High order low-bit Sigma-Delta quantization for fusion frames

TL;DR

This paper develops high order low-bit Sigma-Delta quantizers tailored for fusion frames, enabling efficient and stable quantization of vector-valued measurements with minimal bits, and demonstrates effective signal reconstruction.

Contribution

It introduces a novel high order low-bit Sigma-Delta quantization scheme for fusion frames, with stability guarantees and practical reconstruction methods.

Findings

Quantizers can be implemented with g bits per measurement.

Stable quantization achieved for fusion frame measurements.

Numerical experiments validate the quantization and reconstruction performance.

Abstract

We construct high order low-bit Sigma-Delta $(\Sigma \Delta)$ quantizers for the vector-valued setting of fusion frames. We prove that these $\Sigma \Delta$ quantizers can be stably implemented to quantize fusion frame measurements on subspaces $W_n$ using $\log_2( {\rm dim}(W_n)+1)$ bits per measurement. Signal reconstruction is performed using a version of Sobolev duals for fusion frames, and numerical experiments are given to validate the overall performance.