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##

Image Generators

The definitions of the image generators are:

- 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;
- gives an image with an intensity gradient along the
-axis within the shape, with zero values outside. The image is
also demeaned and has partial volume modelled multiplicatively
(i.e.
, which is then demeaned);
- and give images with intensity gradients along the
- and -axes respectively;
- 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 and 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 terms ( and up).

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

The image generator can also be written in matrix form:

where represents an by matrix and is a column
vector of length (typically , but will be changed - see
later). Note that in this form implicitly encodes information
about the transformation and the underlying shapes,
. 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.

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