Solving variational problems and partial differential equations mapping between manifolds via the closest point method

Many applications from fields such as mathematical physics, image processing, computer vision and medical imaging require computation of maps between manifolds. A numerical framework is introduced for solving variational problems and partial differential equations that map from a source manifold M to a target manifold N . The numerics rely on the closest point representations of M and N. Using the closest point representation produces simple algorithms for handling complex surface geometries, since standard Cartesian numerical methods can be used. The framework is illustrated with harmonic maps and a straightforward algorithm is given for this case. Harmonic maps are important in applications such as texture mapping, brain image regularization and colour image denoising. Moreover, the harmonic mapping energy is part of numerous energy functionals. The algorithm is justified theoretically and shown to be first order accurate. It is implemented in two applications: removing noise from texture maps and colour image enhancement.