The New Family of Cracked Sets and the Image Segmentation Problem Revisited

Abstract

The object of this paper is to introduce the new family of cracked sets which yields a compactness result in the W ^1,p-topology associated with the oriented distance function and to give an original application to the celebrated image segmentation problem formulated by Mumford and Shah [21]. The originality of the approach is that it does not require a penalization term on the length of the segmentation and that, within the set of solutions, there exists one with minimum density perimeter as defined by Bucur and Zolesio in [3]. This theory can also handle N-dimensional images. The paper is completed with several variations of the problem with or without a penalization term on the length of the segmentation. In particular, it revisits and recasts the earlier existence theorem of Bucur and Zolesio [3] for sets with a uniform bound or a penalization term on the density perimeter in the W^1,p-framework.