Prior Distribution

The prior distribution is used to bias the global transformations (changes in translation and scale) and local deformations that can be applied to a prototype template. In contrast to the work of Jain et al. [80], rotation is not taken into account as we assume that this is represented in the probabilistic template.

$\psi^{s,\xi,d}$ denotes a deformation of the original template $\psi^0$ . This deformation is performed by locally deforming the template by a set of parameters $\xi$ , scaling the local deformation by a factor of $s$ , and translating the scaled version along the $x$ and $y$ directions by an amount $d = (dx,dy)$ .

Assuming that translations and scale sizes have equal probability^4.1, and using Eq.  for the deformation probability, the prior distribution results in:

$$Pr(s,d,\xi) = K \exp\{-\frac{1}{2\sigma^2} \sum_{k=1}^N (\xi_k-1)^2\}$$ (4.7)

where $K$ is a normalization factor. Intuitively, a deformed template with a geometric shape similar to the prototype template is favoured, regardless of its size and location in the image.