Chapter 6 - Numerical Methods for High-Dimensional Warps

Abstract

This chapter presents numerical methods for the calculation of high dimensional warps which elastically map one brain volume into spatial alignment with a different volume. The fundamental problem in brain warping is to define the class of admissible spatial transformations that must be sufficiently broad to enable a reference anatomy to fit all subject anatomies, and to develop efficient, automated algorithms for the calculation of the appropriate transformation. The finite difference method, which operates directly on the motion equations, is easy to code and computationally fast; however, the finite element method can systematically address more complicated problems. For problems with regular geometries, standard implementations of the finite different methods are simple to program and run quickly. For images of anatomy, where informative features are not only sparse but distributed along interfaces of highly irregular shape, a mesh topology that subdivides the problem domain accordingly would substantially reduce the complexity of the matching calculation. Given such a mesh, the associated difference equations can be constructed in a systematic and efficient way using the finite element method. The Newton–Raphson process used in the chapter to linearize the similarity potential is the basis for iteratively solving the problems made nonlinear by the existence of large deformations or by the nonlinearity in the constitutive relations.