Computational complexity of combinatorial surfaces
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
Parametrization and smooth approximation of surface triangulations
Computer Aided Geometric Design
Level set diagrams of polyhedral objects
Proceedings of the fifth ACM symposium on Solid modeling and applications
Topology matching for fully automatic similarity estimation of 3D shapes
Proceedings of the 28th annual conference on Computer graphics and interactive techniques
Optimally cutting a surface into a disk
Proceedings of the eighteenth annual symposium on Computational geometry
Edgebreaker: a simple compression for surfaces with handles
Proceedings of the seventh ACM symposium on Solid modeling and applications
Cutting 3D freeform objects with genus-n into single boundary surfaces using topological graphs
Proceedings of the seventh ACM symposium on Solid modeling and applications
Proceedings of the 29th annual conference on Computer graphics and interactive techniques
Efficient computation of the topology of level sets
Proceedings of the conference on Visualization '02
Seamster: inconspicuous low-distortion texture seam layout
Proceedings of the conference on Visualization '02
Computing contour trees in all dimensions
Computational Geometry: Theory and Applications - Fourth CGC workshop on computional geometry
Optimal System of Loops on an Orientable Surface
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Computer Aided Geometric Design
Loops in reeb graphs of 2-manifolds
Proceedings of the nineteenth annual symposium on Computational geometry
SMI '03 Proceedings of the Shape Modeling International 2003
Global conformal surface parameterization
Proceedings of the 2003 Eurographics/ACM SIGGRAPH symposium on Geometry processing
Removing excess topology from isosurfaces
ACM Transactions on Graphics (TOG)
Fair morse functions for extracting the topological structure of a surface mesh
ACM SIGGRAPH 2004 Papers
Feature-based surface parameterization and texture mapping
ACM Transactions on Graphics (TOG)
ABF++: fast and robust angle based flattening
ACM Transactions on Graphics (TOG)
Greedy optimal homotopy and homology generators
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Spectral surface quadrangulation
ACM SIGGRAPH 2006 Papers
Planar parameterization for closed 2-manifold genus-1 meshes
SM '04 Proceedings of the ninth ACM symposium on Solid modeling and applications
Setting the boundary free: a composite approach to surface parameterization
SGP '05 Proceedings of the third Eurographics symposium on Geometry processing
Multidimensional Size Functions for Shape Comparison
Journal of Mathematical Imaging and Vision
Technical Section: Discrete Laplace-Beltrami operators for shape analysis and segmentation
Computers and Graphics
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Given a triangulated surface M with arbitrary genus, the set of its cut-graphs depends on the underlying topology and the selection of a specific one should be guided by the surface geometry and targeted applications. Most of the previous work on this topic uses mesh traversal techniques for the evaluation of the geodesic metric, and therefore the cut-graphs are influenced by the mesh connectivity. Our solution is to build up the cut-graph on the iso-contours of a function f:M-R, that cut the topological handles of M, and on the completion of the cut-graph on the planar domain. In the planar domain, geodesic curves are defined by line segments whose counterparts on M, with respect to a diffeomorphism @f:M-R^2, are smooth approximations of geodesic paths. Our method defines a family of cut-graphs of M which can target different applications, such as global parameterization with respect to different criteria (e.g., minimal length, minimization of the parameterization distortion, or interpolation of points as required by remeshing and texture mapping) or the calculation of polygonal schemes for surface classification. The proposed approach finds a cut-graph of an arbitrary triangle mesh M with n vertices and b boundary components in O((b-1)n) time if M has 0-genus, and O(n(log(n)+2g+b-1)) time if g=1. The associated polygonal schema is reduced if g=0, and it has a constant number of redundant edges otherwise.