Approximation by Γ-convergence of a curvature-depending functional in Visual Reconstruction

where Ω ⊂ R2 is a bounded open set (the image domain), H1 denotes the one-dimensional Hausdorff measure, and g ∈ L∞(Ω) is the input image. The functional has to be minimized over all closed sets C ⊂ Ω and all u ∈ C1(Ω \ C). The function u represents a denoised approximation of the input image g, and C represents the set of boundaries of the segmentation. The Mumford and Shah variational model can be extended to several visual reconstruction problems (see March [29]): computation of depth from stereo images (Shah [39]), computation of optical flow (Nesi [34]), shape from shading (Shah [40]). The existence of minimizers of GMS has been proved independently by Dal Maso, Morel and Solimini [21] and De Giorgi, Carriero and Leaci [23] using the compactness and lower semicontinuity theorems of Ambrosio [3]. Mumford and Shah [33] studied the properties of minimizers (u,C) of GMS assuming that C is a finite union of simple C1,1 curves meeting ∂Ω and meeting each other only at their endpoints. They proved that the vertices of C may only be: (i) triple points where three curves meet with equal angles; (ii) points on the boundary of Ω where one curve meets ∂Ω perpendicularly; (iii) ‘crack-tips’ where a curve ends and meets nothing. In Computer Vision, the constraints imposed on the segmentations highlighted by such results constitute a drawback of the variational model. In particular, corners and T -junctions, which are relevant features for pattern recognition, are distorted. Since the length measure is not sensitive to corners and junctions, to allow for such singularities in the segmentations it is necessary to consider curvature-depending energies. Functionals