LEARNING SPARSE AND LOW-RANK PRIORS FOR IMAGE RECOVERY VIA ITERATIVE REWEIGHTED LEAST SQUARES MINIMIZATION

Abstract

We introduce a novel optimization algorithm for image recovery under learned sparse and low-rank constraints, which we parameterize as weighted extensions of the ℓ p p -vector and S p p Schatten-matrix quasi-norms for 0 < p ≤ 1, respectively. Our proposed algorithm generalizes the Iteratively Reweighted Least Squares (IRLS) method, used for signal recovery under ℓ 1 and nuclear-norm constrained minimization. Further, we interpret our overall minimization approach as a recurrent network that we then employ to deal with inverse low-level computer vision problems. Thanks to the convergence guarantees that our IRLS strategy offers, we are able to train the derived reconstruction networks using a memory-efficient implicit back-propagation scheme, which does not pose any restrictions on their effective depth. To assess our networks' performance, we compare them against other existing reconstruction methods on several inverse problems, namely image deblurring, super-resolution, demosaicking and sparse recovery. Our reconstruction results are shown to be very competitive and in many cases outperform those of existing unrolled networks, whose number of parameters is orders of magnitude higher than that of our learned models. The code is available at this link.

1. INTRODUCTION

With the advent of modern imaging techniques, we are witnessing a significant rise of interest in inverse problems, which appear increasingly in a host of applications ranging from microscopy and medical imaging to digital photography, 2D&3D computer vision, and astronomy (Bertero & Boccacci, 1998 ). An inverse imaging problem amounts to estimating a latent image from a set of possibly incomplete and distorted indirect measurements. In practice, such problems are typical illposed (Tikhonov, 1963; Vogel, 2002) , which implies that the equations relating the underlying image with the measurements (image formation model) are not adequate by themselves to uniquely characterize the solution. Therefore, in order to recover approximate solutions, which are meaningful in a statistical or physical sense, from the set of solutions that are consistent with the image formation model, it is imperative to exploit prior knowledge about certain properties of the underlying image. Among the key approaches for solving ill-posed inverse problems are variational methods (Benning & Burger, 2018) , which entail the minimization of an objective function. A crucial part of such an objective function is the regularization term, whose role is to promote those solutions that fit best our prior knowledge about the latent image. Variational methods have also direct links to Bayesian methods and can be interpreted as seeking the penalized maximum likelihood or the maximum a posteriori (MAP) estimator (Figueiredo et al., 2007) , with the regularizer matching the negative log-prior. Due to the great impact of the regularizer in the reconsturction quality, significant research effort has been put in the design of suitable priors. Among the overwhelming number of existing priors in the literature, sparsity and low-rank (spectral-domain sparsity) promoting priors have received considerable attention. This is mainly due to their solid mathematical foundation and the competitive results they achieve (Bruckstein et al., 2009; Mairal et al., 2014) . Nowdays, thanks to the advancements of deep learning there is a plethora of networks dedicated to image reconstruction problems, which significantly outperform conventional approaches. Nevertheless, they are mostly specialized and applicable to a single task. Further, they are difficult to analyze and interpret since they do not explicitly model any of the well-studied image properties,

