 
Summary: A VARIATIONAL METHOD IN IMAGE RECOVERY #
GILLES AUBERT + AND LUMINITA VESE #
SIAM J. NUMER. ANAL. c
# 1997 Society for Industrial and Applied Mathematics
Vol. 34, No. 5, pp. 19481979, October 1997 016
Abstract. This paper is concerned with a classical denoising and deblurring problem in image
recovery. Our approach is based on a variational method. By using the LegendreFenchel transform,
we show how the nonquadratic criterion to be minimized can be split into a sequence of halfquadratic
problems easier to solve numerically. First we prove an existence and uniqueness result, and then we
describe the algorithm for computing the solution and we give a proof of convergence. Finally, we
present some experimental results for synthetic and real images.
Key words. image processing, LegendreFenchel transform, partial di#erential equations, cal
culus of variations
AMS subject classifications. 35J, 49J, 65N
PII. S003614299529230X
1. Introduction. An important problem in image analysis is the reconstruc
tion of an original image f describing a real scene from an observed image p. The
transformation (or degradation) connecting f to p is in general the result of two phe
nomena. The first phenomenon is deterministic and is related to the mode of image
acquisition (for example, the computation of integral projections in tomography) or to
