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SIGGRAPH 2011 / ACM Transactions on Graphics
Abstract: We introduce a new method for computing conformal transformations of triangle meshes in R3. Conformal maps are desirable in digital geometry processing because they do not exhibit shear, and therefore preserve texture fidelity as well as the quality of the mesh itself. Traditional discretizations consider maps into the complex plane, which are useful only for problems such as surface parameterization and planar shape deformation where the target surface is flat. We instead consider maps into the quaternions, which allows us to work directly with surfaces sitting in R3. In particular, we introduce a quaternionic Dirac operator and use it to develop a novel integrability condition on conformal deformations. Our discretization of this condition results in a sparse linear system that is simple to build and can be used to efficiently edit surfaces by manipulating curvature and boundary data, as demonstrated via several mesh processing applications.
Slides - from presentation at SIGGRAPH 2011 in Vancover, BC.
Oberwolfach Report - from MFO workshop on Trends in Mathematical Imaging and Surface Processing
NEW Conformal Willmore Flow - applications of spin transformations to surface fairing.
Note that Windows and Lion versions are currently several times slower than the Mac OS X version due to difficulty with CHOLMOD.
Note that the first two distributions use only a basic (read: slow) conjugate gradient solver with diagonal preconditioning, which makes them several times slower than the "fast" distribution. However, CHOLMOD takes some work to build and link (especially on Windows), so I'd suggest starting with one of the simpler distributions. Each distribution includes a Makefile for Mac/Linux/Cygwin and a VisualStudio project for Windows.
The authors thank Mirela Ben-Chen and Fabian Aiteanu for comparison data, Fernando de Goes for his Green Coordinates implementation, and Jessica Pfeilsticker for illuminating discussions on spin dynamics. Example meshes are courtesy of Autodesk, Luxology, 3D Universe, David Bommes, and Chris Legasse; cat clip art was created by Jon Phillips and Gerald Ganson. This research was partially funded by a Google PhD Fellowship, the Center for the Mathematics of Information at Caltech, the IAS at TU München, DFG Research Center Matheon, DFG Research Unit Polyhedral Surfaces and BMBF project GEOMEC.
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