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Surface Reconstruction using Radial Basis Functions

This paper explores the application of Radial Basis Functions (RBF) for 3D surface reconstruction from point clouds generated by scanning technologies. We discuss various methods including Hierarchical Floating RBFs, Least-Squares Hermite RBFs with adaptive sampling, and Voronoi-based reconstruction techniques. The challenges related to reducing redundancy, optimizing center positions, and selecting the appropriate distance functions are addressed. Our findings highlight the trade-offs between computational speed and the quality of reconstructed surfaces, providing insights for further research in computer graphics.

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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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