Areas of No Interest

Some areas of the underlying object that are known to give rise to signal may not be explicitly modelled (e.g. cortex if only deep brain structures are of interest fot the modelling). In this case it is necessary to include these `areas of no interest' in the model with sufficient flexibility so that they can match any observed image intensities well without adversely affecting the fit of the model parameters of interest.

One way to model these areas is to include them in the model as very small, independent subshapes. For example, if the deep brain structures were of interest, then these would cover large areas of the image with relatively few parameters (e.g. mean intensity and intensity gradients) while the areas outside this can be tesselated with small (say 0.1mm$^3$ sized) voxels (subshapes), each with an independent mean intensity. See figure 3 for a 1D example. With this model, whatever the intensities were in these areas in the measured image, $Y$, the extra subshape parameters would fit exactly (i.e. there would be no error, or residuals, in these areas, which could otherwise degrade the overall image fit).

In terms of the model above, this would introduce three types of extra parameters: extra null parameters, parameters of no interest and also degenerate parameters. The extra null parameters $\alpha_{null}$ are those associated with all subshapes that are outside the field of view of the measured image. The parameters associated with voxels inside the field of view comprise both parameters of no interest, $\alpha_{un}$, and degenerate parameters, $\alpha_{deg}$. The degeneracy occurs because typically many small subshapes will affect the same voxel, and no others, leading to columns in $G$ that are identical. For marginalisation, only a single parameter (at most) can be associated with each voxel in the area of no interest, and so most subshapes will be degenerate and can, with appropriate reparameterisation, be treated in the same way as null parameters.

All parameters associated with areas of no interest are taken to have a flat prior, to enable them to match any intensity with equal probability.