A new proof of regularity for¶two-shaded image segmentations

In this paper, we consider minimizing the Mumford-Shah functional over two-valued functions in the plane, which is equivalent to minimizing over characteristic functions. Existence of minimizers is straightforward and we show that any minimizing set is essentially open, has a boundary with finitely many connected components, and each component is C ¹. The relatively quick proof does not rely on quasi-minimal surface or varifold theory, but on uniform estimates on the regularity of the boundary. Received: 4 February 1997 / Revised version: 4 February 1998