Image Formation Model

• Shape model: $S$ consists of $K$ shapes: $S_k$ , $k = 1, \ldots, K$. (e.g. ventricles, putamen, hippocampus, ...)

• Transformation: $T$ can be any spatial transformation within a specified set. The set can include non-linear warps with arbitrarily large number of Degrees Of Freedom. This transformation applies to all of the shapes, $S_k$.

• Image generation model:
  $$G(T(S)) = \sum_{m,k} \alpha_{m,k} G_m(T(S_k))$$
  where $\alpha_{m,k}$ are intensity scaling parameters and $G_m(\cdots)$ generates the $m\,$th basis image of a shape (see section 2.3).

• Image measurement model:
  $$Y = G(T(S)) + \epsilon$$
  where $\epsilon$ is a spatially uncorrelated Gaussian noise process with $E\{\epsilon\} = 0$, $\ensuremath{\mathrm{Cov}}\{\epsilon\} = \sigma^2 I$.

Note that all images ($\epsilon$, $G_m(T(S_k))$, $Y$, etc) are treated as vectors of length $N$, where $N$ is the number of voxels in the image.