Moving Least Squares Coordinates
Transcript of Moving Least Squares Coordinates
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