Optimal transportation plays a fundamental role in many fields in engineering and medicine, including surface parameterization in graphics, registration in computer vision, and generative models in deep learning. For quadratic distance cost, optimal transportation map is the gradient of the Brenier potential, which can be obtained by solving the Monge-Ampère equation. Furthermore, it is induced to a geometric convex optimization problem. The Monge-Ampère equation is highly non-linear, and during the solving process, the intermediate solutions have to be strictly convex. Specifically, the accuracy of the discrete solution heavily depends on the sampling pattern of the target measure. In this work, we propose a self-adaptive sampling algorithm which greatly reduces the sampling bias and improves the accuracy and robustness of the discrete solutions. Experimental results demonstrate the efficiency and efficacy of our method.

Skull registration plays a fundamental role in forensic science and is crucial for craniofacial reconstruction. The complicated topology, lack of anatomical features, and low quality reconstructed mesh make skull registration challenging. In this work, we propose an automatic skull registration method based on the discrete uniformization theory, which can handle complicated topologies and is robust to low quality meshes. We apply dynamic Yamabe flow to realize discrete uniformization, which modifies the mesh combinatorial structure during the flow and conformally maps the multiply connected skull surface onto a planar disk with circular holes. The 3D surfaces can be registered by matching their planar images using harmonic maps. This method is rigorous with theoretic guarantee, automatic without user intervention, and robust to low mesh quality. Our experimental results demonstrate the efficiency and efficacy of the method.