CHAPTER 6

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6 Huffman Coding Based Image Compression Using Complex Wavelet Transform

103

6.1 Introduction 103

6.2 Compression Techniques 104

6.2.1 Lossless compression 105

6.2.2 Lossy compression 105

6.2.3 Entropy 105

6.3 Wavelet Transform based compression technique 106

6.3.1 Dual Tree Complex Wavelet Transform 107

6.3.2 Thresholding in Image Compression 107

6.4 Introduction to Huffman coding 108

6.4.1 Huffman Coding 108

6.4.2 Huffman Decoding 110

6.5 DT-CWT Based Compression Algorithm 111

6.6 Results and Discussion 112

6.6.1 Performance Metrics 112

6.7 Summary 120

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CHAPTER 6

HUFFMAN CODING BASED IMAGE COMPRESSION USING COMPLEX

WAVELET TRANSFORM

6.1 INTRODUCTION

Image compression is one of the most significant application of the wavelet

transform. Capon was one of the first person to introduce the compression of normal

images by run length encoding[97]. Today there is no change in the basic

methodology of compression , but higher compression ratios could be attained.

In this chapter the investigator discussed about the need of compression in

section 6.1, types of compression techniques under section 6.2, and wavelet transform

based compression in section 6.3. Huffman coding is explained in section 6.4 and

proposed complex wavelet transform based image compression algorithm using

huffman coding is discussed along with results and discussion under section 6.5 and

6.6 respectively.

The goal of compression algorithm is to eliminate redundancy in the data i.e.

the compression algorithms calculates which data is to be considered to recreate the

original image along with the data to be removed [98]. By eliminating redundant

(duplicate) information, the image can be compressed. The three types of

redundancies are coding redundancy, inter pixel redundancy and psycho visual

redundancy. Best possible code words are used to reduce coding redundancy. The

correlation among the pixels results in inter pixel redundancy. Visually unimportant

information lead to psycho visual redundancy.

The purpose of compression system is to shrink the number of bits to the

possible extent, while keeping the visual quality of the reconstructed image as close

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to the original image. Fig 6.1 shows the basic block diagram of a general image

compression system. It consists of an encoder block and a decoder block shown in

the fig: 6.1.

Fig 6.1 General image compression system

In the fig:6.1, the encoder block reduces different redundancies of the input

image. In the first stage, the mapper converts the input image into another set-up

designed to eliminate inter pixel redundancy. In the second stage, a quantized block

reduces the precision of the first stage output in accordance with a predefined

decisive factor. In the final stage, a symbol encoder maps the quantized output in

accordance with a predefined criterion. In the decoder block , an inverse mapper

along with a symbol decoder perform the opposite operations of the encoder block.

An inverse quantizer is not considered since quantization is irretrievable [1].

6.2 COMPRESSION TECHNIQUES

Depending on the possibility of reconstruction of clear-cut original image,

compression techniques are classified into lossless and lossy compression techniques.

Encoder

Original Image f(x ,y)

Mapper Quantizer Symbol coder

Decoder

Symbol decoder

Compressed data

Inverse Mapper

𝑓(𝑥, 𝑦)

Reconstructed Image

Compressed data for storage

and transmission

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6.2. 1 Lossless compression

In lossless technique, the original image can be perfectly recovered from the

compressed (encoded) image. In this technique each and every bit of information is

important. Lossless image compression scheme is used in the applications, where no

loss of data is required such as in medical imaging.

6.2.2 Lossy compression

Higher compression ratios are possible in lossy techniques by reducing some

more amount of redundancies. Lossy techniques are extensively used in most of the

applications like internet browsing and video processing, because the quality of the

reconstructed image is not important. Here a perfect restoration of the image is not

desirable. [99]

6.2.3 Entropy

To categorize various compression methods, it is essential to distinguish the

entropy of the image. A low frequency and highly correlated image which has low

entropy can be compressed more by any compression technique. A compression

algorithm designed for a particular application may not perform good for other

applications . The entropy ‘H’ can be calculated as:

𝐻 = −∑ 𝑃(𝑘)𝑙𝑜𝑔2[𝑃(𝑘)]𝐺−1𝑘=0 …(6.1)

Where G is gray levels and P(k) is probability of gray level k. P(k) can be calculated

by knowing the frequency[h(k)] of gray level k in an image of size M×N as

𝑃(𝑘) = ℎ(𝑘)𝑀.𝑁

…(6.2)

The investigator concentrates on wavelet based transform coding image

compression algorithms. The coefficients obtained from the discrete image transform,

which make little contribution to the information can be eliminated. In Discrete

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Cosine Transform (DCT) based compression method, the input image is split into 8×8

blocks and each block is transformed separately. However this method introduces

blocking artifacts, and also higher compression ratios are not achieved. But the

wavelet transform produces no artifacts and it can be applied to entire image rather

than blocks.

6.3 WAVELET TRANSFORM BASED COMPRESSION TECHNIQUE

In DWT, the most significant information corresponds to high amplitudes

and less significant information corresponds to low amplitudes. Compression can be

accomplished by neglecting the least significant information. The wavelet transforms

make possible in attaining higher compression ratios along with high-quality of re-

construction.

Wavelets are also used in mobile applications, denoising, edge detection,

speech recognition, feature extraction, real time audio-video applications, biomedical

imagining, orthogonal divisional multiplexing. So the wavelet transform popularity is

increasing due to its ability to decrease distraction in the reconstruction signal.

Fig 6.2a : Transform based compression procedure.

Fig 6.2b : Transform based decompression procesdure.

Fig. 6.2 shows the general transform based image compression standard. The

forward and inverse transforms along with encoding and decoding processes are

shown in fig: 6.2 . DWT and IDWT are appropriate transforms used in wavelet

Original Image

Forward

Transform

Encode Transform

Values

Compressed Image

Compressed- Image

Decode Transform

Values

Inverse Transform

Reconstructed Image

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transform based image compression [76]. Discrete Cosine Transform (DCT) is used in

JPEG algorithm.

6.3.1 Dual Tree-Complex Wavelet Transform (DT-CWT)

Complex wavelets are not widely used in image processing applications due to

the difficulty in designing complex wavelet filters. Dr. Nick Kingsbury of Cambridge

university projected a dual tree arrangement of CWT to overcome the limitations in

standard DWT. DT-CWT employ two trees to produce the real and imaginary parts

of wavelet coefficients with real filter set separately. This Transform allows practical

usage of complex wavelets in image processing [19][20][38][100]. DT-CWT can be

used for a variety of applications like image compression, denoising, enhancement,

inpainting and image restoration etc.,

DT-CWT is a structure of DWT which produce complex coefficients by

means of a two set of wavelet filters to achieve the real and imaginary coefficients.

The purpose of usage of complex wavelets is that it presents a high grade of shift

invariance and good directionality. The DT-CWT applied to the rows and columns of

the image results in six complex high pass detailed sub-images and two low-pass sub-

images at each level. The subsequent stage iterate on only low pass sub-image. In 2D

DWT, the three high pass sub-bands result in 00 ,450 , and 900 orientations. Whereas

CWT has six high pass sub-bands at each level which are oriented at ±150,± 450 and

±750.

6.3.2 Thresholding in Image Compression

The wavelet coefficients obtained are near to zero value for some kind of

signals. Thresholding can revise these coefficients to generate more zeros. With hard

thresholding many coefficients are tend to zero. Actual compression of signal is not

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achieved though the wavelet analysis and thresholding are performed. Standard

entropy coding methods like huffman coding allow to compress the data by assigning

short codes for more frequently occurring symbols and large code words for other

symbols. By proper encoding, the signal requires less memory space for storage, and

takes less time during transmission.

A good threshold value is considered to adjust the energy preservation and

quantity of zeros. Energy is lost and high compression is achieved if higher threshold

value is chosen. Thresholding can be selected locally or globally. In global threshold

same threshold is applied to each subband, whereas local thresholding involve usage

of various threshold values for every subband.[68].

6.4 INTRODUCTION TO HUFFMAN CODING

Huffman codes developed by D.A. Huffman in 1952 are optimal codes that

map one symbol to one code word. Huffman coding is one of the well-liked technique

to eliminate coding redundancy. It is a variable length coding and is used in lossless

compression. Huffman encoding assigns smaller codes for more frequently used

symbols and larger codes for less frequently used symbols [101]. Variable-length

coding table can be constructed based on the probability of occurrence of each source

symbol. Its outcome is a prefix code that expresses most common source symbols

with shorter sequence of bits. The more probable of occurrence of any symbol the

shorter is its bit representation.

6.4.1 Huffman Coding

Huffman coding generate the smallest probable number of code symbols for

any one source symbol. The source symbol is likely to be either intensities of an

image or intensity mapping operation output. In the first step of huffman procedure,

a chain of source reductions by arranging the probabilities in descending order and

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merging the two lowest probable symbols to create a single compound symbol that

replaces in the successive source reduction. This procedure is repeated continuously

upto two probabilities of two compound symbols are only left[1]. This process with

an example is illustrated in Table 6.1a.

In Table 6.1a the list of symbols and their corresponding probabilities are

placed in descending order. A compound symbol with probability ‘0.1’ is obtained by

merging the least probabilities ‘0.06’ and ‘0.04’. This is located in source reduction

column ‘1’ . Once again the probabilities of source symbols are arranged in

descending order and the procedure is continued until only two probabilities are

recognized. These probabilities observed at far right are ‘ 0.6’ and ‘0.4’ shown in the

Table 6.1a.

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In the second step, huffman procedure is to prepare a code for each compact

source starting with smallest basis and operating back to its original source. As shown

in the table 6.1b, the code symbols 0 and 1 are assigned to the two symbols of

probabilities ‘0.6’ and ‘0.4’ on the right. The source symbol with probability 0.6 is

generated by merging two symbols of probabilities ‘ 0.3’ and ‘0.3’ in the reduced

source. So to code both of these symbols the code symbol 0 is appended along with 0

and 1 to make a distinction from one another. This procedure is repeated for each

reduced source symbol till the original source code is attained [99]. The final code

become visible at the far-left in table 6.1b. The average code length is defined as the

product of probability of the symbol and number of bits utilized to represent the

same symbol.

Laverage = (0.4)(1) +(0.3)(2) + (0.1)(3) + (0.1)(4) + (0.06)(5) + (0.04)(5)

= 2.2 bits per symbol .

The source entropy calculated is

𝐻 = −�𝑃(𝑘)𝑙𝑜𝑔2[𝑃(𝑘)] = 2.14 𝑏𝑖𝑡𝑠/𝑠𝑦𝑚𝑏𝑜𝑙𝐿

𝑘=0

Then the efficiency of Huffman code 2.14/2.2 = 0.973.

6.4.2 Huffman Decoding

In huffman coding the optimal code is produced for a set of symbols. A

unique error less decoding scheme is achieved in a simple look-up-table approach.

Sequence of huffman encoded symbols can be deciphered by inspecting the individual

symbols of the sequence from left-to-right. Using the binary code present in Table

6.1b, a left-to-right inspection of the encoded string ‘1010100111100’ unveils the

decoded message as “a2 a3 a1 a2 a2 a6”. Hence with huffman decoding process the

compressed data of the image can be decompressed.

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6.5 DT-CWT BASED COMPRESSION ALGORITHM

The investigator proposed an huffman coding based 2D-DT-CWT image

compression algorithm. The following is the basic procedure for implementing the

proposed algorithm.

o The input image to be compressed is considered.

o The input image is decomposed into wavelet coefficients ‘w’ using DT-

CWT.

o The detailed wavelet coefficients ‘w’ are modified using thresholding.

o Huffman encoding is applied to compress the data.

o The ratio of original data size to compressed data size ( Compression ratio) is

calculated.

o Huffman decoding and inverse DT-CWT is applied to reconstruct the

decompressed image.

o MSE and PSNR are computed to test the quality of the decompressed image.

Fig:6.3 illustrates the compression algorithm using DT-CWT along with

Huffman coding.

Fig 6.3 Block diagram of compression algorithm using DT-CWT

ORIGINAL

IMAGE

DUAL TREE

CWT

THRES -

HOLDING

HUFFMAN

ENCODING

RECONSTRUCTED

IMAGE

HUFFMAN

DECODING

INVERSE DUAL TREE

CWT

PSNR

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6.6 RESULTS AND DISCUSSION

As shown in fig:6.3, the input image is decomposed into wavelet coefficients

by using complex wavelet transform. The coefficients obtained are applied to

thresholding. The thresholded coefficients are coded by huffman encoding. To re-

form the decompressed image, the decompression algorithm converts the compressed

data through huffman decoding followed by conversion of wavelet coefficients into a

time domain signal using inverse dual tree complex wavelet transform.

6.6.1 Performance Metrics

Mean square error, root mean square error, peak signal to noise ratio,

compression ratio and bits per pixel are the metrics used for evaluating the

performance of image processing methods. MSE and RMSE can be calculated by

considering the cumulative squared error between the compressed and the original

image. PSNR is the measure of peak error. Compression ratio is the ratio of the

original file to the compressed file and bits per pixel is the ratio of compressed file in

bits to the original file in pixels. The mathematical formulae is determined as

RMSE = � 1M.N

∑ ∑ (X(i, j) − Y(i, j))2 Nj=1

Mi=1 …(6.3)

Where, M and N are width and height of an image , X and Y are original and

processed images respectively.

𝑃𝑆𝑁𝑅 = 10 log102552

𝑀𝑆𝐸 …(6.4)

𝐶𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 ( 𝐶𝑅) = 𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝐼𝑚𝑎𝑔𝑒 𝑠𝑖𝑧𝑒𝐶𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑒𝑑 𝑖𝑎𝑚𝑔𝑒 𝑠𝑖𝑧𝑒 …(6.5)

𝐵𝑃𝑃 = 1𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 …(6.6)

In general a large value of PSNR is good. It means that the signal information

is more than the noise (error). A lower value of MSE transforms to a higher value of

PSNR. So a better compression method with high PSNR (low MSE ) and, with a

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good compression ratio can be implemented. The same method may not perform well

for all types of images. The proposed algorithm is tested on different images like a

gray colored ‘cameraman’ image, a colored ‘lena’ image and a ‘medical’ image.

For cameraman image of size 256 x 256, different parameters like the original

image size, compressed image size, CR, BPP, PSNR and RMS error for various

threshold values are tabulated. Table 6.2a and Table 6.2b shows parameter values

obtained using proposed and existing methods.

Table 6.2a: Different parameter values at different thresholds for a gray color cameraman image using proposed method.

Parameter TH=6 TH=10 TH=20 TH=30 TH=40 TH=50 TH=60

Original File 65240 65240 65240 65240 65240 65240 65240 Compressed

File Size 10783 10114 9320 8125 8505 8310 8165

Compression Ratio (CR) 6.05 6.45 7 7.43 7.67 7.85 7.99

Bits Per Pixel (BPP) 1.32 1.24 1.14 1.07 1.04 1.01 1

Peak Signal to Noise Ratio

(PSNR) 40.32 36.25 31.15 28.99 27.43 26.23 25.28

RMS error 2.43 3.9 6.92 8.93 10.74 12.34 13.74

Table 6.2b: Different parameter values at different thresholds for a gray color cameraman image using existing method(EZW and huffman encoding [62])

Parameter TH=6 TH=10 TH=30 TH=60

Original File 65240 65240 65240 65240

Compressed File Size 11186 10870 9944 8437 Compression Ratio

(CR) 5.83 6.00 6.56 7.73

Bits Per Pixel (BPP) 2.52 1.48 0.74 0.33 Peak Signal to Noise

Ratio (PSNR) 33.36 33.37 33.16 32.22

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Chart 6.1: Threshold V/s Compression ratio for cameraman image

Chart 6.2: Threshold V/s PSNR for cameraman image

Chart 6.3: Threshold V/s BPP for cameraman image

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From the tabulated results, it is evident that the proposed huffman coding

based 2D-DT-CWT algorithm presents outstanding results. A higher compression

ratio can be attained by selecting an appropriate threshold value.

Threshold V/s Compression ratio , Threshold V/s PSNR and Threshold V/s

BPP graphs have been depicted in the Charts 6.1, 6.2 and 6.3. respectively. Here the

curves are related to the existing and proposed methods. The existing method is based

on DWT with EZW and huffman encoding. The proposed method is based on only

huffman coding with 2D-DT-CWT. It is apparent that the proposed method gives

better performance in compression ratio, BPP and PSNR values compared to the

existing method.

Table 6.3:Different parameter values at different thresholds for lenaRGB image

Parameter TH=6 TH=10 TH=20 TH=30 TH=40 TH=50 TH=60

Original File 786488 786488 786488 786488 786488 786488 786488

Compressed File Size 111876 107150 94986 90297 87777 85580 83936

Compression Ratio (CR) 7.03 7.34 8.28 8.71 8.96 9.19 9.37

Bits Per Pixel (BPP) 3.41 3.26 2.89 2.75 2.67 2.611 2.56

Peak Signal to Noise Ratio

(PSNR) 40.32 36.67 32.2 29.81 28.25 27.41 26.26

RMS error 3.01 3.4 5.8 7.64 9.15 10.39 11.52

Different parameters like original image size, compressed image size, CR,

BPP, PSNR and RMS error for various thresholds are calculated and tabulated in the

Table 6.3 for lenaRGB image of size 256 x 256.

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The original lenaRGB image and retrieved images for different thresholding

values ( TH = 10, 20, 30, 40, 50) and their corresponding PSNR values are shown in

fig:6.5_a, fig:6.5_b, fig:6.5_c, fig:6.5_d, fig:6.5_e, and fig:6.5_f respectively.

Magnetic Resonance Imaging(MRI) is useful for showing abnormalities of the

brain such as hemorrhage, stroke, tumor, multiple sclerosis etc., Signal processing is

required to detect and decode the abnormalities in MRI imaging.

Table 6.4:Different parameter values at different thresholds for a medical image

Parameter TH=6 TH=10 TH=20 TH=30 TH=40 TH=50 TH=60

Original File 17912 17912 17912 17912 17912 17912 17912 Compressed File

Size 3589 3373 2847 2626 2494 2424 2372

Compression Ratio (CR) 4.99 5.31 6.29 6.82 7.18 7.39 7.55

Bits Per Pixel (BPP) 4.8 4.51 3.81 3.51 3.34 3.24 3.17

Peak Signal to Noise Ratio

(PSNR) 40.32 35.89 30.96 28.31 26.4 25.07 24.06

RMS error 3.3 4.09 7.21 9.79 12.2 14.22 15.97

Different parameters like original image size, compressed image size, CR,

BPP, PSNR and RMS error for various thresholds are calculated and tabulated in the

Table 6.4 for medical image of size 256 x 256.

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Cameraman image Image Size: 256*256

Fig 6.4(a)Input image Fig 6.4(d)Threshold=30,PSNR= 28.99

Fig 6.4(b)Threshold=10,PSNR= 36.25 Fig 6.4(e)Threshold=50,PSNR=26.23

Fig6.4(c)Threshold=20,PSNR= 31.15 Fig 6.4(f)Threshold=60,PSNR= 25.28

Fig 6.4 Illustration of 2D-DT-CWT based image compression for various thresholds of cameraman image

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lenaRGB image Image Size: 256*256

Fig 6.5(a)Input image Fig 6.5(d)Threshold=30,PSNR=29.81

Fig 6.5(b)Threshold=10,PSNR=36.29 Fig 6.5(e)Threshold=40,PSNR =28.25

Fig 6.5(c)Threshold=20,PSNR= 32.20 Fig 6.5(f)Threshold=50,PSNR=27.41

Fig 6.5 Illustration of DTCWT based image compression for various Thresholds of LenaRGB image

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Medical image Image Size: 256*256

Fig 6.6(a)Input image Fig 6.6(d)Threshold=30,PSNR= 28.31

Fig 6.6(b)Threshold=10,PSNR= 35.89 Fig 6.6(e)Threshold=40,PSNR= 26.40

Fig 6.6(c)Threshold=20,PSNR=30.96 Fig 6.6(f)Threshold=50,PSNR= 25.57

Fig 6.6 Illustration of DTCWT based image compression for various Thresholds of medical image

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Original medical image and retrieved images for different thresholding

values ( TH = 10, 20, 30, 40, 50) and their corresponding PSNR values are shown in

fig:6.6_a, fig:6.6_b, fig:6.6_c, fig:6.6_d, fig:6.6_e, and fig:6.6_f respectively.

The proposed huffman based 2D-DT-CWT image compression technique

outperforms in terms of good compression ratio, better BPP and higher PSNR . The

algorithm is examined on different standard images and it is investigated that the

proposed image compression method gives consistent results compared to the results

obtained from existing method[62]. It is also identified that the results obtained by

the proposed method are as good as the compression method using embedded zero-

tree wavelet (EZW) and huffman coding

6.7 SUMMARY

In this chapter huffman coding and decoding is explained. Huffman coding

based complex wavelet transform image compression algorithm is implemented. The

results are compared with existing embedded zero-tree wavelet (EZW) along with

huffman coding method. From the investigation results it is evident that a slight better

compression ratio is achieved with single encoding method compared to two encoding

methods used in the existing method. In the next chapter conclusions and further

scope for improvements are explained.