Unified Theory for Recovery of Sparse Signals in a General Transform Domain

TL;DR

This paper develops a unified convex programming framework for robustly recovering sparse signals in any transform domain, extending compressed sensing theory to more general models and demonstrating improved sampling strategies.

Contribution

It introduces a comprehensive theory for sparse signal recovery in general transform domains, applicable to various acquisition and sparsity models, with extensions for structured sparsity and incoherence.

Findings

Recovery guarantees depend on measurement and transform properties.

Variable density sampling improves recovery performance.

Numerical results outperform existing sampling strategies.

Abstract

Compressed sensing provided a data-acquisition paradigm for sparse signals. Remarkably, it has been shown that practical algorithms provide robust recovery from noisy linear measurements acquired at a near optimal sampling rate. In many real-world applications, a signal of interest is typically sparse not in the canonical basis but in a certain transform domain, such as wavelets or the finite difference. The theory of compressed sensing was extended to the analysis sparsity model but known extensions are limited to specific choices of sensing matrix and sparsifying transform. In this paper, we propose a unified theory for robust recovery of sparse signals in a general transform domain by convex programming. In particular, our results apply to general acquisition and sparsity models and show how the number of measurements for recovery depends on properties of measurement and sparsifying transforms. Moreover, we also provide extensions of our results to the scenarios where the atoms in the transform has varying incoherence parameters and the unknown signal exhibits a structured sparsity pattern. In particular, for the partial Fourier recovery of sparse signals over a circulant transform, our main results suggest a uniformly random sampling. Numerical results demonstrate that the variable density random sampling by our main results provides superior recovery performance over known sampling strategies.