On the role of total variation in compressed sensing - structure dependence

TL;DR

This paper investigates how total variation regularization aids in recovering gradient sparse signals from noisy Fourier samples in compressed sensing, emphasizing the importance of non-uniform, structure-dependent sampling strategies.

Contribution

It provides new recovery guarantees for non-uniform random sampling based on the signal's sparsity structure, extending theoretical understanding in compressed sensing.

Findings

Recovery guarantees depend on the signal's sparsity structure.

Non-uniform sampling improves reconstruction stability.

Sampling set design is crucial for effective compressed sensing.

Abstract

This paper considers the use of total variation regularization in the recovery of approximately gradient sparse signals from their noisy discrete Fourier samples in the context of compressed sensing. It has been observed over the last decade that a reconstruction which is robust to noise and stable to inexact sparsity can be achieved when we observe a highly incomplete subset of the Fourier samples for which the samples have been drawn in a random manner. Furthermore, in order to minimize the cardinality of the set of Fourier samples, the sampling set needs to be drawn in a non-uniform manner and the use of randomness is far more complex than the notion of uniform random sampling often considered in the theoretical results of compressed sensing. The purpose of this paper is to derive recovery guarantees in the case where the sampling set is drawn in a non-uniform random manner. We will show how the sampling set is dependent on the sparsity structure of the underlying signal.