Sparsity promoting reconstructions via hierarchical prior models in diffuse optical tomography

TL;DR

This paper introduces hierarchical prior models for diffuse optical tomography to enhance sparse and sharp-edged reconstructions, leveraging hyperpriors to adaptively promote sparsity and improve localization and contrast.

Contribution

It formulates and evaluates hierarchical prior models with hyperpriors for nonlinear DOT, improving edge sharpness and sparsity in reconstructions compared to traditional priors.

Findings

Hierarchical models improve edge sharpness and contrast.

Hyperprior-based hyperparameter selection enhances reconstruction quality.

Numerical simulations demonstrate better localization of optical parameters.

Abstract

Diffuse optical tomography (DOT) is a severely ill-posed nonlinear inverse problem that seeks to estimate optical parameters from boundary measurements. In the Bayesian framework, the ill-posedness is diminished by incorporating {\em a priori} information of the optical parameters via the prior distribution. In case the target is sparse or sharp-edged, the common choice as the prior model are non-differentiable total variation and $\ell^1$ priors. Alternatively, one can hierarchically extend the variances of a Gaussian prior to obtain differentiable sparsity promoting priors. By doing this, the variances are treated as unknowns allowing the estimation to locate the discontinuities. In this work, we formulate hierarchical prior models for the nonlinear DOT inverse problem using exponential, standard gamma and inverse-gamma hyperpriors. Depending on the hyperprior and the hyperparameters, the hierarchical models promote different levels of sparsity and smoothness. To compute the MAP estimates, the previously proposed alternating algorithm is adapted to work with the nonlinear model. We then propose an approach based on the cumulative distribution function of the hyperpriors to select the hyperparameters. We evaluate the performance of the hyperpriors with numerical simulations and show that the hierarchical models can improve the localization, contrast and edge sharpness of the reconstructions.