Variational methods in image processing

Abstract

The purpose of this thesis is to investigate the mathematical properties of some models which are currently used in image processing. Generalizing an approach by S.J. Osher, L. Rudin and E. Fatemi, we decompose an image f ∈ L 2 (R 2) as a sum u + v where u belongs to some functional Banach space E while v belongs to L 2 (R 2). The Banach space E is aimed at modeling the objects contained in the given image and the optimal decomposition minimizes the energy J(u) = u E + λ f − u 2 2. The main difficulty is to choose an adapted Banach space E. The common choices are E =Ḃ 1,1 1 (R 2) which leads to the well-known Donoho's wavelet thresholding or E = BV (R 2) the space of functions of bounded variations. The latter choice is the Osher Rudin Fatemi algorithm. These two choices are suffering from severe drawbacks. In the first case, sharp edges are erased. The second choice does not lead to a wavelet thresholding. That is why we propose E =Ḃ 1,∞ 1 (R 2) which yields sharp edges and is given by wavelet thresholding. This is the the two first parts of the thesis. In the third part, we investigate the mathematical properties of the Osher-Vese newest algorithm which keeps track of the textured components.

Modeling Oscillatory Components with The Homogeneous Spaces BM^-α and W^-α,p*

“…Meyer's G-model is approximated and studied in [51]- [52], [39], [6], [5], [8], [38], [22], [53], [12], [29], [27], among others. Meyer's E model was studied and discussed in [7], [18] and [26].…”

“…Other related decomposition models using wavelets are by I. Daubechies and G. Teschke [14], R. Coifman and D. Donoho [13], J. L. Starck, M. Elad, and D. Donoho [42], F. Malgouyres [33], S. Lintner and F. Malgouyres [32], A. Haddad and Y. Meyer [22], A. Haddad, [23], or J. Gilles [20].…”

The Total Variation Regularized L1 Model for Multiscale Decomposition

“…We give a short proof below based on Lemma 7.1. A similar approach is also used in [28] to derive the G-norm related properties for the ROF model [39].…”

Image Regularity and Fidelity Measure with a Two‐Modality Potential Function

“…Meyer [2] shows that the ROF model is ineffective in cartoon-texture decomposition because the 2 norm is not a good measure of the texture, yet the TV is effective in measuring the cartoon component. To overcome the ineffectiveness of the 2 norm in measuring the texture component, Meyer [2] and Haddad and Meyer [16] proposed using the -norm, Vese and Osher [17] approximated the -norm by the div( ) norm, Osher et al [18] proposed using the −1 norm, Lieu and Vese [19] proposed using the more general − norm, and Le and Vese [20] proposed using the div(BMO) norm to measure the texture component. However, the models involving these norms are difficult to solve.…”