Research Statement Teaching Statement Resume

Ali Haddad, CAM Assistant Professor Department of Mathematics University of California, Los Angeles Los Angeles, CA 90095-1555 Office: MS 7360 Phone: +1-310-825-4746 E-mail: ahaddad@math.ucla.edu

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. Fameti, we decompose an image f of L^2 as a sum u+v where u belongs to somme functional Banach space E while v belongs to L^2. The Banach space is aimed at modeling the objects contained in the given image and the optimal decomposition minimizes the energy J(u)=||u||_E+\lambda||f-u||^2_2. The main difficulty is to choose an adapted Banach space E. The common choice are E=\dot{B}^{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=\B1inf(\R^2) which yields sharp edges and is given by wavelet thresholding. This is 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.

STABILITY IN A CLASS OF VARIATIONAL METHODS
A.HADDAD. ACHA, Special Issue on Mathematical Imaging. Volume 23, Issue 1, Pages 57-73 (July 2007)

Abstract:

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 as a sum u+v where u belongs to and v belongs to . The Banach space BV is aimed at modeling the objects contained in the given image. the optimal decomposition minimizes the energy . We denote this optimal solution. After recalling some properties of that optimal decomposition, we pro ve the stability of the mapping Φ. Moreover, we generalize the stability result to other models where the Ban ach space BV is replaced by other functional Banach spaces E.

Abstract:

In this article, we investigate some mathematical properties of a new algorithm proposed by Meyer to improve the Rudin–Osher–Fatemi model (ROF) in order to separate objects and textures contained in an image. He pointed out the crucial role played by a certain norm called the G-norm or “dual norm,” denoted , and the main drawback for the ROF model: any image is considered to have a textured component. We are then interested in minimizing the functional u_BV+λv _*. The main Theorem 6.1 is about invariance and stability properties of the new algorithm. It was first implemented by Osher and Vese. In particular, we point out the role played by particular functions called extremal functions and characterize them.

Abstract:

In this work, we compare two models that were proposed to improve the well-known model of Rudin, Osher, and Fatemi. The first one, proposed by Meyer, is the BV-G model, and the second one is the BV-L^1 model. We state the similitudes between both models. In particular, we prove a characterization theorem for optimal decompositions for the BV-L^1 model. We then compare these models in the particular case of radial functions.

Abstract:

Several algorithms have been proposed to unveil the geometrical structure of a given image. We will focus on the ROF algorithm designed by Leonid Rudin, Stanley Osher, and Emad Fatemi. The ROF model can be used to detect the objects which are contained in an image and also to analyse its textured components. We then comment on some improved versions of the ROF algorithm.