In this article I compile a list of good papers and tutorials related to MRFs, CRFs and DRFs. Hopefully you will find it useful.

Recently I have been interested in conditional random fields (CRFs) for image modeling/labeling. I had really difficult time finding good materials to read. In this post, I would like to dedicate to people who are having a difficult time understanding CRFs, particularly, for image classification. My goal is to save your time by pointing you out to some good and useful materials, so that you don’t have to waste a lot of time like I did in past few weeks.

You might come up with some questions like what are the differences between CRF vs Bayesian networks (BNs) or between CRF vs  MRF? What are the advantages of CRFs which are discriminative models over generative models like MRF and BN? What are the relationships between CRFs and other fundamental statistics models e.g. logistic regression and log-linear model? and most importantly…I’m a newbie..where should I get started?

Here are the list of materials:

1. Log-linear Models and Conditional Random Fields by Charles Elkanhttp://videolectures.net/cikm08_elkan_llmacrf/ . I think this should be the first material you might want to learn from. The instructor did a really good job giving the overview of fundamental topics on statistics, e.g. maximum likelihood, logistic regression, log-linear model, then connect the idea to  CRF at the end. However, in this lecture, there is not much connection between CRFs and other graphical models.
2. Discriminative Random Fields (IJCV paper) by Sanjiv Kumar and Martial Hebert. For me, this is the best paper  talking about CRFs for image classification/labeling. The paper discusses about MRF, BN in brief, then points out the main problems using those models, and shows how CRF can solve the existing problems.
3. Models for Learning Spatial Interactions in Natural Images for Context-Based Classification (PhD thesis) by Sanjiv Kumar. If you like the paper [2] above and would like to see more detail of how to derive some learning formula, then you might want to see the PhD thesis of this paper which provides a lot mode details and images for better understanding.
4. An Introduction to Conditional Random Fields for Relational Learning by Sutton, C., McCallum, A. (tutorial paper). This is a good and pretty long tutorial paper. What I like in this paper is that the paper motivates readers by some good examples especially in natural language processing which is a good application to show the power of CRFs. Another good thing is that this paper shows some connections between CRFs and some other graphical models.

At some point, you might feel that CRFs are closely related to MRFs. For those who are not familiar to MRFs, there are some good books and papers I would lkike to recommend:

1. Markov random field modeling in image analysis by Stan Z. Li — This might be the best book on MRFs so far as it explains almost everything about MRFs in considerable details ranging from Gibbs random fields, MRFs, CRFs, DRFs, energy functions, smoothness constrains, learning algorithms, inference algorithms, etc. I really recommend this book if you have TIME to read it. However, this book seems to focus more on theoretical than real example aspect.
2. Image processing: dealing with texture by Maria Petrou & Pedro García Sevilla — I like this book because there are a lot of good examples on how MRFs, energy functions, etc look like in practice. This would be a good book to read parallel to the book from Stan Z. Li.
3. Image analysis, random fields, and dynamic Monte Carlo methods: a mathematical introduction by Gerhard Winkler
4. Markov Random Field Models: A Bayesian Approach to Computer Vision Problems (technical report) by Gerda Kamberova. — This is a free, good , and concise report on MRFs or computer vision.