· References:
(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
· 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