Multiple Description Lattice Vector Quantization Research Articles

Index Mapping for Bit-Error Resilient Multiple Description Lattice Vector Quantizer

In conventional multiple description coding (MDC), two descriptions of a source are generated and sent over ON/OFF channels. In this paper, we are interested in exploiting the redundancy built in MDC to additionally confer robustness against other channel errors. In particular, we consider a multiple description lattice vector quantizer (MDLVQ) whose output (a pair of side lattice points) is mapped to a pair of binary indexes and each index is sent over a binary channel. One channel is noiseless, while the other is noisy. Thus, at the decoder, one description is received error-free, while the other may carry bit errors. Then the decoder uses the error-free description as side information to improve the reconstruction. The effectiveness of the decoder in alleviating the impact of bit errors depends on the mapping $\gamma $ of side lattice points to binary indexes. We propose the design of a structured bit-error resilient mapping $\gamma $ . For this, the set of side lattice points is first partitioned using Voronoi regions of an appropriate coarse lattice. Next a good linear channel code is selected, each Voronoi region is assigned a coset of this channel code, and the side lattice points within each Voronoi region are mapped to binary sequences in the corresponding coset. In addition, we argue that the performance of $\gamma $ is improved by assigning cosets close in Hamming distance to neighboring Voronoi regions, and propose a technique to achieve this goal. We derive a lower bound on the error correction performance of the proposed mapping $\gamma $ in terms of the performance of the channel code $C$ used in its construction. Interestingly, we prove that, as the rate of the MDLVQ grows to infinity, the mapping $\gamma $ becomes as good as the code $C$ in correcting bit errors. Simulation results show the significant superiority of the proposed index mapping versus random mappings.

On the Design of Symmetric Entropy-constrained Multiple Description Scalar Quantizer with Linear Joint Decoders

This paper addresses the design of symmetric entropy-constrained multiple description scalar quantizers (EC-MDSQ) with linear joint decoders, i.e., where some of the decoders compute the reconstruction by averaging the reconstructions of individual descriptions. Thus, the use of linear decoders reduces the space complexity at the decoder since only a subset of the codebooks needs to be stored. The proposed design algorithm locally minimizes the Lagrangian, which is a weighted sum of the expected distortion and of the side quantizers’ rates. The algorithm is inspired by the EC-MDSQ design algorithm proposed by Vaishampayan and Domaszewicz, and it is adapted from two to $K$ descriptions. Differently from the aforementioned work, the optimization of the reconstruction values can no longer be performed separately at the decoder optimization step. Interestingly, we show that the problem is a convex quadratic optimization problem, which can be efficiently solved. Moreover, the generalization of the encoder optimization step from two to $K$ descriptions increases drastically the amount of computations. We show how to exploit the special form of the cost function conferred by the linear joint decoders to significantly reduce the time complexity at this step. We compare the performance of the proposed design with multiple description lattice vector quantizers (MDLVQ) and with the multiple description scheme based on successive refinement and unequal erasure protection (UEP). Our experiments show that the proposed approach outperforms MDLVQ with dimension 1 quantization, as expected. Additionally, when more codebooks are added our scheme even beats MDLVQ with quantization dimension approaching $\infty $ , for rates sufficiently high. Furthermore, the proposed approach is also superior to UEP with dimension 1 quantization when the rates are low.

Optimized Multiple Description Lattice Vector Quantization Coding for 3D Depth Image

Multiple Description (MD) coding is a promising alternative for the robust transmission of information over error-prone channels. Lattice vector quantization (LVQ) is a significant version of MD techniques to design an MD image coder. However, different from the traditional 2D texture image, the 3D depth image has its own special characteristics, which should be taken into account for efficient compression. In this paper, an optimized MDLVQ scheme is proposed in view of the characteristics of 3D depth image. First, due to the sparsity of depth image, the image blocks can be classified into edge blocks and smooth blocks, which are encoded by different modes. Furthermore, according to the boundary contents in edge blocks, the step size of LVQ can be regulated adaptively for each block. Experimental results validate the effectiveness of the proposed scheme, which show better rate distortion performance compared with the conventional MDLVQ.

Flexible Symmetric Multiple Description Lattice Vector Quantizer With <inline-formula> <tex-math notation="TeX">$L\geq 3$</tex-math></inline-formula> Descriptions

In the previous work on multiple description lattice vector quantizers (MDLVQs) with L ≥ 3 descriptions, once the central and side lattice codebooks are fixed, the decoding quality is determined for all numbers k of received descriptions. Therefore, it is not possible to achieve tradeoffs between the quality of reconstruction for different values of k, 1 k ≤ L - 1. This paper proposes a flexible MDLVQ capable of overcoming the above drawback. For this, a different reconstruction method is employed and a heuristic index assignment (IA) algorithm, which uses L - 2 parameters to control the distortions for 2 k ≤ L - 1, is developed. Experimental results show that the proposed MDLVQ, in addition to achieving the desired tradeoffs, significantly outperforms the classic MD scheme based on unequal erasure protection. The second contribution of this paper is a structured IA for the case of L = 3 and the derivation of the corresponding expressions of the distortions at high resolution. The proposed IA has a simple mechanism for controlling the tradeoff between the reconstruction quality for k = 1, 2. The IA is able to achieve a wide range of distortion values, while keeping the product of the distortions for k = 1, 2 the same as in the prior work.

Asymptotical Analysis of Several Multiple Description Scenarios with L≥ 3 Descriptions

This work compares theoretically the performance of several representative practical multiple description (MD) frameworks with L≥ 3 symmetric descriptions. The first scenario is the classic unequal erasure protection (UEP) scheme using a successively refinable code (SRC) and Reed-Solomon codes. The second scenario is an improvement upon UEP by applying domain partitioning and permuted Reed-Solomon codes. The third scenario uses a finer partitioning and erasure correction via repetition codes. Additionally, the MD lattice vector quantizer and another recent MD scheme are considered in the comparison. The aforementioned MD schemes are compared in terms of the expected squared error asymptotically achievable as the rate R of a description approaches ∞, assuming independent description losses. Our analysis reveals that the improvement of the second scenario upon the first one when R→ ∞ can reach up to 1.68 dB, but it approaches 0 as the description loss rate p goes to 0 and L approaches ∞. Additionally, we find that the first two schemes outperform the third one with an unbounded gain when R→ ∞, as p→ 0 or L→ ∞. Further, we show that the first three scenarios achieve unbounded improvements over the other two as R→ ∞ and p→ 0. On the other hand, we point out that some of the results of our asymptotic analysis rely on strong assumptions and therefore an experimental validation is needed before applying them to practical situations.

Multiple description lattice vector quantization using multiple A4 quantizers

This paper presents a new Multiple Description Lattice Vector Quantization (MDLVQ) based on A4 lattice quantizers for multiple description coding abbreviated as MDLVQ-A4. The Coinciding similar A4 sublattices are used to quantize the input stream for multiple description coding system. The use of multiple coinciding quantizers eliminates the labeling function of the traditional schemes. Experimental results of MDLVQ-A4 scheme for image coding show a higher performance in terms of PSNR of the side decoders as compared to the renowned techniques of image coding for several test images.

High-Rate Analysis of Symmetric L-Channel Multiple Description Coding

This paper studies the tight rate-distortion bound for L-channel symmetric multiple-description coding of a scalar Gaussian source with two levels of receivers. Each of the first-level receivers obtains κ of the L descriptions (κ <; L). The second-level receiver obtains all L descriptions. We find that if the central distortion (corresponding to the second-level receiver) is much smaller than the side distortion (corresponding to the first-level receivers), the product of a function of the side distortions and the central distortion is asymptotically independent of the redundancy between the descriptions. Using this property, we analyze the asymptotic behavior of a practical multiple-description lattice vector quantizer (MDLVQ). Our analysis includes the treatment of the MDLVQ system from a new geometric viewpoint, which results in an expression for the side distortions using the normalized second moment of a sphere of higher dimensionality than the quantization space. The expression of the distortion product derived from the lower bound is then applied as a criterion to assess the performance loss of the considered MDLVQ system. In principle, the efficiency of other practical MD systems can also be evaluated using the derived distortion product.

$M$-Description Lattice Vector Quantization: Index Assignment and Analysis

In this paper, we investigate the design of symmetric entropy-constrained multiple description lattice vector quantization (MDLVQ), more specifically, MDLVQ index assignment. We consider a fine lattice containing clean similar sublattices with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> -similarity. Due to the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> -similarity of the sublattices, an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -fraction lattice can be used to regularly partition the fine lattice with smaller Voronoi cells than a sublattice does. With the partition, the MDLVQ index assignment design can be translated into a transportation problem in operations research. Both greedy and general algorithms are developed to pursue optimality of the index assignment. Under high-resolution assumption, we compare the proposed schemes with other relevant techniques in terms of optimality and complexity. Following our index assignment design, we also obtain an asymptotical close-form expression of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> -description side distortion. Simulation results on coding different sources of Gaussian, speech and image are presented to validate the effectiveness of the proposed schemes.

Index assignment for 3-description lattice vector quantization based on A2 lattice

Multiple description lattice vector quantization (MDLVQ) is an effective technique to realize multiple description coding. In this paper, we investigate a 3-description lattice vector quantization (LVQ) design using A 2 lattice, which finds a 3-tuple combination of sublattice points to represent each fine lattice point, based on two principles that each edge of the triangle formed by the 3-tuple points needs to be as short as possible, and the gravity center of the triangle is as close as possible to the fine lattice point. Following a delicate sublattice partition, a well-designed construction and mapping of each fine lattice point to a 3-tuple of sublattice points is developed to minimize the side distortion. The proposed index assignment is shown to achieve smaller distortion (up to 0.58 dB for Gaussian source) than some other 3-description index assignments. Compared with a 2-description LVQ, the 3-description LVQ achieves better expected rate-distortion performance in most cases of high description loss rates, while it also exhibits more graceful coding results with significantly smaller side distortions. Numerical analysis on Gaussian source and the simulations on image source validate the effectiveness of our proposed index assignment scheme.

Optimized Multiple Description Lattice Vector Quantization for Wavelet Image Coding

Multiple description (MD) coding is a promising alternative for robust transmission of information over non-prioritized and unpredictable networks. In this paper, an effective MD image coding scheme is introduced based on the MD lattice vector quantization (MDLVQ) for the wavelet transformed images. In view of the characteristics of wavelet coefficients in different frequency subbands, MDLVQ is applied in an optimized way, including an appropriate construction of wavelet coefficient vectors, the optimization of MDLVQ encoding parameters such as the choice of sublattice index values and the quantization accuracy for different subbands. More importantly, optimized side decoding is employed to predict lost information based on inter-vector correlation and an alternative transmission way for further reducing side distortion. Experimental results validate the effectiveness of the proposed scheme with better performance than some other tested MD image codecs including that based on optimized MD scalar quantization.

N-channel entropy-constrained multiple-description lattice vector quantization

In this paper we derive analytical expressions for the central and side quantizers which, under high-resolutions assumptions, minimize the expected distortion of a symmetric multiple-description lattice vector quantization (MD-LVQ) system subject to entropy constraints on the side descriptions for given packet-loss probabilities. We consider a special case of the general n-channel symmetric multiple-description problem where only a single parameter controls the redundancy tradeoffs between the central and the side distortions. Previous work on two-channel MD-LVQ showed that the distortions of the side quantizers can be expressed through the normalized second moment of a sphere. We show here that this is also the case for three-channel MD-LVQ. Furthermore, we conjecture that this is true for the general n-channel MD-LVQ. For given source, target rate and packet-loss probabilities we find the optimal number of descriptions and construct the MD-LVQ system that minimizes the expected distortion. We verify theoretical expressions by numerical simulations and show in a practical setup that significant performance improvements can be achieved over state-of-the-art two-channel MD-LVQ by using three-channel MD-LVQ.

Multiple description vector quantization with a coarse lattice

A multiple description (MD) lattice vector quantization technique for two descriptions was previously introduced in which fine and coarse codebooks are both lattices. The encoding begins with quantization to the nearest point in the fine lattice. This encoding is an inherent optimization for the decoder that receives both descriptions; performance can be improved with little increase in complexity by considering all decoders in the initial encoding step. The altered encoding relies only on the symmetries of the coarse lattice. This allows us to further improve performance without a significant increase in complexity by replacing the fine lattice codebook with a nonlattice codebook that respects many of the symmetries of the coarse lattice. Examples constructed with the two-dimensional (2-D) hexagonal lattice demonstrate large improvement over time sharing between previously known quantizers.

Asymmetric multiple description lattice vector quantizers

We consider the design of asymmetric multiple description lattice quantizers that cover the entire spectrum of the distortion profile, ranging from symmetric or balanced to successively refinable. We present a solution to a labeling problem, which is an important part of the construction, along with a general design procedure. The high-rate asymptotic performance of the quantizer is also studied. We evaluate the rate-distortion performance of the quantizer and compare it to known information-theoretic bounds. The high-rate asymptotic analysis is compared to the performance of the quantizer.