Jan Maes Adhemar Bultheel · (Paul Dierckx, 1997) Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme Powell–Sabin B-splines with control

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


Outline

Section I Powell–Sabin splines

Section II Spherical Powell–Sabin splines

Section III Spline wavelets from the lifting scheme


Powell–Sabin splines


Bernstein–Bézier representation

=⇒

Pierre Étienne Bézier (1910-1999)


Stitching together Bézier triangles

=⇒

No C1 continuity at the red curve


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


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 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)



Three locally supported basis functions per vertex



The control triangle is tangent to the PS spline surface.



It ‘controls’ the local shape of the spline surface.


Spherical Powell–Sabin splines


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 ∈ ∆}


Spherical Powell–Sabin splines

s(vi) = fi , Dgi s(vi) = fgi , Dhi s(vi) = fhi , ∀vi ∈ ∆

has a unique solution in S12(∆PS)


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 ∈ ∆


Spherical B-splines with control triangles


Multiresolution analysis with√

3-refinement

∆PS0 ⊂ ∆PS1 ⊂ · · · ⊂ ∆

PSj ⊂ · · ·

S12(∆PS0 ) ⊂ S

12(∆

PS1 ) ⊂ · · · ⊂ S

12(∆

PSj ) ⊂ · · ·



3-refinement

Sj+1 = Sj ⊕Wj

Large triangles control S0Small triangles control W0Local edit

Resolution level 0



3-refinement

Sj+1 = Sj ⊕Wj

Large triangles control S0Small triangles control W0Local edit

Resolution level 1


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

]


The lifting scheme

forward lifting inverse lifting


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


The update step

Problems

Uj not sparse⇒ Ψj no local supportFix Uj sparse⇒ Ψj local supportWant stability⇒ need 1 vanishing moment for Ψj



The update step

Problems

Want local support⇒ Uj sparseFix Uj sparse⇒ Ψj local supportWant stability⇒ need 1 vanishing moment for Ψj



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



The update step

Problems


i.e. Φ̃j has to reproduce constants



The update step

Problems


An extra linear constraint



Spherical Powell–Sabin spline wavelets

3 wavelets per vertex


Applications

−→

Spherical scattereddata

Spherical PS spline surfacewith multiresolution structure


Applications

Compression

Original 26%


Applications

Denoising

With noise Denoised


Applications

Multiresolution editing

Coarse level edit Fine level edit


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

Documents

Jan Maes Adhemar Bultheel · (Paul Dierckx, 1997) Powell–Sabin splines Spherical Powell–Sabin splines Spline wavelets from the lifting scheme Powell–Sabin B-splines with control