Moving Least Squares Coordinates

Josiah Manson and Scott SchaeferTexas 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

iii fxbxF )()(ˆ

Barycentric Coordinates

• Polygon Domain• Boundary Interpolation• Basis Functions

n

iii fxbxF )()(ˆ

Barycentric Coordinates

• Polygon Domain• Boundary Interpolation• Basis Functions

n

iii fxbxF )()(ˆ

Barycentric Coordinates

• Polygon Domain• Boundary Interpolation• Basis Functions

n

iii fxbxF )()(ˆ

Barycentric Coordinates

• Polygon Domain• Boundary Interpolation• Basis Functions

n

iii fxbxF )()(ˆ

Barycentric Coordinates

• Polygon Domain• Boundary Interpolation• Basis Functions

n

iii fxbxF )()(ˆ

Barycentric Coordinates

• Polygon Domain• Boundary Interpolation• Basis Functions• Linear Precision

n

iii pLxbxL )()()(

Barycentric Coordinates

• Polygon Domain• Boundary Interpolation• Basis Functions• Linear Precision

n

iii pLxbxL )()()(

Barycentric Coordinates

• Polygon Domain• Boundary Interpolation• Basis Functions• Linear Precision

n

ii 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]

Smooth

Closed-FormPositive

Poly. Precision

Poly. Boundary

Derivatives

Wachspress

Concave Shapes

Mean Val.Pos. Mean Val.

Max Entropy

Moving Least Sqr.Hermite MVC

Harmonic

Open Boundaries

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

iii

CpFCpV

CxVxF )()(ˆ 1

)1()( 211 xxxV

Fit a Polynomial to Points

Fit a Polynomial to Points

Interpolating Points

Interpolating Points

21

),(px

pxW

2

1 )()(),(argmin n

iiii

CpFCpVpxW

CxVxF )()(ˆ 1

Interpolating Points

Interpolating Line Segments

2,

1,)1()(i

ii P

PtttP

Interpolating Line Segments

2,

1,)1()(i

ii P

PtttP

2,

1,)1()(i

ii F

FtttF

Interpolating Line Segments

2,

1,)1()(i

ii P

PtttP

2)(

)('),(

tPx

tPtxW

i

ii

2,

1,)1()(i

ii F

FtttF

Interpolating Line Segments

2,

1,)1()(i

ii P

PtttP

dttFCtPVtxWn

iiii

C

21

0

1 )())((),(argmin

2)(

)('),(

tPx

tPtxW

i

ii

2,

1,)1()(i

ii F

FtttF

Line Basis Functions

dttPVtPVtxWAn

iii

Tii

1

0

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

dttFtPVtxWACn

iii

Tii

1

0

1 )())((),(

Line Basis Functions

CxVxF )()(ˆ 1

Line Basis Functions

dttFtPVtxWAxVn

iii

Tii

1

0

11 )())((),()(

CxVxF )()(ˆ 1

Line Basis Functions

dttFtPVtxWAxVn

iii

Tii

1

0

11 )())((),()(

CxVxF )()(ˆ 1

dtF

FtttPVtxWAxV

i

in

ii

Tii

2,

1,1

0

11 )1))(((),()(

Line Basis Functions

dttFtPVtxWAxVn

iii

Tii

1

0

11 )())((),()(

CxVxF )()(ˆ 1

dtF

FtttPVtxWAxV

i

in

ii

Tii

2,

1,1

0

11 )1))(((),()(

2,

1,1

0

11 )1))(((),()(

i

in

ii

Tii F

FdttttPVtxWAxV

Line Basis Functions

dttFtPVtxWAxVn

iii

Tii

1

0

11 )())((),()(

CxVxF )()(ˆ 1

dtF

FtttPVtxWAxV

i

in

ii

Tii

2,

1,1

0

11 )1))(((),()(

2,

1,1

0

11 )1))(((),()(

i

in

ii

Tii F

FdttttPVtxWAxV

2,

1,2,1, )()(

i

in

iii F

FxBxB

Polygon Basis Functions

2,

1,2,1, )()()(ˆ

i

in

iii F

FxBxBxF

Polygon Basis Functions

2,

1,2,1, )()()(ˆ

i

in

iii F

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

iii F

FxBxBxF

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

n

iii fxbxF )()(ˆ

2,11, iii FFf

Polynomial Boundary Values

2,

1,)1()(i

ii F

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

22121212 xxxxxxxV

) 1( ) (32

22

11

12

21

31

22 2 1

21 2 1 3x x x x x x x x x x x x x V

Polynomial Precision

Linear Quadratic

Interpolation of Derivatives

dttFCtPVtxWn

iiii

C

21

0

1 )())((),(argmin

Interpolation of Derivatives

dttFCtGtxW iii

21

0

,1 )()(),(

dttFCtPVtxWn

iiii

C

21

0

1 )())((),(argmin

Interpolation of Derivatives

dttFCtGtxW iii

21

0

,1 )()(),(

dttFCtPVtxWn

iiii

C

21

0

1 )())((),(argmin

))((

))(()()(

1

1

,1

2

1

tPV

tPVtPtG

ix

ixii

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

ii P

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

iii pxbx )(

3D Deformation

n

iii pxbx )(

n

iii pxbx ˆ)(ˆ

Conclusion

• New family of barycentric coordinates– Controlled by parameter α– Polynomial boundaries– Polynomial precision– Derivative interpolation– Open polygons– Closed-form

Extension to 3D

dsdttsFCtsPVtsxWn

i

t

iiiC

21

0 0

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

Constant Precision

1)( xL

n

iii pLxbxL )()()(

n

ii xb )(1