Keenan Crane
Robust Fairing via Conformal Curvature Flow
SIGGRAPH 2013 / ACM Transactions on Graphics
Keenan Crane Ulrich Pinkall Peter Schröder
Caltech TU Berlin Caltech
We present a formulation of Willmore flow for triangulated surfaces that permits extraordinarily large time steps and naturally preserves the quality of the input mesh. The main insight is that Willmore flow becomes remarkably stable when expressed in curvature space—we develop the precise conditions under which curvature is allowed to evolve. The practical outcome is a highly efficient algorithm that naturally preserves texture and does not require remeshing during the flow. We apply this algorithm to surface fairing, geometric modeling, and construction of constant mean curvature (CMC) surfaces. We also present a new algorithm for length-preserving flow on planar curves, which provides a valuable analogy for the surface case.
PDF, 18.2MB
Topology Joke
Thanks to the hard work of Henry Segerman one of our flows has been realized in ceramic, and can be ordered from Shapeways. A nice writeup of this work was put together by Debra Thimmesch.
This research was supported by a Google PhD Fellowship, the Hausdorff Research Institute for Mathematics, BMBF Research Project GEOMEC, SFB / Transregio 109 "Discretization in Geometry and Dynamics," and the TU M√ľnchen Institute for Advanced Study, funded by the German Excellence Initiative. Meshes provided by the Stanford Computer Graphics Laboratory and the AIM@SHAPE Shape Repository.
@article{Crane:2013:RFC, author = {Crane, Keenan and Pinkall, Ulrich and Schr\"{o}der, Peter}, title = {Robust Fairing via Conformal Curvature Flow}, journal = {ACM Trans. Graph.}, volume = {32}, issue = {4}, year = {2013}, publisher = {ACM}, address = {New York, NY, USA}, }
Conformal flow from the Stanford bunny to a round sphere, roughly 200 frames.
High-quality triangulation with texture coordinates and fixed connectivity.
Wavefront OBJ, 250MB
A detailed frog flows to a round sphere in only three large, explicit time steps (top). Meanwhile, the quality of the triangulation (bottom) is almost perfectly preserved.
Left: original surface. Center: standard fairing distorts texture, even with tangential smoothing. Right: conformal fairing preserves texture while producing a pleasing geometric shape.
Our flow remains stable for any time step h < 1, even on a highly irregular mesh with minimum edge length less than 0.2% of the mesh diameter (left). For time steps h > 1, curvature exhibits oscillatory behavior characteristic of forward Euler (bottom).
By augmenting the descent direction we achieve a wide variety of flows. Here we apply frequency-space filters to smooth out features at different scales.
Duck with nontrivial topology (left). Center: unconstrained flow yields distortion of both geometry and texture. Right: a simple linear constraint prevents distortion.
Topologist's view a of a coffee cup. Top: Sphere inversions neither distort angles nor increase Willmore energy, but distort area in undesirable ways. Bottom: a simple linear constraint prevents unnecessary inversion, and improves area distortion by flowing toward a desired metric.
We also describe a simple algorithm for length-preserving flow on curves, which provides a valuable analogy for the surface case. Here the silhouette of a bunny (top left) flows to a perfectly round circle (top right) while preserving the length of edges (bottom), visualized as rigid chain links.
Our flow prohibits sharp cusps, ultimately flowing to the smoothest curve of equal turning number. Here a 1D duck with winding number zero rapidly unwinds to become a smooth figure eight.