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Unformatted text preview: Multiple Contour Finding and Perceptual Grouping as a set of Energy Minimizing Paths Laurent D. COHEN and Thomas DESCHAMPS CEREMADE, UMR 7534, Universite Paris-Dauphine 75775 Paris cedex 16, France firstname.lastname@example.org Philips Recherche France, Medical Imaging Systems Group, 51 rue Carnot, 92156 Suresnes, France email@example.com Abstract. We address the problem of finding a set of contour curves in an image. We consider the problem of perceptual grouping and contour completion, where the data is a set of points in the image. A new method to find complete curves from a set of contours or edge points is presented. Our approach is an extension of previous work on finding a set of contours as minimal paths between end points using the fast marching algorithm. Given a set of key points, we find the pairs of points that have to be linked and the paths that join them. We use the saddle points of the minimal action map. The paths are obtained by backpropagation from the saddle points to both points of each pair. We also propose an extension of this method for contour completion where the data is a set of connected components. We find the minimal paths between each of these components, until the complete set of these regions is connected. The paths are obtained using the same backpropagation from the saddle points to both components. Keywords:Perceptual grouping, salient curve detection, active contours, minimal paths, fast marching, level sets, weighted distance, reconstruction, energy mini- mization, medical imaging. 1 Introduction Since their introduction, active contours  have been extensively used to find the contour of an object in an image through the minimization of an energy. In order to get a set of contours of different objects, we need many active contours to be initialized on the image. The level sets paradigm [15,1] allows changes in topology. It enables to get multiple contours by starting with a single one. However, it does not give satisfying results when there are gaps in the data since the contour may propagate into a hole and then split into several curves where only one contour is desired. This is the problem en- countered with perceptual grouping when we try to group a set of incomplete contours. For example, in a binary image of a shape with holes and spurious edge points like the ones in figure 1, human vision can easily recover the missing boundaries, remove the spurious ones and complete the curves. Perceptual grouping is an old issue in computer vision. It has been approached more recently with energy methods [17,11,18]. These methods make use of a criteria for saliency of a curve component or for each point of 2 the image. This saliency measure is based indirectly on a second order regularization snake-like energy () of a path containing the point. However, the final curves are generally obtained in a second step as ridge lines of the saliency criteria after thresh- olding. In  a similarity between snakes and stochastic completion field is reported.olding....
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