Meshless parameterization and surface reconstruction

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Meshless parameterization and surface
reconstruction

• Reporter Lincong Fang
• 16th May, 2007

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Parameterization

• Problem Given a surface S in R3, find a
  one-to-one function f D-gt R3, D R2, such
  that the image of D is S.

D
f
S
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Surface Reconstruction

• Problem Given a set of unorganized points,
  approximate the underlying surface.

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Related Works

• Surface reconstruction
• Delaunay / Voronoi based
• Implicit methods
• Provable
• Parameterization for organized point set

f
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Mesh Parameterization

• There are many papers

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Meshless Parameterization
f
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Papers

• Meshless parameterization and surface
  reconstruction
• Michael S. Floater, Martin Reimers, CAGD 2001
• Meshless parameterization and B-spline surface
  approximation
• Michael S. Floater, in The Mathematics of
  Surfaces IX, Springer-Verlag (2000)
• Efficient Triangulation of point clouds using
  floater parameterization
• Tim Volodine, Dirk Roose, Denis Vanderstraeten,
  Proc. of the Eighth SIAM Conference on Geometric
  Design and Computing
• Triangulating point clouds with spherical
  topology
• Kai Hormann, Martin Reimers, Proceedings of.
  Curve and Surface Design, 2002
• Meshing point clouds using spherical
  parameterization
• M. Zwicker, C. Gotsman, Eurographics 2004
• Meshing genus-1 point clouds using discrete
  one-forms
• Geetika Tewari, Craig Gotsman, Steven J. Gortler,
  Computers Graphics 2006
• Meshless thin-shell simulation based on global
  conformal parameterization
• Xiaohu Guo, Xin Li, Yunfan Bao, Xianfeng Gu, Hong
  Qin, IEEE ToV and CG 2006

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Basic Idea

• Given X(x1, x2,, xn) in R3, compute
• U (u1, u2,, un) in R2
• Triangulate U
• Obtain both a triangulation and a
  parameterization for X

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Compute U

• Assumptions
• X are samples from a 2D surface
• Topology is known
• Desirable property
• Points closed by in U are close by in X

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• Michael S. Floater
• Professor at the Department of Informatics (IFI)
  of the University of Oslo, and member of the
  Center of Mathematics for Applications(CMA),
  Norway.
• Editor of the journal Computer Aided Geometric
  Design.

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• Martin Reimers
• Postdoctor
• CMA, University of Olso, Norway

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• Meshless parameterization and surface
  reconstruction
• Authors
• Michael S. Floater
• Martin Reimers
• Computer Aided Geometric Design 2001
• Main reference Parameterization and smooth
  approximation of surface triangulations, Michael
  S. Floater, CAGD 1997

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Convex Contraints

• Boundary condition map boundary of X to points
  on a unit circle
• If xjs are neighbors of xi then require ui to be
  a strictly convex combination of ujs
• Solve resulting linear system Au b

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Identify Boundary

• Use natural boundary
• (given as part of the data)
• Choose a boundary manually
• Compute boundary
• Identify boundary points
• Order boundary points curve reconstruction

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Compute Boundary

• Identify boundary points
• Order boundary points

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Neighbors and Weights

• Ball neighborhoods
• Radius is fixed
• K nearest neighborhoods
• Weights
• Uniform weights
• Reciprocal distance weights
• Shape preserving weights

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Uniform Weights

• Uniform weights
• (minimizing )
• If Ni ?i Nk ?k, then uiuk

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Reciprocal Distance Weights

• Weights
• Observation
• Minimizing
• Chord parameterization for curves
• Distinct parameter points
• Well behaved triangulation

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Shape Preserving Weights
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Experiments
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CPU Usage

• Reciprocal distance weights
• Shape preserving weights

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Effect of Noise
Noise added Reciprocal distance weight
No Noise
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• Meshless parameterization and B-spline surface
  approximation
• Author
• Michael S. Floater
• in The Mathematics of Surfaces IX, R. Cipolla and
  R. Martin (eds.), Springer-Verlag (2000)

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Meshless Parameterization
Point set
Meshless parameterization
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Triangulation
Delaunay triangulation
Surface triangulation
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Reparameterization
Shape-preserving parameterization
Spline surface
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Retriangulation
Delaunay retriangulation
Surface retriangulation
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Example
Point set
Triangulation
Spline surface
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Example
Point set
Triangulation
Spline surface
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• Tim Volodine
• PhD student, research assistant
• K.U. Leuven, Belgium

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• Dirk Roose
• Professor
• Department of Computer Science, Faculty of
  Applied Sciences, Head of the research group
  Scientific Computing
• K.U.Leuven, Belgium

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• Denis Vanderstraeten
• Director of Research and IPR at Metris
• J2EE Business Analyst / Software Engineer
• Belgium

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• Efficient triangulation of point clouds using
  Floater Parameterization
• Authors
• Tim Volodine
• Dirk Roose
• Denis Vanderstraeten
• Proc. of the Eighth SIAM Conference on Geometric
  Design and Computing
• Main reference Mean value coordinates, Michael
  S. Floater, CAGD 2003

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Boundary Extraction
Boundary points
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Order Boundary Points
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Mean Value Weight
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Experiments
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• Kai Hormann
• Assistant professor
• Department of informatics, Computer graphics
  group
• Clausthal University of Technology, Germany

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• Triangulating point clouds with spherical
  topology
• Authors
• Kai Hormann
• Martin Reimers
• Proceedings of. Curve and Surface Design 2002

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Spherical Topology
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Partition
Shortest path Correspond to the edges of D
12 nearest neighbors
Point set
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Partition
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Reconstruction of one subset
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Optimization
Optimizing 3D triangulations using discrete
curvature analysis Dyn N., K. Hormann, S.-J.
Kim, and D. Levin
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• Matthias Zwicker
• Assistant Professor
• Computer Graphics Laboratory
• University of California, San Diego, USA

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• Craig Gotsman
• Professor
• Department of Computer Science
• Harvard University

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• Meshing point clouds using spherical
  parameterization
• Authors
• Matthias Zwicker
• Craig Gotsman
• Eurographics Symposium on Point-Based Graphics
  2004
• Main references
• Fundamentals of spherical parameterization for 3d
  meshes
• Gotsman C., Gu X., Sheffer A. SiG 2003
• Computing conformal structures of surfaces
• Gu X., Yau S.-T. Communications in Information
  and Systems 2002

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Spherical parameterization
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Spherical Parameterization
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O(n2) Complexity
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• Geetika Tewari
• Graduate Student
• Computer Science, Division of Engineering and
  Applied Sciences
• Harvard University

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• Steven J. Gortler
• Co-Director of Undergraduate Studies in Computer
  Science
• Harvard University

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• Meshing genus-1 point clouds using discrete
  one-forms
• Authors
• Geetika Tewari
• Craig Gotsman
• Steven J. Gortler
• Computers Graphics 2006
• Main references
• Computing conformal structures of surfaces
• Gu X., Yau S.-T. Communications in Information
  and Systems 2002
• Discrete one-forms on meshes and applications to
  3D mesh parameterization
• Gortler SJ, Gotsman C, Thurston D. CAGD 2006

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Discrete one-forms
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Discrete one-forms
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Seamless local parameterization
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MCB Minimal Cycle Basis
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MCB Cycles on a KNNG
One Forms on Arbitrary Graph
MCB Minimal cycle basis
O(E3) time
Trivial cycle
Nontrivial cycle
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One-forms on the KNNG
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Parameterize subgraphs
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Example
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Summary

• Disk topology
• Fast and efficient
• Complex topology
• Slow
• Other Methods
• More applications
• Surface fitting
• Ect.

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Thank you!!!