Applied Mathematics and Optimization

, Volume 44, Issue 2, pp 131–161

A Study in the BV Space of a Denoising—Deblurring Variational Problem

DOI: 10.1007/s00245-001-0017-7

Cite this article as:
Vese, L. Appl Math Optim (2001) 44: 131. doi:10.1007/s00245-001-0017-7


In this paper we study, in the framework of functions of bounded variation, a general variational problem arising in image recovery, introduced in [3]. We prove the existence and the uniqueness of a solution using lower semicontinuity 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 Γ -convergence. Finally, we discretize the problems by finite differences schemes and we present several numerical results for signal and image reconstruction.

Key words. Variational methods, Elliptic/ parabolic PDEs, Functions of bounded variation, Convex functions of measures, Duality, Relaxation, Maximal monotone operators, Γ -Convergence, Finite differences scheme, Signal and image processing. AMS Classification. 35, 49, 65. 

Copyright information

© 2001 Springer-Verlag New York Inc.

Authors and Affiliations

  • L. Vese
    • 1
  1. 1.Department of Mathematics, University of California, Los Angeles, 405 Hilgard Avenue, Los Angeles, CA 90095, USA US

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