Sparsity estimation in compressive sensing with application to MR images

TL;DR

This paper introduces a Bayesian hierarchical model-based estimator for unknown sparsity levels in compressive sensing, demonstrating its theoretical properties and practical effectiveness on MRI data for improved image recovery.

Contribution

It proposes the first statistically grounded sparsity estimator for compressive sensing, validated through theoretical proofs and real MRI data applications.

Findings

Estimator is unbiased and asymptotically normal.

Simulation results confirm theoretical properties.

Real MRI data application shows practical utility.

Abstract

The theory of compressive sensing (CS) asserts that an unknown signal $\mathbf{x} \in \mathbb{C}^N$ can be accurately recovered from $m$ measurements with $m\ll N$ provided that $\mathbf{x}$ is sparse. Most of the recovery algorithms need the sparsity $s=\lVert\mathbf{x}\rVert_0$ as an input. However, generally $s$ is unknown, and directly estimating the sparsity has been an open problem. In this study, an estimator of sparsity is proposed by using Bayesian hierarchical model. Its statistical properties such as unbiasedness and asymptotic normality are proved. In the simulation study and real data study, magnetic resonance image data is used as input signal, which becomes sparse after sparsified transformation. The results from the simulation study confirm the theoretical properties of the estimator. In practice, the estimate from a real MR image can be used for recovering future MR images under the framework of CS if they are believed to have the same sparsity level after sparsification.