Image Generators

The definitions of the image generators are:

• $G_1$ gives an image with an intensity 1.0 for all voxels totally within the shape, 0.0 for those outside and the partial volume overlap fraction otherwise;
• $G_2$ gives an image with an intensity gradient along the $x$-axis within the shape, with zero values outside. The image is also demeaned and has partial volume modelled multiplicatively (i.e.  $G_2(x,y,z) = x G_1(x,y,z)$, which is then demeaned);
• $G_3$ and $G_4$ give images with intensity gradients along the $y$- and $z$-axes respectively;
• $G_5$ and above can represent other expected image signals (e.g. quadratic trends, common artifacts, etc.) but will not be used in this report

Note that $G_2, G_3$ and $G_4$ represent the terms that model the combined effects of bias field and spatially varying tissue parameters. These are effectively the first terms in the Taylor series expansion of the general case for bias field and spatially varying parameters. For shapes with large spatial extent this approximation may not be sufficiently accurate, in which case it is possible to include higher-order Taylor series terms (quadratic, cubic, etc.) via extra $G$ terms ($G_5$ and up).

These generators are functions, which when applied to a transformed shape, $T(S_k)$, produce an image containing $N$ values. The images are then reshaped into vectors of length $N$, and assembled into a single $G$ matrix for use in later sections.

The image generator can also be written in matrix form:

$$G(T(S)) = \sum_{m,k} \alpha_{m,k} G_m(T(S_k))$$ (5)

$$= G \alpha$$ (6)

where $G$ represents an $N$ by $D$ matrix and $\alpha$ is a column vector of length $D$ (typically $D = MK$, but will be changed - see later). Note that in this form $G$ implicitly encodes information about the transformation $T$ and the underlying shapes, $S_k$. However, as neither of these will be marginalised, this dependency is not important for any of the following derivations and will therefore be left implicit.