%0 Report
%D 2013
%T Minimal partitions and image classification using a gradient-free perimeter approximation
%A Samuel Amstutz
%A Nicolas Van Goethem
%A Antonio AndrĂ© Novotny
%K Image classification, deblurring, optimal partitions, perimeter approximation
%X In this paper a new mathematically-founded method for the optimal partitioning of domains, with applications to the classification of greyscale and color images, is proposed. Since optimal partition problems are in general ill-posed, some regularization strategy is required. Here we regularize by a non-standard approximation of the total interface length, which does not involve the gradient of approximate characteristic functions, in contrast to the classical Modica-Mortola approximation. Instead, it involves a system of uncoupled linear partial differential equations and nevertheless shows $\Gamma$-convergence properties in appropriate function spaces. This approach leads to an alternating algorithm that ensures a decrease of the objective function at each iteration, and which always provides a partition, even during the iterations. The efficiency of this algorithm is illustrated by various numerical examples. Among them we consider binary and multilabel minimal partition problems including supervised or automatic image classification, inpainting, texture pattern identification and deblurring.
%I SISSA
%G en
%U http://hdl.handle.net/1963/6976
%1 6964
%2 Mathematics
%4 1
%# MAT/05 ANALISI MATEMATICA
%$ Submitted by Nicolas Van Goethem (vangoeth@sissa.it) on 2013-06-27T11:35:17Z
No. of bitstreams: 1
Image.pdf: 1182883 bytes, checksum: 9ca6dcd03d7fd901b0d843d0062dfc74 (MD5)
%0 Report
%D 2012
%T Topological sensitivity analysis for high order elliptic operators
%A Samuel Amstutz
%A Antonio AndrĂ© Novotny
%A Nicolas Van Goethem
%K Topological derivative, Elliptic operators, Polarization tensor
%X The topological derivative is defined as the first term of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of a singular domain perturbation. It has applications in many different fields such as shape and topology optimization, inverse problems, image processing and mechanical modeling including synthesis and/or optimal design of microstructures, fracture mechanics sensitivity analysis and damage evolution modeling. The topological derivative has been fully developed for a wide range of second order differential operators. In this paper we deal with the topological asymptotic expansion of a class of shape functionals associated with elliptic differential operators of order 2m, m>=1. The general structure of the polarization tensor is derived and the concept of degenerate polarization tensor is introduced. We provide full mathematical justifications for the derived formulas, including precise estimates of remainders.
%I SISSA
%G en
%U http://hdl.handle.net/1963/6343
%1 6272
%2 Mathematics
%4 1
%# MAT/05 ANALISI MATEMATICA
%$ Submitted by Nicolas Van Goethem (vangoeth@sissa.it) on 2012-12-14T17:45:48Z
No. of bitstreams: 1
Degenerate_sub.pdf: 479352 bytes, checksum: eefab8b4e64b0ef0c7d7f0e16d529008 (MD5)