Accelerated quadratic proxy for geometric optimization.

We present the Accelerated Quadratic Proxy (AQP) - a simple first-order algorithm for the optimization of geometric energies defined over triangular and tetrahedral meshes. The main stumbling block of current optimization techniques used to minimize geometric energies over meshes is slow convergence due to ill-conditioning of the energies at their minima. We observe that this ill-conditioning is in large part due to a Laplacian-like term existing in these energies. Consequently, we suggest to locally use a quadratic polynomial proxy, whose Hessian is taken to be the Laplacian, in order to achieve a preconditioning effect. This already improves stability and convergence, but more importantly allows incorporating acceleration in an almost universal way, that is independent of mesh size and of the specific energy considered. Experiments with AQP show it is rather insensitive to mesh resolution and requires a nearly constant number of iterations to converge; this is in strong contrast to other popular optimization techniques used today such as Accelerated Gradient Descent and Quasi-Newton methods, e.g., L-BFGS. We have tested AQP for mesh deformation in 2D and 3D as well as for surface parameterization, and found it to provide a considerable speedup over common baseline techniques.