# What Is the Difference Between MRF and Total Variation in Noise Removal?

I have a general question in image processing. I have a noisy image. I would like to classify the noisy image into some regions. Two famous approaches I can use are:

1. MRF/Gibbs MRF: Model the spatial dependence between neighborhood pixels.
2. Total Variation: Key idea may be based on smallest variation of the image.

My question is: Could you tell me what is different between two approaches for noise removal? What is the key idea of TV? Which one is better? Thanks.

These are two different concepts that you talk about. First, MRF gives you a framework to do discrete optimization of problems, which respect the Markovian property, that is a pixel is conditioned only on the neighboring ones (roughly stated). Typical applications include binary or multi-class labeling problems. Total variation on the other hand, is generally used as a regularization by adding the integral of the absolute gradient of the signal/image to the energy functional. This helps to neglect irrelevant detail and focus on important ones.

You cannot say one is better than the other, as they are not exactly contradictory things. It depends on the application and the energy function you use in the MRFs.

An Introduction to Total Variation for Image Analysis is a good total variation tutorial to begin with. Also, Chambolle provides a link between total variation and binary MRFs, leading to new algorithms in the work entitled Total Variation Minimization and a Class of Binary MRF Models:

Total Variation Minimization and a Class of Binary MRF Models, Antonin Chambolle, 2005

• Thank Tolga Birdal. it is very good knowledge. Summarize, could I said as "Gibbs distribution, which based on MRF model, represents the spatial dependence between neighborhood pixels to remove noise image. Meanwhile, TV uses a absolute gradient of the image in the energy minimization process." Is it clear to write the different between MRF and TV in short sentence? – user3051460 Mar 18 '16 at 17:25
• I wouldn't see this as a good way to put it. Read a little bit so that you capture the idea behind them. – Tolga Birdal Mar 19 '16 at 13:17
• Thanks. Could you help me write a short sentence which summarize the key idea of each method in noise removal? – user3051460 Mar 19 '16 at 13:18
• Is this a homework question? – Tolga Birdal Mar 19 '16 at 13:19
• No. It is report project. I need write a short/ overview sentence which show the different between them – user3051460 Mar 19 '16 at 13:19