## Optimal surface parameterization using inverse curvature map

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Venue: | Transactions on Visualization and Computer Graphics |

Citations: | 9 - 2 self |

### BibTeX

@ARTICLE{Yang_optimalsurface,

author = {Yong-liang Yang and Junho Kim and Feng Luo and Shi-min Hu and Xianfeng Gu},

title = {Optimal surface parameterization using inverse curvature map},

journal = {Transactions on Visualization and Computer Graphics},

year = {},

pages = {2008}

}

### OpenURL

### Abstract

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.

### Citations

2143 |
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Citation Context ...ose k is an interior point of k, also an optimum for an energy form EðuÞ, then all the components of rkE corresponding to the singular vertices are equal. Proof. The proof is based on the KKT theorem =-=[38]-=-. If k is an optimum of EðkÞ, then rkE? k. Suppose vi is a nonsingular vertex, the normal to the hyperplane fki 0g is ei, the normal to the plane f P vj2S kj 2 ðMÞg is d, where di 0 for nonsingu... |

293 | Geometry images
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Citation Context ...sometrically embed the mesh on the plane using the circle packing metric obtained from the optimization. We first compute a cut on the mesh to slice it to an open topological disk. Several algorithms =-=[39]-=-, [40] can be applied directly. Then, we embed the open mesh isometrically onto the plane using the optimal circle packing metric. For meshes with less than 30k faces, we select a face near to the cen... |

266 | Parametrization and smooth approximation of surface triangulations
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Citation Context ...y to approximate the Cauchy-Riemann equation; Desbrun et al. [5] optimized Dirichlet energy. Variations of harmonic energies were also optimized using discrete Laplace-Beltrami operators in [6], [7], =-=[8]-=-, [9], [10], and [11]. More general energy forms can be found in [12], [13], [14], [15], [16], and [17]. Most linear methods apply a convex Dirichlet-type boundary. Virtual boundaries were applied in ... |

178 | K.: Surface parameterization: a tutorial and survey
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Citation Context ... and it has broad applications in graphics. Parameterizations introduce distortions between the original surface and its planar image, which can be separated into angle distortion and area distortion =-=[1]-=-, [2], [3]. In theory, angle distortion can be eliminated completely by conformal mapping, but it is impossible for conformal mappings to further eliminate area distortion completely, except for devel... |

139 |
The Geometry and Topology of Three Manifolds
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Citation Context ...pings. Another characteristic of conformal mapping is that it can map infinitesimal circles to infinitesimal circles and preserve their intersection angles. This inspired the circle packing method in =-=[24]-=-. Circle packings and circle patterns replace infinitesimal circles with finite circles. In the limit of refinement, the continuous conformal maps are recovered [25]. Collins and Stephenson [26] have ... |

139 |
The Ricci Flow on Surfaces
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Citation Context ... [25]. Collins and Stephenson [26] have implemented circle packing in their software CirclePack which only considers combinatorics. The connection between circle packing and smooth surface Ricci flow =-=[27]-=- was discovered in [28]. The discrete Ricci flow method was introduced in [29] for hyperbolic parameterization. Kharevych et al. [30] provided conformal parameterizations for arbitrary genus types by ... |

71 | Feature-based surface parameterization and texture mapping. Technical report, Georgia Institute of Technology, 2003. GVU Tech Report 03-29. PARAMETERIZATION CURVATURE MAP TEXTURE MAPPING PARAMETER CRACKS? REMESHING [1] (a) Harmonic map of Eck et al. [5]:
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Citation Context ...s of harmonic energies were also optimized using discrete Laplace-Beltrami operators in [6], [7], [8], [9], [10], and [11]. More general energy forms can be found in [12], [13], [14], [15], [16], and =-=[17]-=-. Most linear methods apply a convex Dirichlet-type boundary. Virtual boundaries were applied in [17] and [18] to absorb distortions introduced by the convex boundary conditions. Alternatively, [4] an... |

62 | Combinatorial ricci flows on surfaces
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(Show Context)
Citation Context ...henson [26] have implemented circle packing in their software CirclePack which only considers combinatorics. The connection between circle packing and smooth surface Ricci flow [27] was discovered in =-=[28]-=-. The discrete Ricci flow method was introduced in [29] for hyperbolic parameterization. Kharevych et al. [30] provided conformal parameterizations for arbitrary genus types by applying circle pattern... |

59 | Variational principles for circle patterns and Koebe’s theorem - BOBENKO, SPRINGBORN - 2004 |

57 | ABF++: Fast and Robust Angle Based Flattening
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Citation Context ...inal mesh and their images on the parameter plane. During the process, the boundary evolves freely to further reduce the distortion. Recently, the method has been improved by several derivative works =-=[22]-=-, [23] in terms of speed and robustness. 1077-2626/08/$25.00 ß 2008 IEEE Published by the IEEE Computer Society Authorized licensed use limited to: Tsinghua University Library. Downloaded on January 4... |

57 | Discrete Conformal Mappings via Circle Patterns
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(Show Context)
Citation Context ...he connection between circle packing and smooth surface Ricci flow [27] was discovered in [28]. The discrete Ricci flow method was introduced in [29] for hyperbolic parameterization. Kharevych et al. =-=[30]-=- provided conformal parameterizations for arbitrary genus types by applying circle patterns based on the variational principle in Bobenko and Springborn [31]. The method in [30] supports very flexible... |

54 | Periodic global parameterization
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Citation Context ...vertices as singular ones in order to minimize the area distortions. Several effective methods have been introduced to select singular vertices, including manual selection [30], vector field analysis =-=[37]-=-, and spectral analysis [32]. We select the singular vertex set in step 3 of algorithm pipeline in Fig. 2. Intuitively, we first let the curvature be uniformly distributed on all vertices, then measur... |

45 |
Mean value coordinates, Computer Aided Geometric Design 20
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- 2003
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Citation Context ... Cauchy-Riemann equation; Desbrun et al. [5] optimized Dirichlet energy. Variations of harmonic energies were also optimized using discrete Laplace-Beltrami operators in [6], [7], [8], [9], [10], and =-=[11]-=-. More general energy forms can be found in [12], [13], [14], [15], [16], and [17]. Most linear methods apply a convex Dirichlet-type boundary. Virtual boundaries were applied in [17] and [18] to abso... |

43 |
de Sturler, Parameterization of faceted surfaces for meshing using angle-based flattening, Engineering with Computers 17 (3
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Citation Context ...l quasiharmonic maps to improve the boundary and reduce the distortion. One of the most prominent characteristics of conformal mapping is that it preserves angles. Angle-based flattening (ABF) method =-=[21]-=- utilizes this property to produce highquality conformal mappings. It derives the discrete conformal mapping by minimizing the ABF energy which is defined as differences between the corner angles of f... |

35 | A circle packing algorithm
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(Show Context)
Citation Context ...od in [24]. Circle packings and circle patterns replace infinitesimal circles with finite circles. In the limit of refinement, the continuous conformal maps are recovered [25]. Collins and Stephenson =-=[26]-=- have implemented circle packing in their software CirclePack which only considers combinatorics. The connection between circle packing and smooth surface Ricci flow [27] was discovered in [28]. The d... |

35 | BUNIN G.: Conformal flattening by curvature prescription and metric scaling
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(Show Context)
Citation Context ...ns ranging from free boundaries to control of the boundary shape via prescribed curvatures. It can further reduce the distortion by incorporating manually selected cone singularities. Ben-Chen et al. =-=[32]-=- introduced a conformal parameterization which automatically determines the locations and target curvatures of the cone singularities. Our work differs from the previous works in the following aspects... |

34 | An adaptable surface parameterization method
- Degener, Meseth, et al.
- 2003
(Show Context)
Citation Context ...nergy. Variations of harmonic energies were also optimized using discrete Laplace-Beltrami operators in [6], [7], [8], [9], [10], and [11]. More general energy forms can be found in [12], [13], [14], =-=[15]-=-, [16], and [17]. Most linear methods apply a convex Dirichlet-type boundary. Virtual boundaries were applied in [17] and [18] to absorb distortions introduced by the convex boundary conditions. Alter... |

31 | A fast and simple stretch-minimizing mesh parameterization
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- 2004
(Show Context)
Citation Context ... Variations of harmonic energies were also optimized using discrete Laplace-Beltrami operators in [6], [7], [8], [9], [10], and [11]. More general energy forms can be found in [12], [13], [14], [15], =-=[16]-=-, and [17]. Most linear methods apply a convex Dirichlet-type boundary. Virtual boundaries were applied in [17] and [18] to absorb distortions introduced by the convex boundary conditions. Alternative... |

28 | Smoothing an Overlay Grid to Minimize Linear Distortion in Texture Mapping
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- 2002
(Show Context)
Citation Context ...approximate the Cauchy-Riemann equation; Desbrun et al. [5] optimized Dirichlet energy. Variations of harmonic energies were also optimized using discrete Laplace-Beltrami operators in [6], [7], [8], =-=[9]-=-, [10], and [11]. More general energy forms can be found in [12], [13], [14], [15], [16], and [17]. Most linear methods apply a convex Dirichlet-type boundary. Virtual boundaries were applied in [17] ... |

23 | Mesh parameterization with a virtual boundary
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(Show Context)
Citation Context ...0], and [11]. More general energy forms can be found in [12], [13], [14], [15], [16], and [17]. Most linear methods apply a convex Dirichlet-type boundary. Virtual boundaries were applied in [17] and =-=[18]-=- to absorb distortions introduced by the convex boundary conditions. Alternatively, [4] and [5] provided parameterizations which require to fix only a few vertices in the parametric domain. Karni et a... |

20 |
Using Particles to Sample and
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Citation Context ...uivalently, the Ricci energy (4) is convex) to ensure the uniqueness of the solution. If the input mesh has too many obtuse angles and skinny triangles, we remesh it using the algorithms described in =-=[35]-=- and [36] to improve the mesh quality. 3.2 Computing the Initial Circle Packing Metric We use a simple method for the initial circle packing metric, described in Algorithm 2. Algorithm 2: Compute the ... |

20 | Robust feature classification and editing
- Lai, Zhou, et al.
(Show Context)
Citation Context ...y, the Ricci energy (4) is convex) to ensure the uniqueness of the solution. If the input mesh has too many obtuse angles and skinny triangles, we remesh it using the algorithms described in [35] and =-=[36]-=- to improve the mesh quality. 3.2 Computing the Initial Circle Packing Metric We use a simple method for the initial circle packing metric, described in Algorithm 2. Algorithm 2: Compute the initial c... |

15 |
Computing Discrete Minimal Surfaces and Their
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- 1993
(Show Context)
Citation Context ...energy to approximate the Cauchy-Riemann equation; Desbrun et al. [5] optimized Dirichlet energy. Variations of harmonic energies were also optimized using discrete Laplace-Beltrami operators in [6], =-=[7]-=-, [8], [9], [10], and [11]. More general energy forms can be found in [12], [13], [14], [15], [16], and [17]. Most linear methods apply a convex Dirichlet-type boundary. Virtual boundaries were applie... |

13 | Variational principles for circle patterns
- Bobenko, Springborn
(Show Context)
Citation Context ...urvatures are constant.) In this paper, we will explain our theoretical results and algorithmic implementations with circle packings. Since circle packing and circle patterns are equivalent in theory =-=[33]-=-, [34], our results can also be explained with the setting of circle patterns (see Appendix B). 1.2 Overview Most of the previous works minimize some energy forms which measure both angle distortion a... |

12 |
MIPS: An efficient global parametrization method. Curve and Surface Design
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Citation Context ...imized Dirichlet energy. Variations of harmonic energies were also optimized using discrete Laplace-Beltrami operators in [6], [7], [8], [9], [10], and [11]. More general energy forms can be found in =-=[12]-=-, [13], [14], [15], [16], and [17]. Most linear methods apply a convex Dirichlet-type boundary. Virtual boundaries were applied in [17] and [18] to absorb distortions introduced by the convex boundary... |

12 |
H.P.: Setting the boundary free: A composite approach to surface parameterization
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Citation Context ...parameterizations which require to fix only a few vertices in the parametric domain. Karni et al. [19] discussed the design of geometrically complex boundary conditions with constraints. Zayer et al. =-=[20]-=- applied discrete tensorial quasiharmonic maps to improve the boundary and reduce the distortion. One of the most prominent characteristics of conformal mapping is that it preserves angles. Angle-base... |

9 |
Linear angle based parameterization
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Citation Context ...esh and their images on the parameter plane. During the process, the boundary evolves freely to further reduce the distortion. Recently, the method has been improved by several derivative works [22], =-=[23]-=- in terms of speed and robustness. 1077-2626/08/$25.00 ß 2008 IEEE Published by the IEEE Computer Society Authorized licensed use limited to: Tsinghua University Library. Downloaded on January 4, 2009... |

4 |
Computing Surface Hyperbolic Structure and
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Citation Context ...ftware CirclePack which only considers combinatorics. The connection between circle packing and smooth surface Ricci flow [27] was discovered in [28]. The discrete Ricci flow method was introduced in =-=[29]-=- for hyperbolic parameterization. Kharevych et al. [30] provided conformal parameterizations for arbitrary genus types by applying circle patterns based on the variational principle in Bobenko and Spr... |

2 |
Free-Boundary Linear Parameterization of 3D
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Citation Context ... absorb distortions introduced by the convex boundary conditions. Alternatively, [4] and [5] provided parameterizations which require to fix only a few vertices in the parametric domain. Karni et al. =-=[19]-=- discussed the design of geometrically complex boundary conditions with constraints. Zayer et al. [20] applied discrete tensorial quasiharmonic maps to improve the boundary and reduce the distortion. ... |

1 |
The Convergence of Circle Packings to the Riemann
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Citation Context ...pired the circle packing method in [24]. Circle packings and circle patterns replace infinitesimal circles with finite circles. In the limit of refinement, the continuous conformal maps are recovered =-=[25]-=-. Collins and Stephenson [26] have implemented circle packing in their software CirclePack which only considers combinatorics. The connection between circle packing and smooth surface Ricci flow [27] ... |

1 |
Variational Principles for Discrete Surfaces. High Education Press, 2007. Authorized licensed use limited to: Tsinghua University Library
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Citation Context ...res are constant.) In this paper, we will explain our theoretical results and algorithmic implementations with circle packings. Since circle packing and circle patterns are equivalent in theory [33], =-=[34]-=-, our results can also be explained with the setting of circle patterns (see Appendix B). 1.2 Overview Most of the previous works minimize some energy forms which measure both angle distortion and are... |

1 |
Yong-Liang Yang received the bachelor’s degree in computer science from Tsinghua University, Beijing, in 2004. He is a PhD student in the Department of Computer Science and Technology, Tsinghua University. His research interests include computer graphics,
- Sheffer, Hart, et al.
- 2002
(Show Context)
Citation Context ...ically embed the mesh on the plane using the circle packing metric obtained from the optimization. We first compute a cut on the mesh to slice it to an open topological disk. Several algorithms [39], =-=[40]-=- can be applied directly. Then, we embed the open mesh isometrically onto the plane using the optimal circle packing metric. For meshes with less than 30k faces, we select a face near to the center of... |