Blind inpainting using l0 and total variation regularization.

In this paper, we address the problem of image reconstruction with missing pixels or corrupted with impulse noise, when the locations of the corrupted pixels are not known. A logarithmic transformation is applied to convert the multiplication between the image and binary mask into an additive problem. The image and mask terms are then estimated iteratively with total variation regularization applied on the image, and l0 regularization on the mask term which imposes sparseness on the support set of the missing pixels.

An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems.

We propose a new fast algorithm for solving one of the standard approaches to ill-posed linear inverse problems (IPLIP), where a (possibly nonsmooth) regularizer is minimized under the constraint that the solution explains the observations sufficiently well. Although the regularizer and constraint are usually convex, several particular features of these problems (huge dimensionality, nonsmoothness) preclude the use of off-the-shelf optimization tools and have stimulated a considerable amount of research. In this paper, we propose a new efficient algorithm to handle one class of constrained problems (often known as basis pursuit denoising) tailored to image recovery applications.

Fast image recovery using variable splitting and constrained optimization.

We propose a new fast algorithm for solving one of the standard formulations of image restoration and reconstruction which consists of an unconstrained optimization problem where the objective includes an l2 data-fidelity term and a nonsmooth regularizer. This formulation allows both wavelet-based (with orthogonal or frame-based representations) regularization or total-variation regularization. Our approach is based on a variable splitting to obtain an equivalent constrained optimization formulation, which is then addressed with an augmented Lagrangian method.