Moving Least Squares Coordinates
Transcript of Moving Least Squares Coordinates
![Page 1: Moving Least Squares Coordinates](https://reader033.fdocuments.in/reader033/viewer/2022051505/586b5a0c1a28abe7148bf991/html5/thumbnails/1.jpg)
Moving Least Squares Coordinates
Josiah Manson and Scott Schaefer
Texas A&M University
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Barycentric Coordinates
• Polygon Domain
0p
1p
2p3p
4p
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Barycentric Coordinates
• Polygon Domain
)(0 tP
)(1 tP
)(2 tP)(3 tP
)(4 tP
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Barycentric Coordinates
• Polygon Domain
0f1f
2f
3f4f
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Barycentric Coordinates
• Polygon Domain
)(0 tF)(1 tF
)(2 tF)(3 tF
)(4 tF
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Barycentric Coordinates
• Polygon Domain
• Boundary Interpolation
)(0 tF)(1 tF
)(2 tF)(3 tF
)(4 tF
)())((ˆ tFtPF ii
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Barycentric Coordinates
• Polygon Domain
• Boundary Interpolation
)(0 tF)(1 tF
)(2 tF)(3 tF
)(4 tF
)())((ˆ tFtPF ii
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Barycentric Coordinates
• Polygon Domain
• Boundary Interpolation
• Basis Functions
0f1f
2f
3f4f
n
i
ii fxbxF )()(ˆ
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Barycentric Coordinates
• Polygon Domain
• Boundary Interpolation
• Basis Functions
n
i
ii fxbxF )()(ˆ
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Barycentric Coordinates
• Polygon Domain
• Boundary Interpolation
• Basis Functions
n
i
ii fxbxF )()(ˆ
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Barycentric Coordinates
• Polygon Domain
• Boundary Interpolation
• Basis Functions
n
i
ii fxbxF )()(ˆ
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Barycentric Coordinates
• Polygon Domain
• Boundary Interpolation
• Basis Functions
n
i
ii fxbxF )()(ˆ
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Barycentric Coordinates
• Polygon Domain
• Boundary Interpolation
• Basis Functions
n
i
ii fxbxF )()(ˆ
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Barycentric Coordinates
• Polygon Domain
• Boundary Interpolation
• Basis Functions
• Linear Precision
n
i
ii pLxbxL )()()(
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Barycentric Coordinates
• Polygon Domain
• Boundary Interpolation
• Basis Functions
• Linear Precision
n
i
ii pLxbxL )()()(
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Barycentric Coordinates
• Polygon Domain
• Boundary Interpolation
• Basis Functions
• Linear Precision
n
i
i xb )(1
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Other Properties
• Desirable Features– Smoothness
– Closed-form solution
– Positivity
• Extended Coordinates– Polynomial Boundary Values
– Polynomial Precision
– Interpolation of Derivatives
– Curved Boundaries
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Applications
• Finite Element Methods [Wachspress 1975]
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• Boundary Value Problems [Ju et al. 2005]
Applications
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• Free-Form Deformations[Sederberg et al. 1986], [MacCracken et al. 1996], [Ju et al. 2005], [Joshi et al. 2007]
Applications
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Applications
• Surface Parameterization
[Hormann et al. 2000], [Desbrun et al. 2002]
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Wachspress
Mean Val.
Pos. Mean Val.
Max Entropy
Moving Least Sqr.
Hermite MVC
Harmonic
Comparison of Methods
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Moving Least Squares Coordinates
• A new family of barycentric coordinates
• Solves a least squares problem
• Solution depends on point of evaluation
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Fit a Polynomial to Points
2
1 )()(argmin n
i
iiC
pFCpV
CxVxF )()(ˆ 1
)1()( 211 xxxV
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Fit a Polynomial to Points
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Fit a Polynomial to Points
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Interpolating Points
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Interpolating Points
2
1),(
pxpxW
2
1 )()(),(argmin n
i
iiiC
pFCpVpxW
CxVxF )()(ˆ 1
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Interpolating Points
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Interpolating Line Segments
2,
1,)1()(
i
i
iP
PtttP
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Interpolating Line Segments
2,
1,)1()(
i
i
iP
PtttP
2,
1,)1()(
i
i
iF
FtttF
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Interpolating Line Segments
2,
1,)1()(
i
i
iP
PtttP
2)(
)('),(
tPx
tPtxW
i
i
i
2,
1,)1()(
i
i
iF
FtttF
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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
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Line Basis Functions
dttPVtPVtxWAn
i
ii
T
ii
1
0
1 ))(())((),(
dttFtPVtxWACn
i
ii
T
ii
1
0
1 )())((),(
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Line Basis Functions
CxVxF )()(ˆ 1
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Line Basis Functions
dttFtPVtxWAxVn
i
ii
T
ii
1
0
1
1 )())((),()(
CxVxF )()(ˆ 1
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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))(((),()(
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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
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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
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Polygon Basis Functions
2,
1,
2,1, )()()(ˆ
i
in
i
iiF
FxBxBxF
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Polygon Basis Functions
2,
1,
2,1, )()()(ˆ
i
in
i
iiF
FxBxBxF
)()()( 2,11, xBxBxb iii
2,11, iii FFf
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Polygon Basis Functions
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Polygon Basis Functions
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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
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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
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Polynomial Boundary Values
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Polynomial Boundary Values
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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
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Polynomial Precision
Linear Quadratic
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Interpolation of Derivatives
dttFCtPVtxWn
i
iiiC
21
0
1 )())((),(argmin
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Interpolation of Derivatives
dttFCtGtxW iii
21
0
,1 )()(),(
dttFCtPVtxWn
i
iiiC
21
0
1 )())((),(argmin
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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
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Interpolation of Derivatives
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Interpolation of Derivatives
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Interpolation of Derivatives
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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
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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
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Comparison to Other Methods
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Comparison to Other Methods
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3D Deformation
n
i
ii pxbx )(
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3D Deformation
n
i
ii pxbx )( n
i
ii pxbx ˆ)(ˆ
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Conclusion
• New family of barycentric coordinates
– Controlled by parameter α
– Polynomial boundaries
– Polynomial precision
– Derivative interpolation
– Open polygons
– Closed-form