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:

• $p(Y\vert T,S,\theta)$ is the image formation/likelihood model where $\theta$ are parameters of the generation process (e.g. noise variance, tissue intensities).
• $p(T,\theta\vert Y,S)$ is the joint posterior probability distribution for the spatial transformation (the quantity of interest) and the unknown generation parameters.
• $p(T\vert Y,S)$ is the marginal posterior distribution, which integrates over the unknown parameters, $\theta$.

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:

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

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

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

and

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

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

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.