7,266 research outputs found
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 -order variation model, which
can outperform the currently popular high order regularization models. There
exist several previous works using total -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 -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
Inexact Bregman iteration with an application to Poisson data reconstruction
This work deals with the solution of image restoration problems by an
iterative regularization method based on the Bregman iteration. Any iteration of this
scheme requires to exactly compute the minimizer of a function. However, in some
image reconstruction applications, it is either impossible or extremely expensive to
obtain exact solutions of these subproblems. In this paper, we propose an inexact
version of the iterative procedure, where the inexactness in the inner subproblem
solution is controlled by a criterion that preserves the convergence of the Bregman
iteration and its features in image restoration problems. In particular, the method
allows to obtain accurate reconstructions also when only an overestimation of the
regularization parameter is known. The introduction of the inexactness in the iterative
scheme allows to address image reconstruction problems from data corrupted by
Poisson noise, exploiting the recent advances about specialized algorithms for the
numerical minimization of the generalized KullbackāLeibler divergence combined with
a regularization term. The results of several numerical experiments enable to evaluat
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