Modeling topological changes in deformable registration

Topological changes are common in brain MR images for aging or disease studies. For deformable registration algorithms, which are formulated as a variational problem and solved by the minimization of certain energy functional, topological changes can cause false deformation in the resulting vector field, and affect algorithm convergence. In this work, we focus on the effect of topological changes on diffeomorphic and inverse-consistent deformable registration algorithms, specifically, diffeomorphic demons and symmetric LDDMM. We first use a simple example to demonstrate the adverse effect of topological changes on these algorithms. Then, we propose an novel framework that can be imposed onto any existing diffeomorphic and inverse-consistent deformable registration algorithm. Our framework renders these registration algorithms robust to topological changes, where the output will consist of two components. The first is a deformation field that presents only the brain structural change which is the expected vector field if the topological change did not exist. The second component is a label map that provides a segmentation of the topological changes appeared in input images.