Sparse Stochastic Processes and Discretization of Linear Inverse Problems

TL;DR

This paper introduces a new discretization framework for linear inverse problems based on sparse stochastic processes, leading to novel MAP estimators that unify and extend existing regularization methods, including nonconvex schemes, with applications in imaging.

Contribution

It develops a statistically-based discretization paradigm using sparse stochastic processes, deriving MAP estimators that encompass and extend traditional regularization techniques.

Findings

The proposed estimators unify Tikhonov and $ ext{l}_1$ regularizations.

The framework enables nonconvex sparsity-promoting regularizations.

Experimental results demonstrate improved performance in imaging applications.

Abstract

We present a novel statistically-based discretization paradigm and derive a class of maximum a posteriori (MAP) estimators for solving ill-conditioned linear inverse problems. We are guided by the theory of sparse stochastic processes, which specifies continuous-domain signals as solutions of linear stochastic differential equations. Accordingly, we show that the class of admissible priors for the discretized version of the signal is confined to the family of infinitely divisible distributions. Our estimators not only cover the well-studied methods of Tikhonov and $\ell_1$-type regularizations as particular cases, but also open the door to a broader class of sparsity-promoting regularization schemes that are typically nonconvex. We provide an algorithm that handles the corresponding nonconvex problems and illustrate the use of our formalism by applying it to deconvolution, MRI, and X-ray tomographic reconstruction problems. Finally, we compare the performance of estimators associated with models of increasing sparsity.