Conditions for Unique Reconstruction of Sparse Signals Using Compressive Sensing Methods

TL;DR

This paper reviews conditions such as spark, restricted isometry, and coherence that guarantee the unique reconstruction of sparse signals from limited measurements, including signals sparse in the DFT domain.

Contribution

It provides a comprehensive review of the conditions ensuring unique sparse signal reconstruction, emphasizing the DFT domain case.

Findings

Conditions like spark, RIP, and coherence are key to guaranteeing unique reconstruction.

Unique reconstruction of DFT-sparse signals is specifically analyzed.

The paper consolidates theoretical criteria for sparse signal recovery.

Abstract

A signal is sparse in one of its representation domain if the number of nonzero coefficients in that domain is much smaller than the total number of coefficients. Sparse signals can be reconstructed from a very reduced set of measurements/observations. The topic of this paper are conditions for the unique reconstruction of sparse signals from a reduced set of observations. After the basic definitions are introduced, the unique reconstruction conditions are reviewed using the spark, restricted isometry, and coherence of the measurement matrix. Uniqueness of the reconstruction of signals sparse in the discrete Fourier domain (DFT), as the most important signal transformation domain, is considered as well.