Multiple-Description Coding by Dithered Delta-Sigma Quantization

Jan Ostergaard, Ram Zamir
2007 2007 Data Compression Conference (DCC'07)  
We address the connection between the multiple-description (MD) problem and Delta-Sigma quantization. The inherent redundancy due to oversampling in Delta-Sigma quantization, and the simple linearadditive noise model resulting from dithered lattice quantization, allow us to construct a symmetric MD coding scheme. We show that the use of a noise shaping filter makes it possible to trade off central distortion for side distortion. Asymptotically as the dimension of the lattice vector quantizer
more » ... order of the noise shaping filter approach infinity, the entropy rate of the dithered Delta-Sigma quantization scheme approaches the symmetric two-channel MD rate-distortion function for a memoryless Gaussian source and MSE fidelity criterion, at any side-to-central distortion ratio and any resolution. In the optimal scheme, the infinite-order noise shaping filter must be minimum phase and have a piece-wise flat power spectrum with a single jump discontinuity. We further show that the optimal noise-shaping filter of any order can be found by solving a set of Yule-Walker equations, and we present an exact rate-distortion analysis for any filter order, lattice vector quantizer dimension and bit rate. An important advantage of the proposed design is that it is symmetric in rate by construction, and there is therefore no need for source splitting. ØSTERGAARD AND ZAMIR: MULTIPLE-DESCRIPTION CODING BY DITHERED DELTA-SIGMA QUANTIZATION 3 bounds to the rate-distortion region for the case of K > 2 channels were presented in [6]-[8] but it is not known whether any of the bounds are tight for K > 2 channels. Practical symmetric MD lattice vector quantization (MD-LVQ) based schemes for two descriptions have been introduced in [9], [10], which in the limit of infinite-dimensional lattices and under high-resolution assumptions, approach the symmetric MD rate-distortion bound. An extension to K ≥ 2 descriptions was presented in [11]-[13]. Asymmetric MD-LVQ allows for unequal side distortions as well as unequal side rates and was first considered in [14], [15] for the case of two descriptions and extended in [13], [16] to the case of K ≥ 2 descriptions. Common for all of the designs [9]-[12], [14]-[16] is that a central quantizer is first applied on the source after which an index-assignment algorithm maps the reconstruction points of the central quantizer to reconstruction points of the side quantizers, which is an idea that was first presented in [17]. To avoid the difficulty of designing efficient index-assignment algorithms, it was suggested in [18] that the index assignments of a two-description system can be replaced by successive quantization and linear estimation. More specifically, the two side descriptions can be linearly combined and further enhanced by a refinement layer to yield the central reconstruction. The design of [18] suffers from a rate loss of 0.5 bit/dim. at high resolution and is therefore not able to achieve the MD rate-distortion bound. Recently, however, this gap was closed by Chen et al. [19] , [20] who recognized that the rate region of the MD problem forms a polymatroid, and showed that the corner points of this rate region can be achieved by successive estimation and quantization. The design of Chen et al. is inherently asymmetric in the description rate since any corner point of a non-trivial rate region will lead to asymmetric rates. To symmetrize the coding rates, it is necessary to break the quantization process into additional stages, which is a method known as "source splitting" (following Urbanke and Rimoldi's rate splitting approach for the multiple access channel). When finite-dimensional quantizers are employed, there is a space-filling loss due to the fact that the quantizer's Voronoi cells are finite dimensional and not completely spherical, [21] , and as such each description suffers a rate loss. The rate loss of the design given in [19] , [20] is that of 2K − 1 quantizers because source splitting is performed by using an additional K − 1 quantizers besides the conventional K side quantizers. In comparison, the designs based on index assignments suffer from a rate loss of only that of K quantizers (actually, for K = 2 the space-filling loss is that of two quantizers having spherical Voronoi August 7, 2007 DRAFT
doi:10.1109/dcc.2007.57 dblp:conf/dcc/OstergaardZ07 fatcat:jw6pnwd4djav5dtblou5z6kgwi