Transfinite mean value interpolation - ifi
Transfinite mean value interpolation Christopher Dyken and Michael Floater Centre of Mathematics for Applications, Department of Informatics, University of Oslo Id: maiatalk.tex,v 1.9 2007/08/23 07:35:58 dyken Exp
Page 1
Transfinite interpolation Given Ω ⊂ R2 , a convex or non-convex set, possibly with holes. Lagrange transfinite interpolation We are given f : ∂Ω → R. Find g : Ω → R that interpolates f on ∂Ω. Hermite transfinite interpolation ∂f We are given f : ∂Ω → R and ∂n : ∂Ω → R. Find g : Ω → R that interpolates f and ∂g ∂n matches
∂f ∂n
on ∂Ω.
I
Lagrange can be solved by solving the harmonic equation
I
Hermite can be solved by solving the biharmonic equation
=⇒ But we want something simpler. . . Page 2
Pseudo-harmonic interpolation [Gordon, Wixom 1974] Let I
x be a point inside the convex set Ω;
I
Q(x, θ) be the infinite line through x in the direction θ.
I
Let p1 (x, θ) and p2 (x, θ) be the two intersections between Q(x, θ) and ∂Ω,
p1(x , θ ) Q (x , θ ) θ Ω x dΩ
then we define Z 2π 1 lerp f (p1 (x, θ)), f (p2 (x, θ)), gGW (x) = 2π 0
p2(x , θ )
kp1 (x,θ)−xk kp1 (x,θ)−p2 (x,θ)k
dθ.
Works only for convex sets. I Evaluation requires numerical integration =⇒ must find intersections for each integration point! I
Page 3
A mean value approach
Let I
I
L(x, θ) be the semi-infinite line originating at x in the direction θ. p(x, θ) be the intersection of L(x, θ) and ∂Ω.
p (x , θ )
L (x , θ ) θ
Ω
x dΩ
and define the “radially linear” function F as I
F (x + r (cos θ, sin θ)) = lerp g (x), f (p(x, θ)),
r kp(x,θ)−xk
.
Page 4
We want F to satisfy the Mean Value property at x. Let Γ be any circle at x with radius r , then Γ Z Ω 1 F (x) = F (z)dz, 2πr Γ whose unique solution is
1 g (x) = φ(x)
Z 0
2π
f (p(x, θ)) dθ, kp(x, θ) − xk
p (x , θ ) r
θ
x
dΩ
Z φ(x) = 0
2π
1 dθ, kp(x, θ) − xk
which is the “angle integral” formula for the MV interpolant g . I I I I I
Generalizes to non-convex sets Evaluation still requires numerical integration. Still must find an intersection for each integration point! How do we differentiate this thing? Luckily, we have two other formulas. . . Page 5
The boundary integral formula [Ju, Schaefer, Warren 2005] Suppose c : [a, b] → R2 is an anticlockwise representation of ∂Ω. Then dθ (c(t) − x) × c0 (t) = dt kc(t) − xk2
c (t )
which gives Z φ(x) = a
b
c0 (t)
(c(t) − x) × kc(t) − xk3
dt,
and g (x) =
θ
Ω x dΩ
1 φ(x)
Z a
b
(c(t) − x) × c0 (t) f (c(t)dt. kc(t) − xk3 Page 6
The polygonal formula Suppose Ω is a polygon with vertices p1 , . . . , pn . Then
p i+1
1 X wi (x)f (pi ), φ(x)
g (x) =
Ω αi
i
α i−1
where φ(x) =
pi
x
X
wi (x),
i
p i−1 dΩ
and wi (x) =
tan(αi−1 (x)/2) + tan(αi (x)/2) . kpi − xk Page 7
We have three formulations for the MV interpolant:
I
The polygonal formula: I I
I
The boundary integral formula I I
I
closed form easy to find expressions for derivatives.
needs adaptive numerical quadrature for evaluation. easy to find expressions for derivatives.
The angle integral I
describes the interpolant along a particular direction.
Page 8
The MV weight function A lot of properties can be deduced from the “weight function” ψ Z 2π 1 1 =1 dθ, ψ(x) = φ(x) kp(x, θ) − xk 0 Z b (c(t) − x) × c0 (t) =1 dt, kc(t) − xk3 a X tan(αi−1 (x)/2) + tan(αi (x)/2) =1 . kpi − xk i
ψ
∂ψ ∂x
∂ψ ∂y Page 9
Minimum principle for ψ
For arbitrary Ω, we have that Z ∆φ(x) = 3 0
2π n(x,θ) X j=1
(−1)j−1 dθ kpj (x, θ) − xk3
from which follows that I
2 2 ∆ψ = ∂ ψ + ∂ ψ ∂x 2 ∂y 2
ψ has no local minima in Ω.
Page 10
Bounds on ψ For all x ∈ Ω we have that 1 dist(x, ∂Ω) ≤ ψ(x) ≤ c dist(x, ∂Ω), 2π 0.5
I
I
c depends on dist(ME , ∂Ω), the distance between ∂Ω and its exterior medial axis If Ω is convex, then c = 12 .
=⇒ For all x ∈ Ω, ψ > 0.
0.4
0.3
0.2
0.1
0 −1
0
1
The plot shows the upper and lower bounds and ψ along a crosssection when Ω is the unit disc. Page 11
Proof of interpolation for the Lagrange MV interpolant If I
f is continuous,
I
∂Ω and any line intersects a bounded number of times,
I
and dist(ME , ∂Ω) > 0
then I
g interpolates f .
Page 12
Normal derivatives of ψ and g If dist(ME , ∂Ω) > 0
and
dist(MI , ∂Ω) > 0,
then, for all y ∈ ∂Ω, I
the inward normal derivative for ψ is ∂ψ 1 (y) = ∂n 2
I
the inward normal derivate for the Lagrange interpolant g is ∂g (y) = ∂n
1 2
Z a
b
(c(t) − y) × c0 (t) f (c(t)) − f (y) dt. 3 kc(t) − xk Page 13
Hermite mean value interpolation In one variable, we have the problem p(xi ) = f (xi )
and
p 0 (xi ) = f 0 (xi ),
i = 0, 1.
One approach of expressing p is p(x) = g0 (x) + ψ(x)g1 (x), where I g0 and g1 are Lagrange interpolants, I ψ vanishes at x0 and x1 and ψ 0 is nonzero at x0 and x1 . Which gives the conditions g0 (xi ) = f (xi )
and
g1 (xi ) =
f 0 (xi ) − g00 (xi ) . ψ 0 (xi ) Page 14
In two variables, we can generalize a similar problem, p(y) = f (y)
and
∂p ∂f (y) = (y), ∂n ∂n
y ∈ ∂Ω.
and let p be on the form p(x) = g0 (x) + ψ(x)g1 (x).
We can use the MV-ψ since ψ(y) = 0
and
∂ψ 1 (y) = , ∂n 2
y ∈ ∂Ω,
and let g0 and g1 be MV Lagrange interpolants.
Page 15
Then, for y ∈ ∂Ω we get the conditions g0 (y) = f (y) , ∂f ∂g0 ∂ψ g1 (y) = (y) − (y) (y). ∂n ∂n ∂n
Page 16
Application: Smooth mappings Reference shape
Computational domain (extended Gordon & Hall)
MV-Lagrange
MV-Hermite
Conjecture: Lagrange interpolation from convex sets to convex sets is always injective. Page 17
Application: WEB-splines [H¨ollig, Reif, Wipper 2001] Idea: Use ψ as a weight function for WEB-splines Parametric circle
Two nested ellipses
Polygon
Piecwise cubic B´ezier curve Page 18
Solution to Poisson’s equation using bicubic web-splines
Using implicit weight function Grid 10 × 8 20 × 16 40 × 32 80 × 64
L2 error 7.3e-02 2.9e-02 1.6e-03 4.4e-05
order 1.31 4.21 5.17
Using MV weight function Grid 10 × 8 20 × 16 40 × 32 80 × 64
L2 error 9.5e-02 4.1e-02 2.4e-03 1.4e-04
order 1.21 4.12 4.01
Page 19
Inhomogenenous Poisson’s equation
True solution
MV Lagrange interpolant
Homogeneous solution
Inhomogeneous solution Page 20
Conclusions I I I
The Lagrange mean value interpolant does in fact interpolate. Constructed a Hermite mean value interpolant. The mean value weight function has nice properties: I I I
positive; C ∞ -smooth; bounded by the distance function: =⇒ a very smooth distance-like function without ridges along the inner medial axis!
I I
I
constant normal derivate; has no local minima in Ω;
The mean value constructions are relatively easy to compute: I I
The polygonal case has a closed form. The boundary integral must be calculated numerically, but: Strong influence of the boundary region closest to the point of evaluation. =⇒ Adaptivity pays off. I Simpler than solving a PDE. I
Page 21
Thank you for listening!
Questions?
Page 22
Page 1
Transfinite interpolation Given Ω ⊂ R2 , a convex or non-convex set, possibly with holes. Lagrange transfinite interpolation We are given f : ∂Ω → R. Find g : Ω → R that interpolates f on ∂Ω. Hermite transfinite interpolation ∂f We are given f : ∂Ω → R and ∂n : ∂Ω → R. Find g : Ω → R that interpolates f and ∂g ∂n matches
∂f ∂n
on ∂Ω.
I
Lagrange can be solved by solving the harmonic equation
I
Hermite can be solved by solving the biharmonic equation
=⇒ But we want something simpler. . . Page 2
Pseudo-harmonic interpolation [Gordon, Wixom 1974] Let I
x be a point inside the convex set Ω;
I
Q(x, θ) be the infinite line through x in the direction θ.
I
Let p1 (x, θ) and p2 (x, θ) be the two intersections between Q(x, θ) and ∂Ω,
p1(x , θ ) Q (x , θ ) θ Ω x dΩ
then we define Z 2π 1 lerp f (p1 (x, θ)), f (p2 (x, θ)), gGW (x) = 2π 0
p2(x , θ )
kp1 (x,θ)−xk kp1 (x,θ)−p2 (x,θ)k
dθ.
Works only for convex sets. I Evaluation requires numerical integration =⇒ must find intersections for each integration point! I
Page 3
A mean value approach
Let I
I
L(x, θ) be the semi-infinite line originating at x in the direction θ. p(x, θ) be the intersection of L(x, θ) and ∂Ω.
p (x , θ )
L (x , θ ) θ
Ω
x dΩ
and define the “radially linear” function F as I
F (x + r (cos θ, sin θ)) = lerp g (x), f (p(x, θ)),
r kp(x,θ)−xk
.
Page 4
We want F to satisfy the Mean Value property at x. Let Γ be any circle at x with radius r , then Γ Z Ω 1 F (x) = F (z)dz, 2πr Γ whose unique solution is
1 g (x) = φ(x)
Z 0
2π
f (p(x, θ)) dθ, kp(x, θ) − xk
p (x , θ ) r
θ
x
dΩ
Z φ(x) = 0
2π
1 dθ, kp(x, θ) − xk
which is the “angle integral” formula for the MV interpolant g . I I I I I
Generalizes to non-convex sets Evaluation still requires numerical integration. Still must find an intersection for each integration point! How do we differentiate this thing? Luckily, we have two other formulas. . . Page 5
The boundary integral formula [Ju, Schaefer, Warren 2005] Suppose c : [a, b] → R2 is an anticlockwise representation of ∂Ω. Then dθ (c(t) − x) × c0 (t) = dt kc(t) − xk2
c (t )
which gives Z φ(x) = a
b
c0 (t)
(c(t) − x) × kc(t) − xk3
dt,
and g (x) =
θ
Ω x dΩ
1 φ(x)
Z a
b
(c(t) − x) × c0 (t) f (c(t)dt. kc(t) − xk3 Page 6
The polygonal formula Suppose Ω is a polygon with vertices p1 , . . . , pn . Then
p i+1
1 X wi (x)f (pi ), φ(x)
g (x) =
Ω αi
i
α i−1
where φ(x) =
pi
x
X
wi (x),
i
p i−1 dΩ
and wi (x) =
tan(αi−1 (x)/2) + tan(αi (x)/2) . kpi − xk Page 7
We have three formulations for the MV interpolant:
I
The polygonal formula: I I
I
The boundary integral formula I I
I
closed form easy to find expressions for derivatives.
needs adaptive numerical quadrature for evaluation. easy to find expressions for derivatives.
The angle integral I
describes the interpolant along a particular direction.
Page 8
The MV weight function A lot of properties can be deduced from the “weight function” ψ Z 2π 1 1 =1 dθ, ψ(x) = φ(x) kp(x, θ) − xk 0 Z b (c(t) − x) × c0 (t) =1 dt, kc(t) − xk3 a X tan(αi−1 (x)/2) + tan(αi (x)/2) =1 . kpi − xk i
ψ
∂ψ ∂x
∂ψ ∂y Page 9
Minimum principle for ψ
For arbitrary Ω, we have that Z ∆φ(x) = 3 0
2π n(x,θ) X j=1
(−1)j−1 dθ kpj (x, θ) − xk3
from which follows that I
2 2 ∆ψ = ∂ ψ + ∂ ψ ∂x 2 ∂y 2
ψ has no local minima in Ω.
Page 10
Bounds on ψ For all x ∈ Ω we have that 1 dist(x, ∂Ω) ≤ ψ(x) ≤ c dist(x, ∂Ω), 2π 0.5
I
I
c depends on dist(ME , ∂Ω), the distance between ∂Ω and its exterior medial axis If Ω is convex, then c = 12 .
=⇒ For all x ∈ Ω, ψ > 0.
0.4
0.3
0.2
0.1
0 −1
0
1
The plot shows the upper and lower bounds and ψ along a crosssection when Ω is the unit disc. Page 11
Proof of interpolation for the Lagrange MV interpolant If I
f is continuous,
I
∂Ω and any line intersects a bounded number of times,
I
and dist(ME , ∂Ω) > 0
then I
g interpolates f .
Page 12
Normal derivatives of ψ and g If dist(ME , ∂Ω) > 0
and
dist(MI , ∂Ω) > 0,
then, for all y ∈ ∂Ω, I
the inward normal derivative for ψ is ∂ψ 1 (y) = ∂n 2
I
the inward normal derivate for the Lagrange interpolant g is ∂g (y) = ∂n
1 2
Z a
b
(c(t) − y) × c0 (t) f (c(t)) − f (y) dt. 3 kc(t) − xk Page 13
Hermite mean value interpolation In one variable, we have the problem p(xi ) = f (xi )
and
p 0 (xi ) = f 0 (xi ),
i = 0, 1.
One approach of expressing p is p(x) = g0 (x) + ψ(x)g1 (x), where I g0 and g1 are Lagrange interpolants, I ψ vanishes at x0 and x1 and ψ 0 is nonzero at x0 and x1 . Which gives the conditions g0 (xi ) = f (xi )
and
g1 (xi ) =
f 0 (xi ) − g00 (xi ) . ψ 0 (xi ) Page 14
In two variables, we can generalize a similar problem, p(y) = f (y)
and
∂p ∂f (y) = (y), ∂n ∂n
y ∈ ∂Ω.
and let p be on the form p(x) = g0 (x) + ψ(x)g1 (x).
We can use the MV-ψ since ψ(y) = 0
and
∂ψ 1 (y) = , ∂n 2
y ∈ ∂Ω,
and let g0 and g1 be MV Lagrange interpolants.
Page 15
Then, for y ∈ ∂Ω we get the conditions g0 (y) = f (y) , ∂f ∂g0 ∂ψ g1 (y) = (y) − (y) (y). ∂n ∂n ∂n
Page 16
Application: Smooth mappings Reference shape
Computational domain (extended Gordon & Hall)
MV-Lagrange
MV-Hermite
Conjecture: Lagrange interpolation from convex sets to convex sets is always injective. Page 17
Application: WEB-splines [H¨ollig, Reif, Wipper 2001] Idea: Use ψ as a weight function for WEB-splines Parametric circle
Two nested ellipses
Polygon
Piecwise cubic B´ezier curve Page 18
Solution to Poisson’s equation using bicubic web-splines
Using implicit weight function Grid 10 × 8 20 × 16 40 × 32 80 × 64
L2 error 7.3e-02 2.9e-02 1.6e-03 4.4e-05
order 1.31 4.21 5.17
Using MV weight function Grid 10 × 8 20 × 16 40 × 32 80 × 64
L2 error 9.5e-02 4.1e-02 2.4e-03 1.4e-04
order 1.21 4.12 4.01
Page 19
Inhomogenenous Poisson’s equation
True solution
MV Lagrange interpolant
Homogeneous solution
Inhomogeneous solution Page 20
Conclusions I I I
The Lagrange mean value interpolant does in fact interpolate. Constructed a Hermite mean value interpolant. The mean value weight function has nice properties: I I I
positive; C ∞ -smooth; bounded by the distance function: =⇒ a very smooth distance-like function without ridges along the inner medial axis!
I I
I
constant normal derivate; has no local minima in Ω;
The mean value constructions are relatively easy to compute: I I
The polygonal case has a closed form. The boundary integral must be calculated numerically, but: Strong influence of the boundary region closest to the point of evaluation. =⇒ Adaptivity pays off. I Simpler than solving a PDE. I
Page 21
Thank you for listening!
Questions?
Page 22
