Direct cortical mapping via solving partial differential equations on implicit surfaces

In this paper, we propose a novel approach for cortical mapping that computes a direct map between two cortical surfaces while satisfying constraints on sulcal landmark curves. By computing the map directly, we can avoid conventional intermediate parameterizations and help simplify the cortical mapping process. The direct map in our method is formulated as the minimizer of a flexible variational energy under landmark constraints. The energy can include both a harmonic term to ensure smoothness of the map and general data terms for the matching of geometric features. Starting from a properly designed initial map, we compute the map iteratively by solving a partial differential equation (PDE) defined on the source cortical surface. For numerical implementation, a set of adaptive numerical schemes are developed to extend the technique of solving PDEs on implicit surfaces such that landmark constraints are enforced. In our experiments, we show the flexibility of the direct mapping approach by computing smooth maps following landmark constraints from two different energies. We also quantitatively compare the metric preserving property of the direct mapping method with a parametric mapping method on a group of 30 subjects. Finally, we demonstrate the direct mapping method in the brain mapping applications of atlas construction and variability analysis.