1. 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 3

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

4. Radial Basis Functions M. Kunerth, P. Omenitsch, G. Sperl 4

5. RBF Surface Reconstruction M. Kunerth, P. Omenitsch, G. Sperl 5

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

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

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

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

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

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

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

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

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

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