A Total Fractional-Order Variation Model for Image Restoration with Non-homogeneous Boundary Conditions and its Numerical Solution

To overcome the weakness of a total variation based model for image restoration, various high order (typically second order) regularization models have been proposed and studied recently. In this paper we analyze and test a fractional-order derivative based total $\alpha$-order variation model, which can outperform the currently popular high order regularization models. There exist several previous works using total $\alpha$-order variations for image restoration; however first no analysis is done yet and second all tested formulations, differing from each other, utilize the zero Dirichlet boundary conditions which are not realistic (while non-zero boundary conditions violate definitions of fractional-order derivatives). This paper first reviews some results of fractional-order derivatives and then analyzes the theoretical properties of the proposed total $\alpha$-order variational model rigorously. It then develops four algorithms for solving the variational problem, one based on the variational Split-Bregman idea and three based on direct solution of the discretise-optimization problem. Numerical experiments show that, in terms of restoration quality and solution efficiency, the proposed model can produce highly competitive results, for smooth images, to two established high order models: the mean curvature and the total generalized variation.Comment: 26 page

An accelerated proximal gradient algorithm for frame-based image restoration via the balanced approach

10.1137/090779437SIAM Journal on Imaging Sciences42573-59

Recent Progress in Image Deblurring

This paper comprehensively reviews the recent development of image deblurring, including non-blind/blind, spatially invariant/variant deblurring techniques. Indeed, these techniques share the same objective of inferring a latent sharp image from one or several corresponding blurry images, while the blind deblurring techniques are also required to derive an accurate blur kernel. Considering the critical role of image restoration in modern imaging systems to provide high-quality images under complex environments such as motion, undesirable lighting conditions, and imperfect system components, image deblurring has attracted growing attention in recent years. From the viewpoint of how to handle the ill-posedness which is a crucial issue in deblurring tasks, existing methods can be grouped into five categories: Bayesian inference framework, variational methods, sparse representation-based methods, homography-based modeling, and region-based methods. In spite of achieving a certain level of development, image deblurring, especially the blind case, is limited in its success by complex application conditions which make the blur kernel hard to obtain and be spatially variant. We provide a holistic understanding and deep insight into image deblurring in this review. An analysis of the empirical evidence for representative methods, practical issues, as well as a discussion of promising future directions are also presented.Comment: 53 pages, 17 figure

Diffusion equations and inverse problems regularization.

The present thesis can be split into two dfferent parts: The first part mainly deals with the porous and fast diffusion equations. Chapter 2 presents these equations in the Euclidean setting highlighting the technical issues that arise when trying to extend results in a Riemannian setting. Chapter 3 is devoted to the construction of exhaustion and cut-o_ functions with controlled gradient and Laplacian, on manifolds with Ricci curvature bounded from below by a (possibly unbounded) nonpositive function of the distance from a fixed reference point, and without any assumptions on the topology or the injectivity radius. The cut-offs are then applied to the study of the fast and porous media diffusion, of Lq-properties of the gradient and of the selfadjointness of Schrödinger-type operators. The second part is concerned with inverse problems regularization applied to image deblurring. In Chapter 5 new variants of the Tikhonov filter method, called fractional and weighted Tikhonov, are presented alongside their saturation properties and converse results on their convergence rates. New iterated fractional Tikhonov regularization methods are then introduced. In Chapter 6 the modified linearized Bregman algorithm is investigated. It is showed that the standard approach based on the block circulant circulant block preconditioner may provide low quality restored images and different preconditioning strategies are then proposed, which improve the quality of the restoration