Surface Reconstruction using Radial Basis Functions

Michael Kunerth, Philipp Omenitsch andGeorg Sperl

1 Institute of Computer Graphicsand Algorithms

Vienna University of Technology

2 <insert 2nd affiliation (institute) here>

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3 <insert 3rd affiliation (institute) here>

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Outline

Problem DescriptionRBF Surface ReconstructionMethods:

Surface Reconstruction Based on Hierarchical Floating Radial Basis FunctionsLeast-Squares Hermite Radial Basis Functions Implicits with Adaptive SamplingVoronoi-based ReconstructionAdaptive Partition of Unity

Conclusion

2M. Kunerth, P. Omenitsch, G. Sperl

Problem Description

3D scanners produce point cloudsFor CG surface representation neededLevel set of implicit functionMesh extraction (e.g. marching cubes)Surface reconstruction with radial basis functions

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Radial Basis Functions

Value depends only on distance from centerFunction satisfies

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RBF Surface Reconstruction

Surface as zero level set of implicit functionWeighted sum of scaled/translated radial basis functions Interpolation vs. approximationSurface extraction

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RBF Surface Reconstruction cont‘d.

Gradients/normals to avoid trivial solutionsCenter reduction (redundancy)Center positions (noise)Partition of unityGlobally supported / compactly supported RBFHierarchical representations

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Hierarchical Floating RBFs

Avoid trivial solution by fitting gradients to normal vectorsAssume a small number of centersCenter positions viewed as own optimization problemRadial 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 unityOctree with multiple levels approximating residual errors

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Least-Squares Hermite RBF

Fit gradients to normalsSubset of points used as centersRadial function: triharmonic function

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Least-Squares Hermite RBF cont‘d.

Adaptive greedy sampling of centersChoose random first centerChoose next center maximizing function residual and gradient difference to nearest already chosen center using the previous set‘s fitted function

Partition of unityOverlapping boxes

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Least-Squares Hermite RBF cont‘d.

Pros:Well distributed centersPreserve local featuresAccurate 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 methodsMain differences:

Which centers should be used?How to optimize existing centers?different distance functions

Smoothing: less noise vs. more detailTradeoff: speed vs. quality

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SourcesY 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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