jan maes adhemar bultheel · (paul dierckx, 1997) powell–sabin splines spherical powell–sabin...
TRANSCRIPT
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
Smooth spline wavelets on the sphere
Jan Maes Adhemar Bultheel
Department of Computer ScienceKatholieke Universiteit Leuven
01 July 2006
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
Outline
Section I Powell–Sabin splines
Section II Spherical Powell–Sabin splines
Section III Spline wavelets from the lifting scheme
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
Powell–Sabin splines
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
Bernstein–Bézier representation
=⇒
Pierre Étienne Bézier (1910-1999)
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
Stitching together Bézier triangles
=⇒
No C1 continuity at the red curve
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
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 Spline wavelets from the lifting scheme
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 Spline wavelets from the lifting scheme
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 Spline wavelets from the lifting scheme
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 Spline wavelets from the lifting scheme
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 Spline wavelets from the lifting scheme
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 Spline wavelets from the lifting scheme
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 Spline wavelets from the lifting scheme
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 Spline wavelets from the lifting scheme
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 Spline wavelets from the lifting scheme
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 Spline wavelets from the lifting scheme
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 Spline wavelets from the lifting scheme
Spherical Powell–Sabin splines
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
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 Spline wavelets from the lifting scheme
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 Spline wavelets from the lifting scheme
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 Spline wavelets from the lifting scheme
Spherical B-splines with control triangles
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
Multiresolution analysis with√
3-refinement
∆PS0 ⊂ ∆PS1 ⊂ · · · ⊂ ∆
PSj ⊂ · · ·
S12(∆PS0 ) ⊂ S
12(∆
PS1 ) ⊂ · · · ⊂ S
12(∆
PSj ) ⊂ · · ·
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
Multiresolution analysis with√
3-refinement
Sj+1 = Sj ⊕Wj
Large triangles control S0Small triangles control W0Local edit
Resolution level 0
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
Multiresolution analysis with√
3-refinement
Sj+1 = Sj ⊕Wj
Large triangles control S0Small triangles control W0Local edit
Resolution level 1
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
Multiresolution analysis with√
3-refinement
Sj+1 = Sj ⊕Wj
Large triangles control S0Small triangles control W0Local edit
Resolution level 1
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
Spline wavelets from the lifting scheme
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
The lifting scheme
Φj = Φj+1Pj
Φj+1 =[Oj+1 N j+1
][Φj Ψj
]= Φj+1
[Pj Qj
] (Wim Sweldens, 1994)Lifting
Ψj = N j+1 − ΦjUj
with Uj the update matrix. We find a relation of the form
[Φj Ψj
]= Φj+1
[Pj
[0j
Ij
]− PjUj
]
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
The lifting scheme
forward lifting inverse lifting
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
The update step
Problems
Semi-orthogonality⇒ Uj not sparseFix Uj sparse⇒ Ψj local supportWant stability⇒ need 1 vanishing moment for Ψj
Remaining orthogonality conditions approximated by leastsquares
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
The update step
Problems
Semi-orthogonality⇒ Uj not sparseFix Uj sparse⇒ Ψj local supportWant stability⇒ need 1 vanishing moment for Ψj
Remaining orthogonality conditions approximated by leastsquares
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
The update step
Problems
Uj not sparse⇒ Ψj no local supportFix Uj sparse⇒ Ψj local supportWant stability⇒ need 1 vanishing moment for Ψj
Remaining orthogonality conditions approximated by leastsquares
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
The update step
Problems
Want local support⇒ Uj sparseFix Uj sparse⇒ Ψj local supportWant stability⇒ need 1 vanishing moment for Ψj
Remaining orthogonality conditions approximated by leastsquares
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
The update step
Problems
Want local support⇒ Uj sparseFix Uj sparse⇒ Ψj local supportWant stability⇒ need 1 vanishing moment for Ψj
Remaining orthogonality conditions approximated by leastsquares
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
The update step
Problems
Want local support⇒ Uj sparseOrthogonalize w.r.t. scaling functions in the update stencil
Want stability⇒ need 1 vanishing moment for Ψj
Remaining orthogonality conditions approximated by leastsquares
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
The update step
Problems
Want local support⇒ Uj sparseOrthogonalize w.r.t. scaling functions in the update stencil
Want stability⇒ need 1 vanishing moment for Ψj
Remaining orthogonality conditions approximated by leastsquares
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
The update step
Problems
Want local support⇒ Uj sparseOrthogonalize w.r.t. scaling functions in the update stencil
i.e. Φ̃j has to reproduce constants
Remaining orthogonality conditions approximated by leastsquares
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
The update step
Problems
Want local support⇒ Uj 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 Spline wavelets from the lifting scheme
The update step
Problems
Want local support⇒ Uj 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 Spline wavelets from the lifting scheme
Spherical Powell–Sabin spline wavelets
3 wavelets per vertex
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
Applications
−→
Spherical scattereddata
Spherical PS spline surfacewith multiresolution structure
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
Applications
Compression
Original 26%
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
Applications
Denoising
With noise Denoised
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
Applications
Multiresolution editing
Coarse level edit Fine level edit
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Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme
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.
M. Lounsbery, T. D. DeRose, and J. Warren. Multiresolution analysis forsurfaces of arbitrary topological type. ACM Trans. Graphics,16(1):34–73, 1997.
J. Maes and A. Bultheel. A hierarchical basis preconditioner for thebiharmonic equation on the sphere. Accepted for publication in IMA J.Numer. Anal., 2006.
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 splinesMultiresolution analysis
Spline wavelets from the lifting schemeThe lifting schemeThe update stepThe waveletsApplicationsReferences