A New Augmented Lagrangian Approach for the L¹ – Mean Curvature Image Denoising

Abstract

Variational models of increasing mathematical complexity are used in Image Denoising. Indeed, Image Denoising problems have renewed the Calculus of Variations from both the theoretical and computational points of view. Among these new variational approaches, those based on the L¹- norm of the mean curvature of the image surface are enjoying a fast growing popularity.

Problem (IDP) being both non-smooth and non-convex its numerical solution is rather challenging. The main goal of this lecture is to discuss an augmented Lagrangian based solution method of IDP, which allows to overcome the two difficulties mentioned above and whose finite element implementation is relatively easy. The resulting algorithm belongs to the Alternating Direction Methods of Multipliers (ADMM) family. Unlike related algorithms developed for the solution of IDP, with the speaker’s approach, the constraints treated by augmentation-duality are all linear, a significant simplification indeed.

The capabilities of his approach are illustrated by the results of numerical experiments.

About the speaker

Prof. Roland Glowinski received his PhD in Mathematics in 1970 from University of Paris VI, where he had then taught for 15 years before becoming Cullen Professor of Mathematics at University of Houston. Recipient of the Seymour Cray Prize, the Grand Prix Marcel Dassault of the French National Academy of Sciences, the SIAM Von Karman Prize and the CFD Award of the American Association for Computational Mechanics, Prof Glowinski is a Fellow of the Society for Industrial and Applied Mathematics and of the American Mathematical Society, a Member of the French National Academy of Sciences, the French National Academy of Technology and the Academia Europaea. He is an Emeritus Professor at University P. & M. Curie in Paris, France, and an Honorary Professor at Fudan University in Shanghai.