New efficient algorithms are presented to design globally optimal two-description quantizers of fixed rate. The optimization objective is to minimize the expected distortion at the receiver side. We formulate the problem as one of shortest path in a directed acyclic graph. The fixed rate requirement puts constraints on the number and type of edges of the shortest path, which leads to an O(K 1 K 2 N 3 ) time design algorithm, where N is the cardinality of the source alphabet, and K 1 , K 2 are the number of codecells, respectively, of the two side quantizers. This complexity is reduced to O(K 1 K 2 N 2 ) by exploiting a so-called Monge property of the objective function. Furthermore, if K 1 = K 2 = K and the two descriptions are subject to the same channel statistics, then the optimal description quantizer design problem can be solved in O(KN 2 ) time. Definitions and NotationsA multiple description quantizer is a set of quantizers, each of which (called a side quantizer), separately, provides a "partial" or coarse description of the source (called side description), and which allow the decoders of side quantizers to collaborate in refining the source representation if few side descriptions are received. Multiple description quantizers have applications in communications over lossy packet-switched networks, sensor networks, communications using antenna diversity, distributed source coding, and distributed storage networks. In this paper we focus on two-description quantizers (2DQ), i.e., the case of two side descriptions. One can always recursively apply a 2DQ to arrive at more descriptions, if so desired. Traditionally, the goal of multiple description quantizer design is to minimize the distortion of the central description (the one obtained when both side descriptions are available), while meeting a constraint on the side distortions. We formulate the problem as one of minimizing an expected distortion of the 2DQ.Let X be a random variable over a finite alphabet A = {x 1 , x 2 , · · · , x N } ⊂ R, x i < x i+1 , 1 ≤ i ≤ N −1. Let the probability mass function (pmf) of X be p i = p(X = x i ), 1 ≤ i ≤ N . A convex subset of the alphabet A is any set c(a, b] = {x i |a < i ≤ b} for some integers a, b, 0 ≤ a ≤ b ≤ N . For any integer K, K < N, a K-level quantizer Q is a partition of the alphabet A in K convex subsets c(q j , q j+1 ] (called codecells), 0 ≤ j ≤ K − 1, where 0 = q 0 < q 1 < · · · q K−1 < q K = N . The quantizer Q can be identified by the above partition of the interval of integers (0, N]. We use interchangeably the terms quantizer and partition. The values q j are called thresholds of the quantizer (or of the partition). The codecell c(q j , q j+1 ] is simply denoted by (q j , q j+1 ].To measure the quantizer performance we consider a distortion function d(x, y), d : R × R → [0, ∞), which is monotone, i.e. for all real values x, y 1 , y 2 , such that x ≤ y 1 < y 2 or x ≥ y 1 > y 2 , the following relation holds:d(x, y 1 ) ≤ d(x, y 2 ).
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