Applied and computational complex analysis. Vol. 3: discrete Fourier analysis—Cauchy integrals—construction of conformal maps---univalent functions
Computer Aided Geometric Design
Fundamentals of spherical parameterization for 3D meshes
ACM SIGGRAPH 2003 Papers
Global conformal surface parameterization
Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Discrete exterior calculus
Cross-parameterization and compatible remeshing of 3D models
ACM SIGGRAPH 2004 Papers
Optimal Global Conformal Surface Parameterization
VIS '04 Proceedings of the conference on Visualization '04
High Resolution Tracking of Non-Rigid 3D Motion of Densely Sampled Data Using Harmonic Maps
ICCV '05 Proceedings of the Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1 - Volume 01
Conformal virtual colon flattening
Proceedings of the 2006 ACM symposium on Solid and physical modeling
Computing surface hyperbolic structure and real projective structure
Proceedings of the 2006 ACM symposium on Solid and physical modeling
Discrete one-forms on meshes and applications to 3D mesh parameterization
Computer Aided Geometric Design
Discrete differential forms and applications to surface tiling
Proceedings of the twenty-second annual symposium on Computational geometry
Graphical Models - Special issue on SPM 05
Meshing genus-1 point clouds using discrete one-forms
Computers and Graphics
Conformal Geometry and Its Applications on 3D Shape Matching, Recognition, and Stitching
IEEE Transactions on Pattern Analysis and Machine Intelligence
Designing quadrangulations with discrete harmonic forms
SGP '06 Proceedings of the fourth Eurographics symposium on Geometry processing
Conformal equivalence of triangle meshes
ACM SIGGRAPH 2008 papers
Globally Optimal Surface Mapping for Surfaces with Arbitrary Topology
IEEE Transactions on Visualization and Computer Graphics
IEEE Transactions on Visualization and Computer Graphics
3D Non-rigid Surface Matching and Registration Based on Holomorphic Differentials
ECCV '08 Proceedings of the 10th European Conference on Computer Vision: Part III
Computing Teichmüller Shape Space
IEEE Transactions on Visualization and Computer Graphics
GMP'08 Proceedings of the 5th international conference on Advances in geometric modeling and processing
Slit map: conformal parameterization for multiply connected surfaces
GMP'08 Proceedings of the 5th international conference on Advances in geometric modeling and processing
Shape analysis of planar objects with arbitrary topologies using conformal geometry
ECCV'10 Proceedings of the 11th European conference on Computer vision: Part V
ACM SIGGRAPH 2011 papers
SMI 2012: Full Canonical conformal mapping for high genus surfaces with boundaries
Computers and Graphics
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Surface parameterization refers to the process of mapping the surface to canonical planar domains, which plays crucial roles in texture mapping and shape analysis purposes. Most existing techniques focus on simply connected surfaces. It is a challenging problem for multiply connected genus zero surfaces. This work generalizes conventional Koebe's method for multiply connected planar domains. According to Koebe's uniformization theory, all genus zero multiply connected surfaces can be mapped to a planar disk with multiply circular holes. Furthermore, this kind of mappings are angle preserving and differ by Möbius transformations. We introduce a practical algorithm to explicitly construct such a circular conformal mapping. Our algorithm pipeline is as follows: suppose the input surface has n boundaries, first we choose 2 boundaries, and fill the other n -- 2 boundaries to get a topological annulus; then we apply discrete Yamabe flow method to conformally map the topological annulus to a planar annulus; then we remove the filled patches to get a planar multiply connected domain. We repeat this step for the planar domain iteratively. The two chosen boundaries differ from step to step. The iterative construction leads to the desired conformal mapping, such that all the boundaries are mapped to circles. In theory, this method converges quadratically faster than conventional Koebe's method. We give theoretic proof and estimation for the converging rate. In practice, it is much more robust and efficient than conventional non-linear methods based on curvature flow. Experimental results demonstrate the robustness and efficiency of the method.