Spatially Adaptive Regularization for Image Restoration

Abstract

Image restoration/reconstruction refers to the estimation of an underlying image from measurements generated by imaging devices. This problem is generally ill-posed since the measurements are corrupted by the physical limitations of the imaging device, and the inherent noise involved in the measurement process. There are three main classes of methods in the current literature. The first class of methods is based on regularization framework that enforces an ad-hoc prior on the restored image. The second class of methods uses regression-based learning paradigms, where a training set of clean images and the corresponding distorted measurements are used to generate a trained prior. The third class of methods adopts trained priors similar to the ones utilized in the second class of methods but within the regularization framework. This third class of methods, the trained regularization methods, are getting increasing attention because of their versatility as regularization methods, while also encompassing natural priors obtained from training. However, the need for training data can limit their applicability. In this thesis, we propose spatially adaptive regularization methods where the adaptation information is retrieved from the measured data that undergoes reconstruction. Due to adaptation, the enforced prior is more natural than the existing regularization methods. At the same time, our methods do not require training data. We summarize our contribution in three parts. In the first part, we propose a novel regularization method that adaptively combines the well-known second-order regularization, called Hessian-Schatten (HSN) norm regularization, and first-order TV (TV-1) functionals with spatially varying weights. The relative weight involved in combining the first- and second-order terms becomes an image, and this weight is determined through the minimization of a composite cost function, without user intervention. Our contributions in this part can be summarized as follows: We construct a composite regulariz...