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Surface Reconstruction using Radial Basis Functions

Surface Reconstruction using Radial Basis Functions. Michael Kunerth, Philipp Omenitsch and Georg Sperl. 1 Institute of Computer Graphics and Algorithms Vienna University of Technology. 2 <insert 2nd affiliation (institute) here> <insert 2nd affiliation (university) here>.

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Surface Reconstruction using Radial Basis Functions

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  1. 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> <insert 2nd affiliation(university) here> 3 <insert 3rd affiliation (institute) here> <insert 3rd affiliation (university) here>

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

  3. 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 3 M. Kunerth, P. Omenitsch, G. Sperl

  4. Radial Basis Functions • Value depends only on distance from center • Function satisfies 4 M. Kunerth, P. Omenitsch, G. Sperl

  5. RBF Surface Reconstruction • Surface as zero level set of implicit function • Weighted sum of scaled/translated radial basis functions • Interpolation vs. approximation • Surface extraction 5 M. Kunerth, P. Omenitsch, G. Sperl

  6. 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 6 M. Kunerth, P. Omenitsch, G. Sperl

  7. 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 7 M. Kunerth, P. Omenitsch, G. Sperl

  8. 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 8 M. Kunerth, P. Omenitsch, G. Sperl

  9. Least-Squares Hermite RBF • Fit gradients to normals • Subset of points used as centers • Radial function: triharmonic function 9 M. Kunerth, P. Omenitsch, G. Sperl

  10. 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 10 M. Kunerth, P. Omenitsch, G. Sperl

  11. Least-Squares Hermite RBF cont‘d. • Pros: • Well distributed centers • Preserve local features • Accurate with few centers • Cons: • Slow / high computational cost 11 M. Kunerth, P. Omenitsch, G. Sperl

  12. Voronoi-based Reconstruction 12 M. Kunerth, P. Omenitsch, G. Sperl

  13. Adaptive Partion of Unity 13 M. Kunerth, P. Omenitsch, G. Sperl

  14. 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 14 M. Kunerth, P. Omenitsch, G. Sperl

  15. 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) 15 M. Kunerth, P. Omenitsch, G. Sperl

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