Welcome to this new course for the fall semester, 2006, developed by Dr. Jens Rittscher from GE Global Research Center and Professor Charles Stewart at Rensselaer. The syllabus is now available, but must be considered only tentative. (We are making this up as we go!) This simple web page provides the lecture notes, which we will try to post one lecture in advance, the homework, including pointers to data sets as well as announcements about corrections and clarifications, and pointers to web resources. Students are invited to make suggestions about additional resources that the class as a whole might find helpful.

- Lecture 1 Random Variables. 28 August 2006.
- Lecture 2 Maximum Likelihood and Bayesian Estimation. 31 August 2006.
- Lecture 3 Non-Parametric Density Estimation. 7 September 2006. This includes a discussion of the use of mutual information in image registration.
- Lecture 4 Gaussian Mixture Models and EM. (Complete as of Sept 11 at 10:45 am, but then revised again by 11:45 am:))
- Lecture 5 (updated Thurs, 9/21) Introduction to Markov Random Fields
- Lecture 6 (updated Monday, 9/25) Sampling methods
- Lecture 7 Approximate Inference
- Lecture 8 Belief Propagation algorithms and application to the stereo correspondence problem. This is just an outline of our reading and in-class discussion.

- Homework 1, due Thursday September 7. The
test images will be available soon.

Notes:- (Friday, Sept 1): On problem 2, you may use the results from problem 5. On problem 4, start by considering the value of sigma (the variance in the data) to be known and then consider sigma unknown.
- (Monday, Sept 4): Here are three images from the Berkeley database to try: image 80 image 88 image 98
- (Tuesday, Sept 12): Graded papers are outside Prof. Stewart's office door.

- Homework 2 (revised 9/07 at 10:25 pm), due
Thursday September 14.

- On problem 2 you may assume you are working with scalars, not vectors.
- (9/19, 8:30am) I have finished grading. The papers will be placed outside my office door soon. A number of students struggled with problems 2 and 3, so we will go over these at the start of class on Thursday.

- Homework 3 due
Thursday September 21.

- (9/19, 7:40am) In problem 3, the correct definition should be

P^c_{i,j} := prob(x_{t+c} = j | x_t = i)

- (9/19, 7:40am) In problem 3, the correct definition should be
- Homework 4 (modified Tuesday 9:30am) due
Monday October 2nd

- The data sets for problem 1 are
em_set1.txt,
em_set2.txt, and
em_set3.txt. The first one is the
easiest. In order to help you debug, here are the statistics
for the first set:
- the means are (-5,3) and (1,3),
- the mixture fractions are pi_1 = .32 and pi_2 = 0.68,
- and the eigenvalues of the covariance matrices (that generated the data) are (2.25,0.25) and (5.0625,0.5625)

- The data sets for problem 1 are
em_set1.txt,
em_set2.txt, and
em_set3.txt. The first one is the
easiest. In order to help you debug, here are the statistics
for the first set:
- Homework 5
- Homework 6

- Mathworld is a hyperlinked dictionary of mathematical terms and topics.
- Perhaps surprisingly, Wikipedia contains many short articles on topics of mathematical interest as well as on the various application areas we are considering.
- The Matrix Cookbook by Peterson and Pederson is a good reference for problems in matrix manipulation, especially those involving derivatives.