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

SLIDE 1

Moving Least Squares Coordinates

Josiah Manson and Scott Schaefer Texas A&M University

SLIDE 2

Barycentric Coordinates

Polygon Domain

p

1

p

2

p

3

p

4

p

SLIDE 3

Barycentric Coordinates

Polygon Domain

) (

0 t

P ) (

1 t

P ) (

2 t

P ) (

3 t

P ) (

4 t

P

SLIDE 4

Barycentric Coordinates

Polygon Domain

f

1

f

2

f

3

f

4

f

SLIDE 5

Barycentric Coordinates

Polygon Domain

) (

0 t

F ) (

1 t

F ) (

2 t

F ) (

3 t

F ) (

4 t

F

SLIDE 6

Barycentric Coordinates

Polygon Domain
Boundary Interpolation

) (

0 t

F ) (

1 t

F ) (

2 t

F ) (

3 t

F ) (

4 t

F

) ( )) ( ( ˆ t F t P F

i i



SLIDE 7

Barycentric Coordinates

Polygon Domain
Boundary Interpolation

) (

0 t

F ) (

1 t

F ) (

2 t

F ) (

3 t

F ) (

4 t

F

) ( )) ( ( ˆ t F t P F

i i



SLIDE 8

Barycentric Coordinates

Polygon Domain
Boundary Interpolation
Basis Functions

f

1

f

2

f

3

f

4

f





n i i i

f x b x F ) ( ) ( ˆ

SLIDE 9

Barycentric Coordinates

Polygon Domain
Boundary Interpolation
Basis Functions





n i i i

f x b x F ) ( ) ( ˆ

SLIDE 10

Barycentric Coordinates

Polygon Domain
Boundary Interpolation
Basis Functions





n i i i

f x b x F ) ( ) ( ˆ

SLIDE 11

Barycentric Coordinates

Polygon Domain
Boundary Interpolation
Basis Functions





n i i i

f x b x F ) ( ) ( ˆ

SLIDE 12

Barycentric Coordinates

Polygon Domain
Boundary Interpolation
Basis Functions





n i i i

f x b x F ) ( ) ( ˆ

SLIDE 13

Barycentric Coordinates

Polygon Domain
Boundary Interpolation
Basis Functions





n i i i

f x b x F ) ( ) ( ˆ

SLIDE 14

Barycentric Coordinates

Polygon Domain
Boundary Interpolation
Basis Functions
Linear Precision





n i i i

p L x b x L ) ( ) ( ) (

SLIDE 15

Barycentric Coordinates

Polygon Domain
Boundary Interpolation
Basis Functions
Linear Precision





n i i i

p L x b x L ) ( ) ( ) (

SLIDE 16

Barycentric Coordinates

Polygon Domain
Boundary Interpolation
Basis Functions
Linear Precision





n i i x

b ) ( 1

SLIDE 17

Other Properties

Desirable Features

– Smoothness – Closed-form solution – Positivity

Extended Coordinates

– Polynomial Boundary Values – Polynomial Precision – Interpolation of Derivatives – Curved Boundaries

SLIDE 18

Applications

Finite Element Methods

[Wachspress 1975]

SLIDE 19

Boundary Value Problems

[Ju et al. 2005]

Applications

SLIDE 20

Free-Form Deformations

[Sederberg et al. 1986], [MacCracken et al. 1996], [Ju et al. 2005], [Joshi et al. 2007]

Applications

SLIDE 21

Applications

Surface Parameterization

[Hormann et al. 2000], [Desbrun et al. 2002]

SLIDE 22

Wachspress Mean Val.

Pos. Mean Val.

Max Entropy Moving Least Sqr. Hermite MVC Harmonic

Comparison of Methods

SLIDE 23

Moving Least Squares Coordinates

A new family of barycentric coordinates
Solves a least squares problem
Solution depends on point of evaluation

SLIDE 24

Fit a Polynomial to Points  

2 1

) ( ) ( argmin  

n i i i C

p F C p V C x V x F ) ( ) ( ˆ

1

 ) 1 ( ) (

2 1 1

x x x V 

SLIDE 25

Fit a Polynomial to Points

SLIDE 26

Fit a Polynomial to Points

SLIDE 27

Interpolating Points

SLIDE 28

Interpolating Points

 2

1 ) , ( p x p x W  

 

2 1

) ( ) ( ) , ( argmin  

n i i i i C

p F C p V p x W C x V x F ) ( ) ( ˆ

1



SLIDE 29

Interpolating Points

SLIDE 30

Interpolating Line Segments

         

2 , 1 ,

) 1 ( ) (

i i i

P P t t t P

SLIDE 31

Interpolating Line Segments

         

2 , 1 ,

) 1 ( ) (

i i i

P P t t t P          

2 , 1 ,

) 1 ( ) (

i i i

F F t t t F

SLIDE 32

Interpolating Line Segments

         

2 , 1 ,

) 1 ( ) (

i i i

P P t t t P

 2

) ( ) ( ' ) , ( t P x t P t x W

i i i

           

2 , 1 ,

) 1 ( ) (

i i i

F F t t t F

SLIDE 33

Interpolating Line Segments

         

2 , 1 ,

) 1 ( ) (

i i i

P P t t t P

  dt

t F C t P V t x W

n i i i i C





2 1 1

) ( )) ( ( ) , ( argmin

 2

) ( ) ( ' ) , ( t P x t P t x W

i i i

           

2 , 1 ,

) 1 ( ) (

i i i

F F t t t F

SLIDE 34

Line Basis Functions

dt t P V t P V t x W A

n i i i T i i





1 1

)) ( ( )) ( ( ) , ( dt t F t P V t x W A C

n i i i T i i

 





1 1

) ( )) ( ( ) , (

SLIDE 35

Line Basis Functions

C x V x F ) ( ) ( ˆ

1



SLIDE 36

Line Basis Functions

dt t F t P V t x W A x V

n i i i T i i

 





1 1 1

) ( )) ( ( ) , ( ) (

C x V x F ) ( ) ( ˆ

1



SLIDE 37

Line Basis Functions

dt t F t P V t x W A x V

n i i i T i i

 





1 1 1

) ( )) ( ( ) , ( ) (

C x V x F ) ( ) ( ˆ

1



dt F F t t t P V t x W A x V

i i n i i T i i

         

 

 2 , 1 , 1 1 1

) 1 ))( ( ( ) , ( ) (

SLIDE 38

Line Basis Functions

dt t F t P V t x W A x V

n i i i T i i

 





1 1 1

) ( )) ( ( ) , ( ) (

C x V x F ) ( ) ( ˆ

1



dt F F t t t P V t x W A x V

i i n i i T i i

         

 

 2 , 1 , 1 1 1

) 1 ))( ( ( ) , ( ) (          



 2 , 1 , 1 1 1

) 1 ))( ( ( ) , ( ) (

i i n i i T i i

F F dt t t t P V t x W A x V

SLIDE 39

Line Basis Functions

dt t F t P V t x W A x V

n i i i T i i

 





1 1 1

) ( )) ( ( ) , ( ) (

C x V x F ) ( ) ( ˆ

1



dt F F t t t P V t x W A x V

i i n i i T i i

         

 

 2 , 1 , 1 1 1

) 1 ))( ( ( ) , ( ) (          



 2 , 1 , 1 1 1

) 1 ))( ( ( ) , ( ) (

i i n i i T i i

F F dt t t t P V t x W A x V

 

        

2 , 1 , 2 , 1 ,

) ( ) (

i i n i i i

F F x B x B

SLIDE 40

Polygon Basis Functions

 

        

2 , 1 , 2 , 1 ,

) ( ) ( ) ( ˆ

i i n i i i

F F x B x B x F

SLIDE 41

Polygon Basis Functions

 

        

2 , 1 , 2 , 1 ,

) ( ) ( ) ( ˆ

i i n i i i

F F x B x B x F ) ( ) ( ) (

2 , 1 1 ,

x B x B x b

i i i 

 

2 , 1 1 , 

 

i i i

F F f

SLIDE 42

Polygon Basis Functions

SLIDE 43

Polygon Basis Functions

SLIDE 44

Polygon Basis Functions

 

        

2 , 1 , 2 , 1 ,

) ( ) ( ) ( ˆ

i i n i i i

F F x B x B x F ) ( ) ( ) (

2 , 1 1 ,

x B x B x b

i i i 

 





n i i i

f x b x F ) ( ) ( ˆ

2 , 1 1 , 

 

i i i

F F f

SLIDE 45

Polynomial Boundary Values

         

2 , 1 ,

) 1 ( ) (

i i i

F F t t t F             

3 , 2 , 1 , 2 2

) ) 1 ( 2 ) 1 ( ( ) (

i i i i

F F F t t t t t F                  

4 , 3 , 2 , 1 , 3 2 2 3

) ) 1 ( 3 ) 1 ( 3 ) 1 ( ( ) (

i i i i i

F F F F t t t t t t t F



SLIDE 46

Polynomial Boundary Values

SLIDE 47

Polynomial Boundary Values

SLIDE 48

Polynomial Precision

) 1 ( ) (

2 1 1

x x x V  ) 1 ( ) (

2 2 2 1 2 1 2 1 2

x x x x x x x V 

) 1 ( ) (

3 2 2 2 1 1 1 2 2 1 3 1 2 2 2 1 2 1 2 1 3

x x x x x x x x x x x x x V 



SLIDE 49

Polynomial Precision

Linear Quadratic

SLIDE 50

Interpolation of Derivatives

  dt

t F C t P V t x W

n i i i i C





2 1 1

) ( )) ( ( ) , ( argmin

SLIDE 51

Interpolation of Derivatives

  dt

t F C t G t x W

i i i 2 1 , 1

) ( ) ( ) , (





 

  dt

t F C t P V t x W

n i i i i C





2 1 1

) ( )) ( ( ) , ( argmin

SLIDE 52

Interpolation of Derivatives

  dt

t F C t G t x W

i i i 2 1 , 1

) ( ) ( ) , (





 

  dt

t F C t P V t x W

n i i i i C





2 1 1

) ( )) ( ( ) , ( argmin

        

    

)) ( ( )) ( ( ) ( ) (

1 1 , 1

2 1

t P V t P V t P t G

i x i x i i

SLIDE 53

Interpolation of Derivatives

SLIDE 54

Interpolation of Derivatives

SLIDE 55

Interpolation of Derivatives

SLIDE 56

Solutions are Closed-Form

For polygons is linear

– and are constant – Polynomial numerator – Denominator quadratic to power 2α – Integrals have closed-form solutions

) (t P

i 

) ( ' t P

i

) (t P

i

SLIDE 57

Curved Boundaries

         

2 , 1 ,

) 1 ( ) (

i i i

P P t t t P             

3 , 2 , 1 , 2 2

) ) 1 ( 2 ) 1 ( ( ) (

i i i i

P P P t t t t t P                  

4 , 3 , 2 , 1 , 3 2 2 3

) ) 1 ( 3 ) 1 ( 3 ) 1 ( ( ) (

i i i i i

P P P P t t t t t t t P



SLIDE 58

Comparison to Other Methods

SLIDE 59

Comparison to Other Methods

SLIDE 60

3D Deformation





n i i i

p x b x ) (

SLIDE 61

3D Deformation





n i i i

p x b x ) (





n i i i

p x b x ˆ ) ( ˆ

SLIDE 62

Conclusion

New family of barycentric coordinates

– Controlled by parameter α – Polynomial boundaries – Polynomial precision – Derivative interpolation – Open polygons – Closed-form