powell--sabin splines on the sphere with applications in...
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Powell–Sabin splines on the sphere withapplications in CAGD
Jan Maes
Department of Computer ScienceKatholieke Universiteit Leuven
Paris, November 17, 2006
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Outline
Section I Powell–Sabin splines
Section II Spherical Powell–Sabin splines
Section III Multiresolution Analysis
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Powell–Sabin splines
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Bernstein–Bézier representation
=⇒
Pierre Étienne Bézier (1910-1999)
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Stitching together Bézier triangles
=⇒
No C1 continuity at the red curve
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
C1 continuity with Powell–Sabin splines
Conformal triangulation ∆
PS 6-split ∆PS
S12(∆PS) = space of PS splines
M.J.D. Powell M.A. Sabin
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
C1 continuity with Powell–Sabin splines
Conformal triangulation ∆
PS 6-split ∆PS
S12(∆PS) = space of PS splines
M.J.D. Powell M.A. Sabin
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
C1 continuity with Powell–Sabin splines
Conformal triangulation ∆
PS 6-split ∆PS
S12(∆PS) = space of PS splines
M.J.D. Powell M.A. Sabin
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
The dimension of S12(∆PS)?
There is exactly one solution s ∈ S12(∆PS) to theHermite interpolation problem
s(Vi) = αi ,
Dxs(Vi) = βi , ∀Vi ∈ ∆, i = 1, . . . ,N.Dys(Vi) = γi ,
The dimension of S12(∆PS) is 3N. Therefore we need 3N basis
functions.
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
The dimension of S12(∆PS)?
There is exactly one solution s ∈ S12(∆PS) to theHermite interpolation problem
s(Vi) = αi ,
Dxs(Vi) = βi , ∀Vi ∈ ∆, i = 1, . . . ,N.Dys(Vi) = γi ,
The dimension of S12(∆PS) is 3N. Therefore we need 3N basis
functions.
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Powell–Sabin B-splines with control triangles
s(x , y) =N∑
i=1
3∑j=1
cijBij(x , y)
Bij is the unique solution to
[Bij(Vk ),DxBij(Vk ),DyBij(Vk )] = [0,0,0] for all k 6= i[Bij(Vi),DxBij(Vi),DyBij(Vi)] = [αij , βij , γij ] for j = 1,2,3
Partition of unity:∑Ni=1
∑3j=1 Bij(x , y) = 1,
Bij(x , y) ≥ 0
(Paul Dierckx, 1997)
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Powell–Sabin B-splines with control triangles
s(x , y) =N∑
i=1
3∑j=1
cijBij(x , y)
Bij is the unique solution to
[Bij(Vk ),DxBij(Vk ),DyBij(Vk )] = [0,0,0] for all k 6= i[Bij(Vi),DxBij(Vi),DyBij(Vi)] = [αij , βij , γij ] for j = 1,2,3
Partition of unity:∑Ni=1
∑3j=1 Bij(x , y) = 1,
Bij(x , y) ≥ 0
(Paul Dierckx, 1997)
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Powell–Sabin B-splines with control triangles
s(x , y) =N∑
i=1
3∑j=1
cijBij(x , y)
Bij is the unique solution to
[Bij(Vk ),DxBij(Vk ),DyBij(Vk )] = [0,0,0] for all k 6= i[Bij(Vi),DxBij(Vi),DyBij(Vi)] = [αij , βij , γij ] for j = 1,2,3
Partition of unity:∑Ni=1
∑3j=1 Bij(x , y) = 1,
Bij(x , y) ≥ 0
(Paul Dierckx, 1997)
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Powell–Sabin B-splines with control triangles
Three locally supported basis functions per vertex
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Powell–Sabin B-splines with control triangles
The control triangle is tangent to the PS spline surface.
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Powell–Sabin B-splines with control triangles
It ‘controls’ the local shape of the spline surface.
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Spherical Powell–Sabin splines
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Spherical spline spaces
P. Alfeld, M. Neamtu, and L. L. Schumaker (1996)
Homogeneous of degree d : f (αv) = αd f (v)Hd := space of trivariate polynomials of degree d that arehomogeneous of degree dRestriction of Hd to a plane in R3 \ {0}⇒ we recover the space of bivariate polynomials∆ := conforming spherical triangulation of the unit sphere S
Srd(∆) := {s ∈ Cr (S) | s|τ ∈ Hd(τ), τ ∈ ∆}
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Spherical Powell–Sabin splines
s(vi) = fi , Dgi s(vi) = fgi , Dhi s(vi) = fhi , ∀vi ∈ ∆
has a unique solution in S12(∆PS)
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
1− 1 connection with bivariate PS splines
⇒ |v |2Bij(v|v |
)⇒
←−
Spherical PS B-spline Bij(v)
piecewise trivari-ate polynomial ofdegree 2 that ishomogeneous ofdegree 2
Restriction to theplane tangent toS at vi ∈ ∆
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
1− 1 connection with bivariate PS splines
Let Ti be the plane tangent to S at vertex viRadial projection:
Riv := v :=v|v |∈ S, v ∈ Ti
Define ∆i as the star of vi in ∆, and let ∆PSi ⊂ ∆PS be its
PS 6-split.
Theorem
Let s ∈ S12(∆PSi ). Let s be the restriction of |v |2s(v/|v |) to Ti .
Then s is in S12(R−1i ∆
PSi ) and
s(vi) = s(vi), Dgi s(vi) = Dgi s(vi), Dhi s(vi) = Dhi s(vi).
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
1− 1 connection with bivariate PS splines
Let Ti be the plane tangent to S at vertex viRadial projection:
Riv := v :=v|v |∈ S, v ∈ Ti
Define ∆i as the star of vi in ∆, and let ∆PSi ⊂ ∆PS be its
PS 6-split.
Theorem
Let s ∈ S12(∆PSi ). Let s be the restriction of |v |2s(v/|v |) to Ti .
Then s is in S12(R−1i ∆
PSi ) and
s(vi) = s(vi), Dgi s(vi) = Dgi s(vi), Dhi s(vi) = Dhi s(vi).
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Spherical B-splines with control triangles
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Applications on a spherical domain
Approximation of a mesh: consider the triangles of the originaltriangular mesh as control triangles of a PS spline.
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Applications on a spherical domain
Compression by smoothing: Decimate a given triangular meshand approximate the decimated mesh.
triangular mesh reduced mesh
(40000 triangles) (5000 triangles)
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Applications on a spherical domain
Compression by smoothing: Decimate a given triangular meshand approximate the decimated mesh.
triangular mesh Powell–Sabin spline
(40000 triangles) (5000 control triangles)
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Applications on a spherical domain
triangular mesh decimated mesh spherical(40000 triangles) (5000 triangles) parameterization
(5000 control triangles) PS spline surface
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Multiresolution analysis
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Multiresolution analysis (1989)
Stéphane Mallat Yves Meyer
A nested sequence of subspaces
S0 ⊂ S1 ⊂ S2 ⊂ · · · ⊂ S` ⊂ · · ·
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Multiresolution analysis (1989)
Stéphane Mallat Yves Meyer
Complement spaces W`
S`+1 = S` ⊕W`
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Multiresolution analysis (1989)
Stéphane Mallat Yves Meyer
A stable basis for the complement space W`
W` = span{ψ`,i}
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Multiresolution analysis
Refine the triangulation ∆ and its PS 6-split ∆PS.
Vi
Rki
Vk
Rjk
Vj
Rij
Zijk
Vi
Rki
Vk
Rjk
Vj
Rij
Zijk
dyadic refinement triadic refinement
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Multiresolution analysis
Refine the triangulation ∆ and its PS 6-split ∆PS.
Vi
Vki
Vk
Vjk
Vj
Vij
Zijk
Vi
Rki
Vk
Rjk
Vj
Rij
Vik
Vki
Vkj
Vjk
Vji
Vij
Vijk
dyadic refinement triadic refinement
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Multiresolution analysis
Refine the triangulation ∆ and its PS 6-split ∆PS.
Vi
Vki
Vk
Vjk
Vj
Vij
Zijk
Vi
Rki
Vk
Rjk
Vj
Rij
Vik
Vki
Vkj
Vjk
Vji
Vij
Vijk
dyadic refinement triadic refinement
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Multiresolution analysis with√
3-refinement
∆PS0 ⊂ ∆PS1 ⊂ · · · ⊂ ∆
PS` ⊂ · · ·
S12(∆PS0 ) ⊂ S
12(∆
PS1 ) ⊂ · · · ⊂ S
12(∆
PS` ) ⊂ · · ·
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Multiresolution analysis with√
3-refinement
∆PS0 ⊂ ∆PS1 ⊂ · · · ⊂ ∆
PS` ⊂ · · ·
S12(∆PS0 ) ⊂ S
12(∆
PS1 ) ⊂ · · · ⊂ S
12(∆
PS` ) ⊂ · · ·
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Multiresolution analysis with√
3-refinement
∆PS0 ⊂ ∆PS1 ⊂ · · · ⊂ ∆
PS` ⊂ · · ·
S12(∆PS0 ) ⊂ S
12(∆
PS1 ) ⊂ · · · ⊂ S
12(∆
PS` ) ⊂ · · ·
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Multiresolution analysis with√
3-refinement
∆PS0 ⊂ ∆PS1 ⊂ · · · ⊂ ∆
PS` ⊂ · · ·
S12(∆PS0 ) ⊂ S
12(∆
PS1 ) ⊂ · · · ⊂ S
12(∆
PS` ) ⊂ · · ·
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Multiresolution analysis with√
3-refinement
S`+1 = S` ⊕W`
Large triangles control S0Small triangles control W0Local edit
Resolution level 0
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Multiresolution analysis with√
3-refinement
S`+1 = S` ⊕W`
Large triangles control S0Small triangles control W0Local edit
Resolution level 1
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Multiresolution analysis with√
3-refinement
S`+1 = S` ⊕W`
Large triangles control S0Small triangles control W0Local edit
Resolution level 1
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
The hierarchical basis
{Vi ∈ ∆`} ⊂ {Vi ∈ ∆`+1}
∆PS` ⊂ ∆PS`+1
S` := S12(∆PS` ), S` ⊂ S`+1
S2 = S0 ⊕W0 ⊕W1
Largest triangles control S0Medium triangles control W0Smallest triangles control W1
PSspline.mpgMedia File (video/mpeg)
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
The hierarchical basis
Basis functions: S` = span{φ`,k : k = 1, . . . ,3N`}
s`(x , y) = φ`c` =N∑̀i=1
3∑j=1
Bij`(x , y)cij`
φ`+1 = [φo`+1 φ
n`+1],
φo`+1 correspond to old reused vertices of level `φn`+1 correspond to the new vertices of level `+ 1
The set of splines
φ0 ∪m⋃
`=1
φn`
forms a hierarchical basis for Sm.
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Wavelets via the lifting scheme
φ` = φ`+1P`φ`+1 =
[φo`+1 φ
n`+1
][φ` ψ`
]= φ`+1
[P` Q`
] (Wim Sweldens, 1994)Lifting
ψ` = φn`+1 − φ`U`
with U` the update matrix. We find a relation of the form[φ` ψ`
]= φ`+1
[P`
[0`I`
]− P`U`
]
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
The update step
Problems
Semi-orthogonality⇒ U` not sparseFix U` sparse⇒ ψ` local supportWant stability⇒ need 1 vanishing moment for ψ`Remaining orthogonality conditions approximated by leastsquares
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
The update step
Problems
Semi-orthogonality⇒ U` not sparseFix U` sparse⇒ ψ` local supportWant stability⇒ need 1 vanishing moment for ψ`Remaining orthogonality conditions approximated by leastsquares
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
The update step
Problems
U` not sparse⇒ ψ` no local supportFix U` sparse⇒ ψ` local supportWant stability⇒ need 1 vanishing moment for ψ`Remaining orthogonality conditions approximated by leastsquares
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
The update step
Problems
Want local support⇒ U` sparseFix U` sparse⇒ ψ` local supportWant stability⇒ need 1 vanishing moment for ψ`Remaining orthogonality conditions approximated by leastsquares
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
The update step
Problems
Want local support⇒ U` sparseFix U` sparse⇒ ψ` local supportWant stability⇒ need 1 vanishing moment for ψ`Remaining orthogonality conditions approximated by leastsquares
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
The update step
Problems
Want local support⇒ U` sparseOrthogonalize w.r.t. scaling functions in the update stencil
Want stability⇒ need 1 vanishing moment for ψ`Remaining orthogonality conditions approximated by leastsquares
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
The update step
Problems
Want local support⇒ U` sparseOrthogonalize w.r.t. scaling functions in the update stencil
Want stability⇒ need 1 vanishing moment for ψ`Remaining orthogonality conditions approximated by leastsquares
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
The update step
Problems
Want local support⇒ U` sparseOrthogonalize w.r.t. scaling functions in the update stencil
i.e. φ̃` has to reproduce constants
Remaining orthogonality conditions approximated by leastsquares
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
The update step
Problems
Want local support⇒ U` sparseOrthogonalize w.r.t. scaling functions in the update stencil
An extra linear constraint
Remaining orthogonality conditions approximated by leastsquares
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
The update step
Problems
Want local support⇒ U` sparseOrthogonalize w.r.t. scaling functions in the update stencil
An extra linear constraint
Remaining orthogonality conditions approximated by leastsquares
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Spherical Powell–Sabin spline wavelets
3 wavelets per vertex
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Applications
−→
Spherical scattereddata
Spherical PS spline surfacewith multiresolution structure
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Applications
Compression
Original 26%
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Applications
Denoising
With noise Denoised
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Applications
Multiresolution editing
Coarse level edit Fine level edit
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Applications
(a) Coarse part of (c) (b) Coarse part of (d)
(c) Original surface (d) Coarse level edit of (c)
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Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Some references
P. Alfeld, M. Neamtu, and L. L. Schumaker. Bernstein–Bézierpolynomials on spheres and sphere-like surfaces. Comput. AidedGeom. Design, 13:333–349, 1996.
P. Dierckx. On calculating normalized Powell–Sabin B-splines. Comput.Aided Geom. Design, 15(1), 61–78, 1997.
J. Maes and A. Bultheel. Modeling sphere-like manifolds with sphericalPowell–Sabin B-splines. Comput. Aided Geom. Design, to appear.
M. Neamtu and L. L. Schumaker. On the approximation order of splineson spherical triangulations. Adv. in Comp. Math., 21:3–20, 2004.
W. Sweldens. The lifting scheme: A construction of second generationwavelets. SIAM J. Math. Anal., 29(2):511–546, 1997.
Powell--Sabin splinesBernstein--BézierThe space of Powell--Sabin splinesB-splines with control triangles
Spherical Powell--Sabin splinesSpherical spline spacesThe space of spherical Powell--Sabin splines
Multiresolution analysisMultiresolution analysisWavelets via the lifting schemeThe update stepThe waveletsApplicationsReferences
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