Keenan Crane - Conformal Geometry Processing

Keenan Crane

Email
Office	329 Annenberg
Phone	(502) 821-3239
Mail	MC 305-16
	California Institute of Technology
	1200 E California Blvd
	Pasadena, CA 91125
Calendar

Home Images Misc. Projects Publications CV Links

Spin Transformations of Discrete Surfaces
SIGGRAPH 2011 / ACM Transactions on Graphics

Keenan Crane
Caltech Ulrich Pinkall
TU Berlin Peter Schröder
Caltech

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.

Paper: Spin Transformations of Discrete Surfaces [errata]

Data: example meshes from paper figures - includes original meshes and their spin transformations in OBJ format (4.2 MB)

Video:

Conference presentation [QuickTime] [YouTube]
Fast Forward [YouTube]
Conformal Willmore flow on rhino head [QuickTime] [YouTube]

Supplemental Material:

Supplemental Information

in practice, conformal deformations can often be computed using a single linear solve instead of an eigenvalue problem; this document shows how and gives some additional timing information. It also describes how to build the eigenvalue problem via a simple facewise construction and gives example code in C++.

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

Executable:

SpinXForm

Mac OS X
Mac OS X (Lion)
Windows

Note that Windows and Lion versions are currently several times slower than the Mac OS X version due to difficulty with CHOLMOD.

Source Code:

Listed below is a reference implementation of the basic algorithm described in the paper. It handles manifold triangle meshes without boundary. Several distributions are available which are increasingly feature-rich and hence have an increasing number of dependencies:

spinxform_commandline.zip - no external dependencies whatsoever; written in strict ISO C++ (should compile anywhere!).
spinxform_opengl.zip - adds visualization via OpenGL/GLUT.
spinxform_fast.zip - adds visualization and replaces basic linear solver with CHOLMOD, but significantly more difficult to build/link.
MATLAB - port to MATLAB courtesy of Jan Hakenberg.
CUDA - a GPU-accelerated version written by Nikos Yiotis. (Currently under development, but will likely get much faster!)

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.

Acknowledgements
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.

Figures:

Left: even for modest displacements, simple normal offsets can severely distort texture. Right: using a spin transformation to apply displacement prevents distortion.
Given an initial surface and desired change in curvature (left) we construct a new surface conformally equivalent to the first one (right). Here green and purple indicate a positive and negative change in curvature, respectively.	Top left: the man in the moon is sculpted by ``painting'' a scalar function (inset) onto a disk. Applying standard image filters to this function achieves a variety of effects while preserving a conformal map to the original disk. Top right: low-pass filter. Bottom left: high-pass filter. Bottom right: unsharp mask.
Spin transformations can be used to compute minimal surfaces in two distinct ways. Top: starting with a curved surface (left) we remove all mean curvature (right). Bottom: starting with a flat surface (left) we prescribe new tangent directions along the boundary without changing mean curvature (right). In both cases the surface is computed directly without an iterative flow.	On the sphere, eigenfunctions of the Dirac operator correspond to relativistic wave functions of an electron orbiting an atomic nucleus, here visualized for the first time.
Giraffe attempts ballet: a textured mesh (left) can be modified arbitrarily and projected onto the nearest conformally equivalent surface (middle), preserving texture fidelity. We can also explicitly modify scale without disturbing texture -- on the right we ask for a much larger head.	The boundary of a surface (left) can be modified without disrupting texture or geometric detail (right).
Conformal Willmore flow on a rhino head: mean curvature shrinks as quickly as possible.