Near-Optimal Encoding for Sigma-Delta Quantization of Finite Frame Expansions

TL;DR

This paper presents an encoding and decoding scheme for Sigma-Delta quantized finite frame expansions that achieves exponential decay in approximation error relative to the number of bits, applicable to various frame types.

Contribution

It introduces a simple, effective encoding algorithm acting on the Sigma-Delta bit stream that guarantees near-optimal exponential error decay for a broad class of finite frames.

Findings

Encoding algorithm achieves exponential error decay.

Applicable to random and piecewise smooth frames.

Decoding involves linear least squares minimization.

Abstract

In this paper we investigate encoding the bit-stream resulting from coarse Sigma-Delta quantization of finite frame expansions (i.e., overdetermined representations) of vectors. We show that for a wide range of finite-frames, including random frames and piecewise smooth frames, there exists a simple encoding algorithm ---acting only on the Sigma-Delta bit stream--- and an associated decoding algorithm that together yield an approximation error which decays exponentially in the number of bits used. The encoding strategy consists of applying a discrete random operator to the Sigma-Delta bit stream and assigning a binary codeword to the result. The reconstruction procedure is essentially linear and equivalent to solving a least squares minimization problem.