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## Genus zero surface conformal mapping and its application to brain surface mapping (2004)

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Venue: | IEEE Transactions on Medical Imaging |

Citations: | 188 - 78 self |

### Citations

600 | Stuetzle Multiresolution analysis for arbitrary meshes
- Eck, DeRose, et al.
- 1995
(Show Context)
Citation Context ...generalize the method for surfaces with boundaries in [7]. For genus zero surfaces, there are five basic approaches to achieve conformal parameterizations. 1. Harmonic energy minimization. Eck et al. =-=[8]-=- introduce the discrete harmonic map, which approximates the continuous harmonic map [9] by minimizing a metric dispersion criterion. Desbrun et al. [10, 11] compute the discrete Dirichlet energy and ... |

407 |
Special Functions and the Theory of Group Representations,
- Vilenkin
- 1968
(Show Context)
Citation Context ...ier coefficient, equal to . Spherical harmonic has an explicit formula where is the associated Legendre function of degree (29) and order , and is a normalization factor. The details are explained in =-=[24]-=-. Once the brain surface is conformally mapped to , the surface can be represented as three spherical functions, , and . The function to is regularly sampled and transformed using the fast spherical h... |

347 | Computing discrete minimal surfaces and their conjugates,
- Pinkall, Juni, et al.
- 1993
(Show Context)
Citation Context ...gy and apply conformal parameterization to interactive geometry remeshing. Pinkall and Polthier compute the discrete harmonic map and Hodge star operator for the purpose of creating a minimal surface =-=[12]-=-. Kanai et al. use a harmonic map for geometric metamorphosis in [13]. Gu and Yau in [6] introduce a non-linear optimization method to compute global conformal parameterizations for genus zero surface... |

339 | High-resolution intersubject averaging and a coordinate system for the cortical surface.
- Fischl, Sereno, et al.
- 1999
(Show Context)
Citation Context ...ng brain data. One way to analyze and compare brain data is to map them into a canonical space while retaining geometric information on the original structures as far as possible [1–5]. Fischl et al=-=. [1]-=- demonstrate that surface based brain mapping can offer advantages over volume based brain mapping, especially when localizing cortical deficits and functional activations. Thompson et al. [4, 5] intr... |

323 | Least squares conformal maps for automatic texture atlas generation”.
- Levy, Petitjean, et al.
- 2002
(Show Context)
Citation Context ...ethod to compute global conformal parameterizations for genus zero surfaces. The optimization is carried out in the tangent spaces of the sphere. 2. Cauchy-Riemann equation approximation. Levy et al. =-=[14]-=- compute a quasiconformal parameterization of topological disks by approximating the CauchyRiemann equation using the least squares method. They show rigorously that the quasi-conformal parameterizati... |

140 | Conformal Surface Parameterization for Texture Mapping
- HAKER, ANGENENT, et al.
- 2000
(Show Context)
Citation Context ...si-conformal parameterization exists uniquely, and is invariant to similarity transformations, independent of resolution, and orientation preserving. 3. Laplacian operator linearization. Haker et al. =-=[3]-=- use a method to compute a global conformal mapping from a genus zero surface to a sphere by representing the Laplace-Beltrami operator as a linear system. 4. Circle packing. Circle packing is introdu... |

139 | Global conformal surface parameterization. In:
- Gu, Yau
- 2003
(Show Context)
Citation Context ...] introduce a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one-forms. They generalize the method for surfaces with boundaries in =-=[7]-=-. For genus zero surfaces, there are five basic approaches to achieve conformal parameterizations. 1. Harmonic energy minimization. Eck et al. [8] introduce the discrete harmonic map, which approximat... |

132 |
Desbrun M. Interactive geometry remeshing
- Alliez, Meyer
(Show Context)
Citation Context ...ns. 1. Harmonic energy minimization. Eck et al. [8] introduce the discrete harmonic map, which approximates the continuous harmonic map [9] by minimizing a metric dispersion criterion. Desbrun et al. =-=[10, 11]-=- compute the discrete Dirichlet energy and apply conformal parameterization to interactive geometry remeshing. Pinkall and Polthier compute the discrete harmonic map and Hodge star operator for the pu... |

129 | FFTs for the 2-Sphere--Improvements and Variations,”
- Healy, Kostelec, et al.
- 2003
(Show Context)
Citation Context ... mapped to , the surface can be represented as three spherical functions, , and . The function to is regularly sampled and transformed using the fast spherical harmonic transformation as described in =-=[25]-=-. Many processing tasks that use the geometric surface of the brain can be accomplished in the frequency domain more efficiently, such as geometric compression, matching, surface denoising, feature de... |

105 | Dynamics of gray matter loss in Alzheimer's disease.
- Thompson, Hayashi, et al.
- 2003
(Show Context)
Citation Context ...rks, and the result of the optimization by a Möbius transformation. Our landmarks consist of a set of sulcal lines that were manually traced on 3-D surface models extracted from individual MRI images =-=[29]-=-. The lines correspond to various sulci, such as the central sulcus, post-central sulcus, pre-central sulcus, etc. The mappings were constrained by all landmarks that occur consistently in the brains ... |

93 | On the geometry and shape of brain sub-manifolds.
- Joshi, Miller, et al.
- 1997
(Show Context)
Citation Context ...he geometric surface of the brain can be accomplished in the frequency domain more efficiently, such as geometric compression, matching, surface denoising, feature detection, and shape analysis [26], =-=[27]-=-. A. Brain Geometry Compression Similar to image compression using Fourier analysis, geometric brain data can be compressed using spherical harmonic analysis [26]. Global geometric information is conc... |

88 | Shape analysis of brain ventricles using SPHARM. In:
- Gerig, Styner, et al.
- 2001
(Show Context)
Citation Context ...re them directly without any registration. Fig. 12 illustrate the shape descriptors for the same brain with different orientations. It is clear that the shape descriptor is totally rotation invariant =-=[28]-=-. The brain surface can be represented as a vector valued function defined on the sphere via conformal mapping of its surface to the surface. The brain surface can then be decomposed in terms of linea... |

82 |
A surface-based technique for warping 3-dimensional images of the brain.
- Thompson, Toga
- 1996
(Show Context)
Citation Context ...for our parameterization purpose. Bakircioglu et al. use spherical harmonics to compute a flow on the sphere in [15] in order to match curves on the brain. Thompson and Toga use a similar approach in =-=[16]-=-. This flow field can be thought of as the variational minimizer of the integral over the sphere of Lu, with L some power of the Laplacian, and u the deformation. This is very similar to the spherical... |

76 |
A numerical solution to the generalized mapmaker’s problem: flattening nonconvex polyhedral surfaces.
- Schwartz, Shaw, et al.
- 1989
(Show Context)
Citation Context ...tegrating and comparing brain data. One way to analyze and compare brain data is to map them into a canonical space while retaining geometric information on the original structures as far as possible =-=[3]��-=-� [9]. Among them, Schwartz et al. [3] and Timsari [7] computed quasi-isometric flat maps of the cerebral cortex. Hurdal et al. [5] and Haker et al. [6] computed quasi-conformal and conformal maps of ... |

67 | Computing conformal structures of surfaces.
- Gu, Yau
- 2002
(Show Context)
Citation Context ...ameterizations have been studied intensively. Most works in conformal parameterizations deal with surface patches homeomorphic to topological disks. For surfaces with arbitrary topologies, Gu and Yau =-=[6]-=- introduce a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one-forms. They generalize the method for surfaces with boundaries in [... |

63 |
Parameterization of faceted surfaces for meshing using angle based flattening,
- Sheffer, Sturler
- 2001
(Show Context)
Citation Context ...3, 16] use a method to compute a global conformal mapping from a genus zero surface to a sphere by representing the Laplace-Beltrami operator as a linear system. 4. Angle based method. Sheffer et al. =-=[17]-=- introduce an angle based flattening method to flatten a mesh to a 2D plane so that it minimizes the relative distortion of the planar angles with respect to their counterparts in the three-dimensiona... |

49 |
Lectures on harmonic maps.
- Schoen, Yau
- 1997
(Show Context)
Citation Context ...e are five basic approaches to achieve conformal parameterizations. 1. Harmonic energy minimization. Eck et al. [8] introduce the discrete harmonic map, which approximates the continuous harmonic map =-=[9]-=- by minimizing a metric dispersion criterion. Desbrun et al. [10, 11] compute the discrete Dirichlet energy and apply conformal parameterization to interactive geometry remeshing. Pinkall and Polthier... |

48 | A framework for computational anatomy.
- Thompson, Toga
- 2002
(Show Context)
Citation Context ... et al. [1] demonstrate that surface based brain mapping can offer advantages over volume based brain mapping, especially when localizing cortical deficits and functional activations. Thompson et al. =-=[4, 5]-=- introduce a mathematical framework based on covariant partial differential equations, and pull-backs of mappings under harmonic flows, to help analyze signals localized on brain surfaces.s2 Xianfeng ... |

47 | Quasiconformally flat mapping the human cerebellum. Medical Image Computing and Computer-Assisted Intervention-MICCAIV99.
- Hurdal, Bowers, et al.
- 1999
(Show Context)
Citation Context ...h to a 2D plane so that it minimizes the relative distortion of the planar angles with respect to their counterparts in the three-dimensional space. 5. Circle packing. Circle packing is introduced in =-=[18]-=-. Classical analytic functions can be approximated using circle packings. But for general surfaces in R 3 , the circle packing method only considers the connectivity but not geometry, so it is not sui... |

47 |
de Sturler, E.: Parameterization of faceted surfaces for meshing using angle based flattening. Engineering with Computers 17
- Sheffer
- 2001
(Show Context)
Citation Context ..., [15] use a method to compute a global conformal mapping from a genus zero surface to a sphere by representing the Laplace-Beltrami operator as a linear system. 4) Angle based method. Sheffer et al. =-=[16]-=- introduce an angle based flattening method to flatten a mesh to a 2D plane so that it minimizes the relative distortion of the planar angles with respect to their counterparts in the three-dimensiona... |

47 | Nondistorting flattening maps and the 3D visualization of colon CT images
- Haker, Angenent, et al.
(Show Context)
Citation Context ...ry, so it is not suitable for our parameterization purpose. 2) Global Conformal Parameterization for Nonzero Genus Closed Surfaces: For genus one surfaces, conformal parameterization is introduced in =-=[20]-=- by adding periodic constraints for harmonic maps defined on the fundamental domain of the surface. It is impossible to generalize the current harmonic mapping method to the high genus case. The probl... |

34 |
Threedimensional geometric metamorphosis based on harmonic maps. The Visual Computer
- Suzuki, Kimura
- 1998
(Show Context)
Citation Context ...hing. Pinkall and Polthier compute the discrete harmonic map and Hodge star operator for the purpose of creating a minimal surface [12]. Kanai et al. use a harmonic map for geometric metamorphosis in =-=[13]-=-. Gu and Yau in [6] introduce a non-linear optimization method to compute global conformal parameterizations for genus zero surfaces. The optimization is carried out in the tangent spaces of the spher... |

34 |
Mapping techniques for aligning sulci across multiple brains. Med Image Anal 8(3):295--309
- Tosun, RettmannME
- 2004
(Show Context)
Citation Context ... current conformal and harmonic surface parameterization methods in Table I. There are some methods applying the Möbius automorphism group to brain conformal mapping. For instance, Tosun et al. [22], =-=[23]-=- used an approach based on Haker’s conformal mapping and employed a Möbius transformation to minimize area distortion and sulcal alignment across multiple brains. B. Basic Idea Suppose are two surface... |

27 | Optimization method for creating semi-isometric flat maps of the cerebral cortex. Medical Imaging 2000: Image Processing, - Timsari, Leahy - 2000 |

24 |
M.I.: Curve matching on brain surfaces using Frenet distances. Human Brain Mapping 6
- Bakircioglu, Grenander, et al.
- 1998
(Show Context)
Citation Context ...king method considers only the connectivity but not the geometry, so it is not suitable for our parameterization purpose. Bakircioglu et al. use spherical harmonics to compute a flow on the sphere in =-=[15]-=- in order to match curves on the brain. Thompson and Toga use a similar approach in [16]. This flow field can be thought of as the variational minimizer of the integral over the sphere of Lu, with L s... |

20 |
Conformal geometry and brain flattening.
- Angenent, Haker, et al.
- 1999
(Show Context)
Citation Context ...nformal parameterization exists uniquely, and is invariant to similarity transformations, independent of resolution, and orientation-preserving. c) Laplacian operator linearization. Haker et al. [6], =-=[17]-=- use a method to compute a global conformal mapping from a genus zero surface to a sphere by representing the Laplace–Beltrami operator as a linear system. d) Angle-based method. Sheffer et al. [18] i... |

19 | Detecting Disease-Specific Patterns of Brain Structure using Cortical Pattern Matching and a Population-Based Probabilistic Brain Atlas.
- Thompson, Mega, et al.
- 2001
(Show Context)
Citation Context ... et al. [1] demonstrate that surface based brain mapping can offer advantages over volume based brain mapping, especially when localizing cortical deficits and functional activations. Thompson et al. =-=[4, 5]-=- introduce a mathematical framework based on covariant partial differential equations, and pull-backs of mappings under harmonic flows, to help analyze signals localized on brain surfaces.s2 Xianfeng ... |

18 | Surface parametrization and shape description
- Brechbühler, Gerig, et al.
- 1992
(Show Context)
Citation Context ... use the geometric surface of the brain can be accomplished in the frequency domain more efficiently, such as geometric compression, matching, surface denoising, feature detection, and shape analysis =-=[26]-=-, [27]. A. Brain Geometry Compression Similar to image compression using Fourier analysis, geometric brain data can be compressed using spherical harmonic analysis [26]. Global geometric information i... |

15 |
Coordinate systems for conformal cerebellar flat maps. NeuroImage 11
- HURDAL, STEPHENSON, et al.
- 2000
(Show Context)
Citation Context ...method to compute a global conformal mapping from a genus zero surface to a sphere by representing the Laplace-Beltrami operator as a linear system. 4. Circle packing. Circle packing is introduced in =-=[2]-=-. Classical analytic functions can be approximated using circle packing. But for general surfaces in R 3 , the circle packing method considers only the connectivity but not the geometry, so it is not ... |

15 | Approximation of conformal structures via circle packing - Stephenson - 1999 |

14 |
Intrinsic parametrizations of surface meshes
- Desbrun, Meyer, et al.
- 2002
(Show Context)
Citation Context ...ns. 1. Harmonic energy minimization. Eck et al. [8] introduce the discrete harmonic map, which approximates the continuous harmonic map [9] by minimizing a metric dispersion criterion. Desbrun et al. =-=[10, 11]-=- compute the discrete Dirichlet energy and apply conformal parameterization to interactive geometry remeshing. Pinkall and Polthier compute the discrete harmonic map and Hodge star operator for the pu... |

14 |
A hemispherical map for the human brain cortex
- Tosun, Prince
(Show Context)
Citation Context ...marize current conformal and harmonic surface parameterization methods in Table I. There are some methods applying the Möbius automorphism group to brain conformal mapping. For instance, Tosun et al. =-=[22]-=-, [23] used an approach based on Haker’s conformal mapping and employed a Möbius transformation to minimize area distortion and sulcal alignment across multiple brains. B. Basic Idea Suppose are two s... |

5 |
Conformal geometry and brain flattening,” MICCAI
- Angenent, Haker, et al.
- 1999
(Show Context)
Citation Context ...si-conformal parameterization exists uniquely, and is invariant to similarity transformations, independent of resolution, and orientation preserving. 3. Laplacian operator linearization. Haker et al. =-=[3, 16]-=- use a method to compute a global conformal mapping from a genus zero surface to a sphere by representing the Laplace-Beltrami operator as a linear system. 4. Angle based method. Sheffer et al. [17] i... |

1 |
Solving variational problems and partial equations mapping into general target manifolds
- Memoli, Sapiro, et al.
- 2002
(Show Context)
Citation Context ...ethod is more accurate, with no regions of large area distortion. It is also more stable and can be readily extended to compute maps between two general manifolds. Finally, we note that Memoli et al. =-=[17]-=- mentioned they were developing implicit methods to compute harmonic maps between general source and target Subject Vertex # Face # Before After A 65,538 131,072 - - B 65,538 131,072 604.134 506.665 C... |

1 |
Invariant conformal mapping of cortical surfaces
- Joshi, Leahy
- 2002
(Show Context)
Citation Context ...tegrating and comparing brain data. One way to analyze and compare brain data is to map them into a canonical space while retaining geometric information on the original structures as far as possible =-=[1, 2, 3, 4, 5, 6]-=-. Fischl et al. [1] demonstrate that surface based brain mapping can offer advantages over volume based brain mapping, especially when localizing cortical deficits and functional activations. Thompson... |

1 |
conformal surface parameterization
- “Global
(Show Context)
Citation Context ...] introduce a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one-forms. They generalize the method for surfaces with boundaries in =-=[7]-=-. For genus zero surfaces, there are five basic approaches to achieve conformal parameterizations. 1) Harmonic energy minimization. Eck et al. [8] introduce the discrete harmonic map, which approximat... |

1 |
Invariant conformal mapping of cortical surfaces,” in Image analysis and understanding data from scientific experiments
- Joshi, Leahy
- 2002
(Show Context)
Citation Context ...the cortical surface to the complex plane. In the resulting mapping, the local shape is preserved and distances and areas are only changed by a scaling factor. Based on Haker et al. [6], Joshi et al. =-=[30]-=- obtained a unique conformal mapping by fixing three point correspondences between two brains. Since stereo projection is involved, there is significant distortion around the north pole areas, which b... |