Surface Reconstruction Using RBF

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Surface Reconstruction Using RBF

• Reporter Lincong Fang
• 11.07.2007

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Surface Reconstruction
sample
Reconstruction
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Surface Reconstruction

• Delaunay/Voronoi
• Alpha shape/Conformal alpha shape
• Crust/Power crust
• Cocone
• Etc.
• Implicit surfaces
• Signed distance function
• Radial basis function(RBF)
• Poisson
• Fourier
• MPU
• Etc.

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Implicit Surface

• Defined by implicit function
• Such as
• Many topics within broad area of implicit
  surfaces

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Implicit Surface

• Mesh independent representation - generate the
  desired mesh when you require it
• Compact representation to within any desired
  precision
• A solid model is guaranteed to produce manifold
  (manufacturable) surface

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Implicit surface

• Tangent planes and normals can be determined
  analytically from the gradient of the implicit
  function

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Implicit surface

• CSG operation

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Implicit surface

• Morphing

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Implicit surface reconstruction
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Introduction to RBF

• Interpolation problem

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Introduction to RBF
J.Duchon. Splines minimizing rotation-invariant
semi-norms in Sololev spaces. In W. Schempp and
K.Zeller, editors, Constructive Theory of
Functions of Several Variables, number 571 in
Lecture Notes in Mathematics, pages 85-100,
Berlin, 1977. Springer-Verlag.
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Introduction to RBF
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Introduction to RBF

• Popular choices for  include
• For fitting functions of three variables, good
  choices include

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Introduction to RBF

• Matrix form

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Introduction to RBF

• Matrix form

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• Reconstruction and representation of 3D objects
  with Radial Basis functions
• J.C.Carr1,2, R.K.Beatson2, J.B.Cherrie1,
  T.J.Mitchell1,2
• W.R.Fright1, B.C.McCallum1, T.R.Evans1
• 1. Applied Research Associates NZ Ltd
• 2. University of Canterbury, New Zealand
• Sig 2001

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Off-surface Points
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RBF Center Reduction
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Greedy Algorithm

• Choose a subset from the interpolation nodes xi
  and fit an RBF only to these.
• Evaluate the residual, ei fi f(xi), at all
  nodes.
• If maxei lt fitting accuracy then stop.
• Else append new centers where ei is large.
• Re-fit RBF and goto 2

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• 544000 points, 80000 centers, accuracy of 510-4

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Noisy
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Hole Filled Non-uniformly
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• Interpolating implicit surfaces from scattered
  surface data using compactly supported radial
  basis functions
• Bryan S. Morse1, Terry S. Yoo2, Penny Rheingans3,
  
• David T.Chen2, K.R. Subramanian4
• 1. Department of CS, Brigham Young University
• 2. National Library of Medicine
• 3. Department of CS and EE, University of
  Maryland Baltimore County
• 4. Department of CS, University of North Carolina
  at Charlotte
• Proceeding of the International Conference on
  Shape Modeling and Applications 2001

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Compactly-supported RBF
H. Wendland. Piecewise polynomial, positive
definite and compactly supported radial functions
of minimal degree. AICM, 4389-396, 1995
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Matrix Form
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Choice of Support Size
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Comparison
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Comparison

• Compactly supported basic functions is much more
  efficient.
• Non-compactly supported basic functions are
  better suited to extrapolation and interpolation
  of irregular, non-uniformly sampled data.

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• Modeling with implicit surfaces that interpolate
• Greg Turk
• GVU Center, College of Computing
• Georgia Institute of Technology
• James F.OBrien
• EECS, Computer Science Division
• University of California, Berkeley

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Modeling
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Interior Constraints
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Matrix Form
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Exterior Constraints
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Normal Constraints
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Normal Constraints
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Example
The interpolating implicit surface defined by
the 800 vertices and their normals
Polygonal surface
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• A Multi-scale Approach to 3D Scattered Data
  Interpolation with Compactly Supported Basis
  Functions
• Yutaka Ohtake, Alexandaer Belyaev, Hans-Peter
  Seidel
• Computer Graphics Group, Max-Planck-Institute for
  informatics
• Germany
• Proceedings of the Shape Modeling International
  2003

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Construct RBF
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Single level Interpolation

• 35K points
• 6 seconds

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Multi-level Interpolation
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Multi-level Interpolation
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Coarse to Fine
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• 3D Scattered Data Approximation with Adaptive
  Compactly Supported Radial Basis Functions
• Yutaka Ohtake, Alexandaer Belyaev, Hans-Peter
  Seidel
• Computer Graphics Group, Max-Planck-Institute for
  informatics
• Germany
• Proceedings of the Shape Modeling International
  2004

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Construct RBF
Base approximation
Local details
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Adaptive PU
Normalized RBF
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Selection of Centers
100
500
1000
2000
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Example
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Compare With Multi-scale
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• Reconstructing Surfaces Using Anisotropic Basis
  Functions
• Huong quynh Dinh, Greg Turk
• Georgia Institute of Technology
• College of Computing
• Greg Slabaugh
• Georgia Institute of Technology
• Scholl of Electrical and Computer Engineering
• Center for Signal and Image Processing
• Computer Vision, Vol 2, 2001, p606-613.

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Basic Function
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Direction of Anisotropy

• Covariance matrix
• Corner point all three eigenvalues are nearly
  equal
• Edge point one strong eigenvalue
• Plane point two eigenvalues are nearly equal
  and larger than the third

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Noisy
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Summary
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Summary
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Thank you !!!