Variational approach to image segmentation

This talk deals with free discontinuity problems related to image segmentation, focussing on the mathematical analysis of Blake & Zisserman functional. Calculus of Variations is the framework where energy minimization and equilibrium notions find a precise language and formalizations by means of variational principles. Image segmentation is a relevant problem both in digital image processing and in the understanding of biological vision. There exist many different way to define the tasks of segmentation (template matching, component labelling, thresholding, boundary detection, quad-trees, texture matching, texture segmentation) and there is no universally accepted notion (optimality criteria for segmentation, analogies and differences between biological and automata perspective in segmentation): here the exposition is confined to some models for decomposing an image field, where is given a function describing the signal intensity associate to each point (typically the light intensity on a screen image). Such purpose has a clear connection with the problem of optimal partitions of a domain minimizing the length of the boundaries. In simple words the segmentation we look for provides a cartoon of the given image satisfying some requirements: the decomposition of the image is performed by choosing a pattern of lines of steepest discontinuity for light intensity, and this pattern will be called segmentation of the image. The variational formalizations of segmentation models provided deeper understanding of image analysis, produced intriguing mathematical questions (some of them still open) and entailed global estimates for geometric quantities in visual and automatic perception at both low and high level vision. We discuss some recent results based on the innovative notion of free discontinuity problem introduced by Ennio De Giorgi. This approach balances carefully signal smoothing and segmentation length. In such framework, modern tools of Geometric Measure Theory and recent developments about minimal surfaces and regularity of extremals in Calculus of Variations allow the study of problems coupling bulk and surface terms: in such context discontinuous (in the mathematical sense) solutions are admissible and sometimes their discontinuities are the main features of the solution