Flatness and finiteness in the Mumford-Shah problem
The paper is concerned with the regularity properties of optimal pairs \((u,K)\) for the Mumford-Shah image segmentation problem. By a blow-up method introduced by A. Bonnet, this problem is closely related to the classification of the global minimizers (i.e., minimizers with respect to compactly supported perturbations which, loosely speaking, do not decrease the number of connected components at infinity). \textit{A. Bonnet} [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 13, No. 4, 485-528 (1996; Zbl 0883.49004)] was able to classify all globally minimizing pairs \((u,K)\) with \(K\) connected, thus proving the Mumford-Shah conjecture for sets \(K\) which locally have finitely many connected components. The classification in the general case is still an open problem. The main result of the paper is the replacement of Bonnet's connectedness assumption by a ``mean flatness'' property, stated in terms of the David and Semmes ``Big Pieces of Lipschitz Graphs'' property. The results are achieved using a Cauchy-type representation formula of \(u\) as a function of \(K\) which has an independent interest and more potential applications.
scientific article; zbMATH DE number 1344439