Appl Math Optim 44:131–161 (2001)
2001 Springer-Verlag New York Inc.
A Study in the BV Space of a Denoising–Deblurring
Department of Mathematics, University of California, Los Angeles,
405 Hilgard Avenue, Los Angeles, CA 90095, USA
Communicated by R. Temam
Abstract. In this paper we study, in the framework of functions of bounded vari-
ation, a general variational problem arising in image recovery, introduced in .
We prove the existence and the uniqueness of a solution using lower semicon-
tinuity results for convex functionals of measures. We also give a new and ﬁne
characterization of the subdifferential of the functional, together with optimality
conditions on the solution, using duality techniques of Temam for the theory of
time-dependent minimal surfaces. We study the associated evolution equation in
the context of nonlinear semigroup theory and we give an approximation result in
continuous variables, using -convergence. Finally, we discretize the problems by
ﬁnite differences schemes and we present several numerical results for signal and
Key Words. Variational methods, Elliptic/parabolic PDEs, Functions of bounded
variation, Convex functions of measures, Duality, Relaxation, Maximal monotone
operators, -Convergence, Finite differences scheme, Signal and image processing.
AMS Classiﬁcation. 35, 49, 65.
This work was done while the author was at the Laboratoire Jean-Alexandre Dieudonn´e, from the
University of Nice - Sophia Antipolis, France.