Transcript Slide 1
Radial Basis Functions
Pedro Teodoro
What For
Radial Basis Functions (RBFs) allows for interpolate/approximate scattered data in nD.
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Scattered Data Interpolation
Reconstruct smoothly, a function S(
x
), given N samples (
x
i , f i ), such that S(
x
i )=f i 3
Radial Basis Function method
S
P
i N
1
i
i
is the
weight
of centre
x
i
(r) is the
basis function P
x
is the Euclidean norm
x
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Global Support Basis Functions
S
i N
1
i
P
r
2
n
r
2
n
1 for 3D These basis functions guarrantes solution for the entire domain 5
Finding the RBF coefficients
Results by solving the following linear System
A P
A P T P
0
c
(||
x i
x j
1,...,
n p x j
( ),
i i
1,..., ,
j
Solution Storage 3 ) 2 )
Slow
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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) •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:
S
i
3
N
1
i
P O
((3 ) )
O N
Slower
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Improvements
FastRBF toolbox uses the Fast Multipole Method algorithm to solve the linear system.
Solution Storage log
N
) 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.
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Improvements (cont)
S
Walder et al (2006) shown that it is possible to obtain na implicit surface without offsurface points, neither normals information.
i N
1
i
i N
1
j d
1
ji
i
j
Solution
O N
Storage
O N
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Improvements (cont)
S
RongJiang et al (2009) assuming that the imput point cloud is oriented (normals information), simplified the work of Walder et al (2006).
i N
1
i
i N
1
n
i t
i
P
Solution Storage 3 ) 2 ) 12
Goal
Implement the greedy algorithm of Carr et al (2001) and of RongJiang et al (2009), to interpolate oriented point clouds… … along with a divide to conquer algorithm based on Partition of Unity (PU) with blending functions to reduce the computational power and storage. Solution Storage ( 2
O k N
) ) 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, 12-17 August 2001.
• Implicit surface Modeling eith a Globally Regularised Basis of Compact Suport C. Walder, B. Scholkopf, O. Chappele, Eurographics 2006.
• Hermite variational implicit surface reconstruction PAN RongJiang, MENG XiangXu, WHANGBO TaegKeun, Science in China Press, 2009.
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Thanks 15