Appl Math Optim 51:163–182 (2005)
2004 Springer Science+Business Media, Inc.
Modeling Very Oscillating Signals.
Application to Image Processing
and Jean-Fran¸cois Aujol
Laboratoire J.A. Dieudonn´e, UMR CNRS 6621,
Universit´e de Nice Sophia-Antipolis, Parc Valrose,
06108 Nice Cedex 2, France
ARIANA, projet commun CNRS/INRIA/UNSA,
INRIA Sophia Antipolis, 2004 route des Lucioles,
BP93, 06902 Sophia Antipolis Cedex, France
Abstract. This article is a companion paper of a previous work  where we
have developed the numerical analysis of a variational model ﬁrst introduced by
Rudin et al.  and revisited by Meyer  for removing the noise and capturing
textures in an image. The basic idea in this model is to decompose an image f
into two components (u + v) and then to search for (u,v) as a minimizer of an
energy functional. The ﬁrst component u belongs to BV and contains geometrical
information, while the second one v is sought in a space G which contains signals
with large oscillations, i.e. noise and textures. In  Meyer carried out his study
in the whole R
and his approach is rather built on harmonic analysis tools. We
place ourselves in the case of a bounded set of R
which is the proper setting for
image processing and our approach is based upon functional analysis arguments. We
deﬁne in this context the space G, give some of its properties, and then study in this
continuous setting the energy functional which allows us to recover the components
u and v. We present some numerical experiments to show the relevance of the model
for image decomposition and for image denoising.
Key Words. Sobolev spaces, Functions of bounded variations, PDEs, Oscilla-
ting patterns, Image decomposition, Convex analysis, Optimization, Calculus of
AMS Classiﬁcation. 35J, 46E, 49J, 68U10.
J.-F. Aujol is now in the Dept. of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA.