Radial Basis Functions Pedro Teodoro

2 What For Radial Basis Functions (RBFs) allows for interpolate/approximate scattered data in nD.

3 Scattered Data Interpolation Reconstruct smoothly, a function S(x), given N samples (x i, f i ), such that S(x i )=f i

Radial Basis Function method 4

Global Support Basis Functions These basis functions guarrantes solution for the entire domain 5

Finding the RBF coefficients Results by solving the following linear System Slow SolutionStorage 6

Closed Curves and Surfaces In case of having point clouds defining a curve or a surface, we want to obtain a distance field, where its isovalues defines the surface, otherwise, the solution would be a constant in all the domain. For that, we define offsurface points and assign: Positive values (outside)Positive values (outside) Negative values (inside)Negative values (inside) 7

Closed Curves and Surfaces (cont) If for every point, we assign two more points (one inside and one outside), the interpolant is: SolutionStorage Slower SolutionStorage 8

Improvements FastRBF toolbox uses the Fast Multipole Method algorithm to solve the linear system. SolutionStorage Feasible but matematically complex and proprietary 9

Improvements (cont) Carr et al (2001) used a greedy algorithm to reduce the necessary centers to approximate a surface to a point cloud within a desire accuracy. 10

Improvements (cont) Walder et al (2006) shown that it is possible to obtain na implicit surface without offsurface points, neither normals information. SolutionStorage 11

Improvements (cont) RongJiang et al (2009) assuming that the imput point cloud is oriented (normals information), simplified the work of Walder et al (2006). SolutionStorage 12

Goal Implement the greedy algorithm of Carr et al (2001) and of RongJiang et al (2009), to interpolate oriented point clouds… SolutionStorage … along with a divide to conquer algorithm based on Partition of Unity (PU) with blending functions to reduce the computational power and storage. 13

Bibliography 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, August Implicit surface Modeling eith a Globally Regularised Basis of Compact Suport C. Walder, B. Scholkopf, O. Chappele, Eurographics Hermite variational implicit surface reconstruction PAN RongJiang, MENG XiangXu, WHANGBO TaegKeun, Science in China Press,

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