Large Deformation Registration via n-dimensional Quasi-conformal Maps

We propose a new method to obtain registration between n-dimensional manifolds with very large deformations. Given a set of landmark correspondences, our algorithm produces an optimal diffeomorphism that matches prescribed landmark constraints. The obtained registration is a n-dimensional quasi-conformal map. The basic idea of the model is to minimize an energy functional with a conformality term and a smoothness term. The conformality term allows the algorithm to produce diffeomorphisms even with very large deformations. We minimize the energy functional using alternating direction method of multipliers (ADMM). The algorithm only involves solving an elliptic problem and a point-wise minimization problem. The time complexity and robustness of the algorithm is independent of the number of landmark constraints. Either Dirichlet or free boundary condition can be enforced, depending on applications. To further speed up the algorithm, the multi-grid method is applied. Experiments are carried out to test our proposed algorithm to compute landmark-matching registration with different landmark constraints. Results show that our proposed model is efficient to obtain a diffeomorphic registration between n-dimensional data with large deformations.