next up previous
Next: Image Formation Model Up: tr05mj1 Previous: Introduction

Problem Formulation

Consider an image, $ Y$, generated by some image generation process, $ G$, from a known (ground truth) object, $ S$, where $ Y$ is not spatially aligned with $ S$, but related spatially by a transformation, $ T$. The objective of the segmentation/registration problem is to recover the spatial transformation, $ T$, which relates $ S$ to $ Y$. That is, find $ T$ such that $ G(T(S))$ and $ Y$ are `most similar'.

A Bayesian formulation of this problem is as follows:

Note that generally $ T$ is parameterised by its own set of parameters, and that $ p(T\vert Y,S)$ is giving the posterior probability for these transformation parameters.

The probabilities are related by:

$\displaystyle p(T,\theta\vert Y,S)$ $\displaystyle \propto$ $\displaystyle p(Y\vert T,S,\theta) \, p(T,\theta\vert S)$ (1)
  $\displaystyle \propto$ $\displaystyle p(Y\vert T,S,\theta) \, p(T\vert S) p(\theta\vert S)$ (2)
  $\displaystyle \propto$ $\displaystyle p(Y\vert T,S,\theta) \, p(T) p(\theta)$ (3)


$\displaystyle p(T\vert Y,S) = \int p(T,\theta\vert Y,S) \, d\theta$ (4)

where $ p(T)$ is the prior probability distribution for the transformation, $ T$, and $ p(\theta)$ is the prior for the image formation parameters, $ \theta$. It is assumed that $ T$ and $ S$ are independent, so that $ p(T\vert S) = p(T)$. Similarly for $ \theta$ and $ S$. The constant of proportionality, $ p(Y\vert S)$, does not depend on $ T$ and hence will be ignored for the remainder of this report.

Note that marginalising over $ \theta$ is the difficult step in calculating $ p(T\vert Y,S)$.

The problem of finding the `best' single segmentation/registration1 is then equivalent to finding the maximum a-posterior probability: the MAP estimate.

That is:

$\displaystyle T_{MAP} = \arg \max_{T} p(T\vert Y,S) = \arg \max_{T} \log\left( p(T\vert Y,S) \right)

Therefore $ p(T\vert Y,S)$, or $ \log(p(T\vert Y,S))$, play an equivalent role to the similarity function in common registration techniques.

next up previous
Next: Image Formation Model Up: tr05mj1 Previous: Introduction