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:

- is the image formation/likelihood model where are parameters of the generation process (e.g. noise variance, tissue intensities).
- is the joint posterior probability distribution for the spatial transformation (the quantity of interest) and the unknown generation parameters.
- is the marginal posterior distribution, which integrates over the unknown parameters, .

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) |

and

(4) |

where is the prior probability distribution for the transformation, , and is the prior for the image formation parameters, . It is assumed that and are independent, so that . Similarly for and . The constant of proportionality, , does not depend on and hence will be ignored for the remainder of this report.

Note that marginalising over is the difficult step in calculating .

The problem of finding the `*best*' single
segmentation/registration^{1} 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.

- Image Formation Model
- Anatomical Information, Intensities and Bias Field
- Image Generators

- Probabilistic Forms