We can approximate the distribution in equation 8 by replacing the discrete labels, , with continuous weights vectors, :

where , is given by equation 10 with where as before is specified by a deterministic mapping between and (the logistic transform, equation 12).

However, instead of equation 11, is now a continuous Gaussian conditionally specified auto-regressive (CAR) or continuous MRF prior (Cressie, 1993) on each of the class maps, i.e. (where } and ) with:

where is an matrix whose (i,j)th element is , , and is the MRF control parameter which controls the amount of spatial regularisation. We set , if and are spatial neighbours and otherwise (where is the geometric mean of the number of neighbours for voxels and ) , giving approximately:

How does this posterior approximation using the continuous class weights vectors instead of class labels allow us to adaptively determine the amount of spatial regularisation? The answer is that is a continuous random variable ranging effectively between and . Therefore, unlike in equation 6, the normalising constant, , in equation 16 is known:

and hence we can adaptively determine the MRF control parameter, . To achieve this becomes a parameter in the model, and equation 14 becomes:

where the prior , is a non-informative conjugate gamma prior:

In summary, we have approximated the discrete labels with vectors of continuous weights which a priori approximate delta functions at 0 and 1. Each vector of continuous weights deterministically corresponds (via the logistic transform) to another vector of continuous weights, which a priori are uniform across the real line. As they are uniform across the real line these vectors of continuous weights can be regularised using a spatial prior (a continuous MRF) for which the amount of spatial regularisation can be determined adaptively.

Figure 2 shows a graphical representation of the discrete labels mixture model (equation 8) and the continuous weights spatial mixture model with adaptive spatial regularisation (equation 14).