Large Block Entropy-Constrained Reflected Residual Vector Quantization

Mohammad Asmat Ullah Khan Department of Electrical Engineering, COMSATS Institute of Information Technology, Islamabad, Pakistan.

mohammadalhan @ yahoo.com

Absrracr - Multispectral imagery and video coding appli- cations benefit from the use of large vector sizes. Other applications also require large vector sizes such as variable dimension Vector Quantizers (VQ) and transform VQ. Entropy- constrained reflected residual vector quantization (EC-RRVQ) is an algorithm that is used to desgin codebooks for image cod- ing with large vector sizes in addtion to high output rate while mainting a very low complexity in terns of computations and memory requirements. EC-RRVQ has several advantages which are important. It can outperform entropy-constrained resid- ual vector quanization (EC-RVQ) in terms of rate-distortion performance, encoder complexity compuatations, and memory. Experimental results indicate that good image repmduction quality can be accomplished at relatively low hit rates. For example, a peak signal-to-noise ratio of 29 dB is obtained for the 512 x 512 image LENA at a bit rate of 0.2 bpp with dimension of 16 x 16.

I . INTRODUCTION

Vector qunatization (VQ) provides superior performance over the scalar quantization(SQ) due to its ability to exploit linear and nonlinear dependence among the vector components and the extra freedom in choosing the multidimensional quantizer cell shapes. These advantages are discusses in depth from from a more quantitative point of view using high rate quantizer theory in Lookabaugh et al.[l]. The authors provided relative gains of VQ over SQ in terms of vector dimension and had shown that larger the domension of a vector will be larger will he the gains provided by the VQ. However, the direct use of VQ on pratical sources suffers from serious complexity barriers and is limited to rather modest vector dimensions.Practica1 Entropy-constrained VQs (EC-VQ) are limited to sizes of 4 x 4. Also, experimental results for both EC-VQ and entropy-purned tree-structured VQ (EPTSVQ) with 8 x 8 do not exist 121. In [3] authors utilized residual vector quantization (RVQ) as large dimensionall block vector quantization for image coding.

Residual vector quantization (RVQ) is a vector quantization (VQ) paradigm which imposes multistage structure in order to reduce the encoding search burden and large codebook memory demands. Due to these attractive properties RVQ systems based on large vector sizes ( such as 8x8, 16x16, and 32x32) were designed. Experiments were performed in [3], and results show that significant gains

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Fig. 1. t a t image LENA at increasing values of rn. The YCC~DC size is 8 x 8.

Rate-distortion perfamancc of EC-RRVQ with 32 stages for the

can be obtained by increasing the vector size. However, they employed multipath (M-search) encoders to reduce encoding burden for large block RVQ systems. An M-search encoder has the disadvantage that it increases the encoder computations by a factor of M. Recently, a reflected RVQ (RRVQ) has been introduced as an alternative to RVQ scheme. Reflected RVQ employs single-path search and has provided better rate-distortion performance over that of RVQ [41,[51.

RRVQ is a multistage structure with binary stage codebooks. The encoder and decoder of an RRVQ performs reflecting operations on the residual vectors between the residual stages. The structure imposes symmetry which makes the single path search of stage codebooks optimal. The direct sum codebook, which is the set of all possible stage codevector sums, of the RRVQ inherits a more ordered structure than that of RVQ direct sum codebook [41 which, in consequence, makes the output entropy of the RRVQ lower compared to the RVQ output entropy [4]. The entropy-constrained RRVQ (EC-RRVQ) maintains a better rate-distortion R ( D ) performance compared to the entropy-constrained RVQ (EC-RVQ) for both synthetic and natural image data [4],[51. In addition, the encoder complexity of the EC-RRVQ depends only on the number of stages [61,[41 rather than on stage codebook size as was the case for EC-UVQ.

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11. ENTROPY-CONSTRAINED REFLECTED RESIDUAL VECTOR QUANTIZATION

As reported in [4],[5], the goal of the EC-RRVQ algorithm is to seek an RRVQ codebook that minimizes the average distortion subject to a constraint on the output entropy of the RRVQ. This is done by iteratively minimizing the Lagrangian given by

where z1 and y(j) are realizations of the source (image pixels) and output (direct sum codevectors) respectively, d(zl,y(j)) is a distortion measure which is assumed here to be the mean-square-error, L ( j ) represents the length associated with a direct sum codevector, and X is the slope of the line supporting the convex hull of the operational R(D) curve. The algorithm starts by designing a fixed codeword lengths RRVQ. i.e. X = 0. Then, using a predetermined sequence of A's, the algorithm designs a set of locally optimal EC-RRVQ codebooks with various bit rates. A more detailed information about the EC-RRVQ can he found in

The results reported in [4],[5] indicates that, for synthetic and image sources, EC-RRVQ outperforms the EC-RVQ in

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Fig. 4. stages for lhe test image LENA at m = 1. The vector size is 16 X 16.

Rate-distortion pedormance of EC-RRVQ and EC-RVQ wilh 64

terms of performance, encoder complexity while maintain the same memory requirements.

111. LARGE BLOCK EC-RRVQ EXPERIMENTS Experimental results are used to study the effect of Markov

&del order, an efficient and useful way of reducing entropy- tables, and comparisons of the performance of EC-RRVQ with those of EC-RVQ. The test image used in the experi- ments was the famous LENA image which is excloded from the training sets. The objective performance measure used in all experiments is the peak signal-to-quantization noise ratio (PSNR). The PSNR is defined as

where N x N is the size of the image and z(i,j) and P ( i , j ) represent the original and coded values, respectively, at the ith row and the j t h coulmn.

A. Blocks of size 8 x 8 The training set for an 8 x 8 vector dimension contained no

more than 500,000 vectors. 32 fixed rate RRVQ stages were designed giving 0.5 bpp as a peak bit rate. The experiments were performed for obtaining satisfactory performance as a function of Markov model order. Fig. 1 shows that for rates between 0.25 and 0.4 bpp, there exits a difference of approximately 1 dB on average between EC-RRVQ with m = 0 and rn = 1. Also, for the same range of rates, ihe difference between EC-RRVQ with m = 1 and m = 2 is roughly 0 - 0.2 dB which indicates that first order Markov model is providing satisfactory rate-distortion performance. In contrast, for rates less than 0.25 bpp, all curves provide similar performance. Therefore, for low bit rates there is no advantage of using Markov orders grater than 0 while for rates grater than 0.25 bpp, m = 1 will be sufficient.

A comparison in Fig. 2 is made between EC-RRVQ and EC-RVQ for the same set of data and dimension of 8 x 8 for m = 1. For rates grater than 0.3 bpp, the EC- RRVQ is outperforming the EC-RVQ by an amount of 1

Fig. 5. Image LENA cded using (a) EC-RVQ at a bit rate of 0.179 bpp with PSNR of 28.03 dB (b) EC-RRVQ at a bit rate of 0.177 bpp with PSNR of 28.15 dB both of dimension 8 x 8 and m = 1.

Fig. 6. Image LENA coded using (a) EC-RVQ at a bit rate of 0.215 bpp with PSNR of 28.39 dB (b) EC-RRVQ at a bit rate of 0.201 bpp with PSNR of 29 dB both of dimension 16 x 16 and m = 1.

dB. In contrast, for rates less than 0.3 bpp, the two curves become almost identical. For subjective comparison, Fig. 5 is provided. It shows the test image LENA coded using (a) EC-RVQ at a bit rate of 0.179 bpp with PSNR of 28.03 dB (b) EC-RRVQ at a bit rate of 0.177 bpp with PSNR of 28.15 dB both of dimension 8 x 8 and m = 1.

E. Blocks ofsize 16 x 16

For the 16 x 16 vector dimension, the training set was composed of 350,000 vectors. The same work as in the preivous subsection is done with vectors of size 16 x 16. For the Markov model order experiment, the peak bit rate was 0.25 bpp meaning 64 fixed rate stages were designed. Fig. 3 shows that for rates between 0.1 and 0.2 bpp, there exits a difference of approximately 0.5 dB on average between EC- RRVQ with m = 0 and m = 1. Also, for the range of rates between 0.18 and 0.2, the difference between EC-RRVQ with

m = 1 and m = 2 is 0.2 dB. In contrast, for rates less than 0.1 bpp, all curves provide similar performance.

A comparison is made between EC-RRVQ and EC-RVQ for the same set of data and dimension of 16 x 16 f o r m = 1. It can be seen from Fig. 4 that for rates grater than 0.12 bpp, the gap between the EC-RRVQ and the EC-RVQ curves is about 0.6 dB on average while for rates less than 0.12 bpp, the two curves will join. For subjective comparison, Fig. 6 is provided. It shows the test image LENA coded using (a) EC-RVQ at a bit rate of 0.215 bpp with PSNR of 28.39 dB (b) EC-RRVQ at a hit rate of 0.201 bpp with PSNR of 29 dB both of dimension 16 x 16 and m = 1.

IV. CONCLUSIONS

In symmary, the EC-RRVQ was used for designing code- books with large vector sizes such as 8 x 8 and 16 x 16. In veiw of the obtained results, EC-RRVQ gave compatitive

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results compared EC-RVQ in terms of rate-distortion per- formance in addtition to encoder computations. The main disadvantage of EC-RRVQ is that it requires large training set vectors to a acheive a good rate-distortion performance. This is due to the fact that the RRVQ does not allow more than two codevectors per stage. Therefore, we are in need of large number of stages for large bit rates.

REFERENCES

[ 11 T.D. Lookabaugh and R.M. Gray, “High-resolution qunatization theory and the vector quantizer advantage,” IEEE Trans. on Information theory, vol. 35, no. 5. pp. 1020-1032, September 1989.

[2] E Kossentini, M.J.T. Smith, and C.F. Barnes, “Image coding using entropy-constarined residual vector quanti- zation,” IEEE transactions on image processing, vol. 4, no. 10, pp. 1349-1356, October 1995.

[31 E Kossentini, M. I. T. Smith, and C. E Barnes, “Large block rvq with multipath searching,” in Proceedings of IEEE Int. Con$ on Acoustics, Speech, and Signal Pmcessing (ICASSP), 1992.

[4] W. A. H. Mousa and M. A. U. Khan, “Design and analysis of entropy-constrained reflected residual vector quantization,” Proceedings of E E E lnt. Con$ on Acaus- tics, Speech, and Signal Processing (ICASSP), vol. 111. pp. 2529-2532, May 2002.

[SI M. A. U. Khan and W. A. H. Mousa, “Image cod- ing using entropy-constrained reflected residual vector quantization,’’ Proceedings of IEEE Int. Con$ on Image Pmcessing ( K I P ) , vol. I, pp. 253-256, Sept. 2002.

[6] C. E Barnes, Residual Quantizers, Ph.D. thesis, Brigham Young University, Provo, Utah, 1989.

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