Paul Zhang, Massachusetts Institute of Technology

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Computation of fields evaluated into various manifolds is a challenging task. Common approaches are to locally linearize these manifolds for the evaluation and optimization of the target energy. We show that this linearization exhibits undesirable artifacts such as inaccurate energy evaluation and slow convergence. Our new approach uses the theory of manifold valued splines to evaluate the true energy of a manifold valued field via subdivision splines. Using sensitivity analysis, gradients of a target energy can be propagated back to control points of the subdivision spline, allowing for more accurate optimization of manifold valued fields. A main challenge for our new approach is the computational cost of evaluating subdivision surfaces. On a manifold, the simple euclidean average is replaced with weighted Frechet means, which are generic nonlinear optimizations. Within a subdivision spline, the number of weighted Frechet means grows exponentially with subdivision layers. We address various techniques used to increase performance of these means.

Abstract Author(s): Paul Zhang