2007 International Conference on Convergence Information Technology
Curvelet-based Image Compression with SPIHT Muhammad Azhar Iqbal, Dr Muhammad Younus Javed, Usman Qayyum Department of Computer Engineering College of Electrical and Mechanical Engineering National University of Sciences and Technology, Rawalpindi, Pakistan
[email protected],
[email protected],
[email protected]
image coding, transform coding have been introduced. Among all these, transform coding is most efficient especially at low bit rate [1]. During past few years wavelet transform has been widely used in the field of image compression and has established its effectiveness because of the well-localized property of its coefficients in both space and frequency domains [2]. Different wavelet-based encoding schemes have also been developed i.e. EZW [3], SPECK [4], and SPIHT [5] etc. that exploit the multi-scale nature of wavelet transform and has proven their significance to improve performance. The basic flaw that wavelet transform exhibits, is its inability to represent edge discontinuities along curves. Less number of coefficients is required in compression process but several wavelet coefficients are used to reconstruct edges properly along the curves. This is due to the reason that in a map of large wavelet coefficients, edges repeat at scale after scale. There was a need of a transform that handle twodimensional singularities along the curves sparsely. This led to the birth of new multi-resolution curvelet transform. Curvelet basis elements possess wavelet basis function qualities but these also oriented at a variety of directions and so represent edge discontinuities and other singularities well than wavelet transform [6]. SPIHT method is not only a simple extension of traditional method for image compression but it also produces good quality images. Moreover, it provides a fully embedded code file, code to exact bit rate and is completely adaptive [7]. SPIHT originally used wavelets and due to its support for multi-resolution encoding/decoding applicable to curvelet transform. The rest of the paper structure is as follows. Section 2 and 3 discuss the theoretical basis of Curvelets and SPIHT respectively. Then, section 4 describes the proposed image compression scheme that is the integration of Digital Curvelet Transform and SPIHT. This is followed by a discussion of the compression results and comparisons with results obtained from previous techniques of compression.
Abstract This paper deals with the implementation of a new compression methodology, which uses curvelet coefficients with SPIHT (Set Partitioning In Hierarchical Trees) encoding scheme. The first phase deals with the transformation of the stimulus image into the curvelet coefficients. The curvelet transform is a new family of multi-scale representation containing the information about the scale and location parameters. Unlike wavelets, it also contains the directional parameters. The orientation selectivity behavior and anisotropic nature of the curvelet transform helps to represent suitably the objects with curves and handles other two-dimensional singularities better than wavelets, which makes it a more proficient transformation for image compression application. During the second phase, a threshold-based selection mechanism has been developed to get prominent coefficients out of different scales. Final phase deals with the application of lossy SPIHT encoding technique on selected significant coefficients. SPIHT exploits the multi-scale nature of curvelet transform and removes the statistical and subjective redundancies. The empirical results on standard test images provide higher PSNR than some of the previous approaches, which strengthen the idea of using curvelet transform instead of wavelet transform in order to get lesser bits to represent more prominent features.
Keywords: Curvelet Coefficients, SPIHT, Multi-resolution Analysis
1. Introduction From early days to now, the basic objective of image compression is the reduction of size for transmission or storage while maintaining suitable quality of reconstructed images. For this purpose many compression techniques i.e. scalar/vector quantization, differential encoding, predictive
0-7695-3038-9/07 $25.00 © 2007 IEEE DOI 10.1109/ICCIT.2007.280
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In the last three to four years, however, curvelets have been redesigned with new mathematical architecture, which is simpler, totally transparent, and easier to use than previous implementations. This new mathematical architecture known as Fast Discrete Curvelet Transfom (FDCvT) [11] suggests two algorithmic strategies, UnequiSpaced Fast Fourier Transform (USFFT) based and Frequency wrapping based FDCvT. Both transforms return curvelet coefficients but are differ by the choice of spatial grid used to translate curvelets at each scale and angle. As Frequency wrapping based FDCvT is faster than USFFT, so it has been selected for proposed research work.
2. Curvelet Transform A new multi-resolution transform developed by Candés and Donoho [8] in 1999 known as curvelet transform as a result of motivation to take away the drawbacks associated with wavelet transform. The transform that is a two-dimensional anisotropic extension of wavelet, originally designed to represent edges and other singularities along curves much more efficiently than traditional wavelet transforms. Although curvelets is an extension of wavelets but there exists a correspondence between curvelet and wavelet subbands. The general rule that represents correspondence between curvelet subband (Cs) and wavelet subband (Ws) is [9]
3. SPIHT (Set Partitioning in Hierarchical Trees)
Cs ļ Ws {2 * Cs, 2 * Cs+1}
SPIHT algorithm as proposed by Amir Said and William Pearlman [5] performs lossy compression in three basic steps (i.e. Initialization, Sorting Pass, Refinement Pass) as shown in Figure 1. It maintains three lists i.e. List of Insignificant Pixels (LIP), List of Significant Pixels (LSP) and List of Insignificant Set (LIS) during its flow. During initialization step, initial value for threshold is determined and initializes LIP with a set containing all the coefficients in lowest subband. Moreover, initially LSP set as an empty list and LIS contains the coordinates of roots of all trees that are ¾ of lowest subband. Next is the iterative sorting pass where members of LIP are processed first and then members of LIS. This step is significance map-encoding step. At the end, elements in LSP further processed in refinement step. In fundamental nature, SPIHT provide binary representation of the integer value of coefficients with header data.
Contrary to wavelets’ isotropic principle where length and width of support frame is of equal size; in curvelet transform the width and length are related by the relation width § length2 that is known as parabolic or anisotropic scaling [10]. Moreover, frame elements in curvelets indexed by scale, location and orientation parameters in contrast to wavelets where elements have only scale and location parameters. The procedural definition of curvelet transform for an object f (image in our case) is actually a combination of four ideas [9], which are briefly reviewed as under:
Subband Decomposition: divide image f into several resolution layers and each layer contains details of different frequencies
f o ( P 0 f , '1 f , ' 2 f ,...) Here P0 and 's (where s t 0) are low pass and high pass filters respectively. P0f is the smooth low-pass layer that efficiently represented by using wavelet base and 'sf are high-pass layers effected by discontinuity curves. At the end, each 'sf layer contains objects near high frequencies with fine details.
Image Transform
Initialization
Coefficients Sorting Pass Bit Stream / Header
Smooth Partitioning: to represent high-pass layers 'sf
Refinement Pass
efficiently, dissect layers into small partitions by defining smooth windows WQ(x1, x2) localized around dyadic squares. This windowing is a nonnegative smooth function and creates ridges of width = 2-2s and length =2-s. Multiplication of 'sf with WQ produces a smooth dissection (hQ) of function into squares. The mathematical form is as follows:
Figure 1: The algorithmic flow of three basic SPIHT steps SPIHT encoding scheme has been selected in our implementation to perform compression with curvelets because of its following features [7]: - It supports multi-scale/multi-resolution encoding or decoding - It provides good quality image, means high PSNR value - Encoding and Decoding process is simple and fast
hQ wQ ' s f Renormalization: each square from previous stage is renormalized into the unit square.
Ridgelet Analysis: each normalized square analysed with discrete ridgelet transform that in turn produce curvelet coefficients.
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vector creation that eases the application of thresholding and quantization steps at these levels.
It is completely self-adaptive, so no training is required Efficient combination with error protection It can encode to exact bit-rate
4. Proposed Scheme The new approach is actually an integration of FDCvT with SPIHT algorithm. The block diagram in Figure 2 depicts the methodological detail of the encoding and decoding process of proposed approach.
(a)
(b)
(c)
Wrapping-Based Curvelet Transform
Image
Select Significant Scale(s)
(d) No
Contains Wedges?
(e)
Figure 3: Display of image curvelet coefficients within different scales
Yes
a) Original Peppers Image Threshold & Quantization within scale
b) Display Log of Curvelet Coefficients. Low frequency coefficients are stored at the center. The Cartesian concentric coronae show the coefficients at different scales; the outer coronae correspond to higher frequencies
Single Vector Creation
SPIHT Encoding Bitstream/header
c)
SPIHT Decoding Reconstructed Image
Display of Low frequency (coarse scale) coefficients
d) Display of single vector coefficients of 2nd level
Wrapping-Based Inverse Curvelet Transform
e) Figure 2: Block Diagram of Proposed Scheme
Display of coefficients
high
frequency
(fine
scale)
Final step in compression process deals with the application of SPIHT encoding scheme to these quantized coefficients that encodes it at variable bit rates. SPIHT algorithm here exploits the multi-scale nature of curvelet transform as with wavelet transform. For higher PSNR value, maximum bits have been assigned to levels having low-level details and lesser bits to levels containing highlevel details.
At first step, Frequency wrapping based FDCvT has been performed at different decomposition levels ranges from 1 to 5; provide curvelet coefficients at different scales/levels (terms used interchangeably). Display of curvelet coefficients within different scales, after the application of wrapping-based FDCvT on ‘Peppers’ image is given in Figure 3. The first and final scale at a specified decomposition level contains coefficients in simple array structure but other levels between these contain information in wedges. This transform application is actually a lossless step in compression process. Secondly, to attain maximum compression, scale level thresholding eliminate whole levels contain insignificant coefficients representing high-level details. Levels 1st and last at a given decomposition level (if take part in compression process) are simply thresholded and quantized as information is not present in wedges. Scales containing information in wedges require an extra step of single
5. Results and Discussion Four grayscale standard test images (i.e. Lena, Barbara, Hill and Peppers) of size 512 x 512 have been taken from World Wide Web for experiments and comparisons. MATLAB 7.0 has been used for the implementation of proposed approach and results have been conducted on Pentium-IV, 3.20 GHz processor with a memory of 512 MB. Different quality metrics i.e. Compression Ratio (CR), Compression Percentage (CP),
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Mean Square Error (MSE), and Peak Signal to Noise Ratio (PSNR) are evaluated to compile compression results. Results have been conducted at different combinations of selected parameters (i.e. decomposition level, maximum reconstruction bytes allocated, and encoding levels) named as Test Specifiers (i.e. SP1, SP2, SP3 etc.) as shown in Table1.
decomposition level decreases the compression ratio but improvement in PSNR is negligible as compared with compression results obtained without its participation. The comparisons of the proposed approach with prior well-known lossy compression techniques (i.e. SPECK, simple Wavelet and Curvelet Transform based compression) are given below. PSNR is the main objective metric selected to carry out comparisons in terms of compression ratio or compression percentage. The Image encoding scheme SPECK (Set Partitioned Embedded bloCK) is different from EZW, SPIHT etc. in a way that it does not use trees instead it uses sets in the form of different blocks. Table 3 presents the comparison between PSNR values of SPECK (taken from [12]) and proposed scheme (taken at 3rd decomposition levels including varied scales for encoding and variable reconstruction bytes allocation) for Lena image in terms of compression ratio.
Table 1: Different combination of selected parameters Parameter Name
Decomposition Level
Encoded Scale(s)
Max. Reconstruction Bytes allocated for compression
SP1
3
1
29241
SP2
3
1,2
104823
SP3
3
1,3
50540
SP4
3
1,2,3
126122
SP5
4
1,2,4
49721
Table 3: Comparison between PSNR of SPECK and Proposed Approach SPECK Compression Ratio 78.50 71.94 49.82 15.35 14.39 13.35
The quality metrics of proposed technique at defined parameters are provided in Table 2.
Table 2: Compression Results at selected parameters
Lena
Barbara
Gold Hill
Peppers
Parameters Selected
CP
CR
MSE
PSNR
SP1 SP2 SP3 SP4 SP5 SP1 SP2 SP3 SP4 SP5 SP1 SP2 SP3 SP4 SP5 SP1 SP2 SP3 SP4 SP5
88.84 60.01 80.72 51.88 81.03 88.84 60.01 80.72 51.88 81.03 88.84 60.01 80.72 51.88 81.03
8.96 2.50 5.18 2.07 5.27 8.96 2.50 5.18 2.07 5.27 8.96 2.50 5.18 2.07 5.27
30.11 20.60 29.32 17.29 38.33 49.56 37.04 48.33 33.61 54.79 40.72 30.12 39.04 24.62 48.03
33.34 35.00 33.45 35.75 32.29 31.17 32.44 31.28 32.86 30.74 32.03 33.34 32.21 34.21 31.31
88.84
8.96
30.27
33.32
60.01 80.72 51.88 81.03
2.50 5.18 2.07 5.27
22.76 29.60 20.13 37.49
34.55 33.41 35.09 32.39
25.51 25.66 25.31 32.12 32.25 32.34
It is clear from Table 3 that proposed approach presents better results than SPECK at low compression ratios. Comparison of wavelet compression with proposed scheme is shown graphically in Figure 4. Wavelet vs Proposed Wavelet
P S N R (d B )
Images
Proposed Compression PSNR Ratio 17.92 31.14 14.34 32.82 11.95 33.23 9.44 33.56 5.80 34.22 2.50 35.00
PSNR
From the contents of Table 2 it becomes clear that maximum compression results (high PSNR at high compression percentage) obtained by encoding scales 1 and 2 at given decomposition level, while remaining scales assumed to be uninvolved in compression process. Encoding process involving last scale at any particular
Proposed
40 35 30 25 20 15 10 5 0 0
10
20
30
40
50
Compression Ratio
Figure 4: Comparison wit2h Wavelet-Based Approach
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The comparison shows that proposed scheme gives better results than wavelet based approach. At low compression ratios, there is no major difference between PSNR values of both schemes, but moving towards higher compression ratios, proposed approach show good results. Comparison of proposed approach (taken at different decomposition levels and scales) with simple curvelet transform compression scheme (i.e. with entropy encoding) for ‘peppers image’ in terms of compression percentage is shown graphically in Figure 5.
Acknowledgements The authors would like to express their gratitude to curvelet.org team; Emmanuel Candes, Laurent Demanet, David Donoho, Lexing Ying for providing curvelet toolbox and Jing Tian for SPIHT toolbox.
References [1] Tian-Hu Yu, Zhihai He, and Sanjit K. Mitra, “Simple and Efficient Wavelet Image Compression”, IEEE Trans. Image Processing, vol.3, pp.174-177, Sep 2000.
Simple Curvelet vs Proposed Simple Curvelet
[2] Martin
Vetterli and Jelena Kovacevic, “Wavelets and Subband Coding,” Prentice Hall Signal Processing Series. Prentice Hall, Englewood Cliffs, New Jersey, 1995.
Proposed
PSNR(dB)
40 35
[3] J. M. Shapiro, “Embedded image coding using zerotrees of
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wavelet coefficients,” IEEE Trans. Signal Processing, vol. 41, pp. 3445-3462, Dec. 1993.
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[4] A. Islam and W. A. Pearlman, “An Embedded and Efficient
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Low-Complexity Hierarchical Image Coder,” Visual Communications and Image Processing 99, Proceedings of SPIE Vol.3653, pp.294-305, Jan 1999
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[5] A. Said and W. A. Pearlman, “A new fast and efficient image codec based on set partitioning in hierarchical trees,” IEEE Trans. Circuits Syst. Video Technol., vol. 6, pp. 243250, Jun 1996.
10 40
50
60
70 80 % Compression
90
100
110
[6] Jean-Luc Starck, Emmanuel J. Candès, and David L. Figure 5: Curvelets PSNR comparison with Proposed scheme
Donoho, “The Curvelet Transform for Image Denoising,”, IEEE Transactions on image processing,, Vol. 11, pp 670684 NO. 6, Jun 2002.
It is evident that more obvious difference between PSNR values found at higher compression percentage where proposed scheme shows much better results than compression achieved with simple curvelets.
[7] William Pearlman, “Set Partitioning in Hierarchical Trees”
6. Conclusion and Future Work
[9] David L.Donoho and Mark R. Duncan, “Digital Curvelet
[online] Available: http://www.cipr.rpi.edu/research/spiht/ ew_codes/spiht@jpeg2k_c97.pdf
[8] Candμes, E. and Donoho, D. (1999) Curvelets. Manuscript. Transform: Strategy, Implementation and Experiments”, Nov 1999.
This paper presents a technique for image compression that is actually integration of multi-scale curvelet transform and SPIHT encoding scheme. The proposed scheme is part of those early efforts that combine curvelets with an encoding scheme to achieve better results. It observes the supremacy of curvelets on wavelet based compression techniques as empirical results achieved with proposed scheme for lossy compression provide high PSNR at high compression ration than some of the previous approaches. The future work will be on integration of some other encoding techniques with curvelets to analyze the effect on compression ratio and PSNR of the input image.
[10] E. J. Cand`es and L. Demanet, “ The curvelet representation of wave propagators is optimally sparse,” Comm. Pure Appl. Math., 58-11 (2005) 1472–1528.
[11] Emmanuel Cand`es, Laurent Demanet, David Donoho and Lexing Ying. “Fast Discrete Curvelet Transform,” Jul 2005.
[12] R.Sudhakar, Ms R Karthiga, S.Jayaraman. “Image Compression using Coding of Wavelet Coefficients – A Survey” [online] Available: http://www.icgst.com/gvip/v6/ p1150511002.html
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