Lectures 11:00-11:50 MWF

Place 1289 Comp S&St

noise brain surface mesh and its regulization via kernel smoothing

The course requirement is minimal and accessible for any graduate students who knows multivariate normal distributions and statistical inference and estimation. All topics will be self contained. Other than undergraduate level mathematical statistics concepts, I will introduce everything from scratch. For homework and projects, you can use any programming language such as R/Splus, MATLAB, JAVA, C++ but the language of instruction will be in MATLAB due to its versatility of image manipulation.

Topics Covered: will not be covered in sequential order.

01. discrete and continuous random fields. simulation of random fields. Hilbert space.

02. signal and image filtering and smoothing.

03. scale space. optimal filtering. bandwidth selection.

04. basics of Riemannian metric tensor geometry for image manipulation.

05. diffusion processes and diffusion equations. self-adjoint operators.

06. finite element methods and other numerical schemes.

07. smoothing scalar, vector and tensor data. smoothing manifolds.

08. stochastic optimization. EM algorithm.

09. Bayesian methods in image analysis. Gibbs sampler. image segmentation.

10. functional data analysis: curve and image registration. deformation of images.

11. shape analysis. directional statistics.

12. statistical inference, estimation and simulation in abstract space and manifolds.

Evaluation:

Under no circumstance except medical or religious reasons, late homeworks or projects will be accepted. It is encouraged to discuss some hard problems with classmates only verbally but no plagiarism of any sort will be permitted. Make sure you understand what is plagiarism. You have an option of choosing

1. 20% class project + 30% independent project (your own research area) + 50% biweekly homeworks.

2. 50% class project + 50% biweekly homeworks.

3. 30% class project + 70% biweekly homeworks.

Independent project should be based on techniques covered in the class after consultation with the instructor. No credit will be given to a project that has no connection with the course material. The first week after spring break, please submit the title and abstract of project you will be doing. There will be about 30 challenging homework problems. You have an option of solving 2/3 of problems. So if you are more theoretical person, you may skip some computation heavy problems. You are required to submit homeworks biweekly. Most of problem will be somewhat open ended and worth 5 points. Solving them correctly will give you 4 points. To get an additional point, impress me by doing something creative and inventive. Time consuming and difficult homework problems will be usually assigned as class projects. To get credit for project, submit typed project report of minimum 7pages (20%),10pages (30%) or 15 pages (50%) excluding figures and computer codes plus a disk containing working program. The deadline for submitting the independent project is one week before the last lecture. For class projects, the day of the last lecture.

Textbook:

Edward R. Dougherty, Random Processes for Image and Signal
Processing, IEEE Press 1999. ISBM: 0-8194-2513-3. It is not necessary
to buy this textbook but I recommend you to have one very good math
book on continuous stochastic processes.

Lecture Notes & Homeworks

Lecture 00. Image Analysis framework

Lecture
01. Gaussian random fields

Lecture
02. Linear operators on fields

Lecture
03. Kernel smoothing I.

Lecture
04. Numerical implementation of kernel smoothing

Lecture
05. Diffusion equations

Lecture
06. Iterated kernel smoothing

Lecture
07. Isotropic kernels in manifolds

Lecture
08. Kernel smoothing in manifolds

Lecture
09. Simulating Gaussian fields

Lecture
10. Hilbert Space.

Lecture
11. Karhunen Loeve expansion I.

Lecture
12. Karhunen Loeve expansion II.

Lecture
13. Kernel Smoothing II.

Lecture
14. Kernel Smoothing III.

Lecture
15. Diffusion smoothing I.

Lecture
16. Diffusion smoothing II.

Lecture
17. Laplace Operator.

Lecture
18. Multiple Comparisions I.

Lecture
19. Multiple Comparisions II.

Lecture
20. Bivariate smoothing on sphere I.

Lecture
21. Bivariate smoothing on sphere II.

Lecture
22. Brownian motion II.

Lecture
23. Maxima of random fields I.

Lecture
24. Maxima of random fields II.

Lecture
25. Curve Modeling I.

Lecture
26. Curve Modeling II.

Lecture
27. Diffusion smoothing on manifolds II.

Lecture
28. Diffusion smoothing on manifolds III.

Lecture
29. Finite element method

Lecture
30. Curve modeling III.

Lecture
31. Curve modeling IV.

Lecture
32. EM algorithm I.

Lecture
33. EM algorithm II.

Lecture
33. EM algorithm III appendum.

Lecture
34. Anisotropic smoothing

Lecture
35. Smoothing periodic functional data.

Lecture
36. Smoothness of spatial noise.

Lecture
37. Anisotropic smoothing II.

Lecture
38. Image registration I.

Lecture
39. EM algorithm IV..

Lecture
40. Image registration II.

Lecture
41. References