Hierarchical Bayesian sparse image reconstruction with application to MRFM

TL;DR

This paper introduces a hierarchical Bayesian model for sparse image reconstruction from noisy linear measurements, effectively capturing sparsity and positivity, with automatic hyperparameter tuning and a Gibbs sampling algorithm, demonstrated on MRFM data.

Contribution

The paper presents a novel hierarchical Bayesian approach with a specialized prior and Gibbs sampling for sparse image reconstruction, providing full posterior distributions for all parameters.

Findings

Effective reconstruction of MRFM images from noisy data

Automatic hyperparameter tuning improves model adaptability

Full posterior distributions offer richer information than point estimates

Abstract

This paper presents a hierarchical Bayesian model to reconstruct sparse images when the observations are obtained from linear transformations and corrupted by an additive white Gaussian noise. Our hierarchical Bayes model is well suited to such naturally sparse image applications as it seamlessly accounts for properties such as sparsity and positivity of the image via appropriate Bayes priors. We propose a prior that is based on a weighted mixture of a positive exponential distribution and a mass at zero. The prior has hyperparameters that are tuned automatically by marginalization over the hierarchical Bayesian model. To overcome the complexity of the posterior distribution, a Gibbs sampling strategy is proposed. The Gibbs samples can be used to estimate the image to be recovered, e.g. by maximizing the estimated posterior distribution. In our fully Bayesian approach the posteriors of all the parameters are available. Thus our algorithm provides more information than other previously proposed sparse reconstruction methods that only give a point estimate. The performance of our hierarchical Bayesian sparse reconstruction method is illustrated on synthetic and real data collected from a tobacco virus sample using a prototype MRFM instrument.