· 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 LIT233
· Office Hours: MWF 4 or by appointment
· Course Outline:
This course is 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. Image Segmentation
(1). Edge based active contour method:
Snake model, Geodesic active contour model, Level set formulation;
(2). Region based active
contour method: Mumford-Shah's
model and CV model, Region based active contour with parametric density
estimation, Region based active contour with non-parametric
density estimation; Maximizing likelihood function (MLE), Maximizing a
posterior estimate (MAP), Non-parametric density estimation.
(3). Hybrid methods:
Combination of edge and regional intensity information, or incorporating prior
information into image segmentation;
(4). Fuzzy/ soft segmentation
methods: Relaxing models and global minimum.
· 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.
· Unit 3: Image
Restoration and Reconstruction
(1).
Subdifferential, Shrinkage operator, Fenchel transform;
(2). First-order gradient method, Iterative shrinkage-thresholding algorithm (ISTA), and fast iterative
shrinkage-thresholding algorithm (FISTA); Operator
splitting;
(3). Bregman
iterative algorithm, Split Bregman, Augmented Lagranging, Alternating minimization algorithm (AMA),
Alternating direction method of multipliers (ADMM), Linearized
Bregman, Bregman operator
splitting (BOS);
(4). First-order primal-dual
algorithms, and Chambolle's method for solving dual
problem;
(5). Primal-dual hybrid
gradient (PDHG) algorithm;
(6). Applications in image denoising,
deblurring, and multi-channel MR image reconstruction with arbitrary
under-sampling pattern.
· 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