Editing Image segmentation (section)

==== Supervised image segmentation using MRF and MAP ====
In terms of image segmentation, the function that MRFs seek to maximize is the probability of identifying a labelling scheme given a particular set of features are detected in the image. This is a restatement of the [[maximum a posteriori estimation]] method.

[[File:MRF neighborhood.png|thumb|right|MRF neighborhood for a chosen pixel]]

The generic algorithm for image segmentation using MAP is given below:
{{ordered list
| Define the neighborhood of each feature (random variable in MRF terms).
<br/>Generally this includes 1st-order or 2nd-order neighbors.
| Set initial probabilities {{math|''P''(''f<sub>i</sub>'')}}> for each feature as 0 or| where {{math|''f<sub>i</sub>'' &isin; &Sigma;}} is the set containing features extracted
<br/>for pixel {{mvar|i}} and define an initial set of clusters.
| Using the training data compute the mean ({{mvar|''&mu;''<sub>''ℓ''<sub>''i''</sub></sub>}}) and variance ({{math|&sigma;<sub>''ℓ''<sub>''i''</sub></sub>}}) for each label. This is termed as class statistics.
| Compute the marginal distribution for the given labeling scheme {{math|''P''(''f<sub>i</sub>''&nbsp;{{!}}&nbsp;''ℓ''<sub>''i''</sub>)}} using [[Bayes' theorem]]
and the class statistics calculated earlier. A Gaussian model is used for the marginal distribution.
:<math>
\frac 1 {\sigma(\ell_i) \sqrt{2\pi} } e^{ -(f_i-\mu(\ell_i))^2/(2\sigma(\ell_i)^2) } \, d\ell_i
</math>
| Calculate the probability of each class label given the neighborhood defined previously.
<br/>[[Clique (graph theory)|Clique]] potentials are used to model the social impact in labeling.
| Iterate over new prior probabilities and redefine clusters such that these probabilities are maximized.
<br/>This is done using a variety of optimization algorithms described below.
| Stop when probability is maximized and labeling scheme does not change.
<br/>The calculations can be implemented in [[Log-likelihood|log likelihood]] terms as well.
}}