Sparse image reconstruction for molecular imaging

TL;DR

This paper develops a novel sparse image reconstruction method tailored for molecular imaging, particularly addressing high-coherence convolution matrices, and demonstrates its superiority over traditional methods like lasso through numerical experiments.

Contribution

It introduces a hybrid thresholding estimator for sparse image reconstruction that handles high-coherence convolution matrices without assuming low coherence, advancing molecular imaging techniques.

Findings

Hybrid estimator outperforms lasso in numerical tests.

Proposed method effectively handles high-coherence convolution matrices.

Stein's unbiased risk estimate (SURE) enables hyperparameter tuning.

Abstract

The application that motivates this paper is molecular imaging at the atomic level. When discretized at sub-atomic distances, the volume is inherently sparse. Noiseless measurements from an imaging technology can be modeled by convolution of the image with the system point spread function (psf). Such is the case with magnetic resonance force microscopy (MRFM), an emerging technology where imaging of an individual tobacco mosaic virus was recently demonstrated with nanometer resolution. We also consider additive white Gaussian noise (AWGN) in the measurements. Many prior works of sparse estimators have focused on the case when H has low coherence; however, the system matrix H in our application is the convolution matrix for the system psf. A typical convolution matrix has high coherence. The paper therefore does not assume a low coherence H. A discrete-continuous form of the Laplacian and atom at zero (LAZE) p.d.f. used by Johnstone and Silverman is formulated, and two sparse estimators derived by maximizing the joint p.d.f. of the observation and image conditioned on the hyperparameters. A thresholding rule that generalizes the hard and soft thresholding rule appears in the course of the derivation. This so-called hybrid thresholding rule, when used in the iterative thresholding framework, gives rise to the hybrid estimator, a generalization of the lasso. Unbiased estimates of the hyperparameters for the lasso and hybrid estimator are obtained via Stein's unbiased risk estimate (SURE). A numerical study with a Gaussian psf and two sparse images shows that the hybrid estimator outperforms the lasso.