STAT992 Statistical Methods in Signal and Image Analysis
Lectures 11:00-11:50 MWF
Place 1289 Comp S&St

noise brain surface mesh and its regulization via kernel smoothing

Course Requirement:
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.

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.

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