Ph.D Thesis

Ph.D StudentKluzner Vladimir
SubjectMinimal Surfaces, Measure-Based Distance Functions and
Image Segmentation
DepartmentDepartment of Applied Mathematics
Supervisors PROFESSOR EMERITUS Gershon Wolansky


One of the primary goals of low level vision is image segmentation: given data g, defined as a function on the "pixel space" B, the objective is to deduce an image u which is composed of sub-domains, wherein the image is basically homogeneous, separated by sharp discontinuities (edges). It has been shown that a large number of algorithms for image segmentation are closely related to Mumford-Shah functional minimization. This functional involves a tradeoff between the image structure, which is a two-dimensional surface, and the contours that surround objects or distinct regions in the image, which are one-dimensional parametric curves. This functional was first suggested and analyzed in its one-dimensional case by Mumford and Shah for gray level images.

The above functional was later extensively studied. In particular, the Gamma-convergence framework was invented to overcome the problem of dealing with objects with different dimensionalities in the same functional. The idea is to approximate the functional by a different, parameter dependent functional that is expected to be more regular. The approximating functional approaches the original one in the limit, while the parameter goes to zero. According to this approach, minimizers of approximating functional approximate the minimizer of original one, while enjoying greater regularity.

In this study we propose an alternative functional to Mumford-Shah's one. The proposed functional is independent of parameterization; it is a geometric functional which is given in terms of the geometry of surfaces representing the data and image in the feature space. The Gamma-convergence technique is merged with the minimal surfaces theory in order to yield a global generalization of the Mumford-Shah segmentation functional. Eventually, we apply the above functional model to the inpainting problem. The concept behind our contribution is a slight modification of the developed metric term. We illustrate the model by various numerical results in one-dimensional and two-dimensional cases.