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