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

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 Bézier function. p 0 p 1 p 2 p 3 t = 0 t = 1 t = 2 t = 3 Light interpolation fast general good looking

Reconstruction : the De Casteljau algorithm b 1 β 2 ( b 0 , b 1 , b 2 ; 1 β 2 ( b 0 , b 1 , b 2 ; 3 4 ) 4 ) β 2 ( b 0 , b 1 , b 2 ; 1 2 ) b 0 b 2 | | t 0 1 1 1 3 4 2 4 Light fast interpolation general good looking

How to generalize Bézier curves to manifolds ? The straight line is a geodesic

How to generalize Bézier curves to manifolds ? The exponential map to construct the geodesic γ ( t ) = Exp x ( t ξ x )

How to generalize Bézier curves to manifolds ? The logarithmic map to determine the starting velocity Log x ( y ) = ξ x

Piecewise interpolation on the sphere Light fast general interpolation good looking

Interpolation on Riemannian manifolds with a C 1 piecewize-Bézier path Pierre-Yves Gousenbourger 8 october 2014

Good-looking curve on the Euclidean space p 4 p 2 p 3 p 0 p 1 Find the optimal position of control points

C 1 -piecewise Bézier interpolation α i α i | | | v i v i p i p i − 1 p i +1 b L i = Exp p i ( − α i v i ) b R i = Exp p i ( α i v i )

Optimal C 1 -piecewise Bézier interpolation Minimization of the mean square acceleration of the path � 1 � 1 � 1 n − 1 � � ¨ � ¨ � ¨ β 0 2 ( α i ; t ) � 2 d t + β i 3 ( α i ; t ) � 2 d t + β n 2 ( α i ; t ) � 2 d t min α i 0 0 0 i =1 � �� � Second order polynomial P ( α i ) ∇ P ( α i ) !

Optimal C 1 -piecewise Bézier interpolation Minimization of the mean square acceleration of the path � 1 � 1 � 1 n − 1 � � ¨ � ¨ � ¨ β 0 2 ( α i ; t ) � 2 d t + β i 3 ( α i ; t ) � 2 d t + β n 2 ( α i ; t ) � 2 d t min α i 0 0 0 i =1 � �� � Second order polynomial P ( α i ) ∼ ( p i − 1 − p i ) T v i × = α i v T i − 1 v i , v T i v i , v T i +1 v i

Optimal C 1 -piecewise Bézier interpolation Minimization of the mean square acceleration of the path � 1 � 1 � 1 n − 1 � � ¨ � ¨ � ¨ β 0 2 ( α i ; t ) � 2 d t + β i 3 ( α i ; t ) � 2 d t + β n 2 ( α i ; t ) � 2 d t min α i 0 0 0 i =1 � �� � Second order polynomial P ( α i ) ∼ ( p i − 1 − p i ) T v i × = α i v T i − 1 v i , v T i v i , v T i +1 v i

Optimal C 1 -piecewise Bézier interpolation Minimization of the mean square acceleration of the path � 1 � 1 � 1 n − 1 � � ¨ � ¨ � ¨ β 0 2 ( α i ; t ) � 2 d t + β i 3 ( α i ; t ) � 2 d t + β n 2 ( α i ; t ) � 2 d t min α i 0 0 0 i =1 � �� � Second order polynomial P ( α i ) ∼ (Log p i ( p i − 1 )) T v i = × α i v T i − 1 v i , v T i v i , v T i +1 v i

Optimal C 1 -piecewise Bézier interpolation Minimization of the mean square acceleration of the path � 1 � 1 � 1 n − 1 � � ¨ � ¨ � ¨ β 0 2 ( α i ; t ) � 2 d t + β i 3 ( α i ; t ) � 2 d t + β n 2 ( α i ; t ) � 2 d t min α i 0 0 0 i =1 � �� � Second order polynomial P ( α i ) ∼ (Log p i ( p i − 1 )) T v i = × α i v T i − 1 v i , v T i v i , v T i +1 v i

Optimal C 1 -piecewise Bézier interpolation Minimization of the mean square acceleration of the path � 1 � 1 � 1 n − 1 � � ¨ � ¨ � ¨ β 0 2 ( α i ; t ) � 2 d t + β i 3 ( α i ; t ) � 2 d t + β n 2 ( α i ; t ) � 2 d t min α i 0 0 0 i =1 � �� � Second order polynomial P ( α i ) ∼ � Log p i ( p i − 1 ) , v i � = × α i � v i − 1 , v i � , � v i , v i � , � v i +1 , v i �

A result on R 2 Light fast general good looking interpolation

Generalization to manifolds : the sphere S 2

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

Generalization to manifolds : morphing of shapes

Conclusions Light fast general good looking interpolation No choice of velocities v i ? (Arnould, Samir, Absil) Application to manifolds of high dimension ?

Any questions ? Interpolation on Riemannian manifolds with a C 1 piecewize-Bézier path Pierre-Yves Gousenbourger 8 october 2014