subtracted from the image to create a residue. This is it-erated to produce subsequent residues. In this method the detail is contained in the resi-due images instead of in the IMFs. Apart from that difference it seems likely that newEMD compression algorithms will be influenced by the work on multi-layered coding.

11.5 Open problemsThe work in this thesis was prompted by a central question: how to find, and use for com-pression, an adaptive decomposition of images. The work has found some answers butgenerated even more questions. Some of the problems raised from the work in this thesisbut still unsolved are listed here.

• Develop the relation between the sequency and empiquency.

• Develop better algorithms for extrema point selection, that can handle saddlepoints.

• Use different sampling rates for horizontal and vertical in the variable sampling.

• Develop the compression algorithms further.

• Develop EMD for texture analysis.

201

Appendix A

In this appendix the 17 samples of Brodatz texture images used in these experiments arepresented.

Figure A.1. The 17 samples of Brodatz texture images used in these experiments.

Appendix A

202

203

Appendix B

This appendix shows further parts of the triplet database. The textures in Figure B.1 areused as test images. Here, cost functions are varied for different typer of filters.

Figure B.1. The images bark, brickwall, cslfleath, from left to right.

Figure B.2. The bark image decomposed with the Haar filter and the cost function is varied. The cost of coding the coefficients of the optimal wavelet packet basis is marked by o and the cost of

coding the coefficients of the corresponding wavelet basis is marked by x.

1 2 3 4 5 6 7 8 92.145

2.15

2.155

2.16

2.165

2.17


haarbark

Appendix B

204

Figure B.3. The best basis for the triplet filter Haar, image bark, and cost function hest, defined in Eq. 4.42). The estimated entropy shows in position 2 in Figure B. 2.

Figure B.4. The best basis for the triplet filter Haar, image bark, and cost function n1, defined in Eq. 4.44). The estimated entropy shows in position 4 in Figure B. 2.

(0,0)

(1,0) (1,1)(1,2)(1,3

(2,0) (2,1) (2,2) (2,3)

(3,0) (3,1) (3,2) (3,3) (3,8) (3,9) (3,10) (3,11)

(4,0) (4,1) (4,2) (4,3) (4,8) (4,9) (4,10) (4,11) (4,32) (4,33) (4,34) (4,35) (4,40) (4,41) (4,42) (4,43)

(5,0)(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)(5,7)(5,8)(5,9)(5,10)(5,11)(5,12)(5,13)(5,14)(5,15)(5,32)(5,33)(5,34)(5,35)(5,36)(5,37)(5,38)(5,39)(5,40)(5,41)(5,42)(5,43)(5,44)(5,45)(5,46)(5,47)(5,128)(5,129)(5,130)(5,131)(5,132)(5,133)(5,134)(5,135)(5,136)(5,137)(5,138)(5,139)(5,140)(5,141)(5,142)(5,143)(5,160)(5,161)(5,162)(5,163)(5,164)(5,165)(5,166)(5,167)(5,168)(5,169)(5,170)(5,171)(5,172)(5,173)(5,174)(5,175)

(0,0)

(1,0) (1,1) (1,2) (1,3

(2,0) (2,1) (2,2) (2,3) (2,8) (2,9)(2,10)(2,11)

(3,0) (3,1) (3,2) (3,3) (3,8) (3,9)(3,10)(3,11) (3,32) (3,33)(3,34)(3,35)

(4,0) (4,1)(4,2)(4,3) (4,8) (4,9)(4,10)(4,11) (4,32) (4,33)(4,34)(4,35) (4,128) (4,129)(4,130)(4,131)

(5,0)(5,1)(5,2)(5,3) (5,32)(5,33)(5,34)(5,35) (5,128)(5,129)(5,130)(5,131)(5,132)(5,133)(5,134)(5,135) (5,512)(5,513)(5,514)(5,515)(5,516)(5,517)(5,518)(5,519)

205

Figure B.5. The bark image decomposed with the filter bior1.5 and the cost function is varied. The cost of coding the coefficients of the optimal wavelet packet basis is marked by o and the cost of


Figure B.6. The best basis for the triplet filter bior1.5, image bark, and cost function log, defined in Eq. 4.43).The estimated entropy shows in position 3 in Figure B. 5.

1 2 3 4 5 6 7 8 92

3

4

5

6

7

8


bior1.5bark

(0,0)

(1,0) (1,1) (1,2) (1,3)

(2,0) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (2,7) (2,8) (2,9) (2,10) (2,11) (2,12) (2,13) (2,14) (2,15)

(3,0)(3,1)(3,2)(3,3)(3,4)(3,5)(3,6)(3,7)(3,8)(3,9)(3,10)(3,11)(3,12)(3,13)(3,14)(3,15)(3,16)(3,17)(3,18)(3,19)(3,20)(3,21)(3,22)(3,23)(3,24)(3,25)(3,26)(3,27)(3,28)(3,29)(3,30)(3,31)(3,32)(3,33)(3,34)(3,35)(3,36)(3,37)(3,38)(3,39)(3,40)(3,41)(3,42)(3,43)(3,44)(3,45)(3,46)(3,47)(3,48)(3,49)(3,50)(3,51)(3,52)(3,53)(3,54)(3,55)(3,56)(3,57)(3,58)(3,59)(3,60)(3,61)(3,62)(3,63

(4,0)(4,1)(4,2)(4,3)(4,4)(4,5)(4,6)(4,7)(4,8)(4,9)(4,10)(4,11)(4,12)(4,13)(4,14)(4,15)(4,16)(4,17)(4,18)(4,19)(4,20)(4,21)(4,22)(4,23)(4,24)(4,25)(4,26)(4,27)(4,28)(4,29)(4,30)(4,31)(4,32)(4,33)(4,34)(4,35)(4,36)(4,37)(4,38)(4,39)(4,44)(4,45)(4,46)(4,47)(4,48)(4,49)(4,50)(4,51)(4,52)(4,53)(4,54)(4,55)(4,56)(4,57)(4,58)(4,59)(4,60)(4,61)(4,62)(4,63)(4,64)(4,65)(4,66)(4,67)(4,68)(4,69)(4,70)(4,71)(4,72)(4,73)(4,74)(4,75)(4,76)(4,77)(4,78)(4,79)(4,80)(4,81)(4,82)(4,83)(4,84)(4,85)(4,86)(4,87)(4,88)(4,89)(4,90)(4,91)(4,92)(4,93)(4,94)(4,95)(4,96)(4,97)(4,98)(4,99)(4,100)(4,101)(4,102)(4,103)(4,104)(4,105)(4,106)(4,107)(4,108)(4,109)(4,110)(4,111)(4,112)(4,113)(4,114)(4,115)(4,116)(4,117)(4,118)(4,119)(4,120)(4,121)(4,122)(4,123)(4,124)(4,125)(4,126)(4,127)(4,128)(4,129)(4,130)(4,131)(4,132)(4,133)(4,134)(4,135)(4,136)(4,137)(4,138)(4,139)(4,140)(4,141)(4,142)(4,143)(4,144)(4,145)(4,146)(4,147)(4,148)(4,149)(4,150)(4,151)(4,152)(4,153)(4,154)(4,155)(4,156)(4,157)(4,158)(4,159)(4,160)(4,161)(4,162)(4,163)(4,164)(4,165)(4,166)(4,167)(4,168)(4,169)(4,170)(4,171)(4,172)(4,173)(4,174)(4,175)(4,176)(4,177)(4,178)(4,179)(4,180)(4,181)(4,182)(4,183)(4,184)(4,185)(4,186)(4,187)(4,188)(4,189)(4,190)(4,191)(4,192)(4,193)(4,194)(4,195)(4,196)(4,197)(4,198)(4,199)(4,200)(4,201)(4,202)(4,203)(4,204)(4,205)(4,206)(4,207)(4,208)(4,209)(4,210)(4,211)(4,212)(4,213)(4,214)(4,215)(4,216)(4,217)(4,218)(4,219)(4,220)(4,221)(4,222)(4,223)(4,224)(4,225)(4,226)(4,227)(4,228)(4,229)(4,230)(4,231)(4,232)(4,233)(4,234)(4,235)(4,236)(4,237)(4,238)(4,239)(4,240)(4,241)(4,242)(4,243)(4,244)(4,245)(4,246)(4,247)(4,248)(4,249)(4,250)(4,251(4,252(4,253(4,25(4,25

(5,0)(5,1)(5,2)(5,3)(5,16)(5,17)(5,18)(5,19)(5,24)(5,25)(5,26)(5,27)(5,32)(5,33)(5,34)(5,35)(5,36)(5,37)(5,38)(5,39)(5,48)(5,49)(5,50)(5,51)(5,52)(5,53)(5,54)(5,55)(5,56)(5,57)(5,58)(5,59)(5,64)(5,65)(5,66)(5,67)(5,72)(5,73)(5,74)(5,75)(5,80)(5,81)(5,82)(5,83)(5,96)(5,97)(5,98)(5,99)(5,100)(5,101)(5,102)(5,103)(5,104)(5,105)(5,106)(5,107)(5,108)(5,109)(5,110)(5,111)(5,112)(5,113)(5,114)(5,115)(5,116)(5,117)(5,118)(5,119)(5,120)(5,121)(5,122)(5,123)(5,124)(5,125)(5,126)(5,127)(5,128)(5,129)(5,130)(5,131)(5,132)(5,133)(5,134)(5,135)(5,144)(5,145)(5,146)(5,147)(5,148)(5,149)(5,150)(5,151)(5,152)(5,153)(5,154)(5,155)(5,156)(5,157)(5,158)(5,159)(5,176)(5,177)(5,178)(5,179)(5,180)(5,181)(5,182)(5,183)(5,184)(5,185)(5,186)(5,187)(5,188)(5,189)(5,190)(5,191)(5,192)(5,193)(5,194)(5,195)(5,196)(5,197)(5,198)(5,199)(5,200)(5,201)(5,202)(5,203)(5,204)(5,205)(5,206)(5,207)(5,208)(5,209)(5,210)(5,211)(5,212)(5,213)(5,214)(5,215)(5,216)(5,217)(5,218)(5,219)(5,220)(5,221)(5,222)(5,223)(5,224)(5,225)(5,226)(5,227)(5,228)(5,229)(5,230)(5,231)(5,232)(5,233)(5,234)(5,235)(5,236)(5,237)(5,238)(5,239)(5,240)(5,241)(5,242)(5,243)(5,244)(5,245)(5,246)(5,247)(5,248)(5,249)(5,250)(5,251)(5,256)(5,257)(5,258)(5,259)(5,260)(5,261)(5,262)(5,263)(5,264)(5,265)(5,266)(5,267)(5,272)(5,273)(5,274)(5,275)(5,276)(5,277)(5,278)(5,279)(5,280)(5,281)(5,282)(5,283)(5,284)(5,285)(5,286)(5,287)(5,288)(5,289)(5,290)(5,291)(5,292)(5,293)(5,294)(5,295)(5,296)(5,297)(5,298)(5,299)(5,300)(5,301)(5,302)(5,303)(5,304)(5,305)(5,306)(5,307)(5,308)(5,309)(5,310)(5,311)(5,312)(5,313)(5,314)(5,315)(5,316)(5,317)(5,318)(5,319)(5,320)(5,321)(5,322)(5,323)(5,324)(5,325)(5,326)(5,327)(5,328)(5,329)(5,330)(5,331)(5,332)(5,333)(5,334)(5,335)(5,336)(5,337)(5,338)(5,339)(5,340)(5,341)(5,342)(5,343)(5,344)(5,345)(5,346)(5,347)(5,352)(5,353)(5,354)(5,355)(5,356)(5,357)(5,358)(5,359)(5,360)(5,361)(5,362)(5,363)(5,364)(5,365)(5,366)(5,367)(5,368)(5,369)(5,370)(5,371)(5,372)(5,373)(5,374)(5,375)(5,376)(5,377)(5,378)(5,379)(5,380)(5,381)(5,382)(5,383)(5,384)(5,385)(5,386)(5,387)(5,388)(5,389)(5,390)(5,391)(5,392)(5,393)(5,394)(5,395)(5,396)(5,397)(5,398)(5,399)(5,400)(5,401)(5,402)(5,403)(5,404)(5,405)(5,406)(5,407)(5,408)(5,409)(5,410)(5,411)(5,412)(5,413)(5,414)(5,415)(5,416)(5,417)(5,418)(5,419)(5,420)(5,421)(5,422)(5,423)(5,424)(5,425)(5,426)(5,427)(5,428)(5,429)(5,430)(5,431)(5,432)(5,433)(5,434)(5,435)(5,436)(5,437)(5,438)(5,439)(5,440)(5,441)(5,442)(5,443)(5,444)(5,445)(5,446)(5,447)(5,448)(5,449)(5,450)(5,451)(5,452)(5,453)(5,454)(5,455)(5,456)(5,457)(5,458)(5,459)(5,460)(5,461)(5,462)(5,463)(5,464)(5,465)(5,466)(5,467)(5,468)(5,469)(5,470)(5,471)(5,472)(5,473)(5,474)(5,475)(5,476)(5,477)(5,478)(5,479)(5,480)(5,481)(5,482)(5,483)(5,484)(5,485)(5,486)(5,487)(5,488)(5,489)(5,490)(5,491)(5,492)(5,493)(5,494)(5,495)(5,496)(5,497)(5,498)(5,499)(5,500)(5,501)(5,502)(5,503)(5,504)(5,505)(5,506)(5,507)(5,508)(5,509)(5,510)(5,511)(5,512)(5,513)(5,514)(5,515)(5,516)(5,517)(5,518)(5,519)(5,528)(5,529)(5,530)(5,531)(5,532)(5,533)(5,534)(5,535)(5,536)(5,537)(5,538)(5,539)(5,540)(5,541)(5,542)(5,543)(5,544)(5,545)(5,546)(5,547)(5,548)(5,549)(5,550)(5,551)(5,552)(5,553)(5,554)(5,555)(5,556)(5,557)(5,558)(5,559)(5,560)(5,561)(5,562)(5,563)(5,564)(5,565)(5,566)(5,567)(5,568)(5,569)(5,570)(5,571)(5,572)(5,573)(5,574)(5,575)(5,576)(5,577)(5,578)(5,579)(5,580)(5,581)(5,582)(5,583)(5,584)(5,585)(5,586)(5,587)(5,588)(5,589)(5,590)(5,591)(5,592)(5,593)(5,594)(5,595)(5,596)(5,597)(5,598)(5,599)(5,600)(5,601)(5,602)(5,603)(5,604)(5,605)(5,606)(5,607)(5,608)(5,609)(5,610)(5,611)(5,612)(5,613)(5,614)(5,615)(5,616)(5,617)(5,618)(5,619)(5,620)(5,621)(5,622)(5,623)(5,624)(5,625)(5,626)(5,627)(5,628)(5,629)(5,630)(5,631)(5,632)(5,633)(5,634)(5,635)(5,636)(5,637)(5,638)(5,639)(5,640)(5,641)(5,642)(5,643)(5,644)(5,645)(5,646)(5,647)(5,648)(5,649)(5,650)(5,651)(5,656)(5,657)(5,658)(5,659)(5,660)(5,661)(5,662)(5,663)(5,664)(5,665)(5,666)(5,667)(5,668)(5,669)(5,670)(5,671)(5,672)(5,673)(5,674)(5,675)(5,676)(5,677)(5,678)(5,679)(5,680)(5,681)(5,682)(5,683)(5,688)(5,689)(5,690)(5,691)(5,692)(5,693)(5,694)(5,695)(5,696)(5,697)(5,698)(5,699)(5,700)(5,701)(5,702)(5,703)(5,704)(5,705)(5,706)(5,707)(5,708)(5,709)(5,710)(5,711)(5,712)(5,713)(5,714)(5,715)(5,716)(5,717)(5,718)(5,719)(5,720)(5,721)(5,722)(5,723)(5,724)(5,725)(5,726)(5,727)(5,728)(5,729)(5,730)(5,731)(5,732)(5,733)(5,734)(5,735)(5,736)(5,737)(5,738)(5,739)(5,740)(5,741)(5,742)(5,743)(5,744)(5,745)(5,746)(5,747)(5,748)(5,749)(5,750)(5,751)(5,752)(5,753)(5,754)(5,755)(5,756)(5,757)(5,758)(5,759)(5,760)(5,761)(5,762)(5,763)(5,764)(5,765)(5,766)(5,767)(5,768)(5,769)(5,770)(5,771)(5,772)(5,773)(5,774)(5,775)(5,776)(5,777)(5,778)(5,779)(5,780)(5,781)(5,782)(5,783)(5,784)(5,785)(5,786)(5,787)(5,788)(5,789)(5,790)(5,791)(5,792)(5,793)(5,794)(5,795)(5,796)(5,797)(5,798)(5,799)(5,800)(5,801)(5,802)(5,803)(5,804)(5,805)(5,806)(5,807)(5,808)(5,809)(5,810)(5,811)(5,812)(5,813)(5,814)(5,815)(5,816)(5,817)(5,818)(5,819)(5,820)(5,821)(5,822)(5,823)(5,824)(5,825)(5,826)(5,827)(5,828)(5,829)(5,830)(5,831)(5,832)(5,833)(5,834)(5,835)(5,836)(5,837)(5,838)(5,839)(5,840)(5,841)(5,842)(5,843)(5,844)(5,845)(5,846)(5,847)(5,848)(5,849)(5,850)(5,851)(5,852)(5,853)(5,854)(5,855)(5,856)(5,857)(5,858)(5,859)(5,860)(5,861)(5,862)(5,863)(5,864)(5,865)(5,866)(5,867)(5,868)(5,869)(5,870)(5,871)(5,872)(5,873)(5,874)(5,875)(5,876)(5,877)(5,878)(5,879)(5,880)(5,881)(5,882)(5,883)(5,884)(5,885)(5,886)(5,887)(5,888)(5,889)(5,890)(5,891)(5,892)(5,893)(5,894)(5,895)(5,896)(5,897)(5,898)(5,899)(5,900)(5,901)(5,902)(5,903)(5,904)(5,905)(5,906)(5,907)(5,908)(5,909)(5,910)(5,911)(5,912)(5,913)(5,914)(5,915)(5,916)(5,917)(5,918)(5,919)(5,920)(5,921)(5,922)(5,923)(5,924)(5,925)(5,926)(5,927)(5,928)(5,929)(5,930)(5,931)(5,932)(5,933)(5,934)(5,935)(5,936)(5,937)(5,938)(5,939)(5,940)(5,941)(5,942)(5,943)(5,944)(5,945)(5,946)(5,947)(5,948)(5,949)(5,950)(5,951)(5,952)(5,953)(5,954)(5,955)(5,956)(5,957)(5,958)(5,959)(5,960)(5,961)(5,962)(5,963)(5,964)(5,965)(5,966)(5,967)(5,968)(5,969)(5,970)(5,971)(5,972)(5,973)(5,974)(5,975)(5,976)(5,977)(5,978)(5,979)(5,980)(5,981)(5,982)(5,983)(5,984)(5,985)(5,986)(5,987)(5,988)(5,989)(5,990)(5,991)(5,992)(5,993)(5,994)(5,995)(5,996)(5,997)(5,998)(5,999)(5,1000(5,1001(5,1002(5,1003(5,1004(5,1005(5,1006(5,1007(5,1008(5,100(5,101(5,101(5,101(5,101(5,101(5,101(5,101(5,101(5,101(5,10(5,10(5,10(5,10(5,10

Appendix B

206

Figure B.7. The best basis for the triplet filter bior1.5, image bark, and cost function n1, defined in Eq. 4.44). The estimated entropy shows in position 4 in Figure B. 5.

Figure B.8. The bark image decomposed with the db4 filter and the cost function is varied. The cost of coding the coefficients of the optimal wavelet packet basis is marked by o and the cost of


(0,0)

(1,0) (1,1) (1,2) (1,3)

(2,0) (2,1) (2,2) (2,3)

(3,0) (3,1) (3,2) (3,3)

(4,0) (4,1) (4,2) (4,3)

(5,0) (5,1) (5,2) (5,3)

1 2 3 4 5 6 7 8 91.5

2

2.5

3

3.5

4

4.5

5


db4bark

207

Figure B.9. The best basis for the triplet filter db4, image bark, and cost function hest, defined in Eq. 4.42). The estimated entropy shows in position 2 in Figure B. 8.

Figure B.10. The best basis for the triplet filter db4, image bark, and cost function log, defined in Eq. 4.43). The estimated entropy shows in position 3 in Figure B. 8.

Figure B.11. The best basis for the triplet filter db4, image bark, and cost function n1, defined in Eq. 4.44). The estimated entropy shows in position 4 in Figure B. 8.

(0,0)

(1,0) (1,1) (1,2) (1,3)

(2,0) (2,1) (2,2) (2,3)

(0,0)

(1,0) (1,1) (1,2) (1,3)

(2,0) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (2,7) (2,8) (2,9) (2,10) (2,11) (2,12) (2,13) (2,14) (2,15)

(3,0)(3,1)(3,2)(3,3)(3,4)(3,5)(3,6)(3,7)(3,8)(3,9)(3,10)(3,11)(3,12)(3,13)(3,14)(3,15)(3,16)(3,17)(3,18)(3,19)(3,20)(3,21)(3,22)(3,23)(3,24)(3,25)(3,26)(3,27)(3,28)(3,29)(3,30)(3,31)(3,32)(3,33)(3,34)(3,35)(3,36)(3,37)(3,38)(3,39)(3,40)(3,41)(3,42)(3,43)(3,44)(3,45)(3,46)(3,47)(3,48)(3,49)(3,50)(3,51)(3,52)(3,53)(3,54)(3,55)(3,56)(3,57)(3,58)(3,59)(3,60)(3,61)(3,62)(3,63

(4,0)(4,1)(4,2)(4,3)(4,4)(4,5)(4,6)(4,7)(4,8)(4,9)(4,10)(4,11)(4,12)(4,13)(4,14)(4,15)(4,16)(4,17)(4,18)(4,19)(4,20)(4,21)(4,22)(4,23)(4,24)(4,25)(4,26)(4,27)(4,28)(4,29)(4,30)(4,31)(4,32)(4,33)(4,34)(4,35)(4,36)(4,37)(4,38)(4,39)(4,40)(4,41)(4,42)(4,43)(4,44)(4,45)(4,46)(4,47)(4,48)(4,49)(4,50)(4,51)(4,52)(4,53)(4,54)(4,55)(4,56)(4,57)(4,58)(4,59)(4,60)(4,61)(4,62)(4,63)(4,64)(4,65)(4,66)(4,67)(4,68)(4,69)(4,70)(4,71)(4,72)(4,73)(4,74)(4,75)(4,76)(4,77)(4,78)(4,79)(4,80)(4,81)(4,82)(4,83)(4,84)(4,85)(4,86)(4,87)(4,88)(4,89)(4,90)(4,91)(4,92)(4,93)(4,94)(4,95)(4,96)(4,97)(4,98)(4,99)(4,100)(4,101)(4,102)(4,103)(4,104)(4,105)(4,106)(4,107)(4,108)(4,109)(4,110)(4,111)(4,112)(4,113)(4,114)(4,115)(4,116)(4,117)(4,118)(4,119)(4,120)(4,121)(4,122)(4,123)(4,124)(4,125)(4,126)(4,127)(4,128)(4,129)(4,130)(4,131)(4,132)(4,133)(4,134)(4,135)(4,136)(4,137)(4,138)(4,139)(4,140)(4,141)(4,142)(4,143)(4,144)(4,145)(4,146)(4,147)(4,148)(4,149)(4,150)(4,151)(4,152)(4,153)(4,154)(4,155)(4,156)(4,157)(4,158)(4,159)(4,160)(4,161)(4,162)(4,163)(4,164)(4,165)(4,166)(4,167)(4,168)(4,169)(4,170)(4,171)(4,172)(4,173)(4,174)(4,175)(4,176)(4,177)(4,178)(4,179)(4,180)(4,181)(4,182)(4,183)(4,184)(4,185)(4,186)(4,187)(4,188)(4,189)(4,190)(4,191)(4,192)(4,193)(4,194)(4,195)(4,196)(4,197)(4,198)(4,199)(4,200)(4,201)(4,202)(4,203)(4,204)(4,205)(4,206)(4,207)(4,208)(4,209)(4,210)(4,211)(4,212)(4,213)(4,214)(4,215)(4,216)(4,217)(4,218)(4,219)(4,220)(4,221)(4,222)(4,223)(4,224)(4,225)(4,226)(4,227)(4,228)(4,229)(4,230)(4,231)(4,232)(4,233)(4,234)(4,235)(4,236)(4,237)(4,238)(4,239)(4,240)(4,241)(4,242)(4,243)(4,244)(4,245)(4,246)(4,247)(4,248)(4,249)(4,250)(4,251(4,252(4,25(4,25(4,25

(5,0)(5,1)(5,2)(5,3)(5,24)(5,25)(5,26)(5,27)(5,28)(5,29)(5,30)(5,31)(5,36)(5,37)(5,38)(5,39)(5,44)(5,45)(5,46)(5,47)(5,48)(5,49)(5,50)(5,51)(5,52)(5,53)(5,54)(5,55)(5,56)(5,57)(5,58)(5,59)(5,60)(5,61)(5,62)(5,63)(5,64)(5,65)(5,66)(5,67)(5,68)(5,69)(5,70)(5,71)(5,72)(5,73)(5,74)(5,75)(5,76)(5,77)(5,78)(5,79)(5,80)(5,81)(5,82)(5,83)(5,84)(5,85)(5,86)(5,87)(5,88)(5,89)(5,90)(5,91)(5,92)(5,93)(5,94)(5,95)(5,96)(5,97)(5,98)(5,99)(5,100)(5,101)(5,102)(5,103)(5,104)(5,105)(5,106)(5,107)(5,108)(5,109)(5,110)(5,111)(5,112)(5,113)(5,114)(5,115)(5,116)(5,117)(5,118)(5,119)(5,120)(5,121)(5,122)(5,123)(5,124)(5,125)(5,126)(5,127)(5,128)(5,129)(5,130)(5,131)(5,132)(5,133)(5,134)(5,135)(5,136)(5,137)(5,138)(5,139)(5,140)(5,141)(5,142)(5,143)(5,144)(5,145)(5,146)(5,147)(5,148)(5,149)(5,150)(5,151)(5,152)(5,153)(5,154)(5,155)(5,156)(5,157)(5,158)(5,159)(5,160)(5,161)(5,162)(5,163)(5,164)(5,165)(5,166)(5,167)(5,168)(5,169)(5,170)(5,171)(5,172)(5,173)(5,174)(5,175)(5,176)(5,177)(5,178)(5,179)(5,180)(5,181)(5,182)(5,183)(5,184)(5,185)(5,186)(5,187)(5,188)(5,189)(5,190)(5,191)(5,192)(5,193)(5,194)(5,195)(5,196)(5,197)(5,198)(5,199)(5,200)(5,201)(5,202)(5,203)(5,204)(5,205)(5,206)(5,207)(5,208)(5,209)(5,210)(5,211)(5,212)(5,213)(5,214)(5,215)(5,216)(5,217)(5,218)(5,219)(5,220)(5,221)(5,222)(5,223)(5,224)(5,225)(5,226)(5,227)(5,228)(5,229)(5,230)(5,231)(5,232)(5,233)(5,234)(5,235)(5,236)(5,237)(5,238)(5,239)(5,240)(5,241)(5,242)(5,243)(5,244)(5,245)(5,246)(5,247)(5,248)(5,249)(5,250)(5,251)(5,252)(5,253)(5,254)(5,255)(5,256)(5,257)(5,258)(5,259)(5,260)(5,261)(5,262)(5,263)(5,264)(5,265)(5,266)(5,267)(5,268)(5,269)(5,270)(5,271)(5,272)(5,273)(5,274)(5,275)(5,276)(5,277)(5,278)(5,279)(5,280)(5,281)(5,282)(5,283)(5,284)(5,285)(5,286)(5,287)(5,288)(5,289)(5,290)(5,291)(5,292)(5,293)(5,294)(5,295)(5,296)(5,297)(5,298)(5,299)(5,300)(5,301)(5,302)(5,303)(5,304)(5,305)(5,306)(5,307)(5,308)(5,309)(5,310)(5,311)(5,312)(5,313)(5,314)(5,315)(5,316)(5,317)(5,318)(5,319)(5,320)(5,321)(5,322)(5,323)(5,324)(5,325)(5,326)(5,327)(5,328)(5,329)(5,330)(5,331)(5,332)(5,333)(5,334)(5,335)(5,336)(5,337)(5,338)(5,339)(5,340)(5,341)(5,342)(5,343)(5,344)(5,345)(5,346)(5,347)(5,348)(5,349)(5,350)(5,351)(5,352)(5,353)(5,354)(5,355)(5,356)(5,357)(5,358)(5,359)(5,360)(5,361)(5,362)(5,363)(5,364)(5,365)(5,366)(5,367)(5,368)(5,369)(5,370)(5,371)(5,372)(5,373)(5,374)(5,375)(5,376)(5,377)(5,378)(5,379)(5,380)(5,381)(5,382)(5,383)(5,384)(5,385)(5,386)(5,387)(5,388)(5,389)(5,390)(5,391)(5,392)(5,393)(5,394)(5,395)(5,396)(5,397)(5,398)(5,399)(5,400)(5,401)(5,402)(5,403)(5,404)(5,405)(5,406)(5,407)(5,408)(5,409)(5,410)(5,411)(5,412)(5,413)(5,414)(5,415)(5,416)(5,417)(5,418)(5,419)(5,420)(5,421)(5,422)(5,423)(5,424)(5,425)(5,426)(5,427)(5,428)(5,429)(5,430)(5,431)(5,432)(5,433)(5,434)(5,435)(5,436)(5,437)(5,438)(5,439)(5,440)(5,441)(5,442)(5,443)(5,444)(5,445)(5,446)(5,447)(5,448)(5,449)(5,450)(5,451)(5,452)(5,453)(5,454)(5,455)(5,456)(5,457)(5,458)(5,459)(5,460)(5,461)(5,462)(5,463)(5,464)(5,465)(5,466)(5,467)(5,468)(5,469)(5,470)(5,471)(5,472)(5,473)(5,474)(5,475)(5,476)(5,477)(5,478)(5,479)(5,480)(5,481)(5,482)(5,483)(5,484)(5,485)(5,486)(5,487)(5,488)(5,489)(5,490)(5,491)(5,492)(5,493)(5,494)(5,495)(5,496)(5,497)(5,498)(5,499)(5,500)(5,501)(5,502)(5,503)(5,504)(5,505)(5,506)(5,507)(5,508)(5,509)(5,510)(5,511)(5,512)(5,513)(5,514)(5,515)(5,516)(5,517)(5,518)(5,519)(5,520)(5,521)(5,522)(5,523)(5,524)(5,525)(5,526)(5,527)(5,528)(5,529)(5,530)(5,531)(5,532)(5,533)(5,534)(5,535)(5,536)(5,537)(5,538)(5,539)(5,540)(5,541)(5,542)(5,543)(5,544)(5,545)(5,546)(5,547)(5,548)(5,549)(5,550)(5,551)(5,552)(5,553)(5,554)(5,555)(5,556)(5,557)(5,558)(5,559)(5,560)(5,561)(5,562)(5,563)(5,564)(5,565)(5,566)(5,567)(5,568)(5,569)(5,570)(5,571)(5,572)(5,573)(5,574)(5,575)(5,576)(5,577)(5,578)(5,579)(5,580)(5,581)(5,582)(5,583)(5,584)(5,585)(5,586)(5,587)(5,588)(5,589)(5,590)(5,591)(5,592)(5,593)(5,594)(5,595)(5,596)(5,597)(5,598)(5,599)(5,600)(5,601)(5,602)(5,603)(5,604)(5,605)(5,606)(5,607)(5,608)(5,609)(5,610)(5,611)(5,612)(5,613)(5,614)(5,615)(5,616)(5,617)(5,618)(5,619)(5,620)(5,621)(5,622)(5,623)(5,624)(5,625)(5,626)(5,627)(5,628)(5,629)(5,630)(5,631)(5,632)(5,633)(5,634)(5,635)(5,636)(5,637)(5,638)(5,639)(5,640)(5,641)(5,642)(5,643)(5,644)(5,645)(5,646)(5,647)(5,648)(5,649)(5,650)(5,651)(5,652)(5,653)(5,654)(5,655)(5,656)(5,657)(5,658)(5,659)(5,660)(5,661)(5,662)(5,663)(5,664)(5,665)(5,666)(5,667)(5,668)(5,669)(5,670)(5,671)(5,672)(5,673)(5,674)(5,675)(5,676)(5,677)(5,678)(5,679)(5,680)(5,681)(5,682)(5,683)(5,684)(5,685)(5,686)(5,687)(5,688)(5,689)(5,690)(5,691)(5,692)(5,693)(5,694)(5,695)(5,696)(5,697)(5,698)(5,699)(5,700)(5,701)(5,702)(5,703)(5,704)(5,705)(5,706)(5,707)(5,708)(5,709)(5,710)(5,711)(5,712)(5,713)(5,714)(5,715)(5,716)(5,717)(5,718)(5,719)(5,720)(5,721)(5,722)(5,723)(5,724)(5,725)(5,726)(5,727)(5,728)(5,729)(5,730)(5,731)(5,732)(5,733)(5,734)(5,735)(5,736)(5,737)(5,738)(5,739)(5,740)(5,741)(5,742)(5,743)(5,744)(5,745)(5,746)(5,747)(5,748)(5,749)(5,750)(5,751)(5,752)(5,753)(5,754)(5,755)(5,756)(5,757)(5,758)(5,759)(5,760)(5,761)(5,762)(5,763)(5,764)(5,765)(5,766)(5,767)(5,768)(5,769)(5,770)(5,771)(5,772)(5,773)(5,774)(5,775)(5,776)(5,777)(5,778)(5,779)(5,780)(5,781)(5,782)(5,783)(5,784)(5,785)(5,786)(5,787)(5,788)(5,789)(5,790)(5,791)(5,792)(5,793)(5,794)(5,795)(5,796)(5,797)(5,798)(5,799)(5,800)(5,801)(5,802)(5,803)(5,804)(5,805)(5,806)(5,807)(5,808)(5,809)(5,810)(5,811)(5,812)(5,813)(5,814)(5,815)(5,816)(5,817)(5,818)(5,819)(5,820)(5,821)(5,822)(5,823)(5,824)(5,825)(5,826)(5,827)(5,828)(5,829)(5,830)(5,831)(5,832)(5,833)(5,834)(5,835)(5,836)(5,837)(5,838)(5,839)(5,840)(5,841)(5,842)(5,843)(5,844)(5,845)(5,846)(5,847)(5,848)(5,849)(5,850)(5,851)(5,852)(5,853)(5,854)(5,855)(5,856)(5,857)(5,858)(5,859)(5,860)(5,861)(5,862)(5,863)(5,864)(5,865)(5,866)(5,867)(5,868)(5,869)(5,870)(5,871)(5,872)(5,873)(5,874)(5,875)(5,876)(5,877)(5,878)(5,879)(5,880)(5,881)(5,882)(5,883)(5,884)(5,885)(5,886)(5,887)(5,888)(5,889)(5,890)(5,891)(5,892)(5,893)(5,894)(5,895)(5,896)(5,897)(5,898)(5,899)(5,900)(5,901)(5,902)(5,903)(5,904)(5,905)(5,906)(5,907)(5,908)(5,909)(5,910)(5,911)(5,912)(5,913)(5,914)(5,915)(5,916)(5,917)(5,918)(5,919)(5,920)(5,921)(5,922)(5,923)(5,924)(5,925)(5,926)(5,927)(5,928)(5,929)(5,930)(5,931)(5,932)(5,933)(5,934)(5,935)(5,936)(5,937)(5,938)(5,939)(5,940)(5,941)(5,942)(5,943)(5,944)(5,945)(5,946)(5,947)(5,948)(5,949)(5,950)(5,951)(5,952)(5,953)(5,954)(5,955)(5,956)(5,957)(5,958)(5,959)(5,960)(5,961)(5,962)(5,963)(5,964)(5,965)(5,966)(5,967)(5,968)(5,969)(5,970)(5,971)(5,972)(5,973)(5,974)(5,975)(5,976)(5,977)(5,978)(5,979)(5,980)(5,981)(5,982)(5,983)(5,984)(5,985)(5,986)(5,987)(5,988)(5,989)(5,990)(5,991)(5,992)(5,993)(5,994)(5,995)(5,996)(5,997)(5,998)(5,999)(5,1000(5,1001(5,1002(5,1003(5,1004(5,1005(5,1006(5,1007(5,100(5,100(5,101(5,101(5,101(5,101(5,101(5,101(5,101(5,101(5,101(5,10(5,10(5,10(5,10(5,10

(0,0)

(1,0) (1,1) (1,2) (1,3)

(2,0) (2,1) (2,2) (2,3) (2,4)(2,5)(2,6)(2,7)(2,8)(2,9)(2,10)(2,11)(2,12)(2,13)(2,14)(2,15

(3,0) (3,1) (3,2) (3,3)(3,4)(3,5)(3,6)(3,7)(3,8)(3,9)(3,10)(3,11)(3,12)(3,13)(3,14)(3,15)

(4,0) (4,1)(4,2)(4,3) (4,8)(4,9)(4,10)(4,11)

(5,0)(5,1)(5,2)(5,3)

Appendix B

208

Figure B.12. The brickwall image decomposed with the db4 filter and the cost function is varied. The cost of coding the coefficients of the optimal wavelet packet basis is marked by o and the cost

of coding the coefficients of the corresponding wavelet basis is marked by x.

Figure B.13. The best basis for the triplet filter db4, image brickwall, and cost function hest, defined in Eq. 4.42). The estimated entropy shows in position 2 in Figure B. 12.

1 2 3 4 5 6 7 8 91.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2


db4brickwall

(0,0)

(1,0) (1,1) (1,2) (1,3)

(2,0) (2,1) (2,2) (2,3)

(3,0) (3,1) (3,2) (3,3)

209

Figure B.14. The best basis for the triplet filter db4, image brickwall, and cost function n1, defined in Eq. 4.44). The estimated entropy shows in position 4 in Figure B. 12.

Figure B.15. The cslfleath image decomposed with the db4 filter and the cost function is varied. The cost of coding the coefficients of the optimal wavelet packet basis is marked by o and the cost


(0,0)

(1,0) (1,1) (1,2) (1,3

(2,0) (2,1) (2,2) (2,3) (2,8) (2,9) (2,10) (2,11)

(3,0) (3,1) (3,2) (3,3)(3,4)(3,5)(3,6)(3,7) (3,8) (3,9) (3,10)(3,11) (3,32)(3,33)(3,34)(3,35) (3,40) (3,41)(3,42)(3,43)

(4,0)(4,1)(4,2)(4,3)(4,4)(4,5)(4,6)(4,7)(4,8)(4,9)(4,10)(4,11) (4,32)(4,33)(4,34)(4,35)(4,40)(4,41)(4,42)(4,43)(4,128)(4,129)(4,130)(4,131)(4,136)(4,137)(4,138)(4,139)(4,160)(4,161)(4,162)(4,163)(4,168)(4,169)(4,170)(4,171)

(5,0)(5,1)(5,2)(5,3)(5,8)(5,9)(5,10)(5,11) (5,32)(5,33)(5,34)(5,35)(5,40)(5,41)(5,42)(5,43) (5,128)(5,129)(5,130)(5,131)(5,136)(5,137)(5,138)(5,139)(5,168)(5,169)(5,170)(5,171) (5,552)(5,553)(5,554)(5,555)(5,640)(5,641)(5,642)(5,643)(5,648)(5,649)(5,650)(5,651)

1 2 3 4 5 6 7 8 92.5

3

3.5

4

4.5

5

5.5

6

6.5


db4cslfleath

Appendix B

210

Figure B.16. The best basis for the triplet filter db4, image cslfleath, and cost function hest, defined in Eq. 4.42). The estimated entropy shows in position 2 in Figure B. 15.

Figure B.17. The best basis for the triplet filter db4, image cslfleath, and cost function log, defined in Eq. 4.43). The estimated entropy shows in position 3 in Figure B. 15.

Figure B.18. The best basis for the triplet filter db4, image cslfleath, and cost function n1, defined in Eq. 4.44). The estimated entropy shows in position 4 in Figure B. 15.

(0,0)

(1,0) (1,1) (1,2) (1,3)

(0,0)

(1,0) (1,1) (1,2) (1,3)

(2,0) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (2,7) (2,8) (2,9) (2,10) (2,11) (2,12) (2,13) (2,14) (2,15)

(3,0)(3,1)(3,2)(3,3)(3,4)(3,5)(3,6)(3,7)(3,8)(3,9)(3,10)(3,11)(3,12)(3,13)(3,14)(3,15)(3,16)(3,17)(3,18)(3,19)(3,20)(3,21)(3,22)(3,23)(3,24)(3,25)(3,26)(3,27)(3,28)(3,29)(3,30)(3,31)(3,32)(3,33)(3,34)(3,35)(3,36)(3,37)(3,38)(3,39)(3,40)(3,41)(3,42)(3,43)(3,44)(3,45)(3,46)(3,47)(3,48)(3,49)(3,50)(3,51)(3,52)(3,53)(3,54)(3,55)(3,56)(3,57)(3,58)(3,59)(3,60)(3,61)(3,62)(3,63

(4,0)(4,1)(4,2)(4,3)(4,4)(4,5)(4,6)(4,7)(4,8)(4,9)(4,10)(4,11)(4,12)(4,13)(4,14)(4,15)(4,16)(4,17)(4,18)(4,19)(4,20)(4,21)(4,22)(4,23)(4,24)(4,25)(4,26)(4,27)(4,28)(4,29)(4,30)(4,31)(4,32)(4,33)(4,34)(4,35)(4,36)(4,37)(4,38)(4,39)(4,40)(4,41)(4,42)(4,43)(4,44)(4,45)(4,46)(4,47)(4,48)(4,49)(4,50)(4,51)(4,52)(4,53)(4,54)(4,55)(4,56)(4,57)(4,58)(4,59)(4,60)(4,61)(4,62)(4,63)(4,64)(4,65)(4,66)(4,67)(4,68)(4,69)(4,70)(4,71)(4,72)(4,73)(4,74)(4,75)(4,76)(4,77)(4,78)(4,79)(4,80)(4,81)(4,82)(4,83)(4,84)(4,85)(4,86)(4,87)(4,88)(4,89)(4,90)(4,91)(4,92)(4,93)(4,94)(4,95)(4,96)(4,97)(4,98)(4,99)(4,100)(4,101)(4,102)(4,103)(4,104)(4,105)(4,106)(4,107)(4,108)(4,109)(4,110)(4,111)(4,112)(4,113)(4,114)(4,115)(4,116)(4,117)(4,118)(4,119)(4,120)(4,121)(4,122)(4,123)(4,124)(4,125)(4,126)(4,127)(4,128)(4,129)(4,130)(4,131)(4,132)(4,133)(4,134)(4,135)(4,136)(4,137)(4,138)(4,139)(4,140)(4,141)(4,142)(4,143)(4,144)(4,145)(4,146)(4,147)(4,148)(4,149)(4,150)(4,151)(4,152)(4,153)(4,154)(4,155)(4,156)(4,157)(4,158)(4,159)(4,160)(4,161)(4,162)(4,163)(4,164)(4,165)(4,166)(4,167)(4,168)(4,169)(4,170)(4,171)(4,172)(4,173)(4,174)(4,175)(4,176)(4,177)(4,178)(4,179)(4,180)(4,181)(4,182)(4,183)(4,184)(4,185)(4,186)(4,187)(4,188)(4,189)(4,190)(4,191)(4,192)(4,193)(4,194)(4,195)(4,196)(4,197)(4,198)(4,199)(4,200)(4,201)(4,202)(4,203)(4,204)(4,205)(4,206)(4,207)(4,208)(4,209)(4,210)(4,211)(4,212)(4,213)(4,214)(4,215)(4,216)(4,217)(4,218)(4,219)(4,220)(4,221)(4,222)(4,223)(4,224)(4,225)(4,226)(4,227)(4,228)(4,229)(4,230)(4,231)(4,232)(4,233)(4,234)(4,235)(4,236)(4,237)(4,238)(4,239)(4,240)(4,241)(4,242)(4,243)(4,244)(4,245)(4,246)(4,247)(4,248)(4,249)(4,250)(4,251(4,252(4,25(4,25(4,25

(5,0)(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)(5,7)(5,12)(5,13)(5,14)(5,15)(5,16)(5,17)(5,18)(5,19)(5,20)(5,21)(5,22)(5,23)(5,24)(5,25)(5,26)(5,27)(5,28)(5,29)(5,30)(5,31)(5,36)(5,37)(5,38)(5,39)(5,40)(5,41)(5,42)(5,43)(5,44)(5,45)(5,46)(5,47)(5,48)(5,49)(5,50)(5,51)(5,52)(5,53)(5,54)(5,55)(5,56)(5,57)(5,58)(5,59)(5,60)(5,61)(5,62)(5,63)(5,64)(5,65)(5,66)(5,67)(5,68)(5,69)(5,70)(5,71)(5,72)(5,73)(5,74)(5,75)(5,76)(5,77)(5,78)(5,79)(5,80)(5,81)(5,82)(5,83)(5,84)(5,85)(5,86)(5,87)(5,88)(5,89)(5,90)(5,91)(5,92)(5,93)(5,94)(5,95)(5,96)(5,97)(5,98)(5,99)(5,100)(5,101)(5,102)(5,103)(5,104)(5,105)(5,106)(5,107)(5,108)(5,109)(5,110)(5,111)(5,112)(5,113)(5,114)(5,115)(5,116)(5,117)(5,118)(5,119)(5,120)(5,121)(5,122)(5,123)(5,124)(5,125)(5,126)(5,127)(5,128)(5,129)(5,130)(5,131)(5,132)(5,133)(5,134)(5,135)(5,136)(5,137)(5,138)(5,139)(5,140)(5,141)(5,142)(5,143)(5,144)(5,145)(5,146)(5,147)(5,148)(5,149)(5,150)(5,151)(5,152)(5,153)(5,154)(5,155)(5,156)(5,157)(5,158)(5,159)(5,160)(5,161)(5,162)(5,163)(5,164)(5,165)(5,166)(5,167)(5,168)(5,169)(5,170)(5,171)(5,172)(5,173)(5,174)(5,175)(5,176)(5,177)(5,178)(5,179)(5,180)(5,181)(5,182)(5,183)(5,184)(5,185)(5,186)(5,187)(5,188)(5,189)(5,190)(5,191)(5,192)(5,193)(5,194)(5,195)(5,196)(5,197)(5,198)(5,199)(5,200)(5,201)(5,202)(5,203)(5,204)(5,205)(5,206)(5,207)(5,208)(5,209)(5,210)(5,211)(5,212)(5,213)(5,214)(5,215)(5,216)(5,217)(5,218)(5,219)(5,220)(5,221)(5,222)(5,223)(5,224)(5,225)(5,226)(5,227)(5,228)(5,229)(5,230)(5,231)(5,232)(5,233)(5,234)(5,235)(5,236)(5,237)(5,238)(5,239)(5,240)(5,241)(5,242)(5,243)(5,244)(5,245)(5,246)(5,247)(5,248)(5,249)(5,250)(5,251)(5,252)(5,253)(5,254)(5,255)(5,256)(5,257)(5,258)(5,259)(5,260)(5,261)(5,262)(5,263)(5,264)(5,265)(5,266)(5,267)(5,268)(5,269)(5,270)(5,271)(5,272)(5,273)(5,274)(5,275)(5,276)(5,277)(5,278)(5,279)(5,280)(5,281)(5,282)(5,283)(5,284)(5,285)(5,286)(5,287)(5,288)(5,289)(5,290)(5,291)(5,292)(5,293)(5,294)(5,295)(5,296)(5,297)(5,298)(5,299)(5,300)(5,301)(5,302)(5,303)(5,304)(5,305)(5,306)(5,307)(5,308)(5,309)(5,310)(5,311)(5,312)(5,313)(5,314)(5,315)(5,316)(5,317)(5,318)(5,319)(5,320)(5,321)(5,322)(5,323)(5,324)(5,325)(5,326)(5,327)(5,328)(5,329)(5,330)(5,331)(5,332)(5,333)(5,334)(5,335)(5,336)(5,337)(5,338)(5,339)(5,340)(5,341)(5,342)(5,343)(5,344)(5,345)(5,346)(5,347)(5,348)(5,349)(5,350)(5,351)(5,352)(5,353)(5,354)(5,355)(5,356)(5,357)(5,358)(5,359)(5,360)(5,361)(5,362)(5,363)(5,364)(5,365)(5,366)(5,367)(5,368)(5,369)(5,370)(5,371)(5,372)(5,373)(5,374)(5,375)(5,376)(5,377)(5,378)(5,379)(5,380)(5,381)(5,382)(5,383)(5,384)(5,385)(5,386)(5,387)(5,388)(5,389)(5,390)(5,391)(5,392)(5,393)(5,394)(5,395)(5,396)(5,397)(5,398)(5,399)(5,400)(5,401)(5,402)(5,403)(5,404)(5,405)(5,406)(5,407)(5,408)(5,409)(5,410)(5,411)(5,412)(5,413)(5,414)(5,415)(5,416)(5,417)(5,418)(5,419)(5,420)(5,421)(5,422)(5,423)(5,424)(5,425)(5,426)(5,427)(5,428)(5,429)(5,430)(5,431)(5,432)(5,433)(5,434)(5,435)(5,436)(5,437)(5,438)(5,439)(5,440)(5,441)(5,442)(5,443)(5,444)(5,445)(5,446)(5,447)(5,448)(5,449)(5,450)(5,451)(5,452)(5,453)(5,454)(5,455)(5,456)(5,457)(5,458)(5,459)(5,460)(5,461)(5,462)(5,463)(5,464)(5,465)(5,466)(5,467)(5,468)(5,469)(5,470)(5,471)(5,472)(5,473)(5,474)(5,475)(5,476)(5,477)(5,478)(5,479)(5,480)(5,481)(5,482)(5,483)(5,484)(5,485)(5,486)(5,487)(5,488)(5,489)(5,490)(5,491)(5,492)(5,493)(5,494)(5,495)(5,496)(5,497)(5,498)(5,499)(5,500)(5,501)(5,502)(5,503)(5,504)(5,505)(5,506)(5,507)(5,508)(5,509)(5,510)(5,511)(5,512)(5,513)(5,514)(5,515)(5,516)(5,517)(5,518)(5,519)(5,520)(5,521)(5,522)(5,523)(5,524)(5,525)(5,526)(5,527)(5,528)(5,529)(5,530)(5,531)(5,532)(5,533)(5,534)(5,535)(5,536)(5,537)(5,538)(5,539)(5,540)(5,541)(5,542)(5,543)(5,544)(5,545)(5,546)(5,547)(5,548)(5,549)(5,550)(5,551)(5,552)(5,553)(5,554)(5,555)(5,556)(5,557)(5,558)(5,559)(5,560)(5,561)(5,562)(5,563)(5,564)(5,565)(5,566)(5,567)(5,568)(5,569)(5,570)(5,571)(5,572)(5,573)(5,574)(5,575)(5,576)(5,577)(5,578)(5,579)(5,580)(5,581)(5,582)(5,583)(5,584)(5,585)(5,586)(5,587)(5,588)(5,589)(5,590)(5,591)(5,592)(5,593)(5,594)(5,595)(5,596)(5,597)(5,598)(5,599)(5,600)(5,601)(5,602)(5,603)(5,604)(5,605)(5,606)(5,607)(5,608)(5,609)(5,610)(5,611)(5,612)(5,613)(5,614)(5,615)(5,616)(5,617)(5,618)(5,619)(5,620)(5,621)(5,622)(5,623)(5,624)(5,625)(5,626)(5,627)(5,628)(5,629)(5,630)(5,631)(5,632)(5,633)(5,634)(5,635)(5,636)(5,637)(5,638)(5,639)(5,640)(5,641)(5,642)(5,643)(5,644)(5,645)(5,646)(5,647)(5,648)(5,649)(5,650)(5,651)(5,652)(5,653)(5,654)(5,655)(5,656)(5,657)(5,658)(5,659)(5,660)(5,661)(5,662)(5,663)(5,664)(5,665)(5,666)(5,667)(5,668)(5,669)(5,670)(5,671)(5,672)(5,673)(5,674)(5,675)(5,676)(5,677)(5,678)(5,679)(5,680)(5,681)(5,682)(5,683)(5,684)(5,685)(5,686)(5,687)(5,688)(5,689)(5,690)(5,691)(5,692)(5,693)(5,694)(5,695)(5,696)(5,697)(5,698)(5,699)(5,700)(5,701)(5,702)(5,703)(5,704)(5,705)(5,706)(5,707)(5,708)(5,709)(5,710)(5,711)(5,712)(5,713)(5,714)(5,715)(5,716)(5,717)(5,718)(5,719)(5,720)(5,721)(5,722)(5,723)(5,724)(5,725)(5,726)(5,727)(5,728)(5,729)(5,730)(5,731)(5,732)(5,733)(5,734)(5,735)(5,736)(5,737)(5,738)(5,739)(5,740)(5,741)(5,742)(5,743)(5,744)(5,745)(5,746)(5,747)(5,748)(5,749)(5,750)(5,751)(5,752)(5,753)(5,754)(5,755)(5,756)(5,757)(5,758)(5,759)(5,760)(5,761)(5,762)(5,763)(5,764)(5,765)(5,766)(5,767)(5,768)(5,769)(5,770)(5,771)(5,772)(5,773)(5,774)(5,775)(5,776)(5,777)(5,778)(5,779)(5,780)(5,781)(5,782)(5,783)(5,784)(5,785)(5,786)(5,787)(5,788)(5,789)(5,790)(5,791)(5,792)(5,793)(5,794)(5,795)(5,796)(5,797)(5,798)(5,799)(5,800)(5,801)(5,802)(5,803)(5,804)(5,805)(5,806)(5,807)(5,808)(5,809)(5,810)(5,811)(5,812)(5,813)(5,814)(5,815)(5,816)(5,817)(5,818)(5,819)(5,820)(5,821)(5,822)(5,823)(5,824)(5,825)(5,826)(5,827)(5,828)(5,829)(5,830)(5,831)(5,832)(5,833)(5,834)(5,835)(5,836)(5,837)(5,838)(5,839)(5,840)(5,841)(5,842)(5,843)(5,844)(5,845)(5,846)(5,847)(5,848)(5,849)(5,850)(5,851)(5,852)(5,853)(5,854)(5,855)(5,856)(5,857)(5,858)(5,859)(5,860)(5,861)(5,862)(5,863)(5,864)(5,865)(5,866)(5,867)(5,868)(5,869)(5,870)(5,871)(5,872)(5,873)(5,874)(5,875)(5,876)(5,877)(5,878)(5,879)(5,880)(5,881)(5,882)(5,883)(5,884)(5,885)(5,886)(5,887)(5,888)(5,889)(5,890)(5,891)(5,892)(5,893)(5,894)(5,895)(5,896)(5,897)(5,898)(5,899)(5,900)(5,901)(5,902)(5,903)(5,904)(5,905)(5,906)(5,907)(5,908)(5,909)(5,910)(5,911)(5,912)(5,913)(5,914)(5,915)(5,916)(5,917)(5,918)(5,919)(5,920)(5,921)(5,922)(5,923)(5,924)(5,925)(5,926)(5,927)(5,928)(5,929)(5,930)(5,931)(5,932)(5,933)(5,934)(5,935)(5,936)(5,937)(5,938)(5,939)(5,940)(5,941)(5,942)(5,943)(5,944)(5,945)(5,946)(5,947)(5,948)(5,949)(5,950)(5,951)(5,952)(5,953)(5,954)(5,955)(5,956)(5,957)(5,958)(5,959)(5,960)(5,961)(5,962)(5,963)(5,964)(5,965)(5,966)(5,967)(5,968)(5,969)(5,970)(5,971)(5,972)(5,973)(5,974)(5,975)(5,976)(5,977)(5,978)(5,979)(5,980)(5,981)(5,982)(5,983)(5,984)(5,985)(5,986)(5,987)(5,988)(5,989)(5,990)(5,991)(5,992)(5,993)(5,994)(5,995)(5,996)(5,997)(5,998)(5,999)(5,1000(5,1001(5,1002(5,1003(5,1004(5,1005(5,1006(5,1007(5,100(5,100(5,101(5,101(5,101(5,101(5,101(5,101(5,101(5,101(5,101(5,10(5,10(5,10(5,10(5,10

(0,0)

(1,0) (1,1) (1,2) (1,3)

(2,0) (2,1) (2,2) (2,3) (2,4)(2,5)(2,6)(2,7)(2,8)(2,9)(2,10)(2,11)(2,12)(2,13)(2,14)(2,15

(3,0) (3,1) (3,2) (3,3)(3,4)(3,5)(3,6)(3,7)(3,8)(3,9)(3,10)(3,11)(3,12)(3,13)(3,14)(3,15) (3,44)(3,45)(3,46)(3,47)

(4,0) (4,1) (4,2) (4,3) (4,4)(4,5)(4,6)(4,7)(4,8)(4,9)(4,10)(4,11) (4,44)(4,45)(4,46)(4,47)

(5,0)(5,1)(5,2)(5,3)(5,8)(5,9)(5,10)(5,11)(5,12)(5,13)(5,14)(5,15) (5,40)(5,41)(5,42)(5,43)

211

Next, filters are varied for different cost functions to show filter effects.

Figure B.19. The cslfleath image decomposed with varying filters and the cost function hest is used. The cost of coding the coefficients of the optimal wavelet packet basis is marked by o and the

cost of coding the coefficients of the corresponding wavelet basis is marked by x.

Figure B.20. The brickwall image decomposed with varying filters, using the cost function hest. The cost of coding the coefficients of the best basis is marked by o and the cost of coding the

coefficients of the corresponding wavelet basis is marked by x.

0 5 10 152

3

4

5

6

7

8hestcslfleath


0 5 10 151

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6hestbrickwall


Appendix B

212

Figure B.21. The Bark image decomposed with varying filters and the cost function log is used. The cost of coding the coefficients of the optimal wavelet packet basis is marked by o and the cost


Figure B.22. The brickwall image decomposed with varying filters and the cost function log is used. The cost of coding the coefficients of the optimal wavelet packet basis is marked by o and the

cost of coding the coefficients of the corresponding wavelet basis is marked by x.

0 5 10 151.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4logbark


0 5 10 151.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3logbrickwall


213

Figure B.23. The cslfleath image decomposed with varying filters and the cost function log is used. The cost of coding the coefficients of the optimal wavelet packet basis is marked by o and the cost


Figure B.24. The Bark image decomposed with varying filters and the cost function n1 is used. The cost of coding the coefficients of the optimal wavelet packet basis is marked by o and the cost


0 5 10 152

2.5

3

3.5

4

4.5

5

5.5logcslfleath


0 5 10 151.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3norm1bark


Appendix B

214

Figure B.25. The brickwall image decomposed with varying filters and the cost function n1 is used. The cost of coding the coefficients of the optimal wavelet packet basis is marked by o and the

cost of coding the coefficients of the corresponding wavelet basis is marked by x

Figure B.26. The cslfleath image decomposed with varying filters and the cost function norm1 is used. The cost of coding the coefficients of the optimal wavelet packet basis is marked by o and the

cost of coding the coefficients of the corresponding wavelet basis is marked by x

0 5 10 151.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2norm1brickwall


0 5 10 152

2.5

3

3.5norm1cslfleath


215

Appendix C

Listing of filter coefficients

'haar'LO_D = 0.7071 0.7071 HI_D =-0.7071 0.7071LO_R = 0.7071 0.7071HI_R = 0.7071 -0.7071

‘db2'LO_D = -0.1294 0.2241 0.8365 0.4830HI_D = -0.4830 0.8365 -0.2241 -0.1294LO_R = 0.4830 0.8365 0.2241 -0.1294HI_R = -0.1294 -0.2241 0.8365 -0.4830

'db4'LO_D = -0.0106 0.0329 0.0308 -0.1870 -0.0280 0.6309 0.7148 0.2304HI_D = -0.2304 0.7148 -0.6309 -0.0280 0.1870 0.0308 -0.0329 -0.0106LO_R = 0.2304 0.7148 0.6309 -0.0280 -0.1870 0.0308 0.0329 -0.0106HI_R = -0.0106 -0.0329 0.0308 0.1870 -0.0280 -0.6309 0.7148 -0.2304

'db6'LO_D = -0.0011 0.0048 0.0006 -0.0316 0.0275 0.0975 -0.1298 -0.2263 0.3153 0.7511

0.4946 0.1115HI_D = -0.1115 0.4946 -0.7511 0.3153 0.2263 -0.1298 -0.0975 0.0275 0.0316 0.0006 -0.0048

-0.0011LO_R = 0.1115 0.4946 0.7511 0.3153 -0.2263 -0.1298 0.0975 0.0275 -0.0316 0.0006 0.0048

-0.0011HI_R = -0.0011 -0.0048 0.0006 0.0316 0.0275 -0.0975 -0.1298 0.2263 0.3153 -0.7511 0.4946

-0.1115

'db8'LO_D = -0.0001 0.0007 -0.0004 -0.0049 0.0087 0.0140 -0.0441 -0.0174 0.1287 0.0005 -0.2840 -0.0158 0.5854 0.6756 0.3129 0.0544HI_D = -0.0544 0.3129 -0.6756 0.5854 0.0158 -0.2840 -0.0005 0.1287 0.0174 -0.0441 -0.0140 0.0087 0.0049 -0.0004 -0.0007 -0.0001LO_R = 0.0544 0.3129 0.6756 0.5854 -0.0158 -0.2840 0.0005 0.1287 -0.0174 -0.0441 0.0140

0.0087 -0.0049 -0.0004 0.0007 -0.0001HI_R = -0.0001 -0.0007 -0.0004 0.0049 0.0087 -0.0140 -0.0441 0.0174 0.1287 -0.0005 -0.2840 0.0158 0.5854 -0.6756 0.3129 -0.0544

Appendix C

216

'bior1.5'LO_D = 0.0166 -0.0166 -0.1215 0.1215 0.7071 0.7071 0.1215 -0.1215 -0.0166 0.0166HI_D = 0 0 0 0 -0.7071 0.7071 0 0 0 0LO_R = 0 0 0 0 0.7071 0.7071 0 0 0 0HI_R = 0.0166 0.0166 -0.1215 -0.1215 0.7071 -0.7071 0.1215 0.1215 -0.0166 -0.0166

'bior2.8'LO_D = 0 0.0015 -0.0030 -0.0129 0.0289 0.0530 -0.1349 -0.1638 0.4626 0.9516 0.4626 -0.1638 -0.1349 0.0530 0.0289 -0.0129 -0.0030 0.0015HI_D = 0 0 0 0 0 0 0 0.3536 -0.7071 0.3536 0 0 0 0

0 0 0 0LO_R = 0 0 0 0 0 0 0 0.3536 0.7071 0.3536 0 0 0 0

0 0 0 0HI_R = 0 -0.0015 -0.0030 0.0129 0.0289 -0.0530 -0.1349 0.1638 0.4626 -0.9516 0.4626

0.1638 -0.1349 -0.0530 0.0289 0.0129 -0.0030 -0.0015

'bior3.5'LO_D = -0.0138 0.0414 0.0525 -0.2679 -0.0718 0.9667 0.9667 -0.0718 -0.2679 0.0525

0.0414 -0.0138 HI_D = 0 0 0 0 -0.1768 0.5303 -0.5303 0.1768 0 0 0 0LO_R = 0 0 0 0 0.1768 0.5303 0.5303 0.1768 0 0 0 0HI_R = -0.0138 -0.0414 0.0525 0.2679 -0.0718 -0.9667 0.9667 0.0718 -0.2679 -0.0525

0.0414 0.0138

'bior6.8'LO_D = 0 0.0019 -0.0019 -0.0170 0.0119 0.0497 -0.0773 -0.0941 0.4208 0.8259 0.4208 -0.0941 -0.0773 0.0497 0.0119 -0.0170 -0.0019 0.0019HI_D = 0 0 0 0.0144 -0.0145 -0.0787 0.0404 0.4178 -0.7589 0.4178 0.0404 -0.0787

-0.0145 0.0144 0 0 0 0LO_R = 0 0 0 0.0144 0.0145 -0.0787 -0.0404 0.4178 0.7589 0.4178 -0.0404 -0.0787

0.0145 0.0144 0 0 0 0HI_R = 0 -0.0019 -0.0019 0.0170 0.0119 -0.0497 -0.0773 0.0941 0.4208 -0.8259 0.4208

0.0941 -0.0773 -0.0497 0.0119 0.0170 -0.0019 -0.0019

'coif1'LO_D = -0.0157 -0.0727 0.3849 0.8526 0.3379 -0.0727HI_D = 0.0727 0.3379 -0.8526 0.3849 0.0727 -0.0157LO_R = -0.0727 0.3379 0.8526 0.3849 -0.0727 -0.0157HI_R = -0.0157 0.0727 0.3849 -0.8526 0.3379 0.0727

'coif2'LO_D = -0.0007 -0.0018 0.0056 0.0237 -0.0594 -0.0765 0.4170 0.8127 0.3861 -0.0674 -0.0415 0.0164HI_D = -0.0164 -0.0415 0.0674 0.3861 -0.8127 0.4170 0.0765 -0.0594 -0.0237 0.0056

217

0.0018 -0.0007LO_R = 0.0164 -0.0415 -0.0674 0.3861 0.8127 0.4170 -0.0765 -0.0594 0.0237 0.0056 -0.0018 -0.0007HI_R = -0.0007 0.0018 0.0056 -0.0237 -0.0594 0.0765 0.4170 -0.8127 0.3861 0.0674 -0.0415 -0.0164

'coif5'LO_D = 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0003 -0.0006 -0.0017 0.0024

0.0068 -0.0092 -0.0198 0.0327 0.0413 -0.1056 -0.0620 0.4380 0.7743 0.4216 -0.0520 -0.0919 0.0282 0.0234 -0.0101 -0.0042 0.0022 0.0004 -0.0002HI_D = 0.0002 0.0004 -0.0022 -0.0042 0.0101 0.0234 -0.0282 -0.0919 0.0520 0.4216 -0.7743 0.4380 0.0620 -0.1056 -0.0413 0.0327 0.0198 -0.0092 -0.0068 0.0024 0.0017 -0.0006 -0.0003 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000LO_R = -0.0002 0.0004 0.0022 -0.0042 -0.0101 0.0234 0.0282 -0.0919 -0.0520 0.4216

0.7743 0.4380 -0.0620 -0.1056 0.0413 0.0327 -0.0198 -0.0092 0.0068 0.0024 -0.0017 -0.0006 0.0003 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000HI_R = 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 -0.0003 -0.0006 0.0017 0.0024

-0.0068 -0.0092 0.0198 0.0327 -0.0413 -0.1056 0.0620 0.4380 -0.7743 0.4216 0.0520 -0.0919 -0.0282 0.0234 0.0101 -0.0042 -0.0022 0.0004 0.0002

'sym2'LO_D = -0.1294 0.2241 0.8365 0.4830HI_D = -0.4830 0.8365 -0.2241 -0.1294LO_R = 0.4830 0.8365 0.2241 -0.1294HI_R = -0.1294 -0.2241 0.8365 -0.4830

'sym4'LO_D = -0.0758 -0.0296 0.4976 0.8037 0.2979 -0.0992 -0.0126 0.0322HI_D = -0.0322 -0.0126 0.0992 0.2979 -0.8037 0.4976 0.0296 -0.0758LO_R = 0.0322 -0.0126 -0.0992 0.2979 0.8037 0.4976 -0.0296 -0.0758HI_R = -0.0758 0.0296 0.4976 -0.8037 0.2979 0.0992 -0.0126 -0.0322

'sym8'LO_D = -0.0034 -0.0005 0.0317 0.0076 -0.1433 -0.0613 0.4814 0.7772 0.3644 -0.0519 -0.0272 0.0491 0.0038 -0.0150 -0.0003 0.0019HI_D = -0.0019 -0.0003 0.0150 0.0038 -0.0491 -0.0272 0.0519 0.3644 -0.7772 0.4814

0.0613 -0.1433 -0.0076 0.0317 0.0005 -0.0034LO_R = 0.0019 -0.0003 -0.0150 0.0038 0.0491 -0.0272 -0.0519 0.3644 0.7772 0.4814 -0.0613 -0.1433 0.0076 0.0317 -0.0005 -0.0034 HI_R = -0.0034 0.0005 0.0317 -0.0076 -0.1433 0.0613 0.4814 -0.7772 0.3644 0.0519 -0.0272 -0.0491 0.0038 0.0150 -0.0003 -0.001

Appendix C

218

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