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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