Low Bit Rate Vector Quantization of Outlier Contaminated Data Based on Shells of Golay Codes

Abstract

In this paper we study how to encode N -long vectors, with N in the range of hundreds, at low bit rates of 0.5 bit per sample or lower. We adopt a vector quantization structure, where an overall gain is encoded with a scalar quantizer and the remaining scaled vector is encoded using a vector quantizer built out by combining smaller (length L) binary codes known to be efficient in filling the space, the important examples discussed here being the Golay codes. Due to the typical nonstationary distribution of the long vectors, a piecewise stationary plus contamination model is assumed. The generic solution is to encode the outliers using GolombRice codes, and for each L-long subvector to encode the vector of absolute values using the nearest neighbor in a certain shell of a chosen binary {0, 1} code, the sign information being transmitted separately. The rate-distortion optimization problem can be very efficiently organized and solved for the unknowns, which include the Hamming weights of the chosen shells for each of the n b = N/L subvectors, and the overall gain g. The essential properties which influence the selection of a certain binary code as a building block are its space filling properties, the number of shells of various Hamming weights (allowing more or less flexibility in the rate-distortion optimization), the closeness of N to a multiple of L, and the existence of fast search of nearest neighbor on a shell. We show results when using the Golay codes for vector quantization on audio coding applications.

Fast search on the shells of Golay codes

“…E g that is the closest to the received r. and then the 11 informational bits are extracted from §.. For some channels, for each transmitted bit Si the receiver does not take immediately trivial hard decisions ri E {O, I}, but instead it can evaluate the probability Pi = P(ri = 0) and P(ri = 1) = 1 -Pi. Then for each Golay codevector 9 E g the probability P(r. = g) = L P(ri = gi) can be used for a better decision, given by the maximum likelihood decoding L argmaxP(r. = g) = argmax L P(ri = gi) [ (1,0), (w, …”

“…For some specific problems, see e.g., [1], the nearest neighbor must be looked for in a particular shell of the codebook g. The nearest neighbor in the Euclidean distance sense minimizes the distance (5) L L L L v; -2 L Vigi + L g; (4) i=l i=l i=l L L(Vi -gi)2 i=l which is equivalent to maximizing L argmax~Es L Vi~i + dH(J).…”

“…ALSO, MULTIPLICATION (IN GF(4)) OF ANY VALID CODEWORD WITH A SCALAR PRODUCES A search on individual shells of Golay codes in terms of compression efficiency[1].…”