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 W1,p-framework.