Abstract
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.
Highlights
Skinning is perhaps the simplest and most popular method for shape deformation
7.1 Baseline For clarity, we refer to the original version of bounded biharmonic weights solved in (6) as Bounded biharmonic weights (BBW)
BBW can be two orders of magnitude slower to compute than BBWA due to the partition of unity constraint, so we primarily compare with BBWA
Summary
The problem of computing skinning weights amounts to designing a partition of unity, or set of functions that sums to 1 at every point on the domain, that satisfies a few properties. A popular method, bounded biharmonic weights [Jacobson et al 2011, 2012b], adds explicit constraints to the problem to prevent these artifacts, yielding a quadratic programming formulation that is considerably slower to solve This prohibits applications where fast iteration or interactive feedback are desired Our method is built on the key insight that second-order elliptic PDEs provide a rich parametric family of weights that automatically satisfy all necessary constraints (§4.3). Within this family, we search for the minimizer of a conventional smoothness energy. Our weights are visually similar to the state-of-the-art with orders-of-magnitude speedup, as confirmed on typical examples in skinning animations (see §7)
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