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33
FINITE ELEMENT EXTERIOR CALCULUS: FROM HODGE THEORY TO NUMERICAL STABILITY
, 2009
"... Abstract. This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to diff ..."
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Cited by 90 (5 self)
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Abstract. This article reports on the confluence of two streams of research, one emanating from the fields of numerical analysis and scientific computation, the other from topology and geometry. In it we consider the numerical discretization of partial differential equations that are related to differential complexes so that de Rham cohomology and Hodge theory are key tools for exploring the wellposedness of the continuous problem. The discretization methods we consider are finite element methods, in which a variational or weak formulation of the PDE problem is approximated by restricting the trial subspace to an appropriately constructed piecewise polynomial subspace. After Hilbert space framework for analyzing the stability and convergence of such discretizations. In this framework, the differential complex is represented by a complex of Hilbert spaces and stability is obtained by transferring Hodge theoretic structures that ensure wellposedness of the continuous problem from the continuous level to the discrete. We show stable discretization is achieved if the finite element spaces satisfy two hypotheses: they can be arranged into a subcomplex of this Hilbert complex, and there exists a bounded cochain projection from that complex to the subcomplex. In the next part of the paper, we consider the most canonical example of the abstract theory, in which the Hilbert complex is the de Rham complex of a domain in Euclidean space. We use the Koszul complex to construct two families of finite element differential forms, show that these can be arranged in subcomplexes of the de Rham complex in numerous ways, and for each construct a bounded cochain projection. The abstract theory therefore applies to give the stability and convergence of finite element approximations of the Hodge Laplacian. Other applications are considered as well, especially the elasticity complex and its application to the equations of elasticity. Background material is included to make the presentation selfcontained for a variety of readers.
Design of tangent vector fields
 ACM Trans. Graph
, 2007
"... Tangent vector fields are an essential ingredient in controlling surface appearance for applications ranging from anisotropic shading to texture synthesis and nonphotorealistic rendering. To achieve a desired effect one is typically interested in smoothly varying fields that satisfy a sparse set of ..."
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Cited by 65 (4 self)
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Tangent vector fields are an essential ingredient in controlling surface appearance for applications ranging from anisotropic shading to texture synthesis and nonphotorealistic rendering. To achieve a desired effect one is typically interested in smoothly varying fields that satisfy a sparse set of userprovided constraints. Using tools from Discrete Exterior Calculus, we present a simple and efficient algorithm for designing such fields over arbitrary triangle meshes. By representing the field as scalars over mesh edges (i.e., discrete 1forms), we obtain an intrinsic, coordinatefree formulation in which field smoothness is enforced through discrete Laplace operators. Unlike previous methods, such a formulation leads to a linear system whose sparsity permits efficient prefactorization. Constraints are incorporated through weighted least squares and can be updated rapidly enough to enable interactive design, as we demonstrate in the context of anisotropic texture synthesis.
Discrete Laplace operators: No free lunch
, 2007
"... Discrete Laplace operators are ubiquitous in applications spanning geometric modeling to simulation. For robustness and efficiency, many applications require discrete operators that retain key structural properties inherent to the continuous setting. Building on the smooth setting, we present a set ..."
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Cited by 59 (1 self)
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Discrete Laplace operators are ubiquitous in applications spanning geometric modeling to simulation. For robustness and efficiency, many applications require discrete operators that retain key structural properties inherent to the continuous setting. Building on the smooth setting, we present a set of natural properties for discrete Laplace operators for triangular surface meshes. We prove an important theoretical limitation: discrete Laplacians cannot satisfy all natural properties; retroactively, this explains the diversity of existing discrete Laplace operators. Finally, we present a family of operators that includes and extends wellknown and widelyused operators.
Mesh Parameterization: Theory and Practice
 SIGGRAPH ASIA 2008 COURSE NOTES
, 2008
"... Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools ..."
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Cited by 56 (5 self)
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Mesh parameterization is a powerful geometry processing tool with numerous computer graphics applications, from texture mapping to animation transfer. This course outlines its mathematical foundations, describes recent methods for parameterizing meshes over various domains, discusses emerging tools like global parameterization and intersurface mapping, and demonstrates a variety of parameterization applications.
Discrete Surface Ricci Flow
 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS
"... This work introduces a unified framework for discrete surface Ricci flow algorithms, including spherical, Euclidean, and hyperbolic Ricci flows, which can design Riemannian metrics on surfaces with arbitrary topologies by userdefined Gaussian curvatures. Furthermore, the target metrics are conform ..."
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Cited by 42 (22 self)
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This work introduces a unified framework for discrete surface Ricci flow algorithms, including spherical, Euclidean, and hyperbolic Ricci flows, which can design Riemannian metrics on surfaces with arbitrary topologies by userdefined Gaussian curvatures. Furthermore, the target metrics are conformal (anglepreserving) to the original metrics. Ricci flow conformally deforms the Riemannian metric on a surface according to its induced curvature, such that the curvature evolves like a heat diffusion process. Eventually, the curvature becomes the user defined curvature. Discrete Ricci flow algorithms are based on a variational framework. Given a mesh, all possible metrics form a linear space, and all possible curvatures form a convex polytope. The Ricci energy is defined on the metric space, which reaches its minimum at the desired metric. The Ricci flow is the negative gradient flow of the Ricci energy. Furthermore, the Ricci energy can be optimized using Newton’s method more efficiently. Discrete Ricci flow algorithms are rigorous and efficient. Our experimental results demonstrate the efficiency, accuracy and flexibility of the algorithms. They have the potential for a wide range of applications in graphics, geometric modeling, and medical imaging. We demonstrate their practical values by global surface parameterizations.
Discrete surface ricci flow: theory and applications
 Mathematics of Surfaces XII. Lecture Notes in Computer Science
, 2007
"... Abstract. Conformal geometry is at the core of pure mathematics. Conformal structure is more flexible than Riemaniann metric but more rigid than topology. Conformal geometric methods have played important roles in engineering fields. This work introduces a theoretically rigorous and practically eff ..."
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Cited by 17 (5 self)
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Abstract. Conformal geometry is at the core of pure mathematics. Conformal structure is more flexible than Riemaniann metric but more rigid than topology. Conformal geometric methods have played important roles in engineering fields. This work introduces a theoretically rigorous and practically efficient method for computing Riemannian metrics with prescribed Gaussian curvatures on discrete surfaces—discrete surface Ricci flow, whose continuous counter part has been used in the proof of Poincare ́ conjecture. Continuous Ricci flow conformally deforms a Riemannian metric on a smooth surface such that the Gaussian curvature evolves like a heat diffusion process. Eventually, the Gaussian curvature becomes constant and the limiting Riemannian metric is conformal to the original one. In the discrete case, surfaces are represented as piecewise linear triangle meshes. Since the Riemannian metric and the Gaussian curvature are discretized as the edge lengths and the angle deficits, the discrete Ricci flow can be defined as the deformation of edge lengths driven by the discrete curvature. The existence and uniqueness of the solution and the convergence of the flow process are theoretically proven, and numerical algorithms to compute Riemannian metrics with prescribed Gaussian curvatures using discrete Ricci flow are also designed. Discrete Ricci flow has broad applications in graphics, geometric modeling, and medical imaging, such as surface parameterization, surface matching, manifold splines, and construction of geometric structures on general surfaces. 1
Usercontrollable Polycube Map for Manifold Spline Construction
"... (a)Usercontrollable polycube map. (b)Polycube Tspline. (c)Tjunctions on polycube spline. (d)Closeup of control points. Figure 1: Polycube spline for the David Body model. (a) The usercontrollable polycube map serves the parametric domain. (b) and (c) Polycube Tsplines obtained via affine struc ..."
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Cited by 9 (3 self)
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(a)Usercontrollable polycube map. (b)Polycube Tspline. (c)Tjunctions on polycube spline. (d)Closeup of control points. Figure 1: Polycube spline for the David Body model. (a) The usercontrollable polycube map serves the parametric domain. (b) and (c) Polycube Tsplines obtained via affine structure induced by the polycube map. Note that our polycube spline is globally defined as a “onepiece” shape representation without any cutting and gluing work except at the finite number of extraordinary points (corners of the polycube). The extraordinary points are colored in green in (b). The red curves on the spline surface (see (c)) highlight the Tjunctions. (d) Closeup of the spline model overlaid by the control points. The original model contains nearly 100K vertices and the polycube Tspline has 9781 control points. The rootmeansquare error is 0.3 % of the model’s main diagonal. Polycube Tspline has been formulated elegantly that can unify Tsplines and manifold splines to define a new class of shape representations for surfaces of arbitrary topology by using polycube map as its parametric domain. In essense, The data fitting quality using polycube Tsplines hinges upon the construction of underlying polycube maps. Yet, existing methods for polycube map construction exhibit some disadvantages. For example, existing approaches
Duals of OrphanFree Anisotropic Voronoi Diagrams are Embedded Meshes
, 2012
"... Given an anisotropic Voronoi diagram, we address the fundamental question of when its dual is embedded. We show that, by requiring only that the primal be orphanfree (have connected Voronoi regions), its dual is always guaranteed to be an embedded triangulation. Further, the primal diagram and its ..."
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Cited by 7 (1 self)
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Given an anisotropic Voronoi diagram, we address the fundamental question of when its dual is embedded. We show that, by requiring only that the primal be orphanfree (have connected Voronoi regions), its dual is always guaranteed to be an embedded triangulation. Further, the primal diagram and its dual have properties that parallel those of ordinary Voronoi diagrams: the primal’s vertices, edges, and faces are connected, and the dual triangulation has a simple, closed boundary. Additionally, if the underlying metric has bounded anisotropy (ratio of eigenvalues), the dual is guaranteed to triangulate the convex hull of the sites. These results apply to the duals of anisotropic Voronoi diagrams of any set of sites, so long as their Voronoi diagram is orphanfree. By combining this general result with existing conditions for obtaining orphanfree anisotropic Voronoi diagrams, a simple and natural condition for a set of sites to form an embedded anisotropic Delaunay triangulation follows.
Slit map: conformal parameterization for multiply connected surfaces
 In Proc. Geometric Modeling and Processing
, 2008
"... Abstract. Surface parameterization is a fundamental tool in geometric modeling and processing. Most existing methods deal with simply connected disks. This work introduces a novel method to handle multiply connected surfaces based on holomorphic oneforms. The method maps genus zero surfaces with ar ..."
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Cited by 3 (1 self)
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Abstract. Surface parameterization is a fundamental tool in geometric modeling and processing. Most existing methods deal with simply connected disks. This work introduces a novel method to handle multiply connected surfaces based on holomorphic oneforms. The method maps genus zero surfaces with arbitrary number of boundaries to an annulus with concentric circular slits. Any two boundaries can be chosen to map to the inner circle and the outer circle, the other boundaries to slits. Equivalently, the surfaces can be mapped to a rectangle with horizontal slits. Compared to existing linear methods that require surface partition, this method is more intrinsic and automatic. Compared to the existing holomorphic oneform method that requires double covering, it is more efficient and has better control over singularities. Compared to the existing Ricci flow method, this one is linear and simpler. The proposed method has many merits. The images of boundaries are parallel line segments. This regularity not only helps improve the accuracy for surface matching with boundaries, but also makes quadremeshing or meshspline conversion conversion convenient. The whole rectangle in texture domain is fully occupied without any gap or overlapping; this improves the packing efficiency for texture mapping. The positions of the slits are completely determined by the surface geometry, which can be treated as the finger print of the surface to classify surfaces by conformal equivalence. The algorithm is thoroughly explained in detail. Experimental results are demonstrated to show the usefulness of the algorithm for multiply connected domains. Key words: conformal parameterization, slits, holomorphic oneform 1