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48
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 33 (2 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.
Greedy routing with guaranteed delivery using ricci flows
 In Proc. of the 8th International Symposium on Information Processing in Sensor Networks (IPSN’09
, 2009
"... Greedy forwarding with geographical locations in a wireless sensor network may fail at a local minimum. In this paper we propose to use conformal mapping to compute a new embedding of the sensor nodes in the plane such that greedy forwarding with the virtual coordinates guarantees delivery. In parti ..."
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Cited by 22 (15 self)
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Greedy forwarding with geographical locations in a wireless sensor network may fail at a local minimum. In this paper we propose to use conformal mapping to compute a new embedding of the sensor nodes in the plane such that greedy forwarding with the virtual coordinates guarantees delivery. In particular, we extract a planar triangulation of the sensor network with nontriangular faces as holes, by either using the nodes ’ location or using a landmarkbased scheme without node location. The conformal map is computed with Ricci flow such that all the nontriangular faces are mapped to perfect circles. Thus greedy forwarding will never get stuck at an intermediate node. The computation of the conformal map and the virtual coordinates is performed at a preprocessing phase and can be implemented by local gossipstyle computation. The method applies to both unit disk graph models and quasiunit disk graph models. Simulation results are presented for these scenarios.
Discrete surface ricci flow: Theory and applications
 In IMA Conference on the Mathematics of Surfaces
, 2007
"... Abstract. Conformal geometry is in 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 effi ..."
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Cited by 15 (3 self)
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Abstract. Conformal geometry is in 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 Poincaré 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.
Polycube splines
, 2008
"... This paper proposes a new concept of polycube splines and develops novel modeling techniques for using the polycube splines in solid modeling and shape computing. Polycube splines are essentially a novel variant of manifold splines which are built upon the polycube map, serving as its parametric dom ..."
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Cited by 13 (5 self)
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This paper proposes a new concept of polycube splines and develops novel modeling techniques for using the polycube splines in solid modeling and shape computing. Polycube splines are essentially a novel variant of manifold splines which are built upon the polycube map, serving as its parametric domain. Our rationale for defining spline surfaces over polycubes is that polycubes have rectangular structures everywhere over their domains, except a very small number of corner points. The boundary of polycubes can be naturally decomposed into a set of regular structures, which facilitate tensorproduct surface definition, GPUcentric geometric computing, and imagebased geometric processing. We develop algorithms to construct polycube maps, and show that the introduced polycube map naturally induces the affine structure with a finite number of extraordinary points. Besides its intrinsic rectangular structure, the polycube map may approximate any original scanned dataset with a very low geometric distortion, so our method for building polycube splines is both natural and necessary, as its parametric domain can mimic the geometry of modeled objects in a topologically correct and geometrically meaningful manner. We design a new data structure that facilitates the intuitive and rapid construction of polycube splines in this paper. We demonstrate the polycube splines with applications in surface reconstruction and shape computing.
Ricci flow for 3D shape analysis
 In Proceedings of ICCV ’07
, 2007
"... Ricci flow is a powerful curvature flow method in geometric analysis. This work is the first application of surface Ricci flow in computer vision. We show that previous methods based on conformal geometries, such as harmonic maps and leastsquare conformal maps, which can only handle 3D shapes with ..."
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Cited by 12 (7 self)
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Ricci flow is a powerful curvature flow method in geometric analysis. This work is the first application of surface Ricci flow in computer vision. We show that previous methods based on conformal geometries, such as harmonic maps and leastsquare conformal maps, which can only handle 3D shapes with simple topology are subsumed by our Ricci flow based method which can handle surfaces with arbitrary topology. Because the Ricci flow method is intrinsic and depends on the surface metric only, it is invariant to rigid motion, scaling, and isometric and conformal deformations. The solution to Ricci flow is unique and its computation is robust to noise. Our Ricci flow based method can convert all 3D problems into 2D domains and offers a general framework for 3D surface analysis. Large nonrigid deformations can be registered with feature constraints, hence we introduce a method that constrains Ricci flow computation using feature points and feature curves. Finally, we demonstrate the applicability of this intrinsic shape representation through standard shape analysis problems, such as 3D shape matching and registration. 1.
Andreev’s theorem on hyperbolic polyhedra
, 2006
"... In 1970, E. M. Andreev published a classification of all threedimensional compact hyperbolic polyhedra having nonobtuse dihedral angles [3, 4]. Given a combinatorial description of a polyhedron, C, Andreev’s Theorem provides five classes of linear inequalities, depending on C, for the dihedral ang ..."
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Cited by 9 (0 self)
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In 1970, E. M. Andreev published a classification of all threedimensional compact hyperbolic polyhedra having nonobtuse dihedral angles [3, 4]. Given a combinatorial description of a polyhedron, C, Andreev’s Theorem provides five classes of linear inequalities, depending on C, for the dihedral angles, which are necessary and sufficient conditions for the existence of a hyperbolic polyhedron realizing C with the assigned dihedral angles. Andreev’s Theorem also shows that the resulting polyhedron is unique, up to hyperbolic isometry. Andreev’s Theorem is both an interesting statement about the geometry of hyperbolic 3dimensional space, as well as a fundamental tool used in the proof for Thurston’s Hyperbolization Theorem for 3dimensional Haken manifolds. It is also remarkable to what level the proof of Andreev’s Theorem resembles (in a simpler way) the proof of Thurston. We correct a fundamental error in Andreev’s proof of existence and also provide a readable new proof of the other parts of the proof of Andreev’s Theorem, because
Intrinsic Geometric Scale Space by Shape Diffusion
"... Abstract—This paper formalizes a novel, intrinsic geometric scale space (IGSS) of 3D surface shapes. The intrinsic geometry of a surface is diffused by means of the Ricci flow for the generation of a geometric scale space. We rigorously prove that this multiscale shape representation satisfies the a ..."
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Cited by 9 (3 self)
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Abstract—This paper formalizes a novel, intrinsic geometric scale space (IGSS) of 3D surface shapes. The intrinsic geometry of a surface is diffused by means of the Ricci flow for the generation of a geometric scale space. We rigorously prove that this multiscale shape representation satisfies the axiomatic causality property. Within the theoretical framework, we further present a featurebased shape representation derived from IGSS processing, which is shown to be theoretically plausible and practically effective. By integrating the concept of scaledependent saliency into the shape description, this representation is not only highly descriptive of the local structures, but also exhibits several desired characteristics of global shape representations, such as being compact, robust to noise and computationally efficient. We demonstrate the capabilities of our approach through salient geometric feature detection and highly discriminative matching of 3D scans. Index Terms—Scale space, feature extraction, geometric flow, Riemannian manifolds. 1
Resilient Routing for Sensor Networks Using Hyperbolic Embedding of Universal Covering Space
"... Abstract—We study how to characterize the families of paths between any two nodes s, t in a sensor network with holes. Two paths that can be deformed to one another through local changes are called homotopy equivalent. Two paths that pass around holes in different ways have different homotopy types. ..."
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Cited by 9 (7 self)
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Abstract—We study how to characterize the families of paths between any two nodes s, t in a sensor network with holes. Two paths that can be deformed to one another through local changes are called homotopy equivalent. Two paths that pass around holes in different ways have different homotopy types. With a distributed algorithm we compute an embedding of the network in hyperbolic space by using Ricci flow such that paths of different homotopy types are mapped naturally to paths connecting s with different images of t. Greedy routing to a particular image is guaranteed with success to find a path with a given homotopy type. This leads to simple greedy routing algorithms that are resilient to both local link dynamics and large scale jamming attacks and improve load balancing over previous greedy routing algorithms. I.
Optimal surface parameterization using inverse curvature map
 Transactions on Visualization and Computer Graphics
"... Abstract—Mesh parameterization is a fundamental technique in computer graphics. The major goals during mesh parameterization are to minimize both the angle distortion and the area distortion. Angle distortion can be eliminated by the use of conformal mapping, in principle. Our paper focuses on solvi ..."
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Cited by 9 (2 self)
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Abstract—Mesh parameterization is a fundamental technique in computer graphics. The major goals during mesh parameterization are to minimize both the angle distortion and the area distortion. Angle distortion can be eliminated by the use of conformal mapping, in principle. Our paper focuses on solving the problem of finding the best discrete conformal mapping that also minimizes area distortion. First, we deduce an exact analytical differential formula to represent area distortion by curvature change in the discrete conformal mapping, giving a dynamic Poisson equation. On a mesh, the vertex curvature is related to edge lengths by the curvature map. Our result shows the map is invertible, i.e., the edge lengths can be computed from the curvature (by integration). Furthermore, we give the explicit Jacobi matrix of the inverse curvature map. Second, we formulate the task of computing conformal parameterizations with least area distortions as a constrained nonlinear optimization problem in curvature space. We deduce explicit conditions for the optima. Third, we give an energy form to measure the area distortions, and show that it has a unique global minimum. We use this to design an efficient algorithm, called free boundary curvature diffusion, which is guaranteed to converge to the global minimum; it has a natural physical interpretation. This result proves the common belief that optimal parameterization with least area distortion has a unique solution and can be achieved by free boundary conformal mapping. Major theoretical results and practical algorithms are presented for optimal parameterization based on the inverse curvature map. Comparisons are conducted with existing methods and using different energies. Novel parameterization applications are also introduced. The theoretical framework of the inverse curvature map can be applied to further study discrete conformal mappings.