Stability in a class of variational methods

The purpose of this work is to investigate the stability property of some models which are currently used in image processing. Following L. Rudin, S.J. Osher and E. Fatemi, we decompose an image f is an element of L-2(R-2) as a sum u + v where u belongs to BV(R-2) and v belongs to L-2(R-2). The Banach space BV is aimed at modeling the objects contained in the given image. the optimal decomposition minimizes the energy J(u) = parallel to u parallel to BV + lambda parallel to f - u parallel to(2)(2). We denote Phi(f) = (u) over bar this optimal solution. After recalling decomposition minimizes the energy J(u) = parallel to u parallel to(BV) + lambda parallel to f - u parallel to(2)(2). some properties of that optimal decomposition, we prove the stability of the mapping (P. Moreover, we generalize the stability result to other models where the Banach space BV is replaced by other functional Banach spaces E. (c) 2007 Elsevier Inc. All rights reserved.