Fig. 2.
The probability of seeing a label image L (b) given an atlas in its reference position (a) is obtained by multiplying the probability of seeing the label image given the optimally deformed atlas (c) with a factor with magnitude less than one, which is a penalty for not actually knowing the deformation field. In the illustration of the atlases, white and black indicate a white matter probability of 1 and 0, respectively. We approximate the penalty factor for not knowing the optimal deformation field by a product of local penalties On, one for each mesh node n. Images (d) and (e) illustrate how this local penalty factor is calculated for the node indicated with number 1 and 2 in images (b) and (c), respectively. The top rows in (d) and (e) provide a magnified view of the local neighborhood around the node under investigation in the label image and the deformed atlas. The left and right images in the middle rows show respectively the prior (before any data is seen) and the posterior (after the data in the top left arrives) distributions of the location of the mesh node. Here, dark indicates high probability density values. Finally, the bottom rows show Gaussian approximations to the priors and posteriors of the middle rows that are used to actually calculate the penalty factors. Each node’s penalty factor essentially quantifies the difference between the prior and the posterior, by comparing each distribution’s MRF energy at the optimal mesh node location and the spread of its Gaussian approximation (see text). As a result, the node shown in (d) incurs a much higher penalty (On ≪ 1) than the node of (e) (On ≈ 1) for not knowing its optimal location. Stated from a data compression point of view, encoding the position of the mesh node requires a high number of bits − log2 On in (d), but ≈0 bits in (e). This reflects the fact that, in contrast to the situation in (e), the position of the node in (d) must be encoded with high precision, because small deviations from its optimal value will result in a large increase in the number of bits required to subsequently encode the labels [top left of (d)]. Note that in reality, the label probabilities in each mesh node are not known either, which gives rise to another penalty factor Rn in each node (see text).