## Globally optimal surface mapping for surfaces with arbitrary topology (2008)

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Venue: | IEEE Trans. on Visualization and Computer Graphics |

Citations: | 4 - 1 self |

### BibTeX

@ARTICLE{Li08globallyoptimal,

author = {Xin Li and Yunfan Bao and Xiaohu Guo and Miao Jin and Xianfeng Gu and Hong Qin},

title = {Globally optimal surface mapping for surfaces with arbitrary topology},

journal = {IEEE Trans. on Visualization and Computer Graphics},

year = {2008}

}

### OpenURL

### Abstract

Abstract — Computing smooth and optimal one-to-one maps between surfaces of same topology is a fundamental problem in graphics and such a method provides us a ubiquitous tool for geometric modeling and data visualization. Its vast variety of applications includes shape registration/matching, shape blending, material/data transfer, data fusion, information reuse, etc. The mapping quality is typically measured in terms of angular distortions among different shapes. This paper proposes and develops a novel quasi-conformal surface mapping framework to globally minimize the stretching energy inevitably introduced between two different shapes. The existing state-of-the-art intersurface mapping techniques only afford local optimization either on surface patches via boundary cutting or on the simplified base domain, lacking rigorous mathematical foundation and analysis. We design and articulate an automatic variational algorithm that can reach the global distortion minimum for surface mapping between shapes of arbitrary topology, and our algorithm is solely founded upon the intrinsic geometry structure of surfaces. To our best knowledge, this is the first attempt towards rigorously and numerically computing globally optimal maps. Consequently, we demonstrate our mapping framework offers a powerful computational tool for graphics and visualization tasks such as data and texture transfer, shape morphing, and shape matching. Index Terms — Quasi-conformal surface mapping, harmonic map, uniformization metric, surface parameterization.

### Citations

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Citation Context ... −v3)+q2(v3 −v1)+q3(v1 −v2))/(2〈p1, p2, p3〉). and Sv = (q1(u3−u2)+q2(u1−u3)+q3(u2−u1))/(2〈p1, p2, p3〉). The larger singular value Γ and smaller singular value γ of the Jacobian are given respectively =-=[38]-=-: � E + G ± Γ, γ = � (E − G) 2 + 4F 2 2 , where E, F, G are terms for the first fundamental form. We compute D ′ on each triangle using D ′ = Γ γ . The maximal value of D ′ of the mapping on the surfa... |

170 | Surface parameterization: a tutorial and survey
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Citation Context ...on to map genus-zero surfaces onto the common spherical domain. These types of techniques are usually based on spherical parameterization techniques [5], [7]–[9] or planar parameterization techniques =-=[10]-=-. Spheres and planar disks are natural domains for computing maps with minimized stretching energy directly. However, they can only serve as intermediate domains when the two surfaces are of genus zer... |

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Citation Context ...efly review some most related work, and refer interested readers to these surveys for details. Angle preservation is typically addressed either from the harmonic point of view (Dirichlet energy) [22]–=-=[24]-=- or from the conformal point of view (Cauchy-Riemann equation) [24], [25]. Most recently, the hyperbolic structure of Riemannian surfaces has been introduced to surface parameterization. Thurston firs... |

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130 |
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Citation Context ...ing a bijective mapping is mostly motivated by the need of blending two shapes. A natural approach is to use some canonical shape such as a sphere or the plane as the intermediate domain. Kent et al. =-=[3]-=- mapped star-shaped surfaces onto spheres, and merged them by clipping one sphere to the other. Kanai et. al. [4] used harmonic map on disk to build correspondence between two genus-zero closed or ope... |

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Citation Context ...ent the meshes into subregions first. For example, in [12] and [13], a common coarse base domain mesh has to be constructed manually by the user with domain knowledge in topological surgery; in [14], =-=[16]-=-–[18], feature points are firstly provided by users, then some automatic subregion tracing algorithms or progressive meshes are applied for coarse base mesh generation. The advantage of these approach... |

118 | Conformal surface parameterization for texture mapping
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Citation Context ...and their constrained spherical parameterization to map genus-zero surfaces onto the common spherical domain. These types of techniques are usually based on spherical parameterization techniques [5], =-=[7]-=-–[9] or planar parameterization techniques [10]. Spheres and planar disks are natural domains for computing maps with minimized stretching energy directly. However, they can only serve as intermediate... |

111 | Cross-Parameterization and Compatible Remeshing of 3D Models
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Citation Context ...ome areas due to under sampling, most likely in high curvature (e.g. sharp feature) areas on S2. In order to capture such geometric details, we simply apply an adaptive remeshing algorithm similar to =-=[17]-=-. We locally modify the connectivity of the mesh using edge split, 9sHarmonic Energy 45 40 35 30 25 20 15 10 5 Greek/4−Torus Knotty/2−Torus 2−Torus/Amphora 2−Torus/Vase Polycube/Buddha RockerArm/Torus... |

104 | Fundamentals of spherical parameterization for 3d meshes
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Citation Context ...and their constrained spherical parameterization to map genus-zero surfaces onto the common spherical domain. These types of techniques are usually based on spherical parameterization techniques [2], =-=[13]-=-, [18], [32] or planar parameterization techniques [12]. Spheres and planar disks are natural domains for computing maps with minimized stretching energy directly. However, they can only serve as inte... |

104 | Spherical parametrization and remeshing
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Citation Context ...their constrained spherical parameterization to map genus-zero surfaces onto the common spherical domain. These types of techniques are usually based on spherical parameterization techniques [5], [7]–=-=[9]-=- or planar parameterization techniques [10]. Spheres and planar disks are natural domains for computing maps with minimized stretching energy directly. However, they can only serve as intermediate dom... |

79 | Globally Smooth Parameterizations With Low Distortion
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Citation Context ...algorithms. Furthermore, the mappings are generally only C 0 continuous across the patch boundaries. In applications such as building domains for splines, a global continuity is critical. The work of =-=[19]-=- addresses the continuity problem by taking into account linear transition functions across patch boundaries. Manifold concept in mapping is introduced in [20], which primarily focuses on topology ins... |

78 | Merging polyhedral shapes with scattered features
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Citation Context ...d surfaces onto spheres, and merged them by clipping one sphere to the other. Kanai et. al. [4] used harmonic map on disk to build correspondence between two genus-zero closed or open surfaces. Alexa =-=[5]-=- wrapped two genus-zero surfaces onto a unit sphere, and computed the mapping by minimizing some distance function. Asirvatham et al. [6] used progressive mesh and their constrained spherical paramete... |

76 | Greedy optimal homotopy and homology generators
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Citation Context ...graph. Computing a special case of cut graphs which pass through a common given base point, called systems of loops, is studied in computational geometry field. One of the stateof-the-art techniques, =-=[34]-=-, used an efficient greedy algorithm to get an optimal cutting loop. We refine their algorithm for our surface cutting. We first briefly describe their algorithm for computing a system of loops L on t... |

73 | Multiresolution mesh morphing
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Citation Context ...y segment the meshes into subregions first. For example, in [12] and [13], a common coarse base domain mesh has to be constructed manually by the user with domain knowledge in topological surgery; in =-=[14]-=-, [16]–[18], feature points are firstly provided by users, then some automatic subregion tracing algorithms or progressive meshes are applied for coarse base mesh generation. The advantage of these ap... |

71 |
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Citation Context ...iew (Cauchy-Riemann equation) [24], [25]. Most recently, the hyperbolic structure of Riemannian surfaces has been introduced to surface parameterization. Thurston firstly introduced circle packing in =-=[26]-=-. An effective algorithm and implementation is presented by Stephenson in [27]. Circle packing has also been generalized to circle patterns [28] and used for surface parameterization in [29]. Hamilton... |

64 | H.: Inter-surface mapping
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Citation Context ... work, we focus on finding stretching-optimized maps between surfaces with non-trivial topology. Approaches for surfaces with non-trivial topology are usually applied through another direction ( [11]–=-=[18]-=-). They typically segment the meshes into subregions first. For example, in [12] and [13], a common coarse base domain mesh has to be constructed manually by the user with domain knowledge in topologi... |

59 | Combinatorial ricci flows on surfaces
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Citation Context ... circle patterns [28] and used for surface parameterization in [29]. Hamilton first introduced Ricci flow on surfaces in [30]. Theoretical results of combinatorial Ricci flow are later generalized in =-=[31]-=-, and applied in surface parameterization fields by [32]. A. Uniformization Metric III. THEORY AND ALGORITHM On a surface, a metric, or Riemannian metric is a tensor that defines inner product on the ... |

56 | Variational principles for circle patterns and Koebe’s theorem
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Citation Context ...rization. Thurston firstly introduced circle packing in [26]. An effective algorithm and implementation is presented by Stephenson in [27]. Circle packing has also been generalized to circle patterns =-=[28]-=- and used for surface parameterization in [29]. Hamilton first introduced Ricci flow on surfaces in [30]. Theoretical results of combinatorial Ricci flow are later generalized in [31], and applied in ... |

54 | Discrete conformal mappings via circle patterns
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Citation Context ...acking in [26]. An effective algorithm and implementation is presented by Stephenson in [27]. Circle packing has also been generalized to circle patterns [28] and used for surface parameterization in =-=[29]-=-. Hamilton first introduced Ricci flow on surfaces in [30]. Theoretical results of combinatorial Ricci flow are later generalized in [31], and applied in surface parameterization fields by [32]. A. Un... |

43 | Mesh parameterization methods and their applications
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(Show Context)
Citation Context ...s on topology instead of geometry, thus is difficult for designing optimization algorithms. Conformal maps have been extensively studied in the literature of the surface parameterization field. [10], =-=[21]-=- provide extensive surveys of state-of-the-art techniques in the field. We only briefly review some most related work, and refer interested readers to these surveys for details. Angle preservation is ... |

38 | Isotropic remeshing of surfaces: A local parametrization approach
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(Show Context)
Citation Context ...a user-specified amount of time are removed from memory on the fly. This is Not Local Parameterization. Note that chart-based approaches have been used in local-parameterization-based remeshing [36], =-=[37]-=-. And our approach is fundamentally different from them in that we are not locally parameterizing these one-ring charts, but directly embedding the pre-computed uniformization metric. Local parameteri... |

37 |
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(Show Context)
Citation Context ... (S1,g1) → (S2,g2). We use their uniformization metrics and compute a harmonic map ¯f : (S1, ¯g1) → (S2, ¯g2). The computational algorithm of hyperbolic harmonic maps is based on theoretic results in =-=[33]-=-. We denote the parameters of S1 on the Poincaré disk as (x, y), the parameter of S2 as (u, v), then the map ¯ f is represented as ¯ f(x, y) = (u(x, y), v(x, y)). The harmonic energy is E( ¯ � f) = S1... |

32 | Feature-based surface decomposition for correspondence and morphing between polyhedra
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(Show Context)
Citation Context ...pology. Approaches for surfaces with non-trivial topology are usually applied through another direction ( [11]–[18]). They typically segment the meshes into subregions first. For example, in [12] and =-=[13]-=-, a common coarse base domain mesh has to be constructed manually by the user with domain knowledge in topological surgery; in [14], [16]–[18], feature points are firstly provided by users, then some ... |

30 |
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(Show Context)
Citation Context ... canonical shape such as a sphere or the plane as the intermediate domain. Kent et al. [3] mapped star-shaped surfaces onto spheres, and merged them by clipping one sphere to the other. Kanai et. al. =-=[4]-=- used harmonic map on disk to build correspondence between two genus-zero closed or open surfaces. Alexa [5] wrapped two genus-zero surfaces onto a unit sphere, and computed the mapping by minimizing ... |

29 | Topological evolution of surfaces
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(Show Context)
Citation Context ...rivial topology. Approaches for surfaces with non-trivial topology are usually applied through another direction ( [11]–[18]). They typically segment the meshes into subregions first. For example, in =-=[12]-=- and [13], a common coarse base domain mesh has to be constructed manually by the user with domain knowledge in topological surgery; in [14], [16]–[18], feature points are firstly provided by users, t... |

27 |
Lectures on Harmonic Maps
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(Show Context)
Citation Context ... Let f : S1 → S2 be a harmonic map between closed oriented surfaces of the same genus with degree equals ±1. And KS2 ≤ 0, then f is a diffeomorphism. Detailed proof can be found in [35], page 187, or =-=[2]-=-, page 15. In our algorithm, the initial map is constructed by matching the fundamental polygons of S1 and S2. Therefore, each point on S2 has a unique pre-image on S1, hence, the degree of the initia... |

26 |
Introduction To Circle Packing
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(Show Context)
Citation Context ...ure of Riemannian surfaces has been introduced to surface parameterization. Thurston firstly introduced circle packing in [26]. An effective algorithm and implementation is presented by Stephenson in =-=[27]-=-. Circle packing has also been generalized to circle patterns [28] and used for surface parameterization in [29]. Hamilton first introduced Ricci flow on surfaces in [30]. Theoretical results of combi... |

23 |
Computing surface hyperbolic structure and real projective structure
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(Show Context)
Citation Context ...ion in [29]. Hamilton first introduced Ricci flow on surfaces in [30]. Theoretical results of combinatorial Ricci flow are later generalized in [31], and applied in surface parameterization fields by =-=[32]-=-. A. Uniformization Metric III. THEORY AND ALGORITHM On a surface, a metric, or Riemannian metric is a tensor that defines inner product on the tangent plane at each point. With the metric, the length... |

22 |
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(Show Context)
Citation Context ... let u : S → R be a globally defined function on S, then e 2u g is another Riemannian metric on S, which is a conformal metric to the original induced Euclidean metric. Riemann uniformization theorem =-=[35]-=- states that for any S, there exists a unique conformal metric, such that it induces Fig. 5. Side-by-side omparison between Distortions of Initial Map (left) and Optimized Map (right). constant Gaussi... |

21 | Topology-driven surface mappings with robust feature alignment
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- 2005
(Show Context)
Citation Context ...itly determined and it gives rise to the mapping complexity for shapes with nontrivial topology. Rigorously speaking, mappings between two given surfaces are classified into infinite homotopy classes =-=[1]-=-. Two maps are isotopic to each other, i.e., belonging to a same homotopy class, only if one can deform to another smoothly. A rigorous surface mapping framework should be able to handle an arbitraril... |

20 | On computing handle and tunnel loops
- Dey, Li, et al.
- 2007
(Show Context)
Citation Context ...y want. We naturally want handles mapped to handles consistently. To get consistent slicing orders of systems of loops, first, we can compute the canonical handle and tunnel loops using the method of =-=[39]-=-; second, with these handle and tunnel loops, we can decide the homotopy class of each closed loop in the system of loops, this pair loops in two systems of loops, providing the consistent slicing ord... |

17 |
transformation by boundary representation interpolation: a recursive approach to establishing face correspondences
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(Show Context)
Citation Context ... this work, we focus on finding stretching-optimized maps between surfaces with non-trivial topology. Approaches for surfaces with non-trivial topology are usually applied through another direction ( =-=[11]-=-–[18]). They typically segment the meshes into subregions first. For example, in [12] and [13], a common coarse base domain mesh has to be constructed manually by the user with domain knowledge in top... |

17 | H.-P.: Dynamic remeshing and applications
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(Show Context)
Citation Context ...d for a user-specified amount of time are removed from memory on the fly. This is Not Local Parameterization. Note that chart-based approaches have been used in local-parameterization-based remeshing =-=[36]-=-, [37]. And our approach is fundamentally different from them in that we are not locally parameterizing these one-ring charts, but directly embedding the pre-computed uniformization metric. Local para... |

15 | Multiresolution Interpolation Meshes - MICHIKAWA, KANAI, et al. - 2001 |

15 |
Computing Discrete Minimal Surfaces and Their
- Pinkall, Polthier
- 1993
(Show Context)
Citation Context ...some of the most related work and refer interested readers to these surveys for details. Angle preservation is typically addressed either from the harmonic point of view (Dirichlet energy) [8], [10], =-=[31]-=- or from the conformal point of view (Cauchy-Riemann equation) [8], [28]. Most recently, the hyperbolic structure of Riemannian surfaces has been introduced to surface parameterization. Thurston first... |

13 | Parameterizing N-holed tori
- Grimm, Hughes
- 2003
(Show Context)
Citation Context ...obal continuity is critical. The work of [19] addresses the continuity problem by taking into account linear transition functions across patch boundaries. Manifold concept in mapping is introduced in =-=[20]-=-, which primarily focuses on topology instead of geometry, thus is difficult for designing optimization algorithms. Conformal maps have been extensively studied in the literature of the surface parame... |

9 | Consistent spherical parameterization
- Asirvatham, Praun, et al.
- 2005
(Show Context)
Citation Context ...pondence between two genus-zero closed or open surfaces. Alexa [5] wrapped two genus-zero surfaces onto a unit sphere, and computed the mapping by minimizing some distance function. Asirvatham et al. =-=[6]-=- used progressive mesh and their constrained spherical parameterization to map genus-zero surfaces onto the common spherical domain. These types of techniques are usually based on spherical parameteri... |

3 |
Computing Surface Hyperbolic Structure and
- Jin, Luo, et al.
- 2006
(Show Context)
Citation Context ...n in [24]. Hamilton first introduced the Ricci flow on surfaces in [19]. Theoretical results of combinatorial Ricci flow are later generalized in [6] and applied in surface parameterization fields in =-=[20]-=-. 3 THEORY AND ALGORITHM 3.1 Uniformization Metric On a surface, a metric, orRiemannian metric is a tensor that defines the inner product on the tangent plane at each point. With the metric, the lengt... |

1 |
On Computing Handle and Tunnel
- Dey, Li, et al.
- 2007
(Show Context)
Citation Context ...y want. We naturally want handles mapped to handles consistently. To get consistent slicing orders of systems of loops, first, we can compute the canonical handle and tunnel loops using the method in =-=[9]-=-; second, with these handle and tunnel loops, we can decide the homotopy class of each closed loop in the system of loops, this pair loops in two systems of loops, providing the consistent slicing ord... |