2258 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 6, JUNE 2009 -Description Lattice Vector Quantization: Index Assignment and Analysis Minglei Liu and Ce Zhu, Senior Member, IEEE Abstract—In this paper, we investigate the design of symmetric entropy-constrained multiple description lattice vector quanti- zation (MDLVQ), more specifically, MDLVQ index assignment. We consider a fine lattice containing clean similar sublattices with -similarity. Due to the -similarity of the sublattices, an -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 -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 Terms—Index assignment, lattice, lattice vector quan- tization, multiple description coding, sublattice, transportation problem. I. INTRODUCTION M ULTIPLE description coding (MDC) is a source coding scheme to combat transmission errors, especially applicable in non-prioritized lossy networks. In an MDC imple- mentation, an encoder generates descriptions (or streams) for one source. The streams are transmitted over separate channels respectively, where each channel may have its own constraints. At the decoder, depending on the number of de- scriptions received correctly, different reconstruction quality will be obtained. To be more specific, if only out of totally streams are received, the reconstruction quality associated with a so-called -description side distortion is expected to be acceptable, and an incremental improvement can be achieved with a reduced side distortion if more streams are received. Manuscript received December 08, 2007; accepted December 17, 2008. First published March 10, 2009; current version published May 15, 2009. The as- sociate editor coordinating the review of this manuscript and approving it for publication was Prof. Christine Guillemot. M. Liu is with School of Communication and Information Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China, and also with School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore (e-mail: LIUM0003@ntu.edu.sg). C. Zhu is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore (e-mail: ECZhu@ntu. edu.sg). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2009.2016873 As all streams are available, the best reconstruction quality is obtained corresponding to the smallest central distortion. As one of the practical MDC schemes, multiple description lattice vector quantization (MDLVQ) is an effective technique to generate two or more representations for a symbol. Symmetric MDLVQ was introduced in [1] and [2] by Servetto, Vaisham- payan and Sloane (known as the SVS technique) for two bal- anced (symmetric) channels, whereas asymmetric multiple de- scription lattice vector quantization (AMDLVQ) was developed for possibly unbalanced (asymmetric) channels [3]. Later this asymmetric construction provided in [3] was improved in the case of two descriptions and further extended to the case of an arbitrary number of descriptions in [4]. For a given fine lattice and similar sublattice 1 with index number 2 , the SVS technique maps each point in the fine lattice to a pair of sublat- tice points, where the key is to design such a mapping (also equivalently known as labeling function or index assignment) that minimizes the one-description (averaged) side distortion . In -description LVQ where , each fine lattice point is indexed by sublattice points which construct an -tuple, given a fine lattice and a similar sublattice. The goal of this index assignment is to minimize -description (averaged) side distor- tion or overall expected distortion. For a finite index number , minimizing side distortions with different may lead to dif- ferent index assignment solutions, while a consistent asymptot- ical solution can be achieved as [6]. It is noted that the MDLVQ design can also be accomplished without using index assignment [7], [8], which will not be discussed here, and our focus in this paper is the index assignment for MDLVQ design. In [6], the analytical expressions for the central and side dis- tortions were derived for symmetric MDLVQ under high-res- olution assumption, where the index assignment problem was approached as a linear assignment problem. Huang and Wu de- veloped a greedy labeling algorithm for MDLVQ in [9]. Given a fine lattice and a sublattice , the main idea is to parti- tion the fine lattice based on its -fraction lattice , 3 and then to label the fine lattice points in each Voronoi cell of the -fraction lattice separately. The advantage of this method is that the -fraction lattice delicately classifies the fine lattice points by its Voronoi cells and all the fine lattice points in a same Voronoi cell can be treated equally, thus reducing the map- ping complexity significantly. The feasibility of using -frac- 1 A sublattice is said to be similar to a fine lattice if can be obtained by scaling, rotating or reflecting [5]. 2 Index number is defined as the number of elements (points) of in each Voronoi cell of . 3 -fraction lattice is defined as [10]. 1053-587X/$25.00 © 2009 IEEE