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Computer Graphics using Radial Basis Functions

Computer Graphics using Radial Basis Functions. C.S. Chen Department of Mathematics University of Southern Mississippi. What is the interpolation problem?. What is the interpolation problem?. Interpolating scattered data with radial basis functions (RBFs).

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Computer Graphics using Radial Basis Functions

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  1. Computer Graphics usingRadial Basis Functions C.S. Chen Department of Mathematics University of Southern Mississippi

  2. What is the interpolation problem?

  3. What is the interpolation problem?

  4. Interpolating scattered data with radial basis functions(RBFs) What is the interpolation problem? • To approximate a real valued function f(x) by s(x) given the • set of values at the distinct points

  5. An RBF takes these points: And gives you this surface:

  6. Farfieldtechnology.com Automatically fitted surfaces  Raw point cloud data *Data courtesy of Stanford computer graphics laboratory

  7. Scattered Data Interpolation • RBF’s are one of the effective solutions to the Scattered Data Interpolation Problem • RBFs can be applied to many areas: • Mesh repair • Surface reconstruction • Range scanning, geographic surveys, medical data • Field Visualization (2D and 3D) • Image warping, morphing, registration

  8. What is RBFs? An RBF is of the form • An RBF is a weighted sum of translations of a radially • symmetric basic function augmented by a polynomial term

  9. What is a basis function? Linear: Cubic: Multiquadrics: Polyharmonic Spines: Gaussian:

  10. Compactly Supported RBFs Define For d=1, For d=2, 3,

  11. Globally Supported RBFs

  12. Compactly Supported RBFs

  13. Surface Reconstruction Scheme Assume that To approximate f by we usually require fitting the given of pairwise distinct centres with the imposed Data set conditions is well-posed if the interpolation matrix is non-singular

  14. Interpolation matrix

  15. Laser Scanners Farfieldtechnolgy.com

  16. Data Modelling Laser scaned point cloud (27,000 points) Fitted surface is described by an RBF consisting of 2,600 terms

  17. Farfield Technology: FastRBF A dragon consisting of 473,000 vertices & 871,000 facets (left) is modelled by a single function consisting of 32,000 terms (right)

  18. Data Modelling

  19. Centre Reduction • Remove redundant centres • Greedy algorithm • Buddha Statue: • 543,652 surface points • 80,518 centres • 5 x 10-4 accuracy

  20. FastRBF • FarFieldTechnology (.com) • Commercial implementation • 3D biharmonic fitter with Fast Multipole Methods • Adaptive Polygonizer that generates optimized triangles • Grid and Point-Set evaluation • Expensive • They have a free demo limited to 30k centres • Use iterative reduction to fit surfaces with more points

  21. Morphing • Turk99 (SIGGRAPH) • 4D Interpolation between two surfaces

  22. Morphing With Influence Shapes

  23. Smoothing • Smooth out noisy range scan data • Repair my rough segmentation

  24. Surface Interpolation with RBFs for Medical Imaging

  25. Incomplete cranial surface Cranial surface interpolated with RBFs

  26. Statue of Liberty • 3,360,300 data points • 402,118 centres • 0.1m accuracy

  27. References: • Reconstruction and representation of 3D objects with radial basis functions, J. C. Carr, R. K. Beatson, J.B. Cherrie T. J. Mitchell, W. R. Fright, B. C. McCallum and T. R. Evans, ACM SIGGRAPH 2001, Los Angeles, CA, pp67-76, 12-17 August 2001 • J. Duchon, Splines minimizing rotation-invariant semi-norms in Sobolev space, in W. Schempp and K. Zeller, editors, Constructive Theory and Functions of Several Variables, #571 in Lecture Notes in Mathematics, p. 85-100, Berlin, 1977, Springer –Verlag.

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