[PPT] - Discrete Thin-Plate Splines for Large Data Sets Linda Stals Steve PowerPoint Presentation

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Outline tps dtps Convergence Results Future Work

Discrete Thin-Plate Splines for Large Data Sets

Linda Stals Steve Roberts

Department of Mathematics Australian National University

July 13, 2006

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Thin Plate Splines Smoothing Splines Radial Basis Functions Discrete Thin Plate Splines Finite Element Approximation System of Equations Convergence Analysis Dirichlet Boundary Conditions Finite Element Convergence Interpolation Error Results Holes 3D Examples Future Work

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Thin-Plate Splines

3D Image Recovery
Finger Print Analysis
Image Warping
Medical Image Analysis
Data Mining

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Thin-Plate Splines

3D Image Recovery
Finger Print Analysis
Image Warping
Medical Image Analysis
Data Mining

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Smoothing Splines

Given a set of attributes vectors x = (x1, x2, · · · , xd)T, build a predictive model y = f (x). y ≈ f (x). To estimate f by a 2nd-order smoothing spline minimise: Jα(f ) = 1 n

n

i=1

(f (x(i)) − y(i))2 + α

Ω
|ν|=2

2 ν

(Dνf (x))2dx,

The first term penalises lack of fit, the second penalises roughness.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Smoothing Splines

Given a set of attributes vectors x = (x1, x2, · · · , xd)T, build a predictive model y ≈ f (x). To estimate f by a 2nd-order smoothing spline minimise: Jα(f ) = 1 n

n

i=1

(f (x(i)) − y(i))2 + α

Ω
|ν|=2

2 ν

(Dνf (x))2dx,

The first term penalises lack of fit, the second penalises roughness.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Smoothing Splines

Given a set of attributes vectors x = (x1, x2, · · · , xd)T, build a predictive model y ≈ f (x). To estimate f by a 2nd-order smoothing spline minimise: Jα(f ) = 1 n

n

i=1

(f (x(i)) − y(i))2 + α

Ω
|ν|=2

2 ν

(Dνf (x))2dx,

The first term penalises lack of fit, the second penalises roughness.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Smoothing Splines

Given a set of attributes vectors x = (x1, x2, · · · , xd)T, build a predictive model y ≈ f (x). To estimate f by a 2nd-order smoothing spline minimise: Jα(f ) = 1 n

n

i=1

(f (x(i)) − y(i))2 + α

Ω
|ν|=2

2 ν

(Dνf (x))2dx,

The first term penalises lack of fit, the second penalises roughness.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Radial Basis Functions

The standard approach is to represent f as a linear combination of radial basis functions f (x) =

M

k=1

aφk(x) + α

n

i=1

wiU(x, x(i)), where φk are monomials of order up to 1 and U are suitable radial basis functions. Favoured method as it gives an analytical solution.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Radial Basis Functions

The standard approach is to represent f as a linear combination of radial basis functions f (x) =

M

k=1

aφk(x) + α

n

i=1

wiU(x, x(i)), where φk are monomials of order up to 1 and U are suitable radial basis functions. Favoured method as it gives an analytical solution.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Radial Basis Functions

2D Eg: U(x, x(i)) = −1 16πr2ln(r).

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Thin Plate Splines

Requires a solution of a dense system of matrices.
System may be ill-conditioned.
Size increases with the number of data points.

Not practical for large data sets.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Thin Plate Splines

Requires a solution of a dense system of matrices.
System may be ill-conditioned.
Size increases with the number of data points.

Not practical for large data sets.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Finite Element Approximation

Represent f as a linear combination of linear finite elements. In vector notation f will be of the form f (x) = b(x)Tc. Minimise Jα over all f of this form Jα(f ) = 1 n

n

i=1

(f (x(i)) − y(i))2 + α

Ω
|ν|=2

2 ν

(Dνf (x))2dx.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Finite Element Approximation

Represent f as a linear combination of linear finite elements. In vector notation f will be of the form f (x) = b(x)Tc. Minimise Jα over all f of this form Jα(f ) = 1 n

n

i=1

(f (x(i)) − y(i))2 + α

Ω
|ν|=2

2 ν

(Dνf (x))2dx.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Finite Element Approximation

Represent f as a linear combination of linear finite elements. In vector notation f will be of the form f (x) = b(x)Tc. Minimise Jα over all f of this form Jα(f ) = 1 n

n

i=1

(f (x(i)) − y(i))2 + α

Ω
|ν|=2

2 ν

(Dνf (x))2dx.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Non-Conforming Finite elements

The smoothing term (derivatives) is not defined for piecewise multi-linear functions. Use non-conforming finite elements. Represent the gradient of f by u = (bTg1, ..., bTgd) where

Ω

∇f (x) · ∇v(x) dx =

Ω

u(x) · ∇v(x) dx, for all piecewise multi-linear function v.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Non-Conforming Finite elements

Ω

∇f (x) · ∇v(x) dx =

Ω

u(x) · ∇v(x) dx, is equivalent to Lc =

d

s=1

Gsgs, where L is a discrete approximation to the negative Laplace

perator and (G1, ..., Gd) is a discrete approximation to the

transpose of the gradient operator.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Non-Conforming Finite elements

Ω

∇f (x) · ∇v(x) dx =

Ω

u(x) · ∇v(x) dx, is equivalent to Lc =

d

s=1

Gsgs, where L is a discrete approximation to the negative Laplace

perator and (G1, ..., Gd) is a discrete approximation to the

transpose of the gradient operator.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Finite Element Approximation

2nd-order smoothing spline: minimise Jα(f ) = 1 n

n

i=1

(f (x(i)) − y(i))2 + α

Ω
|ν|=2

2 ν

(Dνf (x))2dx.

Finite element approximation: minimise Jα(c, g1, g2, · · · , gd) = 1 n

n

i=1

(f (x(i)) − y(i))2 + α

d

s=1

gT

s Lgs,

subject to Lc =

d

s=1

Gsgs.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Finite Element Approximation

2nd-order smoothing spline: minimise Jα(f ) = 1 n

n

i=1

(f (x(i)) − y(i))2 + α

Ω
|ν|=2

2 ν

(Dνf (x))2dx.

Finite element approximation: minimise Jα(c, g1, g2, · · · , gd) = 1 n

n

i=1

(f (x(i)) − y(i))2 + α

d

s=1

gT

s Lgs,

subject to Lc =

d

s=1

Gsgs.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

2D Formulation

2nd-order smoothing spline: minimise Jα(f ) = 1 n

n

i=1

(f (x(i)) − y(i))2 + α

Ω
∂2

1f (x)

2 + 2 (∂1∂2f (x))2 +

∂2

2f (x)

2 dx, Jα(c, g1, g2) = 1 n

n

i=1

(b(x(i))Tc − y(i))2 + α

Ω

∇bT(x)g1.∇bT(x)g1 + ∇bT(x)g2.∇bT(x)g2 dx

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

2D Formulation

2nd-order smoothing spline: minimise Jα(f ) = 1 n

n

i=1

(f (x(i)) − y(i))2 + α

Ω
∂2

1f (x)

2 + 2 (∂1∂2f (x))2 +

∂2

2f (x)

2 dx, Jα(c, g1, g2) = 1 n

n

i=1

(b(x(i))Tc − y(i))2 + α

Ω

∇bT(x)g1.∇bT(x)g1 + ∇bT(x)g2.∇bT(x)g2 dx

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

2D Formulation

Minimise: Jα(c, g1, g2) = cTAc − 2dTc + y2/n + α(gT

1 Lg1 + gT 2 Lg2)

subject to Lc = G1g1 + G2g2. Where A = 1 n

n

i=1

b(x(i))b(x(i))T, and d = 1 n

n

i=1

b(x(i))y(i).

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Discrete System

    A L αL −G T

1

αL −G T

2

L −G1 −G2         c g1 g2 w     =     d     −     h1 h2 h3 h4     , w is a Lagrange multiplier. The vectors h1, · · · , h4 store the boundary information.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Conjugate Gradient

Eliminate all the variables except g1 and g2 to give αL αL

+

G T

1

G T

2

L−1AL−1

G1 G2 g1 g2

=

G T

1 L−1d

G T

2 L−1d

−
h2
h3
,

αdiag(L) + K TKg = d c = L−1 G1g1 + G2g2 − h4

.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Conjugate Gradient

Eliminate all the variables except g1 and g2 to give αL αL

+

G T

1

G T

2

L−1AL−1

G1 G2 g1 g2

=

G T

1 L−1d

G T

2 L−1d

−
h2
h3
,

αdiag(L) + K TKg = d c = L−1 G1g1 + G2g2 − h4

.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Preconditioned Conjugate Gradient

Current preconditioner M = L−1 L−1

.
large α: works well
small α: help.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Preconditioned Conjugate Gradient

Current preconditioner M = L−1 L−1

.
large α: works well
small α: help.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Lagrange Multiplier

Recall the discrete system     A L αL −G T

1

αL −G T

2

L −G1 −G2         c g1 g2 w     =     d     −     h1 h2 h3 h4     , where w is a Lagrange multiplier. We use Dirichlet boundary conditions as L−1 is unique, although Neumann is also possible. What is the Dirichlet boundary value for w?

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Lagrange Multiplier

Recall the discrete system     A L αL −G T

1

αL −G T

2

L −G1 −G2         c g1 g2 w     =     d     −     h1 h2 h3 h4     , where w is a Lagrange multiplier. We use Dirichlet boundary conditions as L−1 is unique, although Neumann is also possible. What is the Dirichlet boundary value for w?

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Lagrange Multiplier

Recall the discrete system     A L αL −G T

1

αL −G T

2

L −G1 −G2         c g1 g2 w     =     d     −     h1 h2 h3 h4     , where w is a Lagrange multiplier. We use Dirichlet boundary conditions as L−1 is unique, although Neumann is also possible. What is the Dirichlet boundary value for w?

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Dirichlet Boundary Conditions

Use Karush-Kuhn-Tucker (KKT) condition with calculus of variations to rewrite weak finite element equations into a system of strong equations. Then ∆ λ(x) = 1 n

n

i=1
f (x) − y(i)

δ(x − x(i)) in Ω, −α∆ u1(x) = ∂1 λ(x) in Ω, −α∆ u2(x) = ∂2 λ(x) in Ω, ∆ f (x) = ∇. u(x) in Ω.

f = minimiser,

u = gradient, λ = lagrange multiplier.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Dirichlet Boundary Conditions

∆ λ(x) = 1 n

n

i=1
f (x) − y(i)

δ(x − x(i)), −α∆ u1(x) = ∂1 λ(x), −α∆ u2(x) = ∂2 λ(x), ∆ f (x) = ∇. u(x).     A L αL −G T

1

αL −G T

2

L −G1 −G2         c g1 g2 w     =     d     −     h1 h2 h3 h4     ,

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Dirichlet Boundary Conditions

Use Karush-Kuhn-Tucker (KKT) condition with calculus of variations to rewrite weak finite element equations into a system of strong equations. Then ∆∆ f (x) = ∆∇. u(x) . . . = −1 α 1 n

n

i=1
f (x) − y(x)
δ(x − x(i)).

Conclusion: Boundary conditions do not matter, always get a minimiser.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Dirichlet Boundary Conditions

Use Karush-Kuhn-Tucker (KKT) condition with calculus of variations to rewrite weak finite element equations into a system of strong equations. Then ∆∆ f (x) = ∆∇. u(x) . . . = −1 α 1 n

n

i=1
f (x) − y(x)
δ(x − x(i)).

Conclusion: Boundary conditions do not matter, always get a minimiser.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Dirichlet Boundary Conditions

Use Karush-Kuhn-Tucker (KKT) condition with calculus of variations to rewrite weak finite element equations into a system of strong equations. Then ∆∆ f (x) = ∆∇. u(x) . . . = −1 α 1 n

n

i=1
f (x) − y(x)
δ(x − x(i)).

Conclusion: Boundary conditions do not matter, always get a minimiser.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Boundary Condition Examples

x(1) = (0.25, 0.25), x(2) = (0.75, 0.25), x(3) = (0.25, 0.75), x(4) = (0.75, 0.75); y(1) = 1, y(2) = 0, y(3) = 0 and y(4) = 1 hf (x) = tps fit. hf (x) = 0. hu = ∇hf (x), hλ(x) = −α∆hf

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Convergence on Smooth Problem

y(i) = fy(x(i)) where ∇4 fy = 0.

f (x) =

fy(x) =

x +

0.5 0.5

2
30
25
20
15
10
10
9
8
7
6
5
4
3
2

log2 of L2 Error log2 of Grid Spacing L2 Error for Model Problem 3 c g1 w interpol

30
25
20
15
10
10
9
8
7
6
5
4
3
2

log2 of L2 Error log2 of Grid Spacing L2 Error for Model Problem 3 O(h) convergence O(h2) convergence

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Convergence on Smooth Problem

y(i) = fy(x(i)) where ∇4 fy = 0.

f (x) =

fy(x) = cosh(2πx1) sin(2πx2)

20
15
10
5

5

10
9
8
7
6
5
4
3
2

log2 of L2 Error log2 of Grid Spacing L2 Error for Model Problem 4 c g1 g2 w interpol

20
15
10
5

5

10
9
8
7
6
5
4
3
2

log2 of L2 Error log2 of Grid Spacing L2 Error for Model Problem 4 O(h) convergence O(h2) convergence

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Further Model Problems - Exponential

exp

−300.65 − x2

2

+ exp
−300.35 − x2

2

Finite element grid of size m = 4225 with different values of α.

0 0.1 0.2 0.3 0.4 0.5 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 Test Problem 5 with alpha = 0.0001 0 0.1 0.2 0.3 0.4 0.5 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 Test Problem 5 with alpha = 0.000001

Boundary conditions: hf = fy, hu = ∇ fy and hλ = −α∆ fy.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Further Model Problems - Sin

sin(4πx1) sin(4πx2) Finite element grid of size m = 4225 with different values of α.

0 0.1 0.2 0.3 0.4 0.5 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1
0.5

0.5 1 Test Problem 6 with alpha = 0.0001 0 0.1 0.2 0.3 0.4 0.5 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1
0.5

0.5 1 Test Problem 6 with alpha = 0.000001

Boundary conditions: hf = fy, hu = ∇ fy and hλ = −α∆ fy.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Interpolation Error

Finite element analysis shows that the interpolation error is given by

(k4 + α)f02

H2 + h2mf 2 Hm +

Cσ2 nαd/(2m) , where

yi = f0(xi) + ǫi, E(ǫ) = 0, s.d. σ,
h is the grid size,
k is spacing between data points, uniform spacing, no holes,
d is the dimension,
C is a constant.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Example Interpolation Error - No Noise

sin(4πx1) sin(4πx2)

(k4 + α)f02

H2 + h2mf 2 Hm +

Cσ2 nαd/(2m) ,

1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 1 1e-09 1e-08 1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 Interpolation Error Smoothing Parameter Interpolation Error for Model Problem 6 with 998001 Data Points m=25 m=81 m=289 m=1089 m=4225 m=16641 m=66049 1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 1 1e-09 1e-08 1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 Interpolation Error Smoothing Parameter Interpolation Error for Model Problem 6 with 998001 Data Points Upper Bound

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Example Interpolation Error - No Noise

sin(4πx1) sin(4πx2)

(k4 + α)f02

H2 + h2mf 2 Hm +

Cσ2 nαd/(2m) ,

1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 1 1e-09 1e-08 1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 Interpolation Error Smoothing Parameter Interpolation Error for Model Problem 6 with 998001 Data Points m=25 m=81 m=289 m=1089 m=4225 m=16641 m=66049 1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 1 1e-09 1e-08 1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 Interpolation Error Smoothing Parameter Interpolation Error for Model Problem 6 with 998001 Data Points Upper Bound

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Example Interpolation Error - Noise

exp

−300.65 − x2

2

+ exp
−300.35 − x2

2

(k4 + α)f02

H2 + h2mf 2 Hm +

Cσ2 nαd/(2m) ,

1e-08 1e-07 1e-06 1e-05 1e-04 0.001 1e-09 1e-08 1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 Interpolation Error Smoothing Parameter Interpolation Error for Model Problem 5 with Noise 5% noise 10% noise 20% noise 1e-08 1e-07 1e-06 1e-05 1e-04 0.001 1e-09 1e-08 1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 Interpolation Error Smoothing Parameter Interpolation Error for Model Problem 5 with Noise O(alpha0.25) divergence

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

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Outline tps dtps Convergence Results Future Work

Example Interpolation Error - Noise

exp

−300.65 − x2

2

+ exp
−300.35 − x2

2

(k4 + α)f02

H2 + h2mf 2 Hm +

Cσ2 nαd/(2m) ,

1e-08 1e-07 1e-06 1e-05 1e-04 0.001 1e-09 1e-08 1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 Interpolation Error Smoothing Parameter Interpolation Error for Model Problem 5 with Noise 5% noise 10% noise 20% noise 1e-08 1e-07 1e-06 1e-05 1e-04 0.001 1e-09 1e-08 1e-07 1e-06 1e-05 1e-04 0.001 0.01 0.1 Interpolation Error Smoothing Parameter Interpolation Error for Model Problem 5 with Noise O(alpha0.25) divergence

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

SLIDE 48

Outline tps dtps Convergence Results Future Work

Sine Example with Holes

y(x, y) = sin(2πx) sin(2πy), such that y(x, y) < 0. n = 179401, m = 4229 with α = 10−6 Boundary: hf (x) = y(x), hu = ∇hf (x), hλ(x) = −α∆hf .

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

SLIDE 49

Outline tps dtps Convergence Results Future Work

Sine Example with Holes

y(x, y) = sin(2πx) sin(2πy), such that y(x, y) < 0. n = 179401, m = 4229 with α = 10−6 Boundary: hf (x) = tps fit, hu = ∇hf (x), hλ(x) = −α∆hf .

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

SLIDE 50

Outline tps dtps Convergence Results Future Work

Sphere Example

Grid - 189 Nodes, α = 10−3.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

SLIDE 51

Outline tps dtps Convergence Results Future Work

Sphere Example

Grid - 68705 Nodes, α = 10−3.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

SLIDE 52

Outline tps dtps Convergence Results Future Work

Semi Sphere Example

Grid - 68705 Nodes, α = 10−3.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

SLIDE 53

Outline tps dtps Convergence Results Future Work

Two Sphere Example

Grid - 68705 Nodes, α = 10−7.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets

SLIDE 54

Outline tps dtps Convergence Results Future Work

What Now?

Adaptive Refinement: Reduce number of times data has to be

read.

Parallel Implementation: Grid v’s data.
Preconditioners for Small α:
Higher Dimensions: Hierarchical, sparse grids.
Finite Element Formulation: Linear operators, different

smoothers, different norms.

Holes: Include a-prior information.

Linda Stals, Steve Roberts Discrete Thin-Plate Splines for Large Data Sets