Colloquium Schedule for Spring 1998 |

Regularity of the singular sets for the Mumford-Shahfunctional in image processing


Guy David
MSRI and Université Paris-Sud

The Mumford-Shah functional is one of the most used tools for image segmentation, but it also gives a very beautiful free boundary problem. Let $\Omega $ be a nice domain in the plane (the screen) and g be a bounded measurable function on $\Omega $. The functional is given by

\begin{displaymath}J(u,K) = \int \int _{\Omega \backslash K } \vert u-g\vert^2 ... ...\int _{\Omega \backslash K } \vert\nabla u\vert^2 + H^1(K) \ , \end{displaymath}

where the authorized competitors are pairs (u,K) such that K is a closed subset of $\Omega $ with finite Hausdorff measure H1(K) and uis a function on $\Omega \backslash K $ with one derivative $\nabla u $ in $L^2(\Omega \backslash K)$.


Minimizers for J are known to exist, and various regularity results for K have been proved, but the conjecture that (after trivial reduction)K is a finite union of C1-curves is still unproved. I will try todiscuss recent progress by A. Bonnet and J.-C. Léger (using blow-ups andthe notion of global minimizers on the plane).