| Summary: | We propose quasi-harmonic weights for interpolating geometric data, which
are orders of magnitude faster to compute than state-of-the-art. Currently,
interpolation (or, skinning) weights are obtained by solving large-scale
constrained optimization problems with explicit constraints to suppress
oscillative patterns, yielding smooth weights only after a substantial amount
of computation time. As an alternative, our weights are obtained as minima
of an unconstrained problem that can be optimized quickly using straightforward numerical techniques. We consider weights that can be obtained as
solutions to a parameterized family of second-order elliptic partial differential equations. By leveraging the maximum principle and careful parameterization, we pose weight computation as an inverse problem of recovering
optimal anisotropic diffusivity tensors. In addition, we provide a customized
ADAM solver that significantly reduces the number of gradient steps; our
solver only requires inverting tens of linear systems that share the same
sparsity pattern. Overall, our approach achieves orders of magnitude acceleration compared to previous methods, allowing weight computation in
near real-time.
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