Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Topological Optimization with the p-Laplacian Operator and an Application in Image Processing | Hindawi Publishing Corporation Boundary Value Problems Volume 2009 Article ID 896813 20 pages doi 2009 896813 Research Article Topological Optimization with the p-Laplacian Operator and an Application in Image Processing Alassane Sy1 2 and Diaraf Seck2 1 Universite de Bambey Bambey Senegal 2 Laboratoire de Mathematique de la Decision et d Analyse Numerique LMDAN Ecole Doctorale de Mathématiques et bnformatique Universite Cheikh Anta Diop de Dakar . 99 000 Dakar-Fann Senegal Correspondence should be addressed to Diaraf Seck dseck@ Received 14 April 2009 Revised 28 June 2009 Accepted 28 July 2009 Recommended by Kanishka Perera We focus in this paper on the theoretical and numerical aspect os image processing. We consider a non linear boundary value problem the p-Laplacian from which we will derive the asymptotic expansion of the Mumford-Shah functional. We give a theoretical expression of the topological gradient as well as a numerical confirmation of the result in the restoration and segmentation of images. Copyright 2009 A. Sy and D. Seck. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction The goal of the topological optimization problem is to find an optimal design with an a priori poor information on the optimal shape of the structure. The shape optimization problem consists in minimizing a functional j Q J Q uq where the function UQ is defined for example on a variable open and bounded subset Q of RN. For e 0 let Qe Q x0 ew be the set obtained by removing a small part x0 ew from Q where x0 e Q and w c RN is a fixed open and bounded subset containing the origin. Then using general adjoint method an asymptotic expansion of the function will be obtained in the following form j Qe j Q f e g X0 of e lirnf e 0 f e 0. e 0 The topological sensitivity g x0 provides information
