TY - RPRT
T1 - Minimal partitions and image classification using a gradient-free perimeter approximation
Y1 - 2013
A1 - Samuel Amstutz
A1 - Nicolas Van Goethem
A1 - Antonio AndrĂ© Novotny
KW - Image classification, deblurring, optimal partitions, perimeter approximation
AB - 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.
PB - SISSA
UR - http://hdl.handle.net/1963/6976
U1 - 6964
U2 - Mathematics
U4 - 1
U5 - MAT/05 ANALISI MATEMATICA
ER -
TY - RPRT
T1 - Topological sensitivity analysis for high order elliptic operators
Y1 - 2012
A1 - Samuel Amstutz
A1 - Antonio AndrĂ© Novotny
A1 - Nicolas Van Goethem
KW - Topological derivative, Elliptic operators, Polarization tensor
AB - 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.
PB - SISSA
UR - http://hdl.handle.net/1963/6343
U1 - 6272
U2 - Mathematics
U4 - 1
U5 - MAT/05 ANALISI MATEMATICA
ER -