Decoding of a Cartesian product set with a constraint on an additive cost; fixed-rate entropy-coded vector quantization

Abstract

The authors consider a discrete set of points A composed of K=|A| elements. A non-negative cost c(a) is associated with each element a/spl isin/A. The n-fold Cartesian product of A is dented as {A}/sup n/. The cost of an n-fold element a=(a/sub 0/, ..., a/sub n-1/)/spl isin/{A}/sup n/ is equal to: c(a)=/spl Sigma//sub i/ c(a/sub i/). The authors select a subset of the n-fold elements, S/sub n//spl isin/{A}/sup n/, with a cost less than or equal to a given value c/sub max/. They refer to A as the constituent subset. They consider another set of n-tuples X/sub n/ denoted as the input set. A non-negative distance is defined between each x=(x/sub 0/, ..., x/sub n-1/)/spl isin/X/sub n/, and each s=(s/sub 0/, ..., s/sub n-1/)/spl isin/S/sub n/. The distance between x/sub i/ and s/sub i/ is denoted as d(x/sub i/,s/sub i/). The distance between x and s is equal to: d(x,s)=/spl Sigma//sub i/d(x/sub i/, s/sub i/). Decoding of an element x/spl isin/X/sub n/ is to find the element s/spl isin/S/sub n/ which has the minimum distance to x. A major application of this decoding problem is in fixed-rate entropy-coded vector quantization where A is the set of reconstruction vectors of a vector quantizer and cost is equivalent to self-information.<>