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Discrete Surface Ricci Flow
 SUBMITTED TO IEEE TVCG
"... 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 40 (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. A 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
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 Industrial Model Predictive Control Technology", Chem. Process Control
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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
A Construction of Rational Manifold Surfaces of Arbitrary Topology and Smoothness from Triangular Meshes
"... Given a closed triangular mesh, we construct a smooth free–form surface which is described as a collection of rational tensor–product and triangular surface patches. The surface is obtained by a special manifold surface construction, which proceeds by blending together geometry functions for each ve ..."
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Cited by 7 (1 self)
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Given a closed triangular mesh, we construct a smooth free–form surface which is described as a collection of rational tensor–product and triangular surface patches. The surface is obtained by a special manifold surface construction, which proceeds by blending together geometry functions for each vertex. The transition functions between the charts, which are associated with the vertices of the mesh, are obtained via subchart parameterization.
Construction of C∞ surfaces from triangular meshes using parametric pseudomanifolds
, 2008
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Computational Conformal Geometry Applied in Engineering Fields
"... Abstract — Computational conformal geometry is an interdisciplinary field, combining modern geometry theories from pure mathematics with computational algorithms from computer science. Computational conformal geometry offers many powerful tools to handle a broad range of geometric problems in engine ..."
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Abstract — Computational conformal geometry is an interdisciplinary field, combining modern geometry theories from pure mathematics with computational algorithms from computer science. Computational conformal geometry offers many powerful tools to handle a broad range of geometric problems in engineering fields. This work summarizes our research results in the past years. We have introduced efficient and robust algorithms for computing conformal structures of surfaces acquired from the real life, which are based on harmonic maps, holomorphic differential forms and surface Ricci flow. We have applied conformal geometric algorithms in computer graphics, computer vision, geometric modeling and medical imaging. I.
ManifoldBased Surfaces with Boundaries
"... We present a manifoldbased surface construction extending the C ∞ construction of Ying and Zorin (2004a). Our surfaces allow for pircewisesmooth boundaries, have usercontrolled arbitrary degree of smoothness and improved derivative and visual behavior. 2flexibility of our surface construction is ..."
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Cited by 1 (0 self)
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We present a manifoldbased surface construction extending the C ∞ construction of Ying and Zorin (2004a). Our surfaces allow for pircewisesmooth boundaries, have usercontrolled arbitrary degree of smoothness and improved derivative and visual behavior. 2flexibility of our surface construction is confirmed numerically for a range of local mesh configurations. Key words: Geometric modeling, manifolds, highorder surfaces