A Low Complexity and Efficient Rate Constrained Wavelet ... - CiteSeerX

EBWIC: A LOW COMPLEXITY AND EFFICIENT RATE CONSTRAINED WAVELET IMAGE CODER Christophe Parisot, Marc Antonini, Michel Barlaud I3S laboratory UPRES-A 6070 of CNRS, University of Nice-Sophia Antipolis Bˆat. Algorithmes/Euclide, 2000 route des Lucioles - BP 121 - 06903 Sophia Antipolis Cedex, France [email protected] ABSTRACT Efficient compression algorithms generally use wavelet transforms. They try to exploit all the signal dependencies that can appear inside and across the different sub-bands of the decomposition. This provides highly complex algorithms that can’t generally be implemented for real-time purposes. However, efficiency of a coding scheme highly depends on bit allocation. In this paper, we present a new wavelet based image coder EBWIC (Efficient Bit allocation Wavelet Image Coder). This algorithm is based on an accurate modelisation of the distorsion-rate curve even at low bit rate. It results that the efficiency of our method is very close to JPEG 2000 with a very low complexity and possible parallelization of the encoding process. The method proposed below has a complexity less than 60 arithmetic operations per pixel (against about 300 for JPEG 2000). Our method can be applied whatever the wavelet transform (quincunx, dyadic...) and the entropy coder are. It can be used for strip-based processing.

2. GLOBAL COMPRESSION SCHEME The global compression scheme that we propose is based on a wavelet transform, a bit allocation to determine the best quantization steps for each sub-band of the wavelet transform, a simple scalar quantization and an entropy coder for each sub-band. This very simple scheme is drawn in Fig.1. Notice that an important aspect of our approach is that the encoding of the sub-band is done separately and can be performed simultaneously on separate ASICs. Because of the simplicity of our coding scheme, we’ll have to compute a very precise bit allocation to get high efficiency.

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1. INTRODUCTION

Fig. 1. Global compression scheme

The whole community has became aware of the economic stakes of compression and works a lot on this problem. In this competition, algorithms become more and more efficient and we can imagine that the last results are no-longer so far from the Shannon theoretical bound. But methods such as SPIHT [1] which was a reference for a long time or the future JPEG 20001 [2], [3] algorithm are very expensive and then, not appropriate for real-time implementation purposes. We propose a new compression scheme which offers both very low complexity and high performance. Our method is well-suited for parallel systems in the sense that subbands of the wavelet transform are quantized and encoded separately after the bit allocation has been computed.

Let’s now describe more precisely each of the components of this compression scheme.

1 Our simulations use the VM7.2 Verification Model of the emerging JPEG 2000 standard This work was supported by CNES (French Space Agency) grant 1/96/CNES/96/0761

2.1. Wavelet transform The aim of the wavelet transform is to decorrelate the source to eliminate its redundancies. For this purpose, we have selected the 9-7 biorthogonal filters [4]. This filter bank is known to be quasi-orthogonal and gives the best results for dyadic sampled images [5]. We use a three level Mallat’s decomposition and apply a one-level dyadic decomposition to the three highest frequency sub-bands. The algorithm is also well-suited for quincunx wavelet transforms [6]. 2.2. Scalar quantization The quantization is performed by a UTQ [7]. Uniform threshold quantizers complexity is very low and those quantiz-

ers are almost optimal for laplacian sources. The complexity cost is reported at the decoding process. The decoded wavelet coefficient is the centro¨ıd of its quantization bin according to the probability density function model of the sub-band considered.

2.3. Entropy coder We assume that wavelet coefficients can be assimilated with a memory-less source if the sub-band is scanned perpendicularly to the edges it represents. Sub-bands of vertical contours are scanned horizontally and sub-bands of horizontal contours are scanned vertically. Diagonal coefficients are arbitrarily scanned horizontally. We want to design a coder with output bit rate as close as possible from the entropy of the quantized coefficients. We suggest to use Run-Length coding followed by an entropy coding of the (run, length) symbols. One can experimentally show that the stability of Huffman tables for the Run-Length coding is high among variations of the probability density function model of the wavelet coefficients. It is then possible in some specific applications to compute and store a library of Huffman tables for future images. We can show that the theoretical bit rate estimated for a RunLength coder followed by an entropy coder is very close to the entropy of the quantized coefficients (Fig.2). Arithmetic coding should have about the same efficiency but is more expensive in computation complexity.

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2.4. Bit allocation This is the crucial point of the method. Our compression scheme does not take into account the correlation that would exist between wavelet coefficients if sub-bands were scanned following the contours. Since sub-bands are processed separately, our method does not use the correlations that exist between the sub-bands of different decomposition levels. The following section shows how to get high performance bit allocation with low complexity. 3. BIT ALLOCATION 3.1. General purpose The purpose of the bit
allocation is to determine the best set  of quantization steps that minimize the average mean squared error at a given bit rate [8]. This constrained problem can be written in the form of Lagrangian operators:     ! 
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where " # and are respectively the mean  squared  error and 5476 #983:; (resp.  the bit rate for the ith sub-band. *   # ) is the quotient of the size of the sub-band divided * by the size of the whole image for a dyadic (resp. quincunx) wavelet  decomposition. = ? is the number of sub-bands and  an optional weight for frequency selection. In high performance compression methods such as SPIHT [1] or EBCOT [3], the bit allocation process can provide different quantization steps in the same sub-band. In the Said and Pearlman algorithm this possibility comes directly from the hierarchical coding tree whereas this possibility comes from the block splitting of sub-bands in EBCOT. This possibility increases the efficiency of the bit allocation because it uses the spatial information of the image in frequency subbands. On the other hand, in both cases, the set of possible quantization steps is finite. Discontinuities in the quantization step between different blocks can generate important block artifacts and other undesirable artifacts on edges that can prevent from a reliable photo-interpretation. For this reasons, we chose to have only one quantization step in each frequency sub-band.

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Fig. 2. Bit rate for a memoryless laplacian source. Solid line: entropy. Dashed line: Run-Length coding

The only way to compute a very efficient bit allocation without pre-quantizing sub-bands is to accurately model the disBA  tribution @ of the wavelet coefficients and use theoretical models for distorsion and bit rate. This distribution can be well-approximated with generalized gaussians [4]. We have @

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   Fig. 3. Bit M L rate   for generalized gaussians with shape paramN O eters K 



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with P QR SUT VXW Y and * H VXW[Y . SZT SUT To
compute the bit rate produced by the quantization step , we need the probability of the quantization level \ .   _/`bac24d eA gf A We have ] ^ \ . `ba 4d @ F Then, the bit rate is obtained with the formula 

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Yex ds 4d  p'r ud t 4d A q T I I . The MSE is then I w v y Z z Y tization level \ is o r ` T ckj { `bac 4d    BA . A  BA g f A " o @  `ba 4d ` j F F

We will use these functions for accurate bit allocation. 3.3. Fast and optimal bit allocation For generalized gaussian distributions, we notice that (1) can be rewritten as


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The algorithm requires tables of several ˆ‹ and functions for different „UXŒ generalized gaussian shape parameters K  . Other values are approximated sampled in  , ˆby Š linear interpolations. Fig.3 and Fig.4 show the and ˆ‹ functions for different generalized gaussians. We obtain very accurate interpolations with only ten points per tabulated function. This implies very low complexity and distorsionrate curves supposed to be as accurate as those used in [9]. The bit allocation process is the following: 1. Lambda 2 , is given. For each sub-band, compute the ˆ$Š bit rate that verifies (3) using the tabulated ˆ‹ functions 2. If the constraint (4) is not verified, compute a new lambda by dichotomy and go to 1 else go to 3 3. For each sub-band, compute , rate functions




using the tabulated bit

Our simulations show that there are generally less than ten iterations before the convergence. Furthermore one can certainly divide this number by two with a rough estimation of lambda for the initialization or in strip-based implementations.

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Fig. 5. PSNR performance for a typical urban satellite image provided by the French Space Agency (CNES)

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Fig. 6. PSNR performance for the JPEG 2000 test image Goldhill 6. REFERENCES

4. COMPLEXITY AND EXPERIMENTAL RESULTS The evaluation of the generalized gaussian parameters costs 4 arithmetic operations per pixel (computation of the 2nd and 4th moment). The bit allocation cost is about 15 arithmetic operations per sub-band and per iteration (only linear interpolations). The 1500 arithmetic operations needed for the convergence are negligible according to the number of pixels of the whole image. We conclude that the bit allocation complexity is less than 5 arithmetic operations per pixel. The global complexity of the coder (including wavelet transform, bit allocation and quantization) is less than 60 arithmetic operations per pixel. Fig.5 and Fig.6 compare PSNR performances of our coder with JPEG 2000. EBWIC reaches almost the same performances than JPEG 2000 and is sometimes better.

5. CONCLUSION The efficiency of EBWIC comes from the bit allocation, that uses exact theoretical models for distorsion-rate curves. This accurate bit allocation allows EBWIC to reach JPEG 2000 performances. However, the complexity of EBWIC is about five times lower than JPEG 2000 and the design of strip-based implementation is easy [10]. The generalization of EBWIC for coders of higher entropy will provide even better results.

[1] A. Said W.A. Pearlman, “A new and efficient image codec based on set partitioning in hierarchical trees,” IEEE Trans. on Circuits and Systems for Video Technology, vol. 6, June 1992. [2] JPEG 2000 project, “Jpeg 2000 verification model 7.2,” http://www.jpeg.org, 2000. [3] D. Taubman, “High performance scalable image compression with ebcot,” Submitted to IEEE Trans. on Image Processing, March, Revised August, 1999. [4] M. Antonini M. Barlaud P. Mathieu I. Daubechies, “Image coding using wavelet transform,” IEEE Trans. on Image Processing, vol. 1, no. 2, pp. 205–220, 1992. [5] J.D. Villasenor B. Bellzer B. Liao, “Wavelet filter evaluation for image compression,” IEEE Trans. on Image Processing, vol. 4, no. 8, pp. 1053–1060, Aug. 1995. [6] A. Gouze M. Antonini M. Barlaud, “Quincunx filtering scheme for image coding,” VCIP, Jan. 1999, San Jose USA. [7] G.J. Sullivan, “Efficient scalar quantization of exponential and laplacian random variables,” IEEE Trans. on Information Theory, vol. 42, pp. 1365–1374, Sept. 1996. [8] K. Ramchandran M. Vetterli, “Best wavelet packet bases in a rate-distorsion sense,” IEEE Trans. on Image Processing, vol. 1, no. 2, pp. 160–176, Apr. 1993. [9] J. H. Kasner M. W. Marcellin B. R. Hunt, “Universal trellis coded quantization,” IEEE Trans. on Image Processing, vol. 8, no. 12, pp. 1677–1687, Dec. 1999. [10] C. Parisot M. Antonini M. Barlaud C. Lambert-Nebout C. Latry G. Moury, “On board strip-based wavelet image coding for future space remote sensing missions,” IGARSS, July 2000, Honolulu, Hawaii.