Approximation, Computation, and Distortion in the Variational Formulation

Abstract

Mumford and Shah [263][264] suggested performing edge detection by minimizing functionals of the form

$$ E(f,K) = \beta {\smallint _\Omega }{\text{ }}{(f - g)^2}dx + {\smallint _{\Omega - K}}|\nabla f{|^2}dx + \alpha |K|,$$

where Ω is the image domain (a rectangle), dx denotes Lebesgue measure, g is the observed grey level image, i.e., a real valued function, f approximates g, K denotes the set of edges (a closed set), | K | is the total length² of K, and β and α are real positive scalars. This approach is a modification of one due to Geman and Geman [132] that uses Markov random fields, which was developed by Marroquin [249] and by Blake and Zisserman [34]. It is referred to as the variational formulation of edge detection.

Research supported by the US Army Research Office under grant ARO DAAL03-92G-0115 (Center for Intelligent Control Systems).