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Abstract:
Functions that optimize Laplacian-based energies have become popular in
geometry processing, e.g. for shape deformation, smoothing, multiscale kernel
construction and interpolation. Minimizers of Dirichlet energies, or solutions
of Laplace equations, are harmonic functions that enjoy the maximum principle,
ensuring no spurious local extrema in the interior of the solved domain occur.
However, these functions are only C0 at the constrained points, which often
causes smoothness problems. For this reason, many applications optimize
higher-order Laplacian energies such as biharmonic or triharmonic. Their
minimizers exhibit increasing orders of continuity but lose the maximum
principle and show oscillations. In this work, we identify characteristic
artifacts caused by spurious local extrema, and provide a framework for
minimizing quadratic energies on manifolds while constraining the solution to
obey the maximum principle in the solved region. Our framework allows the user
to specify locations and values of desired local maxima and minima, while
preventing any other local extrema. We demonstrate our method on the smoothness
energies corresponding to popular polyharmonic functions and show its
usefulness for fast handle-based shape deformation, controllable color
diffusion, and topologically-constrained data smoothing.
