## QuadCover – Surface Parameterization using Branched Coverings. (2007)

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Venue: | COMPUT. GRAPH. FORUM |

Citations: | 63 - 7 self |

### BibTeX

@MISC{Kälberer07quadcover–,

author = {Felix Kälberer and Matthias Nieser and Konrad Polthier},

title = {QuadCover – Surface Parameterization using Branched Coverings.},

year = {2007}

}

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### Abstract

We introduce an algorithm for automatic computation of global parameterizations on arbitrary simplicial 2-manifolds whose parameter lines are guided by a given frame field, for example by principal curvature frames. The parameter lines are globally continuous, and allow a remeshing of the surface into quadrilaterals. The algorithm converts a given frame field into a single vector field on a branched covering of the 2-manifold, and generates an integrable vector field by a Hodge decomposition on the covering space. Except for an optional smoothing and alignment of the initial frame field, the algorithm is fully automatic and generates high quality quadrilateral meshes.

### Citations

238 | Illustrating smooth surfaces
- Hertzmann, Zorin
- 2000
(Show Context)
Citation Context ...tion vs. vector field straightness. 4.3. Aligning the frame fields To smooth the frame field, and to extend the principal curvature directions into umbilic regions, we adapted the smoothing method in =-=[HZ00]-=-. The frame Xi in each triangle Ti is rotated by an angle αi, such that adjacent frames align best with respect to the following smoothing energy. The energy operates on the angles between frames of p... |

115 | Global conformal surface parameterization
- GU, YAU
(Show Context)
Citation Context ...erent previous approaches to surface parameterization and, more general, the generation of quad and quad dominant meshes from given triangle meshes. One of the first impressive works is by Gu and Yau =-=[GY03]-=- who construct global conformal parameterizations of surfaces with arbitrary genus. The resulting parameter lines minimize angle distortion but may have a rather large metric distortion. Furthermore, ... |

109 |
Riemann Surfaces
- Farkas, Kra
- 1980
(Show Context)
Citation Context ...matchings independently at every edge and permit much more flexible topological structures of parameter lines. 2.2. Branched covering spaces First, recall some definitions about Riemann surfaces, see =-=[FK80]-=-, [Ful95], [Jos02]. Definition 1 Let M be a Riemann surface. A (branched) covering M ′ of M is a Riemann surface with a local homeomorphism π : M ′ → M, where for each p ′ ∈ M ′ there exists a neighbo... |

74 | Anisotropic filtering of non-linear surface features
- HILDEBRANDT, POLTHIER
- 2004
(Show Context)
Citation Context ...lic regions, such that they align with the curvature directions in their neighborhood (see Section 4.3). Discrete principal curvature directions and values can be calculated as proposed in [CSM03] or =-=[HP04]-=-. Note that we deal with curvatures given on triangles, not on vertices. 3.1. Algorithm overview The algorithm is split into two parts, the preprocessing part and the main algorithm. In the preprocess... |

73 | Spectral surface quadrangulation - DONG, BREMER, et al. - 2006 |

72 |
Designing quadrangulations with discrete harmonic forms
- TONG, ALLIEZ, et al.
- 2006
(Show Context)
Citation Context ...focus on creating coarse quad-dominant meshes in [MK06] by approximating the surface with very few patches, which are then individually subdivided into quads. Tong, Alliez, Cohen-Steiner, and Desbrun =-=[TACSD06]-=- use harmonic one-forms for surface parameterization. They enlarge the space of harmonic one-forms by allowing additional singular points on the surface. The extended cut graph increases the homology ... |

58 | Harmonic functions for quadrilateral remeshing of arbitrary manifolds
- Dong, Kircher, et al.
- 2005
(Show Context)
Citation Context ...al curvature directions are from Alliez et al. [ACSD ∗ 03]. They were extended by Marinov and Kobbelt [MK04], and base on the integration of curvature lines on the surface. Dong, Kircher, and Garland =-=[DKG05]-=- presented an algorithm which traces isolines in two conjugate harmonic vector fields. Marinov and Kobbelt focus on creating coarse quad-dominant meshes in [MK06] by approximating the surface with ver... |

55 | Discrete conformal mappings via circle patterns
- Kharevych, Springborn, et al.
- 2006
(Show Context)
Citation Context ...nes at given surface features. The method of Boier-Martin, Rushmeier, and Jin [BMRJ04] clusters the surface into macropatches and parameterizes each surface patch. Kharevych, Springborn, and Schröder =-=[KSS06]-=- find a conformal parameterization via circle patterns. In contrast to Gu and Yau, they use conesingularities to increase the flexibility of purely conformal mappings. Dong et al. [DBG ∗ 06] compute t... |

53 | Identifying vector fields singularities using a discrete hodge decomposition
- Polthier, Preu
- 2003
(Show Context)
Citation Context ...ise constant and curl free vector field on M ′ and each curl free vector field can be expressed as gradient field of a function in ˆSr(M ′ ), thus ∇ ˆSr(M ′ ) = {curl free vector fields on M ′ }, see =-=[PP03]-=-. Parameterization function. A symmetric covering function can be projected to a function ϕ : M → R 2 by taking the values of layer 0 and 1 in each point p ∈ Ui: ϕi(p) = (ϕ ′ (τ 0 Ui (p)),ϕ′ (τ 1 Ui (... |

50 | Periodic global parameterization - RAY - 2006 |

46 |
Direct anisotropic quad-dominant remeshing
- MARINOV, KOBBELT
(Show Context)
Citation Context ...o the cone singularities in [KSS06]. Early approaches for quadrangular remeshing guided by principal curvature directions are from Alliez et al. [ACSD ∗ 03]. They were extended by Marinov and Kobbelt =-=[MK04]-=-, and base on the integration of curvature lines on the surface. Dong, Kircher, and Garland [DKG05] presented an algorithm which traces isolines in two conjugate harmonic vector fields. Marinov and Ko... |

35 |
MORVAN J.-M.: Restricted Delaunay triangulations and normal cycle
- COHEN-STEINER
(Show Context)
Citation Context ...eld in umbilic regions, such that they align with the curvature directions in their neighborhood (see Section 4.3). Discrete principal curvature directions and values can be calculated as proposed in =-=[CSM03]-=- or [HP04]. Note that we deal with curvatures given on triangles, not on vertices. 3.1. Algorithm overview The algorithm is split into two parts, the preprocessing part and the main algorithm. In the ... |

35 |
Algebraic Topology, A First Course
- Fulton
- 1995
(Show Context)
Citation Context ...s independently at every edge and permit much more flexible topological structures of parameter lines. 2.2. Branched covering spaces First, recall some definitions about Riemann surfaces, see [FK80], =-=[Ful95]-=-, [Jos02]. Definition 1 Let M be a Riemann surface. A (branched) covering M ′ of M is a Riemann surface with a local homeomorphism π : M ′ → M, where for each p ′ ∈ M ′ there exists a neighborhood U ′... |

28 |
Parameterization of triangle meshes over quadrilateral domains
- BOIER-MARTIN, RUSHMEIER, et al.
(Show Context)
Citation Context .... Furthermore, the space of global conformal parameterizations is too rigid to allow a local alignment of the parameter lines at given surface features. The method of Boier-Martin, Rushmeier, and Jin =-=[BMRJ04]-=- clusters the surface into macropatches and parameterizes each surface patch. Kharevych, Springborn, and Schröder [KSS06] find a conformal parameterization via circle patterns. In contrast to Gu and Y... |

23 | Automatic and interactive mesh to t-spline conversion - LI, RAY, et al. - 2006 |

22 |
Compact Riemann Surfaces
- Jost
(Show Context)
Citation Context ...dently at every edge and permit much more flexible topological structures of parameter lines. 2.2. Branched covering spaces First, recall some definitions about Riemann surfaces, see [FK80], [Ful95], =-=[Jos02]-=-. Definition 1 Let M be a Riemann surface. A (branched) covering M ′ of M is a Riemann surface with a local homeomorphism π : M ′ → M, where for each p ′ ∈ M ′ there exists a neighborhood U ′ ∋ p ′ wi... |

12 | A robust two-step procedure for quad-dominant remeshing
- MARINOV, KOBBELT
(Show Context)
Citation Context ... surface. Dong, Kircher, and Garland [DKG05] presented an algorithm which traces isolines in two conjugate harmonic vector fields. Marinov and Kobbelt focus on creating coarse quad-dominant meshes in =-=[MK06]-=- by approximating the surface with very few patches, which are then individually subdivided into quads. Tong, Alliez, Cohen-Steiner, and Desbrun [TACSD06] use harmonic one-forms for surface parameteri... |

7 |
N-symmetry direction fields on surfaces of arbitrary genus
- RAY, VALLET, et al.
- 2008
(Show Context)
Citation Context ...ng surface. The branch points of our covering space lie at the branch points of the given frame field. They are conceptually similar to the singular points with fractional index used by [TACSD06] and =-=[RVLL06]-=-. Instead of restricting ourselves to harmonic frame fields, we allow an additional divergence part and seek the optimal parameterization in the larger space of locally integrable frame fields, enabli... |