Frame Permutation Quantization

TL;DR

Frame permutation quantization (FPQ) is a novel vector quantization method using finite frames, offering more rate options and improved performance over scalar quantization, with algorithms for reconstruction and analysis of properties.

Contribution

Introduces FPQ, a new quantization technique with multiple rate options, and develops algorithms and theoretical analysis for its reconstruction and performance.

Findings

FPQ outperforms scalar quantization in certain scenarios.

Reconstruction error decays as 1/M^4 with frame size.

Algorithms based on linear and quadratic programming are effective.

Abstract

Frame permutation quantization (FPQ) is a new vector quantization technique using finite frames. In FPQ, a vector is encoded using a permutation source code to quantize its frame expansion. This means that the encoding is a partial ordering of the frame expansion coefficients. Compared to ordinary permutation source coding, FPQ produces a greater number of possible quantization rates and a higher maximum rate. Various representations for the partitions induced by FPQ are presented, and reconstruction algorithms based on linear programming, quadratic programming, and recursive orthogonal projection are derived. Implementations of the linear and quadratic programming algorithms for uniform and Gaussian sources show performance improvements over entropy-constrained scalar quantization for certain combinations of vector dimension and coding rate. Monte Carlo evaluation of the recursive algorithm shows that mean-squared error (MSE) decays as 1/M^4 for an M-element frame, which is consistent with previous results on optimal decay of MSE. Reconstruction using the canonical dual frame is also studied, and several results relate properties of the analysis frame to whether linear reconstruction techniques provide consistent reconstructions.