Powell--Sabin splines on the sphere with applications in...

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


Outline

Section I Powell–Sabin splines

Section II Spherical Powell–Sabin splines

Section III Multiresolution Analysis


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 ∈ ∆


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).


Spherical B-splines with control triangles


Applications on a spherical domain

Approximation of a mesh: consider the triangles of the originaltriangular mesh as control triangles of a PS spline.



Compression by smoothing: Decimate a given triangular meshand approximate the decimated mesh.

triangular mesh reduced mesh

(40000 triangles) (5000 triangles)



Compression by smoothing: Decimate a given triangular meshand approximate the decimated mesh.

triangular mesh Powell–Sabin spline

(40000 triangles) (5000 control triangles)



triangular mesh decimated mesh spherical(40000 triangles) (5000 triangles) parameterization

(5000 control triangles) PS spline surface


Multiresolution analysis


Multiresolution analysis (1989)

Stéphane Mallat Yves Meyer

A nested sequence of subspaces

S0 ⊂ S1 ⊂ S2 ⊂ · · · ⊂ S` ⊂ · · ·




Complement spaces W`

S`+1 = S` ⊕W`




A stable basis for the complement space W`

W` = span{ψ`,i}



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



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


Multiresolution analysis with√

3-refinement

∆PS0 ⊂ ∆PS1 ⊂ · · · ⊂ ∆

PS` ⊂ · · ·

S12(∆PS0 ) ⊂ S

12(∆

PS1 ) ⊂ · · · ⊂ S

12(∆

PS` ) ⊂ · · ·



3-refinement

S`+1 = S` ⊕W`

Large triangles control S0Small triangles control W0Local edit

Resolution level 0



3-refinement

S`+1 = S` ⊕W`

Large triangles control S0Small triangles control W0Local edit

Resolution level 1


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)


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.


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 − φÙ`

with U` the update matrix. We find a relation of the form[φ` ψ`

]= φ`+1

[P`

[0Ì`

]− PÙ`

]


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


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


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


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


The update step

Problems


i.e. φ̃` has to reproduce constants

Remaining orthogonality conditions approximated by leastsquares


The update step

Problems


An extra linear constraint

Remaining orthogonality conditions approximated by leastsquares


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


Applications

(a) Coarse part of (c) (b) Coarse part of (d)

(c) Original surface (d) Coarse level edit of (c)


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

Powell--Sabin splines on the sphere with applications in...

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