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Manifold ParameterizationLei Zhang, Ligang Liu, Zhongping Ji, Guojin Wang
Department of MathematicsZhejiang University
Accepted as regular paper by CGI2006
OverviewParameterizationLeast-squares MeshManifold ParameterizationSimilar destination, different waySimilar way, different destination
ReferenceO. Sorkine and D. Cohen-Or. Least-squares meshes. In Proceedings of Shape Modeling International, 2004.Lei Zhang, Ligang Liu, Zhongping Ji and Guojin Wang. Manifold Parameterization. Accepted as regular paper by Computer Graphics International, 2006.V. Kraevoy and A. Sheffer. Cross-Parameterization and Compatible Remeshing of 3D Models. SIGGRAPH, 2004.
ReferenceM. Paone and Andrew Yuen. Mesh Fitting to Points. Projects Presentations (PPT), Simon Fraser University, Canada.K. D. Cheng, W. P. Wang, H. Qin, K. K. Wong, H. P. Yang and Y. Liu. Fitting Subdivision Surfaces to Unorganized Point Data Using SDM. Proceedings of the 12th Pacific Conference on Computer Graphics and Applications, 2004.
OverviewParameterizationLeast-squares MeshManifold ParameterizationSimilar destination, different waySimilar way, different destination
ParameterizationConceptParameterization is a one-to-one mapping from a triangular mesh surface onto a suitable domain.
MeshDomain
Planar ParameterizationSelect a plane as the parameterization domain for an open mesh
Spherical ParameterizationSelect a sphere as the parameterization domain for 0-genus mesh E. Praun and H. Hoppe, SIGGRAPH 04
Manifold ParameterizationSelect a surface as parameterization domain for another surfaceMeshDomain
Manifold Parameterization
OverviewParameterizationLeast-squares MeshManifold ParameterizationSimilar destination, different waySimilar way, different destination
Least-squares Meshes
O. Sorkine, D. Cohen-OrTel Aviv University
Proceeding of Shape Modeling International 2004
IntroductionMesh??ConnectivityMesh surfaceGeometry=+
IntroductionLeast-squares meshUsing a set of control points, approximate the original mesh surface by its connectivity graph.
Introduction19851 vertices200 control points1000 control points3000 control points
Least-squares meshesVertex conditions-Smooth condition L(vi)=0, vi all vertices-Geometry condition vj=cj, cj constraint L-Laplacian operatorViVj
Least-squares meshesLaplacian EquationSmooth conditionGeometry conditionvj=cj
Least-squares meshesExample
Least-squares meshesEquation SolutionThe system is solved in least-squares sense.
A is sparse, and equation can be solved by TAUCS library quite fast.
Weighted Least-squares meshesHigher weights for control pointsconstraints
Weighted Least-squares meshes
OverviewParameterizationLeast-squares MeshManifold ParameterizationSimilar destination, different waySimilar way, different destination
OverviewParameterizationLeast-squares MeshManifold ParameterizationSimilar destination, different waySimilar way, different destination
Cross-Parameterization and Compatible Remeshing of 3D ModelsV. Kraevoy and A. Sheffer
SIGGRAPH 2004
IntroductionGiven two mesh M1 and M2, obtain correspondence via base meshes.f1f2f1 F f2-1FM1M2B1B2
Main StepsConstruct topologically identical path layoutsNo interior intersectionCyclical order
Main StepsGet topologically identical base mesh
Main StepsMap patch layout to base meshMean value parameterizationf1f2
Main StepsConstruct mapping between base meshBarycentric coordinateF
Main StepsResult parameterizationf1f2f1 F f2-1FM1M2B1B2
Examples
ConclusionIndirectBoring path layout searchingTime-consuming
OverviewParameterizationLeast-squares MeshManifold ParameterizationSimilar destination, different waySimilar way, different destination
Mesh Fitting to PointsM. Paone and A. YuenSupervisor: Richard (Hao) Zhang
Simon Fraser University, Canada
Project Report
Fitting Subdivision Surfaces to Unorganized Point Data Using SDMK. D. Cheng, W.P. Wang, H. Qin, K. K. Wong, H. P. Yang and Y. Liu
PG 04
IntroductionReconstruction of smooth surface from point cloudsTool: Loop subdivision surfaceMeasure: SD (Squared Distance)H. Pottmann and M. Hofer. Geometry of the Squared Distance Function to Curves and Surfaces. Visualization and Mathematics III, Springer, 2003.
Loop SubdivisionEdge-VertexVertex-Vertex
Squared Distance
Main StepsNormalizationTarget data points are scaled to .
Main StepsNormalizationPre-computationCompute distance field and curvatures at all data points.
Main StepsNormalizationPre-computationInitial meshUse Marching Cubes to obtain an initial control mesh.
Main StepsNormalizationPre-computationInitial meshSamplingGet sample points on limit surface.J. Stam. Evaluation of Loop Subdivision Surfaces. SIGGRAPH99, course.
Main StepsNormalizationPre-computationInitial meshSamplingOptimizationSDM error function:
Main StepsNormalizationPre-computationInitial meshSamplingOptimizationError evaluationMaximum approximation error:Average error:
Main StepsNormalizationPre-computationInitial meshSamplingOptimizationError evaluationInsert new control points to regions of large errors.
Examples
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