,
the minimum of the following functional is achieved:
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
Ceremade, Univ. Paris-Dauphine Place du Maréchal de Lattre 75775 Paris Cedex 16 — France | SISSA 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.