Index compressed tree-structured vector quantisation

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Abstract

This paper introduces a novel coding scheme based on Tree-Structured Vector Quantisation (TSVQ) scheme for image compression. The genealogical relationship among the indices of the neighbouring blocks generated from vector quantisation is exploited to improve the coding performance of TSVQ. The proposed coding scheme provides about 3.5 dB improvement over the basic TSVQ scheme and outperforms VQ schemes with memory and JPEG coding standard at low bit-rates. In addition its performance is comparable with address VQ but with much less complexity.

Introduction

Vector quantisation (VQ) is an attractive data compression technique because of its firm information-theoretic basis and decoder simplicity 1, 7, 10, 13. The application of conventional Full Search VQ in image compression suffers from encoder computational complexity when large codebooks are considered; a solution is to apply a tree-structured VQ (TSVQ). However, the performance of TSVQ at low bit-rates is poor because it does not take advantage of image block dependency. TSVQ with memory, such as predictive TSVQ and predictive pruned TSVQ, has been proposed to alleviate this problem at the expense of increased design complexity 3, 6, 9.

High correlation in natural images exhibits itself among the indices of vector quantised image blocks and this observation has led to the group of methods that effect an improvement of the coding performance of simple VQ schemes by exploiting the correlation of the image block indices. For example, there are coding schemes in which indices correlation have been exploited by lossless 5, 11, 12(address VQ) or lossy 14, 15compression strategies. Lossy index compression strategies 14, 15are based on predicting the indices and may result in very unpleasant discontinuties which can be easily noticed in the visually important areas of the image such as edges 14, 15. Lossless compression of the indices is a combination of a VQ scheme and a lossless index compression scheme. Therefore, such schemes retain the subjective quality of the encoded image when compared to the corresponding VQ scheme. In address VQ, the indices obtained from memoryless vector quantisation of the image are losslessly compressed by using special codebooks (address codebook). In the form in which address VQ was proposed, it enjoyed the simplicity of memoryless VQ but suffers from the complexity involved in the index compression [6].

The indices obtained by TSVQ have an interesting property that enables them to give information about the relationship between two quantised image blocks. If two image blocks are highly correlated, they may have an identical index, or the same ancestors; for example identical parents, or grand-parents depending on the quantisation levels. The existence of high inter-block correlation in natural images results in having neighbouring blocks with the same genealogy. In other words the neighbouring blocks of a TSVQ quantised image might have the same predecessor up to a particular stage of the codebook tree map. This characteristic can be used to compress the indices; if the indices of two neighbouring blocks belong to the same generation, the common part of their indices need not be transmitted.

This paper introduces a method, index compressed TSVQ (IC-TSVQ), to exploit the genealogical relation between the image block indices obtained from a TSVQ. IC-TSVQ is based on the fact that neighbouring blocks mostly belong to the same family, and there is no need to transmit or store the family identification for all of them. This means that IC-TSVQ partitions the image into groups of neighbouring blocks belonging to the same family on the basis of the TSVQ codebook.

TSVQ codebook can be considered as a union of a group of small TSVQ codebooks (subtree codebooks); the subtree codebooks require fewer bits to represent their members. If the indices of some neighbouring blocks are the children of a subtree, some bit saving can be achieved by indicating the sub-tree and transmitting or storing the common part of the indices. The amount of bit saving is about as much as the difference between the average rate of the original TSVQ tree and the subtree plus some extra overhead to represent the set of neighbouring blocks.

The image blocks from low activity image areas are highly correlated while those from high activity image areas are less correlated. Thus, using the proposed approach will lead to a variable rate bit assignment. The quantised versions of blocks from low activity areas are more likely to have an identical ancestry than the blocks from high activity areas. This means that the low activity blocks require a smaller subtree for their representation than the high activity blocks. The indices of blocks from low activity areas can be reconstructed with fewer bits when compared with the indices of blocks from high activity areas.

The basic idea of IC-TSVQ is similar to variable rate VQ, where the bits are allocated to blocks depending on their activity. The differences are the image adaptivity of IC-TSVQ, and the fact that the codebook of the VQ used in IC-TSVQ is fixed rate. The variable bit assignment procedure of IC-TSVQ is borne out of considering the characteristic of the image block indices in the encoding procedure, rather than designing a universal-based codebook variable rate VQ, such as entropy constrained VQ [4], pruned TSVQ [9]or greedy tree growing TSVQ [16]. IC-TSVQ first assigns bits uniformly, then this uniform bit allocation is changed into a variable one based on the characteristic of the image block location. A block located in a busy area requires more bits than a block located in a smooth area of the image. Of course, blocks located in active areas of natural images are normally active. Previous variable rate coders allocate bits to each image block based on its activity and disregard the characteristics of the neighbouring blocks 4, 9, 16, 17.

IC-TSVQ is in fact a combination of a memoryless VQ scheme with a lossless index compression suitable for the TSVQ scheme. Address VQ 5, 11, 12is a technique similar to IC-TSVQ from the lossless index compression viewpoint. The difference between IC-TSVQ and address VQ is the method of index compression. IC-TSVQ exploits the genealogical relationship of the indices, while address VQ exploits the indices dependency by finding a special form of pattern for neighbouring block indices from a pre-generated set of indices pattern (address codebook). The approach employed in address VQ gives opportunity to transmit one symbol instead of the indices of four or sixteen indices.

The main problems of address VQ are the high computational requirement of codebook generation, pattern of indices search procedure, and the amount of memory required to store the codebooks in comparison to Full Search VQ [6]. Nasrabadi and Feng [12]solved the computational complexity of the codebook generation, but their solution created other problems such as the synchronisation between the encoder and decoder, and the computational complexity of reordering the codebook at the receiver and transmitter during the encoding of each block [12]. In spite of all the enumerated problems, the high performance of address VQ, in terms of PSNR at low bit-rates, makes it feasible to apply VQ at low bit-rates. The PSNR of address VQ was quoted as 30.6 dB at 0.256 bpp for the image Lena [12].

The rest of this paper is organised as follows. Section 2presents the genealogical characteristics of indices, and Section 3introduces the method of index transmission or storage based on the indices' genealogical characteristics. Section 4gives the rate formulae for IC-TSVQ and methods to improve its compression ratio. Section 5introduces the extended version of IC-TSVQ for Full Search VQ, and Section 6presents the simulation results and discussion.

Section snippets

Characteristics of indices obtained from TSVQ

This section investigates the characteristics of the indices obtained from a quantised image by TSVQ at low bit-rates. It is shown that the high correlation among image pixels exhibits itself among the indices of the quantised image. An implication of this feature is the high probability of having neighbouring blocks with identical index or genealogy. This investigation has been carried out on two sample images displayed in Fig. 1Fig. 2; similar results are obtained with other test images.

Index transmission or storage

The new scheme for TSVQ requires transmitting two groups of information, a map that shows the genealogical relationship of the image block indices, and the bit required to show the genealogical differences. The map of genealogical relationship indicates the subtree to which the neighbouring blocks belong (in other words it gives the common part of the tree shared by neighbouring blocks). The genealogical differences are required to reconstruct the complete index of the image.

Rate of IC-TSVQ

The required information rate for IC-TSVQ consists of the information for the map of genealogical relationships and the genealogical differences. This section gives the required rate for each part.

IC-TSVQ based on Full Search VQ

The indices of images obtained by Full Search VQ do not have the genealogical information as described in TSVQ. The application of the new method in the case of Full Search VQ requires a method of giving characteristics similar to the indices obtained from TSVQ to those obtained from Full SearchVQ. In other words there is a need to give a tree shape to a Full Search VQ codebook, but use the Full Search VQ scheme to quantise the images.

In TSVQ, two children of a tree node are those, which have

Simulation results and discussion

This section presents the simulation results and discussion of the new scheme for TSVQ and Full Search VQ based on the virtual tree structure. The results are based on a 512×512 size image (Lena) and a block size of 4×4. The effects of the block size are discussed as well. The PSNR has been calculated by the following formula:PSNR=10log102552mse.The result for IC-TSVQ has been compared with TSVQ and TSVQ when the indices are Huffman encoded. The same comparison has been performed for Full

Conclusion

A novel method of image coding, IC-TSVQ, capable of providing significant results at low bit-rate is proposed. This method compresses the indices of quantised images based on the fact that neighbouring blocks, in natural images, are highly correlated, and this correlation exhibits itself among the indices of neighbouring blocks in a way that neighbouring blocks are mapped onto a small subset of the original VQ codebook; the size of the small codebook depends on the activity of the neighbouring

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