IMAGE CODING WITH CONTOURLET/WAVELET

TRANSFORMS AND SPIHT ALGORITHM

An Experimental Study

Sławomir Nowak and Przemysław Głomb

Institute of Theoretical and Applied Informatics, Polish Academy of Science, Bałtycka 5, Gliwice, Poland

Keywords: Contourlet transform, Wavelet transform, SPIHT algorithm, Image coding, Transmission errors.

Abstract: We investigate the error resilience of the image coded with the wavelet/contourlet transform and the SPIHT

algorithm. We experimentally verify the behaviour for two scenarios: partial decoding (as with scalable

coding/transmission) and random sequence errors (as with transmission errors). Using a number of image

quality metrics, we analyze the overall performance, as well as differences between each transform. We

observe that error difference between transforms varies with length of decoded sequence.

1 INTRODUCTION

Image coding theory forms a foundation of today's

multimedia technologies: internet web pages, digital

cameras, medical diagnostics to name a few. The

objective of image coding is to reduce the

transmission/storage requirements, by exploiting

statistical structure of images and human visual

perception properties.

Current paradigm for image compression

involves using a transform for recoding pixel data,

then bit packing of the coefficients' data into a coded

bitstream. A JPEG – 2000 standard (JPEG, 2000), is

a practical formalization, focusing on wavelet

transform and EBCOT (embedded block coding

with optimized truncation) bit ordering algorithm.

Other bit ordering algorithms have also been

proposed, most notably SPIHT (set partitioning in

hierarchical trees), described in (Said and Pearlman,

1996). Recently, an alternative to wavelet transform

has been suggested, in the form of contourlet

transform (Do and Vetterli, 2005), which has better

property of representing directional image

information, an important requirement for image

coding applications.

In many applications we expect the image to be

reconstructed from the coded bitstream with errors,

which can result either from the coding system

(lossy compression) or transmission medium

(packet loss or errors). The former is analyzed as

rate-distortion theory, and focuses on relation of

bitstream length to image quality (with respect to

some predefined metric). The latter is studied by

observing distortion introduced by errors in coded

bitstream. While there are methods that improve

resilience of the bitstream, i.e. using error

correcting codes (Kim et al, 2004), or post-process

the image to conceal/remove errors (Kung et al,

2006), the quantitative and qualitative analysis of

coding system errors of both types is the key issue

in design of a compression system.

While the statistical analysis of the contourlet

transform indeed suggests it is better in

representing image information (Po and Do, 2006),

and thus possibly better suited for image coding

systems, the exact gain of using it in place of

wavelet transform is still an open issue. (Belbachir

and Goebel, 2005) present a straightforward

approach to compress images with contourlets, but

provide only brief results with limited error

discussion. (Esakkirajan et al, 2006) give a more

detailed analysis including different filters and

wavelet to contourlet comparisons. Their approach

(multistage vector quantization) is not suitable for

analysis scalability errors; they also don't

investigate error resilience. They report small gain

for contourlet transform. (Eslami and Radha, 2004)

focus on combining contourlets after wavelets with

SPIHT coding. They report better results with their

approach than with wavelets alone. (Liu and Liu,

2006) also focus on combination of wavelets and

contourlets in sophisticated scheme. They report

improved performance of contourlet over wavelet,

Nowak S. and Głomb P. (2009).

IMAGE CODING WITH CONTOURLET/WAVELET TRANSFORMS AND SPIHT ALGORITHM - An Experimental Study .

In Proceedings of the First International Conference on Computer Imaging Theory and Applications, pages 37-42

DOI: 10.5220/0001767400370042

 SciTePress

with their scheme providing additional gains. The

results are provided for selected bit rates.

Our experiments were motivated by the need for

more throughout, than reported above, evaluation of

practical difference of coding with wavelets and

contourlets. The objectives were defined as:

 compare wavelets and contourlets side by

side, as much as possible;

 provide detailed analysis of scalability, in the

form of partial reconstruction of the coded

stream;

 provide analysis of bit stream perturbations

(consistent with network errors, i.e. streaming

in wireless network);

 use a range of error metrics, for qualitative

evaluation of introduced errors.

Our findings are:

 the wavelet-SPIHT method works better when

the near-lossless operations are demanded

(smaller sequences sizes), whereas when a

given level of quality loss is acceptable, the

contourlet method gives better image quality

for smaller sequence size;

 it is possible to parameterize the SPIHT algorithm

to obtain images at definite quality level;

 when certain level of image quality loss is

accepted, some subsequences (refinement

parts) of the output sequence can be treated as

low-priority data.

This work is organized as follows: second

section presents our approach and experimental

framework, third section presents experiment results,

last section presents conclusions.

2 THE METHOD

We investigate a coding systems consisting of two

elements: image transform and bit coder. For

transform, we use interchangeably wavelet

transform (implemented as standard 9/7 filter bank,

as in (JPEG, 2000)) and contourlet transform

(implemented with surfacelet transform, using (Lu

and Do, 2007)). For bit coding, we use SPIHT

algorithm (Said and Pearlman, 1996). The usage is

motivated by good results reported, but also by

difficulties with extending other algorithms, like

EBCOT, to contourlets (Głomb et al, 2007). The

implementation was done in C++.

We use a number of metrics to measure image

distortion. Mean square error (MSE) and Peak signal

to noise ratio (PSNR) are defined as:

PSNR

log10=

(1)

where:

∑∑

−

⋅

jifjif

MSE

))],(),(([

(2)

k – number of image colors minus 1;

N,M – sizes of image;

f(i,j) – input image;

’

(i,j) – output image.

We also use average per pixel error (denoted

PERPX), and measures related to edge degradation.

The latter are included as it has been observed (Al-

Otum, 1998) that edge degradation in video coding

is an important component of human quality

perception. For measurement of the latter, we first

use a Sobel edge operator for reference and distorted

image, then measure MSE and NCC (normalized

cross correlation) of the edge images.

The aim of work was to evaluate the image

quality as a function of decoded sequence size and to

determine the resistance of the output, linear

sequence to errors.

3 EXPERIMENTAL RESULTS

Within the experiments we use the typical set of

images (Baboon, Barbara, Boat, Goldhill, Lena,

Peppers), universally used in digital images studies.

3.1 Image Quality as a Function of the

Decoded Sequence Size

The SPIHT algorithm codes more important

coefficient bits first. While decoding, image quality

increases progressively. Because of this, the graph of

image quality as a function of the size of decoded

sequence is nonlinear.

During experiments the process of decoding

output sequence was being stopped at specific point,

expressed in percent of the whole sequence size. The

single steep was 2[%].

By comparing output images to the original ones,

the qualities of output images were evaluated and

the graph of image quality was obtained. Each graph

expresses the average results of each measurement

method for the whole set of images.

The size of output sequences for contourlet and

wavelet decompositions were different. The sizes for

contourlet method sequences are considerably

larger. We calculated the value γ (3).

IMAGAPP 2009 - International Conference on Imaging Theory and Applications

68,2

...

methwaveletforseqofsizeavrg

methconturletforseqofsizeavrg

(3)

In order to compare both methods we place

contourlet and wavelet methods’ graphs together on

each chart. 100[%] on X axis means 100[%] of

a wavelet sequence and 100/γ = 100/2,68 = 37,3[%]

of a contourlet sequence. Results are presented on

Figure 2. Sample images are presented on Figure 1.

Figure 1: Sample images (Barbara.jpg) for conturlet and

wavelet methods, after decoding 2[%], 20[%], 50[%] of

the output SPIHT sequence.

3.1.1 Discussion

Presented results let us evaluate the methods and

make a preliminary classification.

The wavelet decomposition together with the

SPIHT algorithm generates smaller sizes of

sequences, in compare to the contourlet ones (by

average factor γ). So, in potential application, when

the lossless transmissions are demanded, the wavelet

method seems to be a better way. But in some cases,

when a given level of quality loss is acceptable, the

contourlet method gives better image quality for

smaller sequence size. One can observe that on

Figure 1, where for MSE, PERPX, PSNR graphs the

distinct cross point occurred. It is also evident, while

comparing images presented on Figure 1.

Results can be useful in case of the

parameterization of the SPIHT algorithm. Based on

the results, approximation function of quality from

the length of sequence, can be calculated and

sequences coding images on definite quality can be

obtained. Examples of the approximation functions

(fitting the EDGE NCC metrics) are presented on

Figure 3.

Figure 2: Image quality as a function of the decoded

sequence size for contourlet and wavelet methods.

3.2 Image Quality as a Function of

Distortions

The SPIHT generates two types of data. The first

Barbara.jpg, Wavelet method

2[%] 20[%] 50[%]

Barbara.jpg, Conturlet method

2[%] 20[%] 50[%]

IMAGE CODING WITH CONTOURLET/WAVELET TRANSFORMS AND SPIHT ALGORITHM - An Experimental

Study

Figure 3: Approximation functions of the EDGE NCC

metric for contourlet and wavelet methods.

comes from the process of hierarchical tree

creation. The second consists of refinement bits.

The output sequence alternately consists of “tree

creation bits” subsequences and “refinement bits”

subsequences.

Loss or error affected by any bit in “tree creation

bits” subsequence makes the further process of

decoding impossible, because the algorithm loses the

correct path through binary trees. Considering e.g.

network streaming, the “tree creation bits”

subsequences need error protection.

The “refinement bits” subsequences affect only

for single pixels and do not need special protection.

After loss or error affected by single bits or even the

whole subsequences, the process of decoding can be

continued. Occurrences of errors within the

subsequence influences to a little degree on the

image quality.

When considering errors within “refinement bits”

subsequences it is important to mention, that the

original size of each subsequence have to be preserved.

In the opposite case the algorithm also loses the correct

path through subsequent stages. We consider rather

errors then a loss of data.

Introducing distortions to the “refinement bits”

subsequences, the graph of image quality as

a function of simplified model of error level is

obtained. During the experiments the probability of

distortion of individual bits varied from 0[%] (no

distortions) to 100[%] (every bit was distorted, and

as a consequence the “refinement bits” subsequences

changed to the inverted form of the original).

As in the previous experiments, the contourlet

and wavelet methods are compared. The qualities

of output images were evaluated and the graph of

image quality (average results) was obtained. To

compare results in direct way the γ value was used.

Results are presented on Figure 4 and sample

images are presented on Figure 5.

Figure 4: Image quality as a function of distortion

probability of individual bits in “refinement bits”

sequence.

3.2.1 Discussion

Obtained results show that bits within the

“refinement bits” subsequences have little influence

on image quality. The “refinement bits”

subsequences constitute average about 33[%] or the

SPIHT output sequence size for contourlet

decomposition method and about 28[%] for wavelet

decomposition method.

Considering transmission of images via network

using dedicated protocols, when certain level of

image quality loss is accepted, one can treat

IMAGAPP 2009 - International Conference on Imaging Theory and Applications

Figure 5: Sample images (Barbara.jpg) for conturlet and

wavelet methods, different probability of distortion of

individual “refinement bits” in sequence.

“refinement bits” subsequences as low-priority data.

In case of a transmission error (individual bits

distortion or even whole block loss) retransmission

of data is not necessary. In certain cases, it is

possible to omit completely the “refinement bits”

subsequences. Only the size of “refinement bits”

subsequence have to be retained.

Replacements for it, the additional, excessive

data can be introduced to the “tree creation bits”

subsequences, which are necessary to follow the

SPIHT algorithm.

4 CONCLUSIONS

The main goal of the work was to compare two

methods of decomposition: contourlet and wavelet

in employment with the SPIHT algorithm and to

investigate their properties. The comparison can be

helpful when new protocols for image transmission

are designed, especially when a network streaming

through noisy environment is considered.

The conclusions are:

 The wavelet-SPIHT method generates smaller

sizes (by average factor γ = 2,68) of

sequences, but the contourlet-SPIHT method

gives better image quality for smaller

sequence size (it should be noted that there are

algorithms producing contourlet

decomposition with less redundancy);

 It is possible to parameterize the SPIHT

algorithm to obtain sequences coding images

at definite level of quality;

 The “refinement bits” subsequences can be

treated as low-priority data. Bits within that

subsequences have little influence on image

quality.

An important observation noted during our

experiments is that while several studies report

contourlet transform to be generally superior to

wavelet transform, we report that the gain is closely

related to image structure. Thus, in general one can

expect a considerable variation of differences

between the transforms depending on image source.

ACKNOWLEDGEMENTS

This work was supported by the Polish Ministry of

Science and Higher Education research project

3 T11C 045 30 “Application of contourlet transform

for coding and transmission stereo-images”.

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IMAGAPP 2009 - International Conference on Imaging Theory and Applications