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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.

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