(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 MAT.0117
· Office Hours: MWF 4 or by appointment
· Course Outline:
This course is mainly devoted to the study of the following two topics: (1). the numerical methods for efficiently solving non-differentiable convex optimization problems arising from image analysis, especially, for solving total variation (TV) regularized large scale and ill-conditioned linear inverse problems in compressed sensing and image reconstruction/recovery, and (2). the statistical methods including nonparametric statistical estimators in data analysis with their applications in image segmentation, registration and reconstruction. All these study will be combined with the expoloration of the new development on image denoising, segmentation and registration. 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 TV regularized non-differentiable convex minimization problems.
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
· Unit 2: Some developments on the above methods with applications in image reconstruction, denoising, and compressed sensing.
1. Barzilai-Borwein steepest descent method
2. Integration of split Bregman or linearized split Bregman with the Barzilai-Borwein method for fast convergence.
3. Multi-channel MR image reconstruction with under-sampled data
· Unit 3: 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 denoising and segmentation
· Unit 4: New development on image segmentation, registration, and diffusion weighted MRI.
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