Segmentation of images by variational methods: a constructive approach. (English) Zbl 0679.68205

Morel, Jean-Michel; Solimini, Sergio

Rev. Mat. Univ. Complutense Madr. 1, No. 1-3, 169-182 (1988).

By an “image” there is understood a real function g on an open rectangle \(R\subset {\mathbb{R}}^ 2\), the value g(x,y) being the “grey- level” at the point (x,y). An “image-segmentation” is a pair (u,B), where B is a finite set of piecewise \(C^ 1\) curves, called “contours”, and u is a real function which is regular on connected components of \(R\setminus B\). In a “good” segmentation (u,B) the curves of B should be the boundaries of homogeneous areas in the image and u a sort of mean of g in the interior of such areas. The authors prove that the minimum of the functional \[ E(u,B):=\| u-g\|_{L^ 2(R)}+length(B) \] is attained at some B.

Reviewer: G.Maeß

MSC:

68U99	Computing methodologies and applications
65D99	Numerical approximation and computational geometry (primarily algorithms)
49J99	Existence theories in calculus of variations and optimal control
65K99	Numerical methods for mathematical programming, optimization and variational techniques

Keywords:

total least squares; variational methods; image processing; computer graphics; image-segmentation