Variable Deformability

The second inhomogeneity paradigm varies the strength of deformation constraint. Regions of the image are then classified as strongly or weakly deformable. In the case of registration modelled on the behaviour of a physical material, the deformability is described by one or more parameters of the material properties - the elasticity of an elastic medium (Sect. 3.3) or the viscosity of a fluid (Sect. 4.3). Allowing these parameters to vary spatially requires the derivation of modified Partial Differential Equations (PDEs) to account for the parameter gradients.

The strength of the deformation parameter is supplied at every position in the source image, using prior information obtained from one of two sources: physical or statistical. In the first case, prior information is available on the deformability of the physical tissues which the images represent. For example, [3] demonstrate the estimation of tissue elasticity using certain scanning protocols, and we assume the rigidity of hard tissue. Basing the variability of the deformation constraint on such physical information is valid only in intra-subject studies, where the registration of the source to the target attempts to reproduce actual physical movements of tissue. A second type of prior information is applicable to and derived from cross-population studies, where the variation in the deformation constraint is a function of the statistical cross-population variability in the shape of each structure in the images. Structures which have been found to display little variance in size and shape across a population of normals will be labelled with a high value of $\mu$, while other areas exhibiting greater variability will be labelled with a low value of $\mu$ and allowed greater deformations in registering to their homologues in the target image. For instance, [4] allows high variability in ventricular and cortical fold regions, and low variability in subcortical structures, in a variable-elasticity algorithm (Sect. 3.3).