The application that motivates this paper is molecular imaging at the atomic level. When discretized at subatomic 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. This 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. 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.

This paper considers the detection of a Markov signal in additive white Gaussian noise (AWGN). Here, the Markov signal is taken to be a certain class of random walk processes. A closed form expression of the likelihood ratio (LR) is derived for a general Markov signal in AWGN. Then, under the conditions of low signal to noise ratio (SNR) and long observation time, necessary conditions are derived for the LR of the random walk to be approximated by a bank of filtered energy (FE) detectors, as well as by a single FE detector. The FE detector is an intuitive way to perform detection; however, it is not necessarily optimal. The results are applicable to the detection of an electron spin in a magnetic resonance force microscopy (MRFM) experiment

Sparse image reconstruction is of interest in the fields of radioastronomy and molecular imaging. The observation is assumed to be a linear transformation of the image, and corrupted by additive white Gaussian noise. We study the usage of sparse priors in the empirical Bayes framework: it permits the selection of the hyperparameters of the prior in a data-driven fashion. Three sparse image reconstruction methods are proposed. A simulation study was performed using a binary-valued image and a Gaussian point spread function. In the range of signal to noise ratios considered, the proposed methods had better performance than sparse Bayesian learning (SBL).