Topological gradient for a fourth order operator used in image analysis

This paper is concerned with the computation of the topological gradient associated to a fourth order Kirchhoff type partial differential equation and to a second order cost function. This computation is motivated by fine structure detection in image analysis. The study of the topological sensitivity is performed both in the cases of a circular inclusion and a crack. Résumé. Ce papier porte sur le calcul du gradient topologique associé à une équation aux dérivées partielles de type Kirchhoff et à une fonction coût d’ordre deux. Ce travail est motivé par la détection de structures fines pour des images 2D et 3D. L’étude de la sensibilité topologique est faite dans les cas d’une inclusion circulaire et d’un crack. 1991 Mathematics Subject Classification. 35J30, 49Q10, 49Q12, 94A08, 94A13. December 9, 2013. Introduction The notion of topological gradient which has been rigorously formalized in [13, 17] for shape optimization problems has a wide range of applications : structural mechanics, optimal design, inverse analysis and more recently image processing [6–8]. Roughly speaking the topological gradient approach performs as follows : let Ω be an open bounded set of R and j(Ω) = J(Ω, uΩ) be a cost function where uΩ is the solution of a given PDE on Ω. For small ǫ ≥ 0, let Ωǫ = Ω\x0 + ǫω where x0 ∈ Ω and ω is a given subset of R. The topological analysis provides an asymptotic expansion of j(Ωǫ) as ǫ → 0. In most cases it takes the form : j(Ωǫ) = j(Ω) + ǫ I(x0) + o(ǫ) (1) I(x0) is called the topological gradient at x0. Thus, in optimal design for example, if we want to minimize j(Ωǫ) it would be preferable to create holes at points x0 where I(x0) is “the most negative”. In practice, we keep points x0 where the topological gradient is less than a given negative threshold. In image processing the choice of the cost function is guided by the aimed application. For example for detection or segmentation problems, we have to choose a cost function which blows up in a neighbourhood of the structure we want to detect. Thus removing from the initial domain such a neighbourhood implies a large variation of the cost function and so a large topological gradient. In [8] the method was applied for edge detection by studying the topological sensitivity of j(Ω) = ∫ Ω |∇uΩ|dx where uΩ is the solution of a Laplace equation. For filament (or point) detection, the problematic is different. Indeed there