Publications

Spectral Conformal Parameterization (pdf)

Patrick Mullen, Yiying Tong, Pierre Alliez and Mathieu Desbrun.
Symposium on Geometry Processing, 2008.

We present a spectral approach to automatically and efficiently obtain discrete free-boundary conformal parameterizations of triangle mesh patches, without the common artifacts due to positional constraints on vertices and without undue bias introduced by sampling irregularity. High-quality parameterizations are computed through a constrained minimization of a discrete weighted conformal energy by finding the largest eigenvalue/eigenvector of a generalized eigenvalue problem involving sparse, symmetric matrices. We demonstrate that this novel and robust approach improves on previous linear techniques both quantitatively and qualitatively.

A Variational Approach to Eulerian Geometry Processing (pdf)

Patrick Mullen, Alexander McKenzie, Yiying Tong and Mathieu Desbrun.
ACM Transactions on Graphics (SIGGRAPH), 2007.

We present a purely Eulerian framework for geometry processing of surfaces and foliations. Contrary to current Eulerian methods used in graphics, we use conservative methods and a variational interpretation, offering a unified framework for routine surface operations such as smoothing, offsetting, and animation. Computations are performed on a fixed volumetric grid without recourse to Lagrangian techniques such as triangle meshes, particles, or path tracing. At the core of our approach is the use of the Coarea Formula to express area integrals over isosurfaces as volume integrals. This enables the simultaneous processing of multiple isosurfaces, while a single interface can be treated as the special case of a dense foliation. We show that our method is a powerful alternative to conventional geometric representations in delicate cases such as the handling of high-genus surfaces, weighted offsetting, foliation smoothing of medical datasets, and incompressible fluid animation.

Supplemental Video (divx)

Data-Dependent Fairing of Subdivision Surfaces (pdf)

Ilja Friedel, Patrick Mullen and Peter Schröder

Solid Modeling, 2003

In this paper we present a new algorithm for solving the data dependent fairing problem for subdivision surfaces, using Catmull-Clark surfaces as an example. Earlier approaches to subdivision surface fairing encountered problems with singularities in the parametrization of the surface. We address these issues through the use of the characteristic map parametrization, leading to well defined membrane and bending energies even at irregular vertices. Combining this approach with ideas from data-dependent energy operators we are able to express the associated nonlinear stiffness matrices for Catmull-Clark surfaces as linear combinations of precomputed energy matrices. This machinery also provides exact, inexpensive gradients and Hessians of the new energy operators. With these the nonlinear minimization problem can be solved in a stable and efficient way using Steihaug's Newton/CG trust-region method. We compare properties of linear and nonlinear methods through a number of examples and report on the performance of the algorithm.