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:
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(1) |
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(2) | |
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(3) |
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(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.