References:
(1). Numerical Optimization by Jorge Nocedal, Stephen J Wright
(2). Measure Theory and Fine Properties of Functions by L.C.Evans and R.F.Gariepy
(3). Mathematical Problems in Image Processing - PDE and the Calculus of Variations by Gilles Aubert and Pierre Kornprobst;
(4). The Handbook of Mathematical Models in Computer Vision by Nikos Paragios, Yunmei Chen, and Olivier Faugeras;

Meeting Time and Room: MWF 5 at LIT223

Office Hours: MWF 4 or by appointment

Course Outline:
This course is mainly devoted to the study of the theory and numerical optimization techniques for solving linear inverse problems with total variation regularization, the theory and computational methods for compressed sensing, and the methods for data analysis using nonparametric statistics. The applications of these methods in image reconstruction, denoising, and segmentation will also be discussed. Students are expected to gain knowledge on mathematical theories, methods, and practical experience in solving real world problems.

Arrangement of the course:

Unit 1: Methods for solving linear inverse problem with total variation regularization.
1. Variable splitting, quadratic penalty, continuation method
2. Bregman iteration and split Bregman
3. Primal-dual algorithms, Chambolle's method, and primal-dual hybrid gradient (PDHG) algorithm
4. Operator splitting, conjugate duality for sub-differential
5. Bregman operator splitting
6. Integration of split Bregman or linearized split Bregman with the Barzilai-Borwein method for fast convergence
7. Applications in image reconstruction, denoising, and compressed sensing.

Unit 2: Theory of total variation and applications in image analysis
1. Introduction to TV norm and space of functions of bounded variation
2. Connection to level-sets problems and regularity of solutions
3. Algorithms for solving TV-like problems
4. Applications in image analysis

Unit 3: Theory and methods for compressed sensing.
1. Sparse representation, approximation, and modeling
2. Dictionary learning and KSVD
3. Applications in image and signal processing

Unit 4: Image analysis using nonparametric statistics
1. Bayesian Inference, posterior, Markov random field, Gibbs random field
2. Nonparametric density estimation, Regression
3. Entropy, Computation of entropy
4. Data analysis, PCA,
5. Applications in image denosing and segmentation

Unit 5: New development on image segmentation, registration, and diffusion weighted MRI.

Grading:
Students will be required to present one to two papers and the projects related to the course content. These projects may be related to problems of particular interest to the individual student. Grades will be assigned on the basis of these projects. Current UF grading policies can be found from the following link http://www.registrar.ufl.edu/catalog/policies/regulationgrades.html