A VECTOR QUANTIZATION APPROACH TO UNIVERSAL NOISELESS CODING AND QUANTIZATION

A two-stage code is a block code in which each block of data is coded in two stages: the first stage codes the identity of a block code amon g a collection of codes, and the second stage codes the data using the identified code, The collection of codes may be noiseless codes, fixe d-rate quantizers, or variable-rate quantizers. We take a vector quant ization approach to two-stage coding, in which the first stage code ca n be regarded as a vector quantizer that ''quantizes'' the input data of length n to one of a fixed collection of block codes, We apply the generalized Lloyd algorithm to the first-stage quantizer, using induce d measures of rate and distortion, to design locally optimal two-stage , codes, On a source of medical images, two-stage variable-rate vector quantizers designed in this way outperform standard (one-stage) fixed -rate vector quantizers by over 9 dB, The tail of the operational dist ortion-rate function of the first-stage quantizer determines the optim al rate of convergence of the redundancy of a universal sequence of tw o-stage codes, We show that there exist two-stage universal noiseless codes, fixed-rate quantizers, and variable-rate quantizers whose per-l etter rate and distortion redundancies converge to zero as (k/2)n(-1) log n, when the universe of sources has finite dimension h. This exten ds the achievability part of Rissanen's theorem from universal noisele ss codes to universal quantizers. Further, we show that the redundanci es converge as O(n(-1)) when the universe of sources is countable, and as O(n(-1+epsilon)) when the universe of sources is infinite-dimensio nal, under appropriate conditions.