Moving Least Squares Coordinates

Josiah Manson and Scott Schaefer

Texas A&M University

Barycentric Coordinates

• Polygon Domain

0p

1p

2p3p

4p

Barycentric Coordinates

• Polygon Domain

)(0 tP

)(1 tP

)(2 tP)(3 tP

)(4 tP

Barycentric Coordinates

• Polygon Domain

0f1f

2f

3f4f

Barycentric Coordinates

• Polygon Domain

)(0 tF)(1 tF

)(2 tF)(3 tF

)(4 tF

Barycentric Coordinates

• Polygon Domain

• Boundary Interpolation

)(0 tF)(1 tF

)(2 tF)(3 tF

)(4 tF

)())((ˆ tFtPF ii

Barycentric Coordinates

• Polygon Domain

• Boundary Interpolation

)(0 tF)(1 tF

)(2 tF)(3 tF

)(4 tF

)())((ˆ tFtPF ii

Barycentric Coordinates

• Polygon Domain

• Boundary Interpolation

• Basis Functions

0f1f

2f

3f4f

n

i

ii fxbxF )()(ˆ

Barycentric Coordinates

• Polygon Domain

• Boundary Interpolation

• Basis Functions

n

i

ii fxbxF )()(ˆ

Barycentric Coordinates

• Polygon Domain

• Boundary Interpolation

• Basis Functions

n

i

ii fxbxF )()(ˆ

Barycentric Coordinates

• Polygon Domain

• Boundary Interpolation

• Basis Functions

n

i

ii fxbxF )()(ˆ

Barycentric Coordinates

• Polygon Domain

• Boundary Interpolation

• Basis Functions

n

i

ii fxbxF )()(ˆ

Barycentric Coordinates

• Polygon Domain

• Boundary Interpolation

• Basis Functions

n

i

ii fxbxF )()(ˆ

Barycentric Coordinates

• Polygon Domain

• Boundary Interpolation

• Basis Functions

• Linear Precision

n

i

ii pLxbxL )()()(

Barycentric Coordinates

• Polygon Domain

• Boundary Interpolation

• Basis Functions

• Linear Precision

n

i

ii pLxbxL )()()(

Barycentric Coordinates

• Polygon Domain

• Boundary Interpolation

• Basis Functions

• Linear Precision

n

i

i xb )(1

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]

• 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

Applications

• Surface Parameterization

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

Wachspress

Mean Val.

Pos. Mean Val.

Max Entropy

Moving Least Sqr.

Hermite MVC

Harmonic

Comparison of Methods

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

2

1 )()(argmin n

i

iiC

pFCpV

CxVxF )()(ˆ 1

)1()( 211 xxxV

Fit a Polynomial to Points

Fit a Polynomial to Points

Interpolating Points

Interpolating Points

2

1),(

pxpxW

2

1 )()(),(argmin n

i

iiiC

pFCpVpxW

CxVxF )()(ˆ 1

Interpolating Points

Interpolating Line Segments

2,

1,)1()(

i

i

iP

PtttP

Interpolating Line Segments

2,

1,)1()(

i

i

iP

PtttP

2,

1,)1()(

i

i

iF

FtttF

Interpolating Line Segments

2,

1,)1()(

i

i

iP

PtttP

2)(

)('),(

tPx

tPtxW

i

i

i

2,

1,)1()(

i

i

iF

FtttF

Interpolating Line Segments

2,

1,)1()(

i

i

iP

PtttP

dttFCtPVtxWn

i

iiiC

21

0

1 )())((),(argmin

2)(

)('),(

tPx

tPtxW

i

i

i

2,

1,)1()(

i

i

iF

FtttF

Line Basis Functions

dttPVtPVtxWAn

i

ii

T

ii

1

0

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

dttFtPVtxWACn

i

ii

T

ii

1

0

1 )())((),(

Line Basis Functions

CxVxF )()(ˆ 1

Line Basis Functions

dttFtPVtxWAxVn

i

ii

T

ii

1

0

1

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

CxVxF )()(ˆ 1

Line Basis Functions

dttFtPVtxWAxVn

i

ii

T

ii

1

0

1

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

CxVxF )()(ˆ 1

dtF

FtttPVtxWAxV

i

in

i

i

T

ii

2,

1,1

0

1

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

Line Basis Functions

dttFtPVtxWAxVn

i

ii

T

ii

1

0

1

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

CxVxF )()(ˆ 1

dtF

FtttPVtxWAxV

i

in

i

i

T

ii

2,

1,1

0

1

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

2,

1,1

0

1

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

in

i

i

T

iiF

FdttttPVtxWAxV

Line Basis Functions

dttFtPVtxWAxVn

i

ii

T

ii

1

0

1

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

CxVxF )()(ˆ 1

dtF

FtttPVtxWAxV

i

in

i

i

T

ii

2,

1,1

0

1

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

2,

1,1

0

1

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

in

i

i

T

iiF

FdttttPVtxWAxV

2,

1,

2,1, )()(i

in

i

iiF

FxBxB

Polygon Basis Functions

2,

1,

2,1, )()()(ˆ

i

in

i

iiF

FxBxBxF

Polygon Basis Functions

2,

1,

2,1, )()()(ˆ

i

in

i

iiF

FxBxBxF

)()()( 2,11, xBxBxb iii

2,11, iii FFf

Polygon Basis Functions

Polygon Basis Functions

Polygon Basis Functions

2,

1,

2,1, )()()(ˆ

i

in

i

iiF

FxBxBxF

)()()( 2,11, xBxBxb iii

n

i

ii fxbxF )()(ˆ

2,11, iii FFf

Polynomial Boundary Values

2,

1,)1()(

i

i

iF

FtttF

3,

2,

1,

22 ))1(2)1(()(

i

i

i

i

F

F

F

tttttF

4,

3,

2,

1,

3223 ))1(3)1(3)1(()(

i

i

i

i

i

F

F

F

F

tttttttF

Polynomial Boundary Values

Polynomial Boundary Values

Polynomial Precision

)1()( 211 xxxV

)1()(2

221

2

1212 xxxxxxxV

)1()( 3

2

2

2

1

1

1

2

2

1

3

1

2

221

2

1213 xxxxxxxxxxxxxV

Polynomial Precision

Linear Quadratic

Interpolation of Derivatives

dttFCtPVtxWn

i

iiiC

21

0

1 )())((),(argmin

Interpolation of Derivatives

dttFCtGtxW iii

21

0

,1 )()(),(

dttFCtPVtxWn

i

iiiC

21

0

1 )())((),(argmin

Interpolation of Derivatives

dttFCtGtxW iii

21

0

,1 )()(),(

dttFCtPVtxWn

i

iiiC

21

0

1 )())((),(argmin

))((

))(()()(

1

1

,1

2

1

tPV

tPVtPtG

ix

ix

ii

Interpolation of Derivatives

Interpolation of Derivatives

Interpolation of Derivatives

Solutions are Closed-Form

• For polygons is linear

– and are constant

– Polynomial numerator

– Denominator quadratic to power 2α

– Integrals have closed-form solutions

)(tPi

)(' tPi

)(tPi

Curved Boundaries

2,

1,)1()(

i

i

iP

PtttP

3,

2,

1,

22 ))1(2)1(()(

i

i

i

i

P

P

P

tttttP

4,

3,

2,

1,

3223 ))1(3)1(3)1(()(

i

i

i

i

i

P

P

P

P

tttttttP

Comparison to Other Methods

Comparison to Other Methods

3D Deformation

n

i

ii pxbx )(

3D Deformation

n

i

ii pxbx )( n

i

ii pxbx ˆ)(ˆ

Conclusion

• New family of barycentric coordinates

– Controlled by parameter α

– Polynomial boundaries

– Polynomial precision

– Derivative interpolation

– Open polygons

– Closed-form