Image Segmentation Using Active Contours: Calculus of Variations or Shape Gradients?

Gilles Aubert 1 Michel Barlaud 2 Olivier Faugeras 3 Stéphanie Jehan-Besson 2
3 ODYSSEE - Computer and biological vision
DI-ENS - Département d'informatique de l'École normale supérieure, CRISAM - Inria Sophia Antipolis - Méditerranée , ENS Paris - École normale supérieure - Paris, Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech
Abstract : We consider the problem of segmenting an image through the minimization of an energy criterion involving region and boundary functionals. We show that one can go from one class to the otherb y solving Poisson's orHelmholtz's equation with well-chosen boundary conditions. Using this equivalence, we study the case of a large class of region functionals by standard methods of the calculus of variations and derive the corresponding Euler-Lagrange equations. We revisit this problem using the notion of a shape derivative and show that the same equations can be elegantly derived without going through the unnatural step of converting the region integrals into boundary integrals. We also define a larger class of region functionals based on the estimation and comparison to a prototype of the probability density distribution of image features and show how the shape derivative tool allows us to easily compute the corresponding Gˆateaux derivatives and Euler-Lagrange equations. Finally we apply this new functional to the problem of regions segmentation in sequences of color images. We briefly describe our numerical scheme and show some experimental results.
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SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2003, 63 (6), pp.2128-2154. 〈10.1137/S0036139902408928〉
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Gilles Aubert, Michel Barlaud, Olivier Faugeras, Stéphanie Jehan-Besson. Image Segmentation Using Active Contours: Calculus of Variations or Shape Gradients?. SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2003, 63 (6), pp.2128-2154. 〈10.1137/S0036139902408928〉. 〈hal-00964649〉

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