Full paper in PDF:

$% J.-M. Morel and S. Solimini, Segmentation of Images by Variational Methods: a
Constructive Approach, Rev. Mat. Complut. 1 (1988), no. 1, 2, 3, 169–182.%$

Segmentation of Images by Variational Methods: a Constructive Approach

Jean-Michel MOREL and Sergio SOLIMINI

Ceremade, Univ. Paris-Dauphine Place du Maréchal de Lattre 75775 Paris Cedex 16 — France | SISSA Viale Miramare 325 35014 Trieste — Italy |

ABSTRACT

We give a constructive proof that given a bounded function on a rectangle , the minimum of the following functional is achieved:

where is a finite set of curves in and is locally constant in . In image processing, can be interpreted as an image ( is the grey level at ) and the curves of as the contours of the image, representing the mean value inside each contour. Following ideas of Mumford and Shah, our proof suggests a method for transforming an image into a cartoon.

1980 Mathematics Subject Classification (1985 revision): 68U10, 68U05, 65D07, 49A99.