Spatial

The prior on $\vec{x}$ is now a spatial prior. In this work we assume a discrete MRF (Besag, 1986; Geman and Geman, 1984). Taking $\vec{\lambda}=\{\phi_x\}$, where $\phi_x$ is the MRF control parameter, which controls the amount of spatial regularisation. We have:

$$p(\vec{x}=\vec{\kappa}\vert\phi_x)\propto f(\phi_x)exp \left(-\frac{\phi_x}{4} \sum_i \sum_{j\in{\cal N}_i} I[x_i \neq x_j] \right)$$ (6)

where ${\cal N}_i$ is the spatial neighbourhood of $i$ (for this we use 26-connectivity in 3-dimensions), $I[x_i \neq x_j]$ is an indicator function (it is one if $x_i\neq x_j$ and is zero otherwise), and $f(\phi_x)$ is some unknown function of $\phi_x$. This prior is identical to the prior used in Zhang et al. (2001); Salli et al. (1999), if the parameter $\phi_x$ is set to one. Usually, $\phi_x$ is hand tuned to work well for particular types of dataset. The ``best'' value for $\phi_x$ will depend on the amount of, and topography of, the different classes. Marroquin et al. (2003) refer to this as the parameter which controls the granularity of the field, and they discuss how the use of different values for this parameter can affect the resulting segmented field. Indeed, we shall demonstrate later how fixing the amount of spatial regularisation to a single value will perform considerably less well than determining it adaptively from the data.

The hyperprior we use on $\phi_x$ is a non-informative Gamma distribution:

$$p(\phi_x\vert\tilde{a}_{\phi_{x}}, \tilde{b}_{\phi_x})=Ga(\phi_x; \tilde{a}_{\phi_{x}}, \tilde{b}_{\phi_x})$$ (7)

Using all of this in equation 1, the posterior becomes:

$$p(\vec{x}=\vec{\kappa},\vec{\theta},\vec{\lambda}\vert\vec{y})\pr... ...\theta_{\kappa_i})\} p(\vec{x}=\vec{\kappa}\vert\phi_x)p(\phi_x)p(\vec{\theta})$$ (8)

Clearly, a fourth mixture model could be considered. This would be a spatial mixture model with global class proportions. However, it is far from clear how we would combine the prior on $x$ in equation 4 with that in equation 6. Anyway, as we shall see in the results, we obtain good global histogram fits with the spatial mixture model specified here without including global class proportions.