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Radial Basis Functions: Introduction and Applications

Grady WrightDept. of MathematicsBoise State University

Application: tsunami modeling and bathymetry● 2004 Indian Ocean Tsunami

● Chennai (Madras) Harbor, east coast of India

Application: bathymetry reconstruction and tsunami'sScattered bathymetry samples surrounding Madras Harbor

Application: bathymetry reconstruction and tsunami'sRBF reconstruction of Madras Harbor bathymetry from scattered samples

Application: cranioplastyX-ray CT scan Extracted depth map of hole

RBF reconstruction of skullRendering of skull with RBF prosthetic

Review of standard interpolation methodsProblem: Given discrete data, find a function that interpolates the data.

f

x

Review of standard interpolation methods

Piecewise linear:

Problem: Given discrete data, find a function that interpolates the data.

f

x

f

x

Review of standard interpolation methods

Piecewise linear: Piecewise cubic (spline):Problem: Given discrete data, find a function that interpolates the data.

f

x

f

x

f

x

Review of standard interpolation methods

Piecewise linear: Piecewise cubic (spline):

Global polynomial:

Problem: Given discrete data, find a function that interpolates the data.

f

x

f

x

f

x

Review of standard interpolation methods

Piecewise linear: Piecewise cubic (spline):

Global polynomial: Trigonometric: periodic data

Problem: Given discrete data, find a function that interpolates the data.

f

x

f

x

f

x

f

x

Radial basis function (RBF) interpolationKey idea: linear combination of translates and rotations of a single radial function:

[∣x1−x1∣ ∣x1−x2∣ ⋯ ∣x1−xN∣∣x2−x1∣ ∣x2−x2∣ ⋯ ∣x2−xN∣

⋮ ⋮ ⋱ ⋮∣xN−x1∣ ∣xN−x2∣ ⋯ ∣xN−xN∣

][ 1

2

⋮N]=[ f 1

f 2

⋮f N] ,

s x =∑j=1

N

j∣x−x j∣ , sxk= f k , k=1, , NInterpolant:

Expansion coeffcients:

A

Radial basis function (RBF) interpolationKey idea: linear combination of translates and rotations of a single radial function:

s x =∑j=1

N

j∣x−x j∣ , sxk= f k , k=1, , NInterpolant:

Expansion coeffcients:

A

[∣x1−x1∣ ∣x1−x2∣ ⋯ ∣x1−xN∣∣x2−x1∣ ∣x2−x2∣ ⋯ ∣x2−xN∣

⋮ ⋮ ⋱ ⋮∣xN−x1∣ ∣xN−x2∣ ⋯ ∣xN−xN∣

][ 1

2

⋮N]=[ f 1

f 2

⋮f N] ,

Radial basis function (RBF) interpolationKey idea: linear combination of translates and rotations of a single radial function:

s x =∑j=1

N

j∣x−x j∣ , sxk= f k , k=1, , NInterpolant:

Expansion coeffcients:

A

[∣x1−x1∣ ∣x1−x2∣ ⋯ ∣x1−xN∣∣x2−x1∣ ∣x2−x2∣ ⋯ ∣x2−xN∣

⋮ ⋮ ⋱ ⋮∣xN−x1∣ ∣xN−x2∣ ⋯ ∣xN−xN∣

][ 1

2

⋮N]=[ f 1

f 2

⋮f N] ,

Radial basis function (RBF) interpolationKey idea: linear combination of translates and rotations of a single radial function:

s x =∑j=1

N

j∣x−x j∣ , sxk= f k , k=1, , NInterpolant:

Expansion coeffcients:

A

[∣x1−x1∣ ∣x1−x2∣ ⋯ ∣x1−xN∣∣x2−x1∣ ∣x2−x2∣ ⋯ ∣x2−xN∣

⋮ ⋮ ⋱ ⋮∣xN−x1∣ ∣xN−x2∣ ⋯ ∣xN−xN∣

][ 1

2

⋮N]=[ f 1

f 2

⋮f N] ,

Radial basis function (RBF) interpolationKey idea: linear combination of translates and rotations of a single radial function:

s x =∑j=1

N

j∣x−x j∣ , sxk= f k , k=1, , NInterpolant:

Expansion coeffcients:

A

[∣x1−x1∣ ∣x1−x2∣ ⋯ ∣x1−xN∣∣x2−x1∣ ∣x2−x2∣ ⋯ ∣x2−xN∣

⋮ ⋮ ⋱ ⋮∣xN−x1∣ ∣xN−x2∣ ⋯ ∣xN−xN∣

][ 1

2

⋮N]=[ f 1

f 2

⋮f N] ,

Radial basis function (RBF) interpolationKey idea: linear combination of translates and rotations of a single radial function:

s x =∑j=1

N

j∣x−x j∣ , sxk= f k , k=1, , NInterpolant:

Expansion coeffcients:

A

[∣x1−x1∣ ∣x1−x2∣ ⋯ ∣x1−xN∣∣x2−x1∣ ∣x2−x2∣ ⋯ ∣x2−xN∣

⋮ ⋮ ⋱ ⋮∣xN−x1∣ ∣xN−x2∣ ⋯ ∣xN−xN∣

][ 1

2

⋮N]=[ f 1

f 2

⋮f N] ,

Radial basis function (RBF) interpolationKey idea: linear combination of translates and rotations of a single radial function:

s x =∑j=1

N

j∣x−x j∣ , sxk= f k , k=1, , NInterpolant:

Expansion coeffcients:

A

[∣x1−x1∣ ∣x1−x2∣ ⋯ ∣x1−xN∣∣x2−x1∣ ∣x2−x2∣ ⋯ ∣x2−xN∣

⋮ ⋮ ⋱ ⋮∣xN−x1∣ ∣xN−x2∣ ⋯ ∣xN−xN∣

][ 1

2

⋮N]=[ f 1

f 2

⋮f N] ,

Radial basis function (RBF) interpolationKey idea: linear combination of translates and rotations of a single radial function:

s x =∑j=1

N

j∣x−x j∣ , sxk= f k , k=1, , NInterpolant:

Expansion coeffcients:

Guaranteed positive-definite for appropriate φ (r)

[∣x1−x1∣ ∣x1−x2∣ ⋯ ∣x1−xN∣∣x2−x1∣ ∣x2−x2∣ ⋯ ∣x2−xN∣

⋮ ⋮ ⋱ ⋮∣xN−x1∣ ∣xN−x2∣ ⋯ ∣xN−xN∣

][ 1

2

⋮N]=[ f 1

f 2

⋮f N] ,

Standard methods in higher dimensions● Tensor products: global polynomials, Fourier, splines

Regular grids Irregular grids Polar grids

Issues: geometric flexibilityBenefits: programming, increasing accuracy (smoothness), higher dimensions

Standard methods in higher dimensions● Tensor products: global polynomials, Fourier, splines

Regular grids Irregular grids Polar grids

Issues: geometric flexibilityBenefits: programming, increasing accuracy (smoothness), higher dimensions

● What happens to global methods for scattered data?

➢ Depending on the nodes, the interpolation problem may be ill-posed.

➢ There may be no solution, one solution, or an infinite number of solutions.

Standard methods in higher dimensions● Tensor products: global polynomials, Fourier, splines

Regular grids Irregular grids Polar grids

Issues: geometric flexibilityBenefits: programming, increasing accuracy (smoothness), higher dimensions

● Scattered data solution: use local methods (splines) and triangulations

Benefits: geometrically flexible

Issues: programming, increasing smoothness (accuracy), higher dimension

RBF interpolation in higher dimensionsKey idea: linear combination of translates and rotations of a single radial function:

[∥x1−x1∥ ∥x1−x2∥ ⋯ ∥x1−xN∥∥x2−x1∥ ∥x2−x2∥ ⋯ ∥x2−xN∥

⋮ ⋮ ⋱ ⋮∥xN−x1∥ ∥xN−x2∥ ⋯ ∥xN−xN∥

][ 1

2

⋮N]=[ f 1

f 2

⋮f N] ,

s x =∑j=1

N

j∥x−x j∥ , sxk = f k , k=1, , NInterpolant:

Expansion coeffcients:

1-D: ∣x−x j∣ > 1-D: ∥x−x j∥2

RBF interpolation in higher dimensionsKey idea: linear combination of translates and rotations of a single radial function:

s x =∑j=1

N

j∥x−x j∥ , sxk = f k , k=1, , NInterpolant:

Expansion coeffcients:

1-D: ∣x−x j∣ > 1-D: ∥x−x j∥2

[∥x1−x1∥ ∥x1−x2∥ ⋯ ∥x1−xN∥∥x2−x1∥ ∥x2−x2∥ ⋯ ∥x2−xN∥

⋮ ⋮ ⋱ ⋮∥xN−x1∥ ∥xN−x2∥ ⋯ ∥xN−xN∥

][ 1

2

⋮N]=[ f 1

f 2

⋮f N] ,

RBF interpolation in higher dimensionsKey idea: linear combination of translates and rotations of a single radial function:

s x =∑j=1

N

j∥x−x j∥ , sxk = f k , k=1, , NInterpolant:

Expansion coeffcients:

1-D: ∣x−x j∣ > 1-D: ∥x−x j∥2

[∥x1−x1∥ ∥x1−x2∥ ⋯ ∥x1−xN∥∥x2−x1∥ ∥x2−x2∥ ⋯ ∥x2−xN∥

⋮ ⋮ ⋱ ⋮∥xN−x1∥ ∥xN−x2∥ ⋯ ∥xN−xN∥

][ 1

2

⋮N]=[ f 1

f 2

⋮f N] ,

RBF interpolation in higher dimensionsKey idea: linear combination of translates and rotations of a single radial function:

s x =∑j=1

N

j∥x−x j∥ , sxk = f k , k=1, , NInterpolant:

Expansion coeffcients:

1-D: ∣x−x j∣ > 1-D: ∥x−x j∥2

[∥x1−x1∥ ∥x1−x2∥ ⋯ ∥x1−xN∥∥x2−x1∥ ∥x2−x2∥ ⋯ ∥x2−xN∥

⋮ ⋮ ⋱ ⋮∥xN−x1∥ ∥xN−x2∥ ⋯ ∥xN−xN∥

][ 1

2

⋮N]=[ f 1

f 2

⋮f N] ,

RBF interpolation in higher dimensionsKey idea: linear combination of translates and rotations of a single radial function:

s x =∑j=1

N

j∥x−x j∥ , sxk = f k , k=1, , NInterpolant:

Expansion coeffcients:

1-D: ∣x−x j∣ > 1-D: ∥x−x j∥2

[∥x1−x1∥ ∥x1−x2∥ ⋯ ∥x1−xN∥∥x2−x1∥ ∥x2−x2∥ ⋯ ∥x2−xN∥

⋮ ⋮ ⋱ ⋮∥xN−x1∥ ∥xN−x2∥ ⋯ ∥xN−xN∥

][ 1

2

⋮N]=[ f 1

f 2

⋮f N] ,

RBF interpolation in higher dimensionsKey idea: linear combination of translates and rotations of a single radial function:

s x =∑j=1

N

j∥x−x j∥ , sxk = f k , k=1, , NInterpolant:

Expansion coeffcients:

1-D: ∣x−x j∣ > 1-D: ∥x−x j∥2

[∥x1−x1∥ ∥x1−x2∥ ⋯ ∥x1−xN∥∥x2−x1∥ ∥x2−x2∥ ⋯ ∥x2−xN∥

⋮ ⋮ ⋱ ⋮∥xN−x1∥ ∥xN−x2∥ ⋯ ∥xN−xN∥

][ 1

2

⋮N]=[ f 1

f 2

⋮f N] ,

RBF interpolation in higher dimensionsKey idea: linear combination of translates and rotations of a single radial function:

s x =∑j=1

N

j∥x−x j∥ , sxk = f k , k=1, , NInterpolant:

Expansion coeffcients:

1-D: ∣x−x j∣ > 1-D: ∥x−x j∥2

[∥x1−x1∥ ∥x1−x2∥ ⋯ ∥x1−xN∥∥x2−x1∥ ∥x2−x2∥ ⋯ ∥x2−xN∥

⋮ ⋮ ⋱ ⋮∥xN−x1∥ ∥xN−x2∥ ⋯ ∥xN−xN∥

][ 1

2

⋮N]=[ f 1

f 2

⋮f N] ,

RBF interpolation in higher dimensionsKey idea: linear combination of translates and rotations of a single radial function:

s x =∑j=1

N

j∥x−x j∥ , sxk = f k , k=1, , NInterpolant:

Expansion coeffcients:

1-D: ∣x−x j∣ > 1-D: ∥x−x j∥2

[∥x1−x1∥ ∥x1−x2∥ ⋯ ∥x1−xN∥∥x2−x1∥ ∥x2−x2∥ ⋯ ∥x2−xN∥

⋮ ⋮ ⋱ ⋮∥xN−x1∥ ∥xN−x2∥ ⋯ ∥xN−xN∥

][ 1

2

⋮N]=[ f 1

f 2

⋮f N] , Guaranteed

positive-definite for appropriate φ (r)

Quick overview of RBF properties

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5

0

0.5

1

1.5

2

2.5

3

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Piecewise smooth φ(r): Infinitely smooth φ(r):

cubic

r3

TP spline

r2 log r

multiquadric

1r2

Gaussian

e−r2

Inverse quadratic

11r2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

linear

r

● Classes and examples of radial functions:

Algebraically accurate interpolants Spectrally accurate interpolants

● Well-posedness: Schoenberg (1938), Duchon (1977), Micchelli (1986)

● Error estimates: Duchon, Buhmann, Jetter, Madych, Narcowich, Nelson, Powell, Schaback, Ward, Wendland, Yoon, etc.

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