Surface Reconstruction using Radial Basis Functions

Report
Surface Reconstruction using
Radial Basis Functions
Michael Kunerth, Philipp Omenitsch and
Georg Sperl
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Institute of Computer Graphics
and Algorithms
2 <insert
Vienna University of Technology
3 <insert
2nd affiliation (institute) here>
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(university) here>
3rd affiliation (institute) here>
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Outline
Problem Description
RBF Surface Reconstruction
Methods:
Surface Reconstruction Based on Hierarchical
Floating Radial Basis Functions
Least-Squares Hermite Radial Basis Functions
Implicits with Adaptive Sampling
Voronoi-based Reconstruction
Adaptive Partition of Unity
Conclusion
M. Kunerth, P. Omenitsch, G. Sperl
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Problem Description
3D scanners produce point clouds
For CG surface representation needed
Level set of implicit function
Mesh extraction (e.g. marching cubes)
Surface reconstruction with radial basis
functions
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Radial Basis Functions
Value depends only on distance from center
Function satisfies  () =  (||)
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RBF Surface Reconstruction
Surface as zero level set of implicit function
Weighted sum of scaled/translated radial
basis functions   = 
=1   () + ()
Interpolation vs. approximation
Surface extraction
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RBF Surface Reconstruction cont‘d.
Gradients/normals to avoid trivial solutions
Center reduction (redundancy)
Center positions (noise)
Partition of unity
Globally supported / compactly supported
RBF
Hierarchical representations
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Hierarchical Floating RBFs
Avoid trivial solution by fitting gradients to
normal vectors
Assume a small number of centers
Center positions viewed as own optimization
problem
Radial function: inverse quadratic function
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Hierarchical Floating RBFs cont‘d.
Floating centers: iterative process of refining
initial guess of centers
Partition of unity
Octree with multiple levels approximating
residual errors
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Least-Squares Hermite RBF
Fit gradients to normals
Subset of points used as centers
Radial function: triharmonic function
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Least-Squares Hermite RBF cont‘d.
Adaptive greedy sampling of centers
Choose random first center
Choose next center maximizing function
residual and gradient difference to nearest
already chosen center using the previous
set‘s fitted function
Partition of unity
Overlapping boxes
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Least-Squares Hermite RBF cont‘d.
Pros:
Well distributed centers
Preserve local features
Accurate with few centers
Cons:
Slow / high computational cost
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Voronoi-based Reconstruction
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Adaptive Partion of Unity
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Conclusion
RBF surface reconstruction methods
Main differences:
Which centers should be used?
How to optimize existing centers?
different distance functions
Smoothing: less noise vs. more detail
Tradeoff: speed vs. quality
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Sources
Y Ohtake, A Belyaev, HP Seidel 3D scattered data approximation with adaptive
compactly supported radial basis functions Shape Modeling Applications, 2004.
Proceedings
Samozino M., Alexa M., Alliez P., Yvinec M.: Reconstruction with Voronoi
Centered Radial Basis Functions. Eurographics Symposium on Geometry
Processing (2006)
Ohtake Y., Belyaev A., Seidel H.-P.: Sparse Surface Reconstruction with Adaptive
Partition of Unity and Radial Basis Functions. Graphical Models (2006)
Poranne R., Gotsman C., Keren D.: 3D Surface Reconstruction Using a
Generalized Distance Function. Computer Graphics Forum (2010)
Süßmuth J., Meyer Q., Greiner G.: Surface Reconstruction Based on Hierarchical
Floating Radial Basis Functions. Computer Graphiks Forum (2010)
Harlen Costa Batagelo and João Paulo Gois. 2013. Least-squares hermite radial
basis functions implicits with adaptive sampling. In Proceedings of the 2013
Graphics Interface Conference (GI '13)
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