On the Regularity of Singular Sets of Minimizers for the Mumford-Shah Energy

Edge detection is an image processing technique for finding the boundaries of objects within images. The Mumford-Shah functional was introduced by Mumford and Shah in 1989 as a variational model for image reconstruction. The most important regularity problem is the famous Mumford-Shah conjecture, which states that (in 2 dimensions) the closure of the jump set can be described as the union of a locally finite collection of injective $C^1$ arcs that can meet only at the endpoints, in which case they have to form triple junctions. If a point is an endpoint of one (and only one) of such arcs, it is called a cracktip. In this talk, I plan to introduce the problem, some of the techniques of the calculus of variations, and survey results concerning the regularity of Mumford-Shah minimizers and their singular sets. If time allows I will discuss more recent developments (based on joint work with Camillo De Lellis and Matteo Focardi). This talk requires no previous knowledge of the problem or of calculus of variations.