Segmentation of Images by Variational Methods: a Constructive Approach
Ceremade, Univ. Paris-Dauphine
Place du Maréchal de Lattre
75775 Paris Cedex 16 — France
Viale Miramare 325
35014 Trieste — Italy
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.