Singular Sets of Minimizers for the Mumford-Shah Functional
Image processing is ranked among the most topical sources of inspiration for recent mathematical analysis. The Mumford Shah functional has been proposed as a model for image segmentation. Given a bounded measurable function g on an n-dimensional domain G (the most important case is n=2) that represents the original image, we look for another function u that represents a simplified image. The balance between simplicity and fidelity is expressed by the functional J, which is composed from three summands. The first part of J is the fidelity, which is represented by the distance of u from g measured by the square of the L2-norm. The second part is the (n-1)-dimensional Hausdorff measure of the singular set K of u. In particular, we believe that the singular set found here describes the boundaries between objects that are displayed on the picture. The function u is allowed to jump across K whereas it is assumed to be smooth outside K. The last summand is the Dirichlet integral of u outside of K, which measures the smoothness of u. The major problem in dimension 2 is the celebrated Mumford-Shah conjecture, which claims that if u minimizes J then the singular set K is a finite union of C1 arcs. This would help very much in understanding the planar case.
Even less complete is the knowledge in the higher dimensional case. The open problems are of a fine nature, which is perhaps not so exciting for technically oriented readers but which stimulates the development of mathematics. The importance of the functional does not consist only of its interpretation but more in the fact that it represents a whole class of free boundary problems whose theory will profit from methods developed for this particular model case. This is also the main motivation of the author. Hence, he does not write a handbook of image segmentation for practitioners, but instead a monograph on the Mumford-Shah Theory.
For the definition of a global minimiser, the fidelity part is omitted (hence this is independent of data). There are four types of global minimisers of the functional: a constant function, a function attaining two values separated by a line, a function attaining three values separated by a propeller (a union of three half-lines emanating from the same point forming angles of 120 degrees), and finally the so called cracktip, a special nonconstant solution with a singular half-line. The Mumford-Shah conjecture is equivalent to the conclusion that each minimiser at each point looks locally like one of these fundamental global minimisers.
This discussion is one of the most important achievements of the book. Although the book is focused on the regularity theory in dimension 2, the existence issue and general dimension problems are also treated. The text is comprehensible for graduate students. The author endows the book with his enthusiasm and the presentation sometimes seems like a fascinating adventure. Readers who know the field will surely profit from the development and find inspiration for new inventions in the theory of free boundary problems. The book has been awarded the Ferran Sunyer i Balaguer 2004 prize.