Consider an image, , generated by some image generation process, , from a known (ground truth) object, , where is not spatially aligned with , but related spatially by a transformation, . The objective of the segmentation/registration problem is to recover the spatial transformation, , which relates to . That is, find such that and are `most similar'.
A Bayesian formulation of this problem is as follows:
Note that generally is parameterised by its own set of parameters, and that is giving the posterior probability for these transformation parameters.
The probabilities are related by:
(1) | |||
(2) | |||
(3) |
(4) |
Note that marginalising over is the difficult step in calculating .
The problem of finding the `best' single segmentation/registration1 is then equivalent to finding the maximum a-posterior probability: the MAP estimate.
That is:
Therefore , or , play an equivalent role to the similarity function in common registration techniques.