Image reconstructions using sparse dictionary representations and implicit, non-negative mappings

TL;DR

This paper introduces two novel methods to solve linear inverse imaging problems by promoting sparsity and non-negativity in solutions using a learned patch dictionary, improving results in tasks like deblurring and superresolution.

Contribution

It proposes two sparsity-promoting methods based on MRNSD that incorporate non-negativity constraints, with one using an $$-regularization and the other a new mapping, advancing inverse problem solutions.

Findings

Methods effectively solve inverse problems with non-negativity and sparsity constraints.

Proposed algorithms outperform standard approaches in numerical experiments.

Applications include deblurring, image completion, CT, and superresolution.

Abstract

Many imaging science tasks can be modeled as a discrete linear inverse problem. Solving linear inverse problems is often challenging, with ill-conditioned operators and potentially non-unique solutions. Embedding prior knowledge, such as smoothness, into the solution can overcome these challenges. In this work, we encode prior knowledge using a non-negative patch dictionary, which effectively learns a basis from a training set of natural images. In this dictionary basis, we desire solutions that are non-negative and sparse (i.e., contain many zero entries). With these constraints, standard methods for solving discrete linear inverse problems are not directly applicable. One such approach is the modified residual norm steepest descent (MRNSD), which produces non-negative solutions but does not induce sparsity. In this paper, we provide two methods based on MRNSD that promote sparsity. In our first method, we add an $\ell_1$-regularization term with a new, optimal step size. In our second method, we propose a new non-negative, sparsity-promoting mapping of the solution. We compare the performance of our proposed methods on a number of numerical experiments, including deblurring, image completion, computer tomography, and superresolution. Our results show that these methods effectively solve discrete linear inverse problems with non-negativity and sparsity constraints.