ABSTRACT
We present in this chapter level set formulations for minimizing the Mumford and Shah functional for segmentation from Chapter 6, as proposed by Chan and Vese in [92], [94], [335], [93], [95] (see also the related work by Tsai, Yezzi, and Willsky [328], Samson et al. [280, 279], Amadieu et al. [17], and Cohen et al. [108, 109]). These works make the link between curve evolution, active contours, and Mumford-Shah segmentation. These models have been proposed by restricting the set of minimizers u to specific classes of functions: piecewise-constant, piecewise-smooth, with the edge set K represented by a union of curves or surfaces that are boundaries of open subsets of Ω. For example, if K is the boundary of an open and bounded subset of Ω, then it can be represented implicitly, as the zero-level line of a level set function φ. Thus the set K as an unknown is replaced by an unknown function φ that defines it implicitly, and the Euler-Lagrange equations with respect to the unknowns can be easily computed and discretized.
