radial basis functions: introduction and applicationswright/courses/m565/rbf_intro... · radial...
TRANSCRIPT
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.