Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.
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Transcript of Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.
![Page 1: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/1.jpg)
Moving Least Squares Coordinates
Josiah Manson and Scott SchaeferTexas A&M University
![Page 2: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/2.jpg)
Barycentric Coordinates
• Polygon Domain
0p
1p
2p3p
4p
![Page 3: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/3.jpg)
Barycentric Coordinates
• Polygon Domain
)(0 tP)(1 tP
)(2 tP)(3 tP
)(4 tP
![Page 4: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/4.jpg)
Barycentric Coordinates
• Polygon Domain
0f1f
2f
3f4f
![Page 5: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/5.jpg)
Barycentric Coordinates
• Polygon Domain
)(0 tF)(1 tF
)(2 tF)(3 tF
)(4 tF
![Page 6: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/6.jpg)
Barycentric Coordinates
• Polygon Domain• Boundary Interpolation
)(0 tF)(1 tF
)(2 tF)(3 tF
)(4 tF
)())((ˆ tFtPF ii
![Page 7: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/7.jpg)
Barycentric Coordinates
• Polygon Domain• Boundary Interpolation
)(0 tF)(1 tF
)(2 tF)(3 tF
)(4 tF
)())((ˆ tFtPF ii
![Page 8: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/8.jpg)
Barycentric Coordinates
• Polygon Domain• Boundary Interpolation• Basis Functions
0f1f
2f
3f4f
n
iii fxbxF )()(ˆ
![Page 9: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/9.jpg)
Barycentric Coordinates
• Polygon Domain• Boundary Interpolation• Basis Functions
n
iii fxbxF )()(ˆ
![Page 10: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/10.jpg)
Barycentric Coordinates
• Polygon Domain• Boundary Interpolation• Basis Functions
n
iii fxbxF )()(ˆ
![Page 11: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/11.jpg)
Barycentric Coordinates
• Polygon Domain• Boundary Interpolation• Basis Functions
n
iii fxbxF )()(ˆ
![Page 12: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/12.jpg)
Barycentric Coordinates
• Polygon Domain• Boundary Interpolation• Basis Functions
n
iii fxbxF )()(ˆ
![Page 13: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/13.jpg)
Barycentric Coordinates
• Polygon Domain• Boundary Interpolation• Basis Functions
n
iii fxbxF )()(ˆ
![Page 14: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/14.jpg)
Barycentric Coordinates
• Polygon Domain• Boundary Interpolation• Basis Functions• Linear Precision
n
iii pLxbxL )()()(
![Page 15: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/15.jpg)
Barycentric Coordinates
• Polygon Domain• Boundary Interpolation• Basis Functions• Linear Precision
n
iii pLxbxL )()()(
![Page 16: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/16.jpg)
Barycentric Coordinates
• Polygon Domain• Boundary Interpolation• Basis Functions• Linear Precision
n
ii xb )(1
![Page 17: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/17.jpg)
Other Properties
• Desirable Features– Smoothness– Closed-form solution– Positivity
• Extended Coordinates– Polynomial Boundary Values– Polynomial Precision– Interpolation of Derivatives– Curved Boundaries
![Page 18: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/18.jpg)
Applications
• Finite Element Methods [Wachspress 1975]
![Page 19: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/19.jpg)
• Boundary Value Problems [Ju et al. 2005]
Applications
![Page 20: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/20.jpg)
• Free-Form Deformations [Sederberg et al. 1986], [MacCracken et al. 1996], [Ju et al. 2005], [Joshi et al. 2007]
Applications
![Page 21: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/21.jpg)
Applications
• Surface Parameterization[Hormann et al. 2000], [Desbrun et al. 2002]
![Page 22: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/22.jpg)
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
![Page 23: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/23.jpg)
Moving Least Squares Coordinates
• A new family of barycentric coordinates• Solves a least squares problem• Solution depends on point of evaluation
![Page 24: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/24.jpg)
Fit a Polynomial to Points
2
1 )()(argmin n
iii
CpFCpV
CxVxF )()(ˆ 1
)1()( 211 xxxV
![Page 25: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/25.jpg)
Fit a Polynomial to Points
![Page 26: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/26.jpg)
Fit a Polynomial to Points
![Page 27: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/27.jpg)
Interpolating Points
![Page 28: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/28.jpg)
Interpolating Points
21
),(px
pxW
2
1 )()(),(argmin n
iiii
CpFCpVpxW
CxVxF )()(ˆ 1
![Page 29: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/29.jpg)
Interpolating Points
![Page 30: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/30.jpg)
Interpolating Line Segments
2,
1,)1()(i
ii P
PtttP
![Page 31: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/31.jpg)
Interpolating Line Segments
2,
1,)1()(i
ii P
PtttP
2,
1,)1()(i
ii F
FtttF
![Page 32: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/32.jpg)
Interpolating Line Segments
2,
1,)1()(i
ii P
PtttP
2)(
)('),(
tPx
tPtxW
i
ii
2,
1,)1()(i
ii F
FtttF
![Page 33: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/33.jpg)
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
![Page 34: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/34.jpg)
Line Basis Functions
dttPVtPVtxWAn
iii
Tii
1
0
1 ))(())((),(
dttFtPVtxWACn
iii
Tii
1
0
1 )())((),(
![Page 35: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/35.jpg)
Line Basis Functions
CxVxF )()(ˆ 1
![Page 36: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/36.jpg)
Line Basis Functions
dttFtPVtxWAxVn
iii
Tii
1
0
11 )())((),()(
CxVxF )()(ˆ 1
![Page 37: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/37.jpg)
Line Basis Functions
dttFtPVtxWAxVn
iii
Tii
1
0
11 )())((),()(
CxVxF )()(ˆ 1
dtF
FtttPVtxWAxV
i
in
ii
Tii
2,
1,1
0
11 )1))(((),()(
![Page 38: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/38.jpg)
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
![Page 39: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/39.jpg)
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
![Page 40: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/40.jpg)
Polygon Basis Functions
2,
1,2,1, )()()(ˆ
i
in
iii F
FxBxBxF
![Page 41: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/41.jpg)
Polygon Basis Functions
2,
1,2,1, )()()(ˆ
i
in
iii F
FxBxBxF
)()()( 2,11, xBxBxb iii
2,11, iii FFf
![Page 42: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/42.jpg)
Polygon Basis Functions
![Page 43: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/43.jpg)
Polygon Basis Functions
![Page 44: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/44.jpg)
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
![Page 45: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/45.jpg)
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
![Page 46: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/46.jpg)
Polynomial Boundary Values
![Page 47: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/47.jpg)
Polynomial Boundary Values
![Page 48: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/48.jpg)
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
![Page 49: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/49.jpg)
Polynomial Precision
Linear Quadratic
![Page 50: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/50.jpg)
Interpolation of Derivatives
dttFCtPVtxWn
iiii
C
21
0
1 )())((),(argmin
![Page 51: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/51.jpg)
Interpolation of Derivatives
dttFCtGtxW iii
21
0
,1 )()(),(
dttFCtPVtxWn
iiii
C
21
0
1 )())((),(argmin
![Page 52: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/52.jpg)
Interpolation of Derivatives
dttFCtGtxW iii
21
0
,1 )()(),(
dttFCtPVtxWn
iiii
C
21
0
1 )())((),(argmin
))((
))(()()(
1
1
,1
2
1
tPV
tPVtPtG
ix
ixii
![Page 53: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/53.jpg)
Interpolation of Derivatives
![Page 54: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/54.jpg)
Interpolation of Derivatives
![Page 55: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/55.jpg)
Interpolation of Derivatives
![Page 56: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/56.jpg)
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
![Page 57: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/57.jpg)
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
![Page 58: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/58.jpg)
Comparison to Other Methods
![Page 59: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/59.jpg)
Comparison to Other Methods
![Page 60: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/60.jpg)
3D Deformation
n
iii pxbx )(
![Page 61: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/61.jpg)
3D Deformation
n
iii pxbx )(
n
iii pxbx ˆ)(ˆ
![Page 62: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/62.jpg)
Conclusion
• New family of barycentric coordinates– Controlled by parameter α– Polynomial boundaries– Polynomial precision– Derivative interpolation– Open polygons– Closed-form
![Page 63: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/63.jpg)
![Page 64: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/64.jpg)
Extension to 3D
dsdttsFCtsPVtsxWn
i
t
iiiC
21
0 0
1 ),()),((),,(argmin
![Page 65: Moving Least Squares Coordinates Josiah Manson and Scott Schaefer Texas A&M University.](https://reader031.fdocuments.us/reader031/viewer/2022012919/56649e695503460f94b66263/html5/thumbnails/65.jpg)
Constant Precision
1)( xL
n
iii pLxbxL )()()(
n
ii xb )(1