jan maes adhemar bultheel · (paul dierckx, 1997) powell–sabin splines spherical powell–sabin...

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

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

Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme

Powell–Sabin splines

Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme

Bernstein–Bézier representation

=⇒

Pierre Étienne Bézier (1910-1999)

Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme

Stitching together Bézier triangles

=⇒

No C1 continuity at the red curve

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

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

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

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.

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.

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)

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)

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)

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

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.

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.

Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme

Spherical Powell–Sabin splines

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 ∈ ∆}

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)

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 ∈ ∆

Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme

Spherical B-splines with control triangles

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 ) ⊂ · · ·

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

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

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

Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme

Spline wavelets from the lifting scheme

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

]

Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme

The lifting scheme

forward lifting inverse lifting

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

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

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

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

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

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

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

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

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

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

Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme

Spherical Powell–Sabin spline wavelets

3 wavelets per vertex

Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme

Applications

−→

Spherical scattereddata

Spherical PS spline surfacewith multiresolution structure

Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme

Applications

Compression

Original 26%

Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme

Applications

Denoising

With noise Denoised

Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme

Applications

Multiresolution editing

Coarse level edit Fine level edit

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