Copyright © 2006 by the Association for Computing Machinery, Inc. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from Permissions Dept, ACM Inc., fax +1 (212) 869-0481 or e-mail permissions@acm.org . © 2006 ACM 0730-0301/06/0700- $5.00 1057 Spectral Surface Quadrangulation Shen Dong ∗ Peer-Timo Bremer ∗ Michael Garland ∗ Valerio Pascucci † John C. Hart ∗ ∗ University of Illinois at Urbana-Champaign † Lawrence Livermore National Laboratory (a) Laplacian eigenfunction (b) Morse-Smale complex (c) Optimized complex (d) Semi-regular remeshing Figure 1: We quadrangulate a given triangle mesh by extracting the Morse-Smale complex of a selected eigenvector of the mesh Laplacian matrix. After optimizing the geometry of the base complex, we remesh the surface with a semi-regular grid of quadrilaterals. Abstract Resampling raw surface meshes is one of the most fundamental operations used by nearly all digital geometry processing systems. The vast majority of this work has focused on triangular remeshing, yet quadrilateral meshes are preferred for many surface PDE prob- lems, especially fluid dynamics, and are best suited for defining Catmull-Clark subdivision surfaces. We describe a fundamentally new approach to the quadrangulation of manifold polygon meshes using Laplacian eigenfunctions, the natural harmonics of the sur- face. These surface functions distribute their extrema evenly across a mesh, which connect via gradient flow into a quadrangular base mesh. An iterative relaxation algorithm simultaneously refines this initial complex to produce a globally smooth parameterization of the surface. From this, we can construct a well-shaped quadrilat- eral mesh with very few extraordinary vertices. The quality of this mesh relies on the initial choice of eigenfunction, for which we de- scribe algorithms and hueristics to efficiently and effectively select the harmonic most appropriate for the intended application. Keywords: quadrangular remeshing, spectral mesh decomposi- tion, Laplacian eigenvectors, Morse theory, Morse-Smale complex 1 Introduction Meshes generated from laser scanning, isosurface extraction and other methods often suffer from irregular element and sampling ar- ∗ {shendong,ptbremer,garland,jch}@uiuc.edu † pascucci@llnl.gov tifacts of the process. Because these problems arise so easily and can hinder the accuracy and efficiency of subsequent operations, the ability to remesh surfaces with well-shaped well-spaced elements is an important tool for mesh processing. Much of the remeshing work in the graphics literature focuses on triangle meshes, though many graphics and scientific applications benefit from good quadrilateral meshes. Such meshes should have as few extraordinary vertices as possible and their elements should have internal angles near 90 ◦ . Quadrilaterals are the preferred prim- itive in several simulation domains, including computational fluid dynamics, where extraordinary points can lead to numerical in- stability [Stam 2003]. Catmull-Clark subdivision of a poor mesh can yield wrinkles [Halstead et al. 1993], and the tensor-product NURBS patches still used in CAD/CAM production software work best on a mesh composed exclusively of quadrilaterals. Further- more, decomposing a surface into well-shaped quadrangles simpli- fies the construction of a texture atlas. We have developed a new approach for building a quadrangular base complex over a triangulated manifold of arbitrary genus. This approach is based on the Morse theorem that for almost all real functions, the Morse-Smale complex (reviewed in §4), consisting of the ridge lines that extend from its saddles to its extrema, forms quadrangular regions. To space these regions evenly over the sur- face, we choose as our real function a shape harmonic of the appro- priate frequency, computed in §3 as an eigenvector of the Laplacian matrix of the input mesh. A new iterative relaxation algorithm de- scribed in §5 simultaneously improves this base mesh layout while computing a globally smooth parameterization used to generate the final semi-regular grid of well-shaped quadrilaterals. The complete spectrum of the mesh defines two families of com- plexes: the primal Morse-Smale and their quasi-dual complexes, a construction we propose in §4.3. The quality of the final mesh is in- timately tied to the choice of complex, a choice we make based on parametric distortion. Section 3 provides a detailed analysis of the Laplacian spectrum, using spectral shifts to efficiently limit com- putation only to the eigenvectors around a desired frequency. The resulting method produces fully conforming semi-regular 1057