(1). Partial Differential Equations by L.C.Evans;
(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, Nikos Paragios, Yunmei Chen, and Olivier Faugeras;
· Meeting Time and Room: MWF 5 at LIT237
· Office Hours: MWF 4 or by appointment
· Course Outline:
This course continuous the course MAP7436 in the Fall of 2011 as an introduction to mathematical imaging. We will study basic mathematical methods for modeling and solving certain fundamental problems in image analysis, such as image segmentation, registration, restoration, and reconstruction. We will focus on the use of variational methods, statistical dependence measures and analysis for modeling. We will also learn several currently developed efficient numerical methods for solving large scale and ill-conditioned linear inverse problems with total variation regularization, and discuss their applications in solving image analysis problems. Students will gain knowledge on mathematical theories, methods, and practical experience in solving real world problems in image analysis.
· Arrangement of the course:
Unit 1: 1. Gradient Based Optimization Methods for Non-smooth and Convex Functional with Applications to Image Restoration and Reconstruction
(1). Gradient based optimization methods for differentiable and convex functional:
Unconstrained minimization, Gradient method with Barzilai-Borwein (BB) step size.
Constrained minimization with linear equality constraints, Method of multiplies, Alternative direction method of multiplies (ADMM).
Constrained minimization with inequality constraints, Karush-Kuhn-Tucker (KKT) conditions.
(2). Convexity, Subdifferential, Shrinkage operator, Fenchel transform;
(3). Bregman iterative algorithm, Split Bregman, Augmented Lagranging, Linearized Bregman, Bregman operator splitting (BOS);
(4). First-order primal-dual algorithms, and Chambolle's method for solving dual problem, Primal-dual hybrid gradient (PDHG). algorithm.
(5). First-order gradient method, Iterative shrinkage-thresholding algorithm (ISTA), and fast iterative shrinkage-thresholding algorithm (FISTA); Operator splitting;
(6). Applications in image denoising, deblurring, and multi-channel MR image reconstruction with arbitrary under-sampling pattern
· Unit 2: Image Registration
(1). Mono-modal image registration: Rigid and deformable segmentation, MLE and MAP for intensity difference, Correlation coefficient (CC), Smoothness of deformation field;
(2). Multi- modal image registration: Joint entropy, Mutual information, Kullback-Leibler (KL) distance, Renyi’s statistical dependence measure, Models based on these measures and their local versions;
(3). Joint segmentation and registration
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