A Smoothness Energy without Boundary Distortion for Curved Surfaces
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Current quadratic smoothness energies for curved surfaces either exhibit distortions near the boundary due to zero Neumann boundary conditions, or they do not correctly account for intrinsic curvature, which leads to unnatural-looking behavior away from the boundary. This leads to an unfortunate trade-off: one can either have natural behavior in the interior, or a distortion-free result at the boundary, but not both. We introduce a generalized Hessian energy for curved surfaces. This energy features the curved Hessian of functions on manifolds as well as an additional curvature term which results from applying the Weitzenbock identity. Its minimizers solve the Laplace-Beltrami biharmonic equation, correctly accounting for intrinsic curvature, leading to natural-looking isolines. On the boundary, minimizers are as-linear-as-possible, which reduces the distortion of isolines at the boundary. We also provide an implementation that enables the use of the Hessian energy for applications on curved surfaces for which current quadratic smoothness energies do not produce satisfying results, and observe convergence in our experiments.
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