Appl Math Optim 41:51–85 (2000)
2000 Springer-Verlag New York Inc.
Asymptotic Behavior of a Geman and McClure Discrete Model
Istituto per le Applicazioni del Calcolo M. Picone, del C.N.R.,
Viale del Policlinico 137, 00161 Rome, Italy
Communicated by D. Kinderlehrer
Abstract. In this paper we consider a class of discrete variational models derived
from a theory of Geman and McClure (see ) and study their asymptotic behavior
when their stepsize tends to zero. It is shown that a result of -convergence toward
a certain functional holds true if a characteristic parameter of these models obeys a
well-deﬁned dependence law upon the stepsize. Under this condition the -limit is
a modiﬁed form of the Mumford–Shah functional.
Key Words. Computer vision, Variational convergence.
AMS Classiﬁcation. 45L05, 49F22.
Variational methods in image segmentation and signal recovery have found a precise
mathematical formulation in some continuous models, among which the functional in-
troduced by Mumford and Shah (see ) certainly plays an important role. The varia-
tional approach to those problems has also produced important discrete models, based on
optimization techniques, which are commonly used in the applications. Among these, a
class of discrete models, derived from a theory of Geman and McClure, provides a way
of ﬁnding the numerical solution to segmentation problems by minimizing the function
−→ R deﬁned by
(u) = h
(u) + ν