Radial Basis Functions

Pedro Teodoro

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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 (xi, fi), such that

S(xi)=fi

Radial Basis Function method

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( ) ( )N

i ii

S P

x x x x

is the of centre

(r) is the

( ) is a low-degree polynomial

is the Euclidean norm

i iweight

basis function

P

x

x

x

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Global Support Basis Functions

These basis functions guarrantes solution for the entire domain

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( ) ( )N

i ii

S P

x x x x

2( ) log( ) for 2Dnr r r

2 1( ) for 3Dnr r

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Finding the RBF coefficients

Results by solving the following linear System

0 0T

A P f

P c

,

,

(|| ||), , 1,...,

( ), 1,..., , 1,..., .

i j i j

i j j i

A x x i j n

P p x i n j l

SlowSlowSolutionSolution

StorageStorage

3( )O N2( )O N

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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 Positive values (outside)(outside)

•Negative Negative values (inside)values (inside)

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Closed Curves and Surfaces (cont)

If for every point, we assign two more points (one inside and one outside), the

interpolant is:

3

1

( ) ( )N

i ii

S P

x x x x

SolutionSolution

StorageStorage

3((3 ) )O N2((3 ) )O N

SlowerSlowerSolutionSolution

StorageStorage

3((3 ) )O N

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Improvements

FastRBF toolbox uses the Fast Multipole Method algorithm to solve the linear system.

( )O N

SolutionSolution

StorageStorage

( log )O N N

Feasible but matematically Feasible but matematically complex and proprietarycomplex and proprietary

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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)

Walder et al (2006) shown that it is possible to obtain na implicit surface without offsurface points, neither normals

information.

SolutionSolution

StorageStorage 2((4 ) )O N

3((4 ) )O N

1 1 1

( )N N d

i i ji i ji i j

S

x x x x x

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Improvements (cont)

RongJiang et al (2009) assuming that the imput point cloud is oriented (normals information),

simplified the work of Walder et al (2006).

SolutionSolution

StorageStorage 2( )O N

3( )O N

1 1

( ) ( )N N

ti i i i

i i

S P

x x x n x x x

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Goal

Implement the greedy algorithm of Carr et al (2001) and of RongJiang et al (2009), to

interpolate oriented point clouds…

SolutionSolution

StorageStorage ( )O kN

2( )O k N

… along with a divide to conquer algorithm based on Partition of Unity (PU) with blending functions to reduce the computational power and storage.

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Bibliography

• Reconstruction and Representation of 3D Objects with Radial Basis FunctionsJ. 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 SuportC. Walder, B. Scholkopf, O. Chappele, Eurographics 2006.

• Hermite variational implicit surface reconstructionPAN RongJiang, MENG XiangXu, WHANGBO TaegKeun, Science in China Press, 2009.

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Thanks

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