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Optimal Möbius Transformations for Information Visualization and Meshing [chapter]

Marshall Bern, David Eppstein
2001 Lecture Notes in Computer Science  
We give linear-time quasiconvex programming algorithms for finding a Möbius transformation of a set of spheres in a unit ball or on the surface of a unit sphere that maximizes the minimum size of a transformed  ...  We apply these results to vertex separation and symmetry display in spherical graph drawing, viewpoint selection in hyperbolic browsing, element size control in conformal structured mesh generation, and  ...  Acknowledgements We thank Hervé Brönnimann, Fan Chung, and Ken Stephenson for helpful discussions.  ... 
doi:10.1007/3-540-44634-6_3 fatcat:z5qun34a25ckxbblodrvucj2ua

Optimal Moebius Transformations for Information Visualization and Meshing [article]

Marshall Bern, David Eppstein
2001 arXiv   pre-print
We give linear-time quasiconvex programming algorithms for finding a Moebius transformation of a set of spheres in a unit ball or on the surface of a unit sphere that maximizes the minimum size of a transformed  ...  We apply these results to vertex separation and symmetry display in spherical graph drawing, viewpoint selection in hyperbolic browsing, element size control in conformal structured mesh generation, and  ...  Conclusions We have identified several applications in information visualization and structured mesh generation for which it is of interest to find a Möbius transformation that optimizes an objective function  ... 
arXiv:cs/0101006v2 fatcat:yvewgrcecvbt3d3mkuolha5sve

Optimal Global Conformal Surface Parameterization for Visualization

Xianfeng Gu, Miao Jin, Yalin Wang, Shin-Tung Yau
2004 Communications in Information and Systems  
Good parameterization is crucial for simulation and visualization. This paper gives an explicit method for finding optimal global conformal parameterizations of arbitrary surfaces.  ...  The method is demonstrated by visualizing the Riemann surface structure of real surfaces represented as triangle meshes.  ...  This is the only means to visually convey conformal information of surfaces.  ... 
doi:10.4310/cis.2004.v4.n2.a1 fatcat:6w2kjmybvjfyrck6ykdlnepi6u

Intrinsic brain surface conformal mapping using a variational method

Yalin Wang, Xianfeng Gu, Tony F. Chan, Paul M. Thompson, Shing-Tung Yau, J. Michael Fitzpatrick, Milan Sonka
2004 Medical Imaging 2004: Image Processing  
We developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one-forms with or without boundaries. 1, 2 For genus zero surfaces, our  ...  We use a mesh structure to represent the brain surface. Further constraints are added to ensure that the conformal map is unique.  ...  We use landmarks to obtain the optimal Möbius transformation. Landmarks are commonly used in brain mapping.  ... 
doi:10.1117/12.534480 dblp:conf/miip/0001GCTY04 fatcat:uixrjosqyrcllbkiwbhtvnh4nm

Animation Quadrangulation on Symmetric Surfaces

Ping Du, Changhe Tu
2016 Innovative Computing Information and Control Express Letters, Part B: Applications  
The experimental results we obtained show better in Hausdorff distance and average angle compared to the state-of-the-art algorithms, while the singularity points are less and the visual aesthetics are  ...  We present a novel method to optimize the cross field in the first stage of quadrangulation pipeline to strike a balance between symmetry and animation.  ...  We would like to thank Daniele Panozzo and Alec Jacobson for the geometry processing framework libigl [15], the authors of [19] for making data of the sequences of triangle meshes on the website [18]  ... 
doi:10.24507/icicelb.07.07.1415 fatcat:pw2e5adysbapvi7w5qbf2teiky

Conformal Surface Morphing with Applications on Facial Expressions [article]

Mei-Heng Yueh, Xianfeng David Gu, Wen-Wei Lin, Chin-Tien Wu, Shing-Tung Yau
2015 arXiv   pre-print
Furthermore, by composition of the M\"obius transformation and the 3D deformation field, the morphing sequence can be generated from the mean curvatures and the conformal factors on a unified mesh structure  ...  By registering the landmarks on the conformal parametric domains, the correspondence of the mean curvatures and the conformal factors for each surfaces can be obtained.  ...  Möbius transformation and the optimal Möbius transformation with the global matching functionFig. 9.  ... 
arXiv:1504.00097v1 fatcat:unscmai3gvfhndwylvooyutufe

Establishing Dense Correspondence of High Resolution 3D Faces via Möbius Transformations

Jian Liu, Quan Zhang, Chaojing Tang
2014 International Journal of Computer and Electrical Engineering  
These correspondent facial feature points are used to generate Möbius transformations and achieve sparse correspondence between 3D faces.  ...  The facial feature points are located by using texture and shape information of 3D faces.  ...  the research in this paper use the BJUT-3D Face Database collected under the joint sponsor of National Natural Science Foundation of China, Beijing Natural Science Foundation Program, Beijing Science and  ... 
doi:10.17706/ijcee.2014.v6.866 fatcat:trqxr3go4rhsnbwway2m3cyssi

Genus Zero Surface Conformal Mapping and Its Application to Brain Surface Mapping

X. Gu, Y. Wang, T.F. Chan, P.M. Thompson, S.-T. Yau
2004 IEEE Transactions on Medical Imaging  
We developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one-forms for surfaces with or without boundaries (Gu and Yau, 2002),  ...  Index Terms-Brain mapping, conformal map, landmark matching, spherical harmonic transformation.  ...  Fig. 9 shows the landmarks, and the result of the optimization by a Möbius transformation.  ... 
doi:10.1109/tmi.2004.831226 pmid:15338729 fatcat:wzezjyymfze7jang2cxcsednie

Genus Zero Surface Conformal Mapping and Its Application to Brain Surface Mapping [chapter]

Xianfeng Gu, Yalin Wang, Tony F. Chan, Paul M. Thompson, Shing-Tung Yau
2003 Lecture Notes in Computer Science  
We developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one-forms for surfaces with or without boundaries (Gu and Yau, 2002),  ...  Index Terms-Brain mapping, conformal map, landmark matching, spherical harmonic transformation.  ...  Fig. 9 shows the landmarks, and the result of the optimization by a Möbius transformation.  ... 
doi:10.1007/978-3-540-45087-0_15 fatcat:xxptcaiytbbjdcxuandqtkj2wm

Domain Coloring on the Riemann Sphere

Ángeles Sandoval-Romero, Antonio Hernández-Garduño
2015 The Mathematica Journal  
The former renders the domain coloring of a generic f(z); the latter is optimized for Möbius transformations for the purpose of dynamic rendering.  ...  Additionally, the package allows dynamic visualization of families of Möbius transformations.  ...  Ricardo Berlanga Z. for encouragement and useful insight on the subject matter of this paper. The first author acknowledges the support of Ixtli-UNAM under project IX100310.  ... 
doi:10.3888/tmj.17-9 fatcat:jhvtbze7wfetzda5ll3viorhjy

Conformal spherical representation of 3D genus-zero meshes

Hongdong Li, Richard Hartley
2007 Pattern Recognition  
Specifically, a genus-zero 3D mesh object is first conformally mapped onto the unit sphere by using a modified discrete conformal mapping, where the modification is based on Möbius Factorization and is  ...  The obtained SH coefficients are further made invariant to translation and rotation, while at the same time retain their completeness, so that the original shape information has been faithfully preserved  ...  Thank editor and anonymous reviewers who give invaluable suggestions. The mesh objects used in this paper are from C. Grimm's homepage (//www.cs.wustl.edu/cmg), and is gratefully acknowledged.  ... 
doi:10.1016/j.patcog.2007.01.021 fatcat:smnrgn2dhjdzjgq6su5qszrame

Discrete Surface Ricci Flow

M. Jin, J. Kim, F. Luo, X. Gu
2008 IEEE Transactions on Visualization and Computer Graphics  
Given a mesh, all possible metrics form a linear space, and all possible curvatures form a convex polytope.  ...  Furthermore, the Ricci energy can be optimized using Newton's method more efficiently. Discrete Ricci flow algorithms are rigorous and efficient.  ...  If the mesh is big, we partition it into patches and flatten each patch, then use Möbius transformations to glue different patches together.  ... 
doi:10.1109/tvcg.2008.57 pmid:18599915 fatcat:z5zka2er35corovmdzazoxdpyy

Generalised cyclidic nets for shape modelling in architecture

Romain Mesnil, Cyril Douthe, Olivier Baverel, Bruno Léger
2017 International Journal of Architectural Computing  
Möbius transformations also map circles to circles, and therefore map circular meshes to circular meshes.  ...  Möbius transformations and Möbius geometry Lines of curvature play thus a crucial role in the discretisation of complex shapes.  ...  Innovative methods of structural optimisation combining form-finding and creases can also arise from this work.  ... 
doi:10.1177/1478077117714917 fatcat:62cgg7ukbvbg3d5suktzop5yue

Discrete Surface Ricci Flow [chapter]

Wei Zeng, Xianfeng David Gu
2013 SpringerBriefs in Mathematics  
Given a mesh, all possible metrics form a linear space, and all possible curvatures form a convex polytope.  ...  Furthermore, the Ricci energy can be optimized using Newton's method more efficiently. Discrete Ricci flow algorithms are rigorous and efficient.  ...  If the mesh is big, we partition it into patches and flatten each patch, then use Möbius transformations to glue different patches together.  ... 
doi:10.1007/978-1-4614-8781-4_4 fatcat:begfg75rhzdw7lsz6mgneufm34

Conformal visualization for partially-immersive platforms

Kaloian Petkov, Charilaos Papadopoulos, Min Zhang, Arie E. Kaufman, Xianfeng Gu
2011 2011 IEEE Virtual Reality Conference  
This partially-immersive visualization results in a substantial loss of visual information that may be acceptable for some applications; however, it becomes a major obstacle for critical tasks that need  ...  We demonstrate our conformal visualization pipeline in the 5-sided CAVE for Immersive Virtual Colonoscopy, as well as architectural and abstract data exploration.  ...  ACKNOWLEDGEMENTS This work has been supported by NSF grants CCF-0448399, CCF-0702699, CNS-0959979, CNS-1016829, IIS-0916235, IIS-0916286, NIH grant R01EB7530 and ONR grant N000140910228.  ... 
doi:10.1109/vr.2011.5759453 pmid:26279083 pmcid:PMC4536849 dblp:conf/vr/PetkovPZKG11 fatcat:b6gbrtsryrhinm4qtcckqcuoaa
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