About interpolation on manifolds... How to interpolate points on - - PowerPoint PPT Presentation

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Sep 08, 2022 266 likes •504 views

About interpolation on manifolds... How to interpolate points on curved spaces ? Light fast general good looking interpolation How to interpolate ? Each segment between two consecutive points is a Bzier function. p 0 p 1 p 2 p 3 t = 0 t

SLIDE 1

About interpolation on manifolds...

SLIDE 2

How to interpolate points on curved spaces ?

Light fast general good looking interpolation

SLIDE 3

How to interpolate ?

Each segment between two consecutive points is a Bézier function. p0 t = 0 p1 t = 1 p2 t = 2 p3 t = 3

Light

fast general good looking

interpolation

SLIDE 4

Reconstruction : the De Casteljau algorithm

b0 b1 b2 | | t 1

1 4 1 2 3 4

β2(b0, b1, b2; 1

β2(b0, b1, b2; 3

Light fast

general good looking

interpolation

SLIDE 5

How to generalize Bézier curves to manifolds ?

The straight line is a geodesic

SLIDE 6

How to generalize Bézier curves to manifolds ?

The exponential map to construct the geodesic

γ(t) = Expx(tξx)

SLIDE 7

How to generalize Bézier curves to manifolds ?

The logarithmic map to determine the starting velocity

Logx(y) = ξx

SLIDE 8

Piecewise interpolation on the sphere

Light fast general

good looking

interpolation

SLIDE 9

Interpolation

n Riemannian manifolds

with a C1 piecewize-Bézier path

Pierre-Yves Gousenbourger 8 october 2014

SLIDE 10

Good-looking curve on the Euclidean space

p0 p1 p2 p3 p4 Find the optimal position of control points

SLIDE 11

C1-piecewise Bézier interpolation

vi vi | | | αi αi pi−1 pi pi+1

bL

i = Exppi(−αivi)

bR

i = Exppi( αivi)

SLIDE 12

Optimal C1-piecewise Bézier interpolation

Minimization of the mean square acceleration of the path

min

αi

1 ¨ β0

2(αi; t)2dt + n−1

1 ¨ βi

3(αi; t)2dt +

1 ¨ βn

2 (αi; t)2dt

Second order polynomial P(αi)

∇P(αi) !

SLIDE 13

Optimal C1-piecewise Bézier interpolation

Minimization of the mean square acceleration of the path

min

αi

1 ¨ β0

2(αi; t)2dt + n−1

1 ¨ βi

3(αi; t)2dt +

1 ¨ βn

2 (αi; t)2dt

Second order polynomial P(αi)

i−1vi, vT i vi, vT i+1vi

× αi = ∼ (pi−1 − pi)Tvi

SLIDE 14

Optimal C1-piecewise Bézier interpolation

Minimization of the mean square acceleration of the path

min

αi

1 ¨ β0

2(αi; t)2dt + n−1

1 ¨ βi

3(αi; t)2dt +

1 ¨ βn

2 (αi; t)2dt

Second order polynomial P(αi)

i−1vi, vT i vi, vT i+1vi

× αi = ∼ (pi−1 − pi)Tvi

SLIDE 15

Optimal C1-piecewise Bézier interpolation

Minimization of the mean square acceleration of the path

min

αi

1 ¨ β0

2(αi; t)2dt + n−1

1 ¨ βi

3(αi; t)2dt +

1 ¨ βn

2 (αi; t)2dt

Second order polynomial P(αi)

i−1vi, vT i vi, vT i+1vi

× αi = ∼ (Logpi(pi−1))Tvi

SLIDE 16

Optimal C1-piecewise Bézier interpolation

Minimization of the mean square acceleration of the path

min

αi

1 ¨ β0

2(αi; t)2dt + n−1

1 ¨ βi

3(αi; t)2dt +

1 ¨ βn

2 (αi; t)2dt

Second order polynomial P(αi)

i−1vi, vT i vi, vT i+1vi

× αi = ∼ (Logpi(pi−1))Tvi

SLIDE 17

Optimal C1-piecewise Bézier interpolation

Minimization of the mean square acceleration of the path

min

αi

1 ¨ β0

2(αi; t)2dt + n−1

1 ¨ βi

3(αi; t)2dt +

1 ¨ βn

2 (αi; t)2dt

Second order polynomial P(αi)

vi−1,vi,vi,vi,vi+1,vi × αi = ∼ Logpi(pi−1),vi

SLIDE 18

A result on R2

Light fast general good looking interpolation

SLIDE 19

Generalization to manifolds : the sphere S2

SLIDE 20

Generalization to manifolds : the special orthogonal group SO(3)

SLIDE 21

Generalization to manifolds : morphing of shapes

SLIDE 22

Conclusions

Light fast general good looking interpolation

No choice of velocities vi ? (Arnould, Samir, Absil) Application to manifolds of high dimension ?

SLIDE 23

Any questions ?

Interpolation

n Riemannian manifolds

with a C1 piecewize-Bézier path

Pierre-Yves Gousenbourger 8 october 2014