High-dimensional image registration using symmetric priors

This paper is about warping a brain image from one subject (the object image) so that it matches another (the template image). A high-dimensional model is used, whereby a finite element approach is employed to estimate translations at the location of each voxel in the template image. Bayesian statistics are used to obtain a maximum a posteriori (MAP) estimate of the deformation field. The validity of any registration method is largely based upon the constraints or, in this instance, priors incorporated into the model describing the transformations. In this approach we assume that the priors should have some form of symmetry, in that priors describing the probability distribution of the deformations should be identical to those for the inverses (i.e., warping brain A to brain B should not be different probabilistically from warping B to A). The fundamental assumption is that the probability of stretching a voxel by a factor of n is considered to be the same as the probability of shrinking n voxels by a factor of n(-1). In the Bayesian framework adopted here, the priors are assumed to have a Gibbs form, where the Gibbs potential is a penalty function that embodies this symmetry. The penalty function of choice is based upon the singular values of the Jacobian having a lognormal distribution. This enforces a continuous one-to-one mapping. A gradient descent algorithm is presented that incorporates the above priors in order to obtain a MAP estimate of the deformations. We demonstrate this approach for the two-dimensional case, but the principles can be extended to three dimensions. A number of examples are given to demonstrate how the method works.