Upload
others
View
4
Download
0
Embed Size (px)
Citation preview
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
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
Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Powell–Sabin splines
Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Bernstein–Bézier representation
=⇒
Pierre Étienne Bézier (1910-1999)
Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Stitching together Bézier triangles
=⇒
No C1 continuity at the red curve
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
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
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
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.
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.
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)
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)
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)
Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Powell–Sabin B-splines with control triangles
Three locally supported basis functions per vertex
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.
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.
Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Spherical Powell–Sabin splines
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 ∈ ∆}
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)
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 ∈ ∆
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).
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).
Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Spherical B-splines with control triangles
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.
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)
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)
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
Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Multiresolution analysis
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` ⊂ · · ·
Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Multiresolution analysis (1989)
Stéphane Mallat Yves Meyer
Complement spaces W`
S`+1 = S` ⊕W`
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}
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
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
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
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` ) ⊂ · · ·
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` ) ⊂ · · ·
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` ) ⊂ · · ·
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` ) ⊂ · · ·
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
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
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
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)
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.
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`
]
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
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
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
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
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
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
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
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
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
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
Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Spherical Powell–Sabin spline wavelets
3 wavelets per vertex
Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Applications
−→
Spherical scattereddata
Spherical PS spline surfacewith multiresolution structure
Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Applications
Compression
Original 26%
Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Applications
Denoising
With noise Denoised
Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis
Applications
Multiresolution editing
Coarse level edit Fine level edit
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)
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