15 research outputs found
Theoretical Foundation of the Weighted Laplace Inpainting Problem
Laplace interpolation is a popular approach in image inpainting using partial
differential equations. The classic approach considers the Laplace equation
with mixed boundary conditions. Recently a more general formulation has been
proposed where the differential operator consists of a point-wise convex
combination of the Laplacian and the known image data. We provide the first
detailed analysis on existence and uniqueness of solutions for the arising
mixed boundary value problem. Our approach considers the corresponding weak
formulation and aims at using the Theorem of Lax-Milgram to assert the
existence of a solution. To this end we have to resort to weighted Sobolev
spaces. Our analysis shows that solutions do not exist unconditionally. The
weights need some regularity and fulfil certain growth conditions. The results
from this work complement findings which were previously only available for a
discrete setup.Comment: 16 pages, 2 Figure
iPiano: Inertial Proximal Algorithm for Non-Convex Optimization
In this paper we study an algorithm for solving a minimization problem
composed of a differentiable (possibly non-convex) and a convex (possibly
non-differentiable) function. The algorithm iPiano combines forward-backward
splitting with an inertial force. It can be seen as a non-smooth split version
of the Heavy-ball method from Polyak. A rigorous analysis of the algorithm for
the proposed class of problems yields global convergence of the function values
and the arguments. This makes the algorithm robust for usage on non-convex
problems. The convergence result is obtained based on the \KL inequality. This
is a very weak restriction, which was used to prove convergence for several
other gradient methods. First, an abstract convergence theorem for a generic
algorithm is proved, and, then iPiano is shown to satisfy the requirements of
this theorem. Furthermore, a convergence rate is established for the general
problem class. We demonstrate iPiano on computer vision problems: image
denoising with learned priors and diffusion based image compression.Comment: 32pages, 7 figures, to appear in SIAM Journal on Imaging Science
A trust region-type normal map-based semismooth Newton method for nonsmooth nonconvex composite optimization
We propose a novel trust region method for solving a class of nonsmooth and
nonconvex composite-type optimization problems. The approach embeds inexact
semismooth Newton steps for finding zeros of a normal map-based stationarity
measure for the problem in a trust region framework. Based on a new merit
function and acceptance mechanism, global convergence and transition to fast
local q-superlinear convergence are established under standard conditions. In
addition, we verify that the proposed trust region globalization is compatible
with the Kurdyka-{\L}ojasiewicz (KL) inequality yielding finer convergence
results. We further derive new normal map-based representations of the
associated second-order optimality conditions that have direct connections to
the local assumptions required for fast convergence. Finally, we study the
behavior of our algorithm when the Hessian matrix of the smooth part of the
objective function is approximated by BFGS updates. We successfully link the KL
theory, properties of the BFGS approximations, and a Dennis-Mor{\'e}-type
condition to show superlinear convergence of the quasi-Newton version of our
method. Numerical experiments on sparse logistic regression and image
compression illustrate the efficiency of the proposed algorithm.Comment: 56 page
Contributions en optimisation topologique : extension de la méthode adjointe et applications au traitement d'images
De nos jours, l'optimisation topologique a été largement étudiée en optimisation de structure, problème majeur en conception de systèmes mécaniques pour l'industrie et dans les problèmes inverses avec la détection de défauts et d'inclusions. Ce travail se concentre sur les approches de dérivées topologiques et propose une généralisation plus flexible de cette méthode rendant possible l'investigation de nouvelles applications. Dans une première partie, nous étudions des problèmes classiques en traitement d'images (restauration, inpainting), et exposons une formulation commune à ces problèmes. Nous nous concentrons sur la diffusion anisotrope et considérons un nouveau problème : la super-résolution. Notre approche semble meilleure comparée aux autres méthodes. L'utilisation des dérivées topologiques souffre d'inconvénients : elle est limitée à des problèmes simples, nous ne savons pas comment remplir des trous ... Dans une seconde partie, une nouvelle méthode visant à surmonter ces difficultés est présentée. Cette approche, nommée voûte numérique, est une extension de la méthode adjointe. Ce nouvel outil nous permet de considérer de nouveaux champs d'application et de réaliser de nouvelles investigations théoriques dans le domaine des dérivées topologiques.Nowadays, topology optimization has been extensively studied in structural optimization which is a major interest in the design of mechanical systems in the industry and in inverse problems with the detection of defects or inclusions. This work focuses on the topological derivative approach and proposes a more flexible generalization of this method making it possible to address new applications. In a first part, we study classical image processing problems (restoration, inpainting), and give a common framework to theses problems. We focus on anisotropic diffusion and consider a new problem: super-resolution. Our approach seems to be powerful in comparison with other methods. Topological derivative method has some drawbacks: it is limited to simple problems, we do not know how to fill holes, ... In a second part, to overcome these difficulties, an extension of the adjoint method is presented. Named the numerical vault, it allows us to consider new fields of applications and to explore new theoretical investigations in the area of topological derivative