PERFORMANCE ANALYSIS OF EMBEDDED-WAVELET IMAGE …shyang/Journal Papers... · 2005-03-14 · PERFORMANCE ANALYSIS OF EMBEDDED ... perceptually weighted scalar quantization, and Huffman

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PERFORMANCE ANALYSIS OF EMBEDDED-WAVELET CODERS

Shih-Hsuan Yang and Wu-Jie Liao

Department of Computer Science and Information Engineering

National Taipei University of Technology

1, Sec. 3, Chung-Hsiao E. Rd.

Email: [email protected]

ABSTRACT

In this paper, we analyze the design issues for the SPIHT (set partitioning in hierarchical

trees) coding, one of the most prestigious embedded-wavelet-based algorithms in the

literature. Equipped with the multiresolution decomposition, progressive scalar

quantization and adaptive arithmetic coding, SPIHT generates highly compact scalable

bitstreams suitable for real-time multimedia applications. The design parameters at each

stage of SPIHT greatly influence its performance in terms of compression efficiency and

computational complexity. We first evaluate two important classes of wavelet filters,

orthogonal and biorthogonal. Orthogonal filters are energy preserving while biorthogonal

linear-phase filters allow symmetric extension across boundary. We investigate the

benefits from energy compaction, energy conservation, and symmetric extension,

respectively. Second, the magnitude of biorthogonal wavelet coefficients may not

faithfully reflect their actual significance. We explore a scaling scheme in quantization

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that minimizes the overall mean square error. The contribution of entropy coding is

measured at last.

1. INTRODUCTION

Compression lays the basis for the processing, transmission, and storage of multimedia

data. A picture is worth a thousand words, but full utilization of pictorial information is

impossible without compression. The most successful image coders adopt the

transform-coding structure shown in Fig. 1. A linear transformation such as the discrete

cosine transform (DCT) or discrete wavelet transform (DWT) converts the pixels into

uncorrelated and condensed transform coefficients. The quantizer adequately divides the

coefficient space into disjoint cells and the transform coefficients are reconstructed by a

representative value within the cell. Quantization is thus lossy in nature; under a specified

distortion requirement it aims to minimize the bit rate (or entropy) of the output symbols.

The entropy coder at the last stage finds the most economical binary representation for

the quantization symbol sequence. The baseline JPEG standard follows this structure that

combines DCT, perceptually weighted scalar quantization, and Huffman coding of

zig-zag scanned symbols. In 1993, Shapiro’s embedded zerotree wavelets (EZW) coding

[1] established a new transform-coding paradigm with DWT, successive scalar

quantization, and arithmetic coding. The essential novelty of EZW is the introduction of

the “zerotrees” (a group of insignificant wavelet coefficients pertaining to the same

spatial location and orientation). Of the various improvements of EZW, the set

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partitioning in hierarchical trees (SPIHT) coding [2] is the most renowned. Because of its

excellent compression performance and implementation elegancy, SPIHT has become

one of the de facto standard coding algorithms in the image

coding/processing/transmission community.

Many researchers have investigated the design issues of EZW, SPIHT, and other

DWT-based image coders. Before the introduction of SPIHT, Villasenor et al. [3]

evaluated the compression efficiency of biorthogonal wavelet filters in terms of the

Holder regularity and the impulse and step response properties, where a simple adaptive

scalar quantization scheme with an optimized bit-allocation procedure was used as the

coding platform. Li et al. [4] examined several wavelet filters and extension methods for

EZW. Adams and Kossentini [5] evaluated a wide range of reversible DWT kernels

under the JPEG2000 framework. Unser and Blu [6] investigated the mathematical

properties of the Daubechies 9/7 and LeGall 5/3 wavelets pertaining to their compression

performance. Woods and Naveen [7] derived the optimal bit allocation for

non-orthogonal transforms. Moulin [8] derived a multiscale relaxation algorithm to

improve the coding performance of non-orthogonal wavelet coding. Liu and Moulin [9]

employed the mutual information to model the interscale and intrascale dependencies

between wavelet coefficients. In [10], Xiong et al. showed that DWT outperforms DCT

within 1 dB under the same embedded coding structure. The parent-child coding gain of

SPIHT was quantified in [11]. He and Mitra [12] presented a unified analysis framework

for the transform coding, where a new rate-distortion model in terms of the zeros of the

quantized coefficients was developed. Finally, more sophisticated quantization schemes

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such as vector quantization (VQ) and trellis-coded quantization (TCQ), have been

incorporated into SPIHT [13]-[15].

This paper comprehensively investigates the factors crucial to the performance of

SPIHT. Although SPIHT was implemented mostly with a conventional set of parameters

(e.g., Daubechies 9/7 wavelet with symmetric data extension), this setup may not be

appropriate for all applications. It is thus important to investigate how to attain the

desired performance with the available resources. Furthermore, this investigation offers

an insight into the modern wavelet coders. In the following sections, we first explore the

essential properties of wavelet transforms, including the orthogonality,

energy-compacting capability, and symmetry. Since a biorthogonal wavelet transform

distorts the magnitude of wavelet coefficients, we examine the effects of a scaling

scheme to the quantization efficiency. The effect of the arithmetic coding is presented at

last.

2. WAVELET TRANSFORMS AND SPIHT

2.1 The SPIHT Coding

The encoding process of SPIHT is summarized in Fig. 2. The DWT converts the pixels

into wavelet coefficients, which are organized as the spatial-orientation trees depicted in

Fig. 3. SPIHT adopts a two-pass scalar deadzone quantizer. The first pass, sorting pass,

identifies the significant coefficients with respect to a threshold and gives their sign. The

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positions of significant coefficients are recorded in the list of significant pixels (LSP).

Insignificant spatial-orientation trees (i.e., zerotrees) are recorded in the list of

insignificant sets (LIS), and the other isolated insignificant coefficients are indexed in the

list of insignificant pixels (LIP). The second pass, refinement pass, narrows the

quantization level by a half for all the entries in the LSP excluding those newly added in

the last iteration. The two passes are repeated with a halved threshold in the next iteration,

until a specified rate or distortion constraint has been reached. An optional arithmetic

coder can be used to generate even more compact bit streams. However, the arithmetic

coding incurs intensive computation and reduces error robustness for the otherwise

elegant SPIHT algorithm [2]. Without being otherwise specified, the coding results given

in this paper are exempt from the arithmetic coding.

2.2 Discrete wavelet transform (DWT)

Transformation plays an essential role in image processing. Transformation can be

regarded as an approximation of a signal with a new set of basis functions. The new

space (called the transform domain) manifests itself in the decorrelating capability,

space-frequency localization, energy compaction, and/or other properties desirable for

further processing. In contrast to the conventional Fourier analysis, the wavelet transform

reveals both transient and stationary characteristics of a signal under a multiresolution

framework [16]. For discrete-time signals, the wavelet transform can be realized with the

filter-bank structure shown in Fig. 4, where hB0B[n] and hB1 B[n] correspond to the scaling

(lowpass) coefficients and wavelet (highpass) coefficients, respectively. A p-level

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decomposition along the lowpass subbands creates p+1 subbands H1, H2, … , Hp and Lp,

where the Lp subband stands for the base (approximation) component and the H

subbands represent the details at various scales. Perfect reconstruction can be built from

the constituent subbands by upsampling and interpolative filtering with the synthesis

filters gB0 B[n] and gB1B[n]. For two-dimensional signals such as images, DWT is mostly

independently performed along rows and columns. Observe a two-level decomposition of

the Lena image shown in Fig. 3(a). The resulting DWT coefficients demonstrate two

important facts that support the zerotree coding. First, an overwhelming majority of

energy concentrates in the lowpass subbands. This property is termed “energy

compaction.” Second, there exists obvious correlation between parent and child nodes.

2.3 Properties of wavelet filters

The multiresolution analysis requires the lowpass filter hB0 B[n] to satisfy the following

scaling equation

2][0 =∑n

nh (1)

An orthonormal wavelet of length N+1 further satisfies

][)1(][ 01 nNhnh n −−= , ][][ 00 nhng −= , ][][ 11 nhng −= (2)

and the energy conservation condition

1][][][][ 21

20

21

20 ==== ∑∑∑∑

nnnnngngnhnh (3)

Biorthogonal wavelets have two sets of complementary bases that satisfy

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][)1(][ 10 nhng n −−= , ][)1(][ 01 nhng n −−= . (4)

Orthonormal transformation implies norm preserving; effective quantization can thus be

directly applied to the transform coefficients. Several near-orthogonality measures for

biorthogonal wavelets have been proposed in the literature, mostly based on the

norm-preserving property [6], [17-19]. Let

,][200 ∑=

nnhw .][2

11 ∑=n

nhw (5)

It can be shown that wB0 B and wB1 B are the weighting factors of the mean-square quantization

error to the lowpass and highpass subbands, respectively [18]. In fact, wB0 B and wB1 B are

related to the Riesz constants upon considering the reconstruction error in the frequency

domain [6, 18]. In this paper, we adopt the near-orthogonality measure (NOM) defined in

[17]:

}.,max{NOM 10 ww= (6)

NOM upper bounds the multiplicative distortion to the quantization error introduced by

non-orthogonality. Clearly, orthogonal wavelets have the NOM value equal to 1. A larger

deviation of NOM from 1 indicates less orthogonality. Recently, a more elaborate model

for measuring orthogonality has been proposed in [19]. Their derivation was based on the

eigen-analysis of the discrete hyper-wavelet transform.

In this paper, we examine a wide variety of wavelet bases commonly referred to in

the image-coding community. According to the data type and orthogonality, these

wavelet bases naturally fall into three categories:

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A. Orthonormal wavelets: Haar (D2), D4, D6, D8.

B. Floating-point biorthogonal filters: 9/7D and 10/18.

C. Integer biorthogonal wavelets: 5/3, 9/7M, 5/11A, 5/11C, 13/7C, 13/7T, and

9/7WY.

Haar is the simplest nontrivial wavelet, which takes the sum and difference of input

samples for approximation and detail, respectively. D4, D6, and D8 are the Daubechies

orthonormal wavelets with compact support and maximal number of vanishing moments

[20]. The 9/7D wavelet is an odd-symmetric filter derived from an orthonormal mother

wavelet [21], while 10/18 is a longer even-symmetric filter [22]. Integer biorthogonal

wavelets [5] are fixed-point approximations to their parent real counterparts; they can be

implemented in the lifting framework without costly floating-point operations [5, 23, 24].

The 9/7WY wavelet [25] is a recently derived integer filter that is similar to 9/7D but

with much less computational burden. Moreover, transformation through integer wavelets

is reversible and suitable for a unified lossy and lossless codec [26]. The 5/3 and 9/7D

filters have been adopted in the JPEG-2000 standard [27].

The analysis filters hB0 B[n] and hB1 B[n] of the wavelets under study are listed in Table 1.

For biorthogonal wavelets, only half of the coefficients are given since the other half can

be deduced from symmetry. The NOM values of the biorthogonal filters are listed in

Table 2. The 9/7D and 9/7WY wavelets are much closer to orthogonality than the others.

For Category C wavelets, the scaling factor 2 in (1) is rescaled to 1 in Table 1 to

facilitate integer operations. We evaluated the time complexity of these wavelet

transforms on PC (Pentium 4). The results are given in Table 3, where the number

indicates the ratio of the required processing time with respect to the simplest Haar

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wavelet with convolution. It should be reminded that this complexity measure is

hardware dependent. No hardware acceleration such as the SIMD (single instruction

multiple data) technique was involved in our evaluation. Nevertheless, it is clear that the

convolution-based 10/18 filter is far more complex than the others even with the

floating-point support of the Pentium CPU. In contrast, the integer biorthogonal wavelets

are very attractive in practice.

3. OPTIMIZED WAVELET TRANSFORMS FOR THE SPIHT CODING

In this section, we investigate the parameters of the SPIHT coding crucial to compression

efficiency. Table 4 lists the coding performance of SPIHT at four bit rates (1/8, 1/4, 1/2,

and 1 bpp). Four 512×512 gray-level images Lena, Baboon, Pepper, and F16 (Fig. 5)

selected from the USC image databases [28] are tested. A 5-level DWT with each of the

aforementioned filters is employed for multiresolution decomposition. The visual quality

is objectively measured by the peak signal-to-noise ratio (PSNR). For an image X = (xB1 B,

x B2 B, …, x BM B) and its distorted version Y = (y B1 B, y B2 B, …, y BM B), the PSNR is computed as follows

.)(1

255log 10PSNR

1

2

2

10

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−=

∑=

M

iii yx

M

(7)

Considerable PSNR gap (up to 3dB) is observed when different wavelets are employed,

especially for smooth images such as Lena and Pepper. Among the examined filters, the

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9/7D, 9/7WY and 10/18 filters generally achieve the best PSNR performance. Filters of

shorter taps compromise on their compression efficiency. It is also noted that the

biorthogonal wavelets are substantially better than the orthogonal ones; the 5/3 filter

outperforms the much more complex D8 filter in many cases.

It was conjectured that energy compaction might be the dominant factor for the

compression performance of wavelet transform. We measure the energy compaction in

two ways, approximation spectral significance (in spectral domain) and reconstruction

error (in spatial domain), which are both listed in Table 5. The approximation spectral

significance is the percentage of the sum of squared coefficients within the approximation

subband (LL5) to the sum of squared coefficients of all subbands. The reconstruction

error is defined as the mean square error (MSE) when only the LL5 subband is decoded

(the other subbands are filled with 0). In the view of the SPIHT coding, the former

primarily affects the formation of zerotrees whereas the latter is related to the

quantization error. The 9/7D and 9/7WY wavelets possess the best approximation

spectral significance while the 10/18 wavelet possesses the least reconstruction error.

This partially explains the excellent performance of these two wavelets. The 9/7D and

9/7WY may also benefit from it’s being near orthonormal [6]. However, the coding

performance is not solely determined by energy compaction and orthogonality. For

example, higher approximation spectral significance of orthogonal wavelets does not

translate into better coding performance.

To further distinguish between orthogonal and biorthogonal wavelets, it is of interest

to know the advantage obtained by symmetric extension. Filtering with finite-length

sequences causes data expansion. A universal solution to this problem is the circular

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convolution together with periodic extension. However, this approach introduces

boundary artifacts; the aliasing components introduced by periodic extension makes

compression less efficient. To circumvent this difficulty, symmetric extension of input

samples have been developed for linear-phase filters [29]. Imposing the linear-phase

constraint, however, usually breaks the orthogonality of the filter. The only real-valued

orthogonal linear-phase wavelet with compact support is the trivial Haar filter. Good

linear-phase biorthogonal FIR wavelets (categories B and C) are thereby designed. Note

that the method of data extension should be in conformity with the type of filter’s

symmetry (Fig. 6). Similar consideration should be also borne in mind for

down-sampling and up-sampling. The results shown in Table 4 were obtained with the

best extension method, i.e., periodic extension for orthogonal wavelets and appropriate

symmetric extension for biorthogonal wavelets. In Table 6, the coding results with both

symmetric and periodic extension are given for comparison. Symmetric extension

provides a substantial edge over periodic extension at low rates for low-activity images

(Lena and Pepper). Compared to the orthogonal wavelets of similar length, the

distinguished 5/3, 9/7D (and 9/7WY) filters stand out not solely owing to symmetry.

Unser and Blu [6] attributed the success of these filters to the better approximation for

smooth regions of images [6].

The number of decomposition levels (p) is another factor that influences the

performance of wavelet-based image coders. The PSNR performance of SPIHT for p = 4

and p = 6 under a similar test environment of Table 4 are given in Tables 7 and 8,

respectively. When a very scarce bit budget is available, encoding down to the

higher-level nodes is beneficial because these nodes refer to a larger area. As a

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consequence, a nontrivial increase is observed from p = 4 to p = 5 at low bit rates.

Moving one level further (from p = 5 to p = 6) makes negligible contribution because the

LL6 subband contains few coefficients relative to the available bit budget.

4. QUANTIZATION AND ENTROPY COING OF SPIHT

Divide-and-conquer is the basic philosophy behind the transform coding. An adequate

transformation manages to remove the inter-pixel correlation and arrange the transform

coefficients in a prioritized order. Simple scalar quantization can thus be effortlessly

performed on the resulting transform coefficients. An optimal encoder produces the most

economical representation of wavelet coefficients in the order of their relative importance.

In SPIHT, the magnitude of wavelet coefficients is the prime indicator of their

significance. However, this indicator may not faithfully reflect the actual importance in

the rate-distortion sense, especially when non-orthogonal transformation is employed.

In this paper, we investigate an optimized quantization scheme by scaling subbands.

Adjusting the magnitude of wavelet coefficients may alter their quantized value and

encoding priority. We define the scaling factor K, by which the coefficients of highpass

filter hB1B[n] are multiplied. A smaller scaling factor implies an emphasis on lowpass

coefficients upon quantization and a better change of forming zerotrees. The best choice

of K makes a compromise between the quantization error and the number of bits required

for specifying the significance map. Table 9 gives the optimal scaling factor (using

exhaustive search) and the resulting PSNR gain (cf. Table 4). For a quantization scheme

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that does not take the parent-child correlation into consideration, scaling can provide

substantial coding gain [30]. However, for SPIHT the PSNR increase is negligible (less

than 0.21 dB) and insensitive to the scaling factor. Our conjecture is that the zerotree

coding of SPIHT has encoded the wavelet coefficients in the order of their relative

importance. Both the approximation and detail information is well preserved even with

the skewed coefficients.

The arithmetic coding can further squeeze the quantization symbols. However, the

arithmetic coding involves more intensive computation. Our simulation under the PC

environment indicates that the overall computation time approximately increases by 50%

with the arithmetic coding. The PSNR gain of the arithmetic coding for SPIHT is listed in

Table 10 (cf. Table 4). A nontrivial but limited PSNR gain (about 0.5 dB) is achieved and

this margin is relatively consistent across the filters.

5. CONCLUSION

We have investigated the factors crucial to the performance of the prestigious SPIHT

coding. We first explored the influence of wavelet filters, data extension types, and

decomposition levels. Second, a scaling scheme for restoring the energy distortion of

wavelet subbands was investigated. Finally, the effect of the entropy coding was

examined. A comprehensive evaluation in terms of PSNR and time complexity was made

at four bit rates for four test images. The results of this paper establish the guidelines for

implementing wavelet-based codecs.

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6. ACKNOWLEDGEMENT

This work was supported by the National Science Council, R. O. China, under the

contract number NSC 92-2218-E-027-016. The authors want to thank the anonymous

reviewers for their helpful comments.

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BIOGRAPHIES

Shih-Hsuan Yang received the B.S. degree in electrical engineering from the National

Taiwan University in 1987. He obtained the M.S. and Ph.D. degrees in electrical

engineering and computer science from the University of Michigan, Ann Arbor, in 1990

and 1994, respectively. Since 1994, he has been a faculty member of the National Taipei

University of Technology, Taiwan. He is currently an associate professor of Computer

Science and Information Engineering. His major research interests include image and

video coding, multimedia transmission, data hiding, and information theory.

Wu-Jie Liao was born in Yunlin, Taiwan, in 1979. He received the B.S. and M.S. degrees

from the National Taipei University of Technology in 2002 and 2004, respectively.

Currently he is working with the Primax Electronics Ltd., Taipei, Taiwan, for developing

multifunction peripherals.

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Table 1. Analysis filters of the wavelets under study. The negative indexes apply to categories B and C. An extra scaling factor of 2 is needed for Category C wavelets to conform to the scaling equation.

Filter Index Category & Name 0 1 (-1) 2 (-2) 3(-3) 4 (-4) 5 (–5) 6(-6) 7(-7) 8(-8)

h B0B[n] 0.7071 0.7071 Haar (D2) h B1B[n] 0.7071 -0.7071

h B0B[n] 0.48296 0.83652 0.22414 -0.12941 D4 h B1B[n] 0.12941 0.22414 -0.83652 0.48296 h B0B[n] 0.33267 0.80689 0.45988 -0.13501 -0.08544 0.03523 D6 h B1B[n] -0.03523 -0.08544 0.13501 0.45988 -0.80689 0.33267 h B0B[n] 0.23038 0.71485 0.63088 -0.02798 -0.18703 0.03084 0.03288 -0.01060

A

D8 h B1B[n] 0.01060 0.03288 -0.03084 -0.18703 0.02798 0.63088 -0.71485 0.23038 h B0B[n] 0.85267 0.37740 -0.11062 -0.02385 0.03783 9/7D h B1B[n] 0.78849 -0.41809 -0.04069 0.06454 h B0B[n] 0.75891 0.07679 -0.15753 8.2e-5 0.02885 B

10/18

h B1B[n] 0.62336 -0.16337 -0.08566 0.01377 0.03083 0.00253 -0.00945 2.7e-6 0.00095h B0B[n] 3/4 1/4 -1/8 5/3

h B1B[n] 1 -1/2 h B0B[n] 23/32 1/4 -1/8 0 1/64 9/7M h B1B[n] 1 -9/16 0 1/16 h B0B[n] 3/4 1/4 -1/8 5/11A h B1B[n] 63/64 -67/128 0 7/256 1/128 -1/256 h B0B[n] 3/4 1/4 -1/8 5/11C h B1B[n] 31/32 -35/64 0 7/128 1/64 -1/128 h B0B[n] 41/64 5/16 -31/256 -1/16 7/128 0 -1/256 13/7C h B1B[n] 1 -9/16 0 1/16 h B0B[n] 87/128 9/32 -63/512 -1/32 9/256 0 -1/512 13/7T h B1B[n] 1 -9/16 0 1/16 h B0B[n] 19/32 43/160 -12/160 -3/160 9/320

C

9/7WY h B1B[n] 9/8 -19/32 -1/16 3/32

Table 2: NOM (near-orthogonality measure) of the biorthogonal wavelets under study

Table 3. Relative time complexity of the wavelet transforms under study. D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY

1.00 1.92 2.92 4.03 3.94 7.58 1.03 1.17 1.39 1.40 1.21 1.26 1.61

9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY 1.040 1.215 1.438 1.347 1.438 1.438 1.310 1.300 1.021

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Table 4. SPIHT’s compression performance with various filters for (a) Lena (b) Baboon (c) Pepper (d) F16 (5-level decomposition).

(a)

(b)

(c)

(d)

Table 5. (1) Approximation spectral significance (1P

stP row with each image) (2)

reconstruction error (2P

ndP row with each image).

D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY97.8 97.4 97.3 97.9 98.0 97.4 95.7 96.7 95.7 95.7 96.7 96.7 98.0

Lena 830 839 822 701 744 659 786 762 776 772 734 743 744 98.9 99.0 98.9 99.0 99.2 98.7 98.0 98.5 98.0 98.0 98.6 98.6 99.2

Baboon 960 922 908 897 886 876 901 895 899 899 885 888 886 94.7 94.7 94.1 95.4 95.7 94.3 92.4 94.0 92.5 92.5 93.6 93.8 95.8

Pepper 1823 1665 1693 1435 1605 1410 1692 1652 1678 1675 1588 1612 1603 98.8 98.8 98.7 98.8 99.0 98.3 97.8 98.3 97.8 97.8 98.3 98.3 99.0

F16 997 928 893 866 889 863 915 907 914 917 886 892 889

bpp D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 27.53 28.99 29.39 29.50 30.53 30.69 30.05 30.32 30.22 30.25 30.51 30.50 30.520.25 30.21 31.88 32.37 32.54 33.59 33.76 32.95 33.36 33.16 33.27 33.53 33.50 33.580.5 33.50 35.27 35.75 35.87 36.75 36.88 36.09 36.56 36.32 36.43 36.72 36.70 36.741.0 37.45 38.94 39.27 39.36 39.92 39.95 39.30 39.60 39.47 39.51 39.77 39.74 39.93




21

Table 6. Coding results for period/symmetric extension (a) Lena (b) Baboon (c) Pepper (d) F16.

(a)

(b)

(c)

(d)

bpp 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY

0.125 30.06 /30.53

30.21 /30.69

29.61 /30.05

29.88 /30.32

29.73 /30.22

29.74 /30.25

30.07 /30.51

30.05 /30.50

30.05 /30.52

0.25 33.22 /33.59

33.27 /33.76

32.61 /32.95

33.00 /33.36

32.84 /33.16

32.93 /33.27

33.16 /33.53

33.14 /33.50

33.21 /33.58

0.5 36.53 /36.75

36.48 /36.88

35.91 /36.09

36.27 /36.56

36.12 /36.32

36.20 /36.43

36.42 /36.72

36.42 /36.70

36.52 /36.74

1.0 39.77 /39.92

39.74 /39.95

39.15 /39.30

39.45 /39.60

39.31 /39.47

39.35 /39.51

39.62 /39.77

39.60 /39.74

39.77 /39.93

bpp 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY

0.125 21.43 /21.49

21.55 /21.60

21.23 /21.29

21.16 /21.26

21.20 /21.26

21.16 /21.21

21.40 /21.45

21.35 /21.42

21.43 /21.49

0.25 22.82 /22.88

22.90 /22.97

22.46 /22.53

22.46 /22.56

22.45 /22.52

22.42 /22.46

22.70 /22.77

22.67 /22.72

22.81 /22.89

0.5 25.02 /25.12

25.08 /25.13

24.49 /24.58

24.59 /24.72

24.53 /24.63

24.52 /24.62

24.82 /24.92

24.80 /24.88

25.04 /25.14

1.0 28.53 /28.62

28.53 /28.60

27.91 /28.00

28.09 /28.19

28.00 /28.09

28.01 /28.09

28.32 /28.40

28.28 /28.36

28.54 /28.63

bpp 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY

0.125 28.63 /29.10

28.67 /29.11

28.22 /28.70

28.46 /28.77

28.33 /28.80

28.35 /28.79

28.67 /28.95

28.66 /28.94

28.62 /29.08

0.25 31.48 /31.79

31.34 /31.72

31.19 /31.46

31.24 /31.57

31.35 /31.54

31.30 /31.48

31.42 /31.68

31.41 /31.67

31.47 /31.78

0.5 33.59 /33.83

33.60 /33.81

33.52 /33.60

33.43 /33.55

33.55 /33.63

33.46 /33.55

33.60 /33.73

33.59 /33.69

33.56 /33.81

1.0 36.08 /36.19

36.08 /36.22

35.81 /35.90

35.80 /35.89

35.82 /35.92

35.72 /35.83

35.99 /36.07

35.95 /36.04

36.07 36.18

bpp 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY

0.125 29.07 /29.28

29.09 /29.26

28.64 /28.85

28.72 /28.97

28.73 /28.93

28.70 /28.92

28.90 /29.10

28.90 /29.09

29.06 /29.26

0.25 32.32 /32.51

32.33 /32.55

31.78 /32.00

32.04 /32.28

32.04 /32.22

32.05 /32.22

32.18 /32.34

32.17 /32.36

32.30 /32.50

0.5 36.21 /36.39

36.21 /36.42

35.72 /35.85

36.00 /36.22

35.90 /36.05

35.91 /36.08

36.11 /36.33

36.16 /36.35

36.20 /36.39

1.0 40.70 /40.85

40.65 /40.85

40.12 /40.25

40.38 /40.57

40.30 /40.44

40.29 /40.45

40.55 /40.70

40.53 /40.70

40.69 /40.85

22

Table 7. SPIHT’s compression performance with 4-level (p = 4) decomposition. (a) Lena

(b) Baboon

(c)Pepper

(d) F16

Table 8. SPIHT’s compression performance with 6-level (p = 6) decomposition. (a) Lena

(b) Baboon

(c) Pepper








23

(d) F16

Table 9. Optimal scaling factor/performance gain relative to Table 4. (a) Lena

(b) Baboon

(c) Pepper

(d) F16

Table 10. PSNR gain of arithmetic coding relative to Table 4. (a) Lena

(b) Baboon


D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 0.9/0.11 0.9/0.02 1.0/0 0.9/0.02 1.1/0.05 1.1/0.03 1.1/0.03 1.1/0.01 1.0/0 1.0/0 0.9/0.02 1.0/0 1.1/0.070.25 0.8/0.03 0.9/0.03 1.0/0 0.9/0.04 1.1/0.05 1.0/0 1.1/0.02 1.1/0.02 1.2/0.04 1.2/0.06 1.0/0 1.1/0.01 1.1/0.050.5 0.8/0.03 0.9/0.03 0.9/0.03 0.9/0.05 1.1/0.05 1.0/0 1.2/0.08 1.0/0 1.2/0.06 1.1/0.01 1.0/0 1.0/0 1.1/0.061.0 0.8/0.01 0.9/0.01 1.0/0 0.9/0.01 1.0/0 1.0/0 1.3/0.11 1.0/0 1.3/0.09 1.3/0.06 0.8/0.02 0.8/0.02 1.0/0

D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 1.1/0.03 1.1/0.03 1.1/0.03 1.1/0.03 1.1/0.02 1.0/0 0.9/0.03 1.0/0 0.9/0.04 0.9/0.04 1.0/0 1.0/0 1.1/0.020.25 0.9/0.06 0.9/0.02 0.9/0.07 0.9/0.06 0.9/0.06 0.9/0.06 0.9/0.07 0.8/0.02 0.9/0.07 0.8/0.08 0.9/0.02 0.8/0.01 0.9/0.060.5 1.1/0.04 1.1/0.03 1.0/0 1.0/0 0.9/0.03 0.8/0.12 0.8/0.16 0.8/0.09 0.8/0.15 0.8/0.12 0.8/0.13 0.8/0.13 0.9/0.011.0 1.1/0.02 1.0/0 1.0/0 1.0/0 1.0/0 1.2/0.08 1.3/0.13 1.2/0.03 1.3/0.1 1.3/0.07 1.2/0.06 1.2/0.06 1.0/0

D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 1.3/0.17 1.3/0.12 0.8/0.19 0.8/0.24 1.0/0 0.9/0.04 1.3/0.05 0.9/0.15 1.3/0.08 1.3/0.07 0.9/0.12 0.9/0.12 1.0/00.25 1.1/0.08 0.8/0.13 0.8/0.18 1.3/0.19 1.0/0 1.0/0 1.0/0 0.9/0.04 0.9/0.02 0.9/0.03 0.9/0.05 0.9/0.05 1.0/00.5 0.7/0.05 0.8/0.12 0.8/0.17 0.8/0.18 0.9/0.09 0.9/0.08 0.9/0.02 0.9/0.14 0.9/0.05 0.8/0.07 0.9/0.09 0.9/0.11 0.9/0.111.0 1.1/0.04 0.8/0.07 1.2/0.11 1.2/0.13 1.1/0.09 1.1/0.06 1.4/0.05 1.1/0.06 1.3/0.03 1.3/0.04 1.1/0.05 1.1/0.06 1.1/0.09

D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 0.8/0.04 0.8/0.1 0.9/0.21 0.9/0.09 1.0/0 0.9/0.09 1.0/0 1.0/0 0.9/0.03 1.0/0 0.9/0.13 0.9/0.09 1.0/00.25 1.3/0.11 0.8/0.05 0.8/0.17 0.8/0.21 1.0/0 0.9/0.04 1.0/0 1.0/0 1.0/0 1.0/0 0.9/0.11 0.9/0.05 1.0/00.5 1.2/0.11 0.8/0.1 0.8/0.16 0.8/0.16 0.9/0.01 1.0/0 1.0/0 1.0/0 1.1/0.04 1.1/0.06 1.0/0 1.0/0 1.0/01.0 0.8/0.02 0.9/0.08 0.9/0.01 0.9/0.05 1.0/0 1.0/0 1.2/0.12 1.3/0.12 1.3/0.13 1.3/0.13 1.3/0.04 1.3/0.03 1.0/0

D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 0.56 0.39 0.42 0.44 0.28 0.42 0.42 0.39 0.40 0.43 0.43 0.40 0.270.25 0.47 0.47 0.44 0.45 0.35 0.34 0.38 0.42 0.45 0.46 0.43 0.41 0.350.5 0.51 0.48 0.44 0.45 0.35 0.32 0.45 0.33 0.43 0.40 0.31 0.32 0.361.0 0.55 0.47 0.46 0.46 0.42 0.42 0.44 0.40 0.41 0.38 0.39 0.40 0.41

D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 0.15 0.15 0.14 0.16 0.20 0.14 0.18 0.18 0.18 0.19 0.19 0.17 0.190.25 0.30 0.34 0.38 0.40 0.39 0.32 0.27 0.29 0.29 0.31 0.31 0.32 0.380.5 0.36 0.36 0.34 0.34 0.40 0.40 0.40 0.36 0.40 0.41 0.38 0.39 0.391.0 0.53 0.49 0.48 0.49 0.56 0.54 0.47 0.53 0.48 0.48 0.52 0.53 0.56

24

(c) Pepper

(d) F16

D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 0.55 0.44 0.49 0.59 0.45 0.48 0.54 0.65 0.56 0.54 0.62 0.59 0.450.25 0.51 0.46 0.49 0.50 0.34 0.34 0.44 0.44 0.44 0.45 0.44 0.42 0.340.5 0.51 0.37 0.40 0.41 0.41 0.39 0.35 0.45 0.40 0.39 0.40 0.43 0.431.0 0.56 0.49 0.51 0.53 0.58 0.55 0.47 0.49 0.48 0.45 0.52 0.52 0.58

D2 D4 D6 D8 9/7D 10/18 5/3 9/7M 5/11A 5/11C 13/7C 13/7T 9/7WY0.125 0.44 0.54 0.63 0.53 0.44 0.58 0.55 0.60 0.58 0.56 0.63 0.60 0.450.25 0.56 0.47 0.60 0.69 0.52 0.59 0.55 0.56 0.55 0.59 0.66 0.60 0.520.5 0.74 0.62 0.75 0.70 0.60 0.56 0.55 0.58 0.58 0.60 0.56 0.55 0.591.0 0.77 0.68 0.58 0.57 0.53 0.51 0.64 0.55 0.59 0.59 0.58 0.54 0.52

25

List of figure captions

Fig. 1. Framework of a transform coder.

Fig. 2. Encoding process of SPIHT.

Fig. 3. (a) Two-level wavelet transform (b) SPIHT’s spatial-orientation trees.

Fig. 4. Filter bank structure of wavelet transform.

Fig. 5. Test images: (a) Lena (b) Baboon (c) Pepper (d) F16.

Fig. 6. Extension types (a) periodic extension (b) odd-symmetric extension (c)

even-symmetric extension (d) anti-symmetric extension.

26

Transformation Quantization Entropy CodingImage

Fig. 1. Framework of a transform coder.

OriginalImage DWT

Sorting Pass RefinementPass

EntropyCoding

Bit Streams

SPIHTQuantization

Fig. 2. Encoding process of SPIHT.

*

(a) (b)

Fig. 3. (a) Two-level wavelet transform (b) SPIHT’s spatial-orientation trees.

h0

h1

D2

D2

Down-samplingby 2

x[n]

LowpassSubband (L1)

HighpassSubband (H1)

y[n]

U2

U2

Up-samplingby 2

g0

g1

+

Analysis System Synthesis System Fig. 4. Filter bank structure of wavelet transform.

27

(a) (b)

(c) (d)

Fig. 5. Test images: (a) Lena (b) Baboon (c) Pepper (d) F16.

(a) (b) (c) (d)

Fig. 6. Extension types (a) periodic extension (b) odd-symmetric extension (c) even-symmetric extension (d) anti-symmetric extension.

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PERFORMANCE ANALYSIS OF EMBEDDED-WAVELET IMAGE …shyang/Journal Papers... · 2005-03-14 · PERFORMANCE ANALYSIS OF EMBEDDED ... perceptually weighted scalar quantization, and Huffman