Tight-frame-like Analysis-Sparse Recovery Using Non-tight Sensing Matrices

TL;DR

This paper introduces a novel analysis-sparse recovery method that leverages non-tight sensing matrices by reformulating the problem within a tight-frame framework, leading to improved performance in compressed sensing tasks.

Contribution

It proposes a new formalism for analysis-sparse recovery with non-tight sensing matrices, including algorithms and deep-unfolded neural network variants, enhancing recovery accuracy and performance bounds.

Findings

Proposed algorithms outperform existing methods in analysis-sparse recovery.

Deep-unfolded neural networks achieve superior image reconstruction quality.

Experimental results demonstrate improved metrics on multiple datasets.

Abstract

The choice of the sensing matrix is crucial in compressed sensing. Random Gaussian sensing matrices satisfy the restricted isometry property, which is crucial for solving the sparse recovery problem using convex optimization techniques. However, tight-frame sensing matrices result in minimum mean-squared-error recovery given oracle knowledge of the support of the sparse vector. If the sensing matrix is not tight, could one achieve the recovery performance assured by a tight frame by suitably designing the recovery strategy? -- This is the key question addressed in this paper. We consider the analysis-sparse l1-minimization problem with a generalized l2-norm-based data-fidelity and show that it effectively corresponds to using a tight-frame sensing matrix. The new formulation offers improved performance bounds when the number of non-zeros is large. One could develop a tight-frame variant of a known sparse recovery algorithm using the proposed formalism. We solve the analysis-sparse recovery problem in an unconstrained setting using proximal methods. Within the tight-frame sensing framework, we rescale the gradients of the data-fidelity loss in the iterative updates to further improve the accuracy of analysis-sparse recovery. Experimental results show that the proposed algorithms offer superior analysis-sparse recovery performance. Proceeding further, we also develop deep-unfolded variants, with a convolutional neural network as the sparsifying operator. On the application front, we consider compressed sensing image recovery. Experimental results on Set11, BSD68, Urban100, and DIV2K datasets show that the proposed techniques outperform the state-of-the-art techniques, with performance measured in terms of peak signal-to-noise ratio and structural similarity index metric.