Coordinates Josiah Manson and Scott Schaefer Texas A&M - PowerPoint PPT Presentation

Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University

Barycentric Coordinates • Polygon Domain p 4 p p 3 2 p 0 p 1

Barycentric Coordinates • Polygon Domain P 3 t ( ) P 2 t ( ) P 4 t ( ) P 1 t ( ) P 0 t ( )

Barycentric Coordinates • Polygon Domain f f 4 3 f 2 f 1 f 0

Barycentric Coordinates • Polygon Domain F 3 t ( ) F 2 t ( ) F 4 t ( ) F 1 t ( ) F 0 t ( )

Barycentric Coordinates • Polygon Domain • Boundary Interpolation F 3 t ( ) F 2 t ( ) F 4 t ( ) F 1 t ( ) F 0 t ( ) ˆ  F ( P ( t )) F ( t ) i i

Barycentric Coordinates • Polygon Domain • Boundary Interpolation F 3 t ( ) F 2 t ( ) F 4 t ( ) F 1 t ( ) F 0 t ( ) ˆ  F ( P ( t )) F ( t ) i i

Barycentric Coordinates • Polygon Domain • Boundary Interpolation f f 4 3 • Basis Functions f 2 f 1 f 0 n  ˆ  F ( x ) b ( x ) f i i i

Barycentric Coordinates • Polygon Domain • Boundary Interpolation • Basis Functions n  ˆ  F ( x ) b ( x ) f i i i

Barycentric Coordinates • Polygon Domain • Boundary Interpolation • Basis Functions n  ˆ  F ( x ) b ( x ) f i i i

Barycentric Coordinates • Polygon Domain • Boundary Interpolation • Basis Functions n  ˆ  F ( x ) b ( x ) f i i i

Barycentric Coordinates • Polygon Domain • Boundary Interpolation • Basis Functions n  ˆ  F ( x ) b ( x ) f i i i

Barycentric Coordinates • Polygon Domain • Boundary Interpolation • Basis Functions n  ˆ  F ( x ) b ( x ) f i i i

Barycentric Coordinates • Polygon Domain • Boundary Interpolation • Basis Functions • Linear Precision n   L ( x ) b ( x ) L ( p ) i i i

Barycentric Coordinates • Polygon Domain • Boundary Interpolation • Basis Functions • Linear Precision n   L ( x ) b ( x ) L ( p ) i i i

Barycentric Coordinates • Polygon Domain • Boundary Interpolation • Basis Functions • Linear Precision n   1 b i x ( ) i

Other Properties • Desirable Features – Smoothness – Closed-form solution – Positivity • Extended Coordinates – Polynomial Boundary Values – Polynomial Precision – Interpolation of Derivatives – Curved Boundaries

Applications • Finite Element Methods [Wachspress 1975]

Applications • Boundary Value Problems [Ju et al. 2005]

Applications • Free-Form Deformations [Sederberg et al. 1986], [MacCracken et al. 1996], [Ju et al. 2005], [Joshi et al. 2007]

Applications • Surface Parameterization [Hormann et al. 2000], [Desbrun et al. 2002]

Comparison of Methods Wachspress Mean Val. Pos. Mean Val. Max Entropy Harmonic Hermite MVC Moving Least Sqr.

Moving Least Squares Coordinates • A new family of barycentric coordinates • Solves a least squares problem • Solution depends on point of evaluation

Fit a Polynomial to Points  V ( x ) ( 1 x x ) 1 1 2 2 argmin  n    V ( p ) C F ( p ) 1 i i C i ˆ  F ( x ) V ( x ) C 1

Fit a Polynomial to Points

Fit a Polynomial to Points

Interpolating Points

Interpolating Points 1  W ( x , p )   2 x p 2 argmin  n    W ( x , p ) V ( p ) C F ( p ) i 1 i i C i ˆ  F ( x ) V ( x ) C 1

Interpolating Points

Interpolating Line Segments   P     i , 1 P ( t ) ( 1 t t )   i P   i , 2

Interpolating Line Segments     P F         i , 1 i , 1 P ( t ) ( 1 t t ) F ( t ) ( 1 t t )     i i P F     i , 2 i , 2

Interpolating Line Segments     P F         i , 1 i , 1 P ( t ) ( 1 t t ) F ( t ) ( 1 t t )     i i P F     i , 2 i , 2 P ' ( t )  i W ( x , t )  i  2 x P ( t ) i

Interpolating Line Segments     P F         i , 1 i , 1 P ( t ) ( 1 t t ) F ( t ) ( 1 t t )     i i P F     i , 2 i , 2 P ' ( t )  i W ( x , t )  i  2 x P ( t ) i 2 1 n   dt   argmin W ( x , t ) V ( P ( t )) C F ( t ) i 1 i i C i 0

Line Basis Functions 1 n   T A W ( x , t ) V ( P ( t )) V ( P ( t )) dt i i i 1 i i 0 1 n     1 T C A W ( x , t ) V ( P ( t )) F ( t ) dt i i i i i 0

Line Basis Functions ˆ  F ( x ) V ( x ) C 1

Line Basis Functions ˆ  F ( x ) V ( x ) C 1 1 n     1 T V ( x ) A W ( x , t ) V ( P ( t )) F ( t ) dt 1 i i i i i 0

Line Basis Functions ˆ  F ( x ) V ( x ) C 1 1 n     1 T V ( x ) A W ( x , t ) V ( P ( t )) F ( t ) dt 1 i i i i i 0   1 F n        i , 1 1 T V ( x ) A W ( x , t ) V ( P ( t ))( 1 t t ) dt   1 i i i F   i i , 2 0

Line Basis Functions ˆ  F ( x ) V ( x ) C 1 1 n     1 T V ( x ) A W ( x , t ) V ( P ( t )) F ( t ) dt 1 i i i i i 0   1 F n        i , 1 1 T V ( x ) A W ( x , t ) V ( P ( t ))( 1 t t ) dt   1 i i i F   i i , 2 0   1 F n        i , 1 1 T V ( x ) A W ( x , t ) V ( P ( t ))( 1 t t ) dt   1 i i i F   i i , 2 0

Line Basis Functions ˆ  F ( x ) V ( x ) C 1 1 n     1 T V ( x ) A W ( x , t ) V ( P ( t )) F ( t ) dt 1 i i i i i 0   1 F n        i , 1 1 T V ( x ) A W ( x , t ) V ( P ( t ))( 1 t t ) dt   1 i i i F   i i , 2 0   1 F n        i , 1 1 T V ( x ) A W ( x , t ) V ( P ( t ))( 1 t t ) dt   1 i i i F   i i , 2 0     F n     i , 1 B ( x ) B ( x )   i , 1 i , 2 F   i i , 2

Polygon Basis Functions     F n   ˆ   i , 1 F ( x ) B ( x ) B ( x )   i , 1 i , 2 F   i i , 2

Polygon Basis Functions     F n   ˆ   i , 1 F ( x ) B ( x ) B ( x )   i , 1 i , 2 F   i i , 2   f F F  i i , 1 i 1 , 2   b ( x ) B ( x ) B ( x )  i i , 1 i 1 , 2

Polygon Basis Functions

Polygon Basis Functions

Polygon Basis Functions     F n   ˆ   i , 1 F ( x ) B ( x ) B ( x )   i , 1 i , 2 F   i i , 2   f F F  i i , 1 i 1 , 2   b ( x ) B ( x ) B ( x )  i i , 1 i 1 , 2 n  ˆ  F ( x ) b ( x ) f i i i

Polynomial Boundary Values   F     i , 1 F ( t ) ( 1 t t )   i F     i , 2 F   i , 1      2 2 F ( t ) ( ( 1 t ) 2 ( 1 t ) t t ) F i i , 2     F   F i , 3   i , 1   F     i , 2 3 2 2 3 F ( t ) ( ( 1 t ) 3 ( 1 t ) t 3 ( 1 t ) t t )   i F   i , 3   F   i , 4 

Polynomial Boundary Values

Polynomial Boundary Values

Polynomial Precision  V ( x ) ( 1 x x ) 1 1 2 2  2 V ( x ) ( 1 x x x x x x ) 2 1 2 1 1 2 2  2 2 3 2 1 1 2 3 V ( x ) ( 1 x x x x x x x x x x x x ) 3 1 2 1 1 2 2 1 1 2 1 2 2 

Polynomial Precision Linear Quadratic

Interpolation of Derivatives 2 1 n   dt   argmin W ( x , t ) V ( P ( t )) C F ( t ) i 1 i i C i 0

Interpolation of Derivatives 2 1 n   dt   argmin W ( x , t ) V ( P ( t )) C F ( t ) i 1 i i C i 0 2   dt 1     W ( x , t ) G ( t ) C F ( t ) i 1 , i i 0