PowellSabin splines on the sphere with applications in CAGD Jan - - PowerPoint PPT Presentation

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

Powell–Sabin splines on the sphere with applications in CAGD

Jan Maes

Department of Computer Science Katholieke 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 S1

2(∆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 S1

2(∆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 S1

2(∆PS) = space of PS splines

M.J.D. Powell M.A. Sabin

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

The dimension of S1

2(∆PS)?

There is exactly one solution s ∈ S1

2(∆PS) to the

Hermite interpolation problem s(Vi) = αi, Dxs(Vi) = βi, ∀Vi ∈ ∆, i = 1, . . . , N. Dys(Vi) = γi, The dimension of S1

2(∆PS) is 3N. Therefore we need 3N basis

functions.

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

The dimension of S1

2(∆PS)?

There is exactly one solution s ∈ S1

2(∆PS) to the

Hermite interpolation problem s(Vi) = αi, Dxs(Vi) = βi, ∀Vi ∈ ∆, i = 1, . . . , N. Dys(Vi) = γi, The dimension of S1

2(∆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 = i [Bij(Vi), DxBij(Vi), DyBij(Vi)] = [αij, βij, γij] for j = 1, 2, 3 Partition of unity: N

i=1

3

j=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 = i [Bij(Vi), DxBij(Vi), DyBij(Vi)] = [αij, βij, γij] for j = 1, 2, 3 Partition of unity: N

i=1

3

j=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 = i [Bij(Vi), DxBij(Vi), DyBij(Vi)] = [αij, βij, γij] for j = 1, 2, 3 Partition of unity: N

i=1

3

j=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) = αdf(v) Hd := space of trivariate polynomials of degree d that are homogeneous of degree d Restriction of Hd to a plane in R3 \ {0} ⇒ we recover the space of bivariate polynomials ∆ := conforming spherical triangulation of the unit sphere S Sr

d(∆) := {s ∈ Cr(S) | s|τ ∈ Hd(τ), τ ∈ ∆}

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

Spherical Powell–Sabin splines

s(vi) = fi, Dgis(vi) = fgi, Dhis(vi) = fhi, ∀vi ∈ ∆ has a unique solution in S1

2(∆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 of degree 2 that is homogeneous of degree 2 Restriction to the plane tangent to S 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 vi Radial projection: Riv := v := v |v| ∈ S, v ∈ Ti Define ∆i as the star of vi in ∆, and let ∆PS

i

⊂ ∆PS be its PS 6-split. Theorem Let s ∈ S1

2(∆PS i

). Let s be the restriction of |v|2s(v/|v|) to Ti. Then s is in S1

2(R−1 i

∆PS

i

) and s(vi) = s(vi), Dgis(vi) = Dgis(vi), Dhis(vi) = Dhis(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 vi Radial projection: Riv := v := v |v| ∈ S, v ∈ Ti Define ∆i as the star of vi in ∆, and let ∆PS

i

⊂ ∆PS be its PS 6-split. Theorem Let s ∈ S1

2(∆PS i

). Let s be the restriction of |v|2s(v/|v|) to Ti. Then s is in S1

2(R−1 i

∆PS

i

) and s(vi) = s(vi), Dgis(vi) = Dgis(vi), Dhis(vi) = Dhis(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 original triangular 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 mesh and 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 mesh and 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 ⊂ · · · ⊂ Sℓ ⊂ · · ·

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

Multiresolution analysis (1989)

Stéphane Mallat Yves Meyer

Complement spaces Wℓ Sℓ+1 = Sℓ ⊕ Wℓ

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

Multiresolution analysis (1989)

Stéphane Mallat Yves Meyer

A stable basis for the complement space Wℓ Wℓ = span{ψℓ,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

∆PS ⊂ ∆PS

1

⊂ · · · ⊂ ∆PS

ℓ

⊂ · · · S1

2(∆PS 0 ) ⊂ S1 2(∆PS 1 ) ⊂ · · · ⊂ S1 2(∆PS ℓ ) ⊂ · · ·

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

Multiresolution analysis with √ 3-refinement

∆PS ⊂ ∆PS

1

⊂ · · · ⊂ ∆PS

ℓ

⊂ · · · S1

2(∆PS 0 ) ⊂ S1 2(∆PS 1 ) ⊂ · · · ⊂ S1 2(∆PS ℓ ) ⊂ · · ·

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

Multiresolution analysis with √ 3-refinement

∆PS ⊂ ∆PS

1

⊂ · · · ⊂ ∆PS

ℓ

⊂ · · · S1

2(∆PS 0 ) ⊂ S1 2(∆PS 1 ) ⊂ · · · ⊂ S1 2(∆PS ℓ ) ⊂ · · ·

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

Multiresolution analysis with √ 3-refinement

∆PS ⊂ ∆PS

1

⊂ · · · ⊂ ∆PS

ℓ

⊂ · · · S1

2(∆PS 0 ) ⊂ S1 2(∆PS 1 ) ⊂ · · · ⊂ S1 2(∆PS ℓ ) ⊂ · · ·

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

Multiresolution analysis with √ 3-refinement

Sℓ+1 = Sℓ ⊕ Wℓ Large triangles control S0 Small triangles control W0 Local edit

Resolution level 0

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

Multiresolution analysis with √ 3-refinement

Sℓ+1 = Sℓ ⊕ Wℓ Large triangles control S0 Small triangles control W0 Local edit

Resolution level 1

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

Multiresolution analysis with √ 3-refinement

Sℓ+1 = Sℓ ⊕ Wℓ Large triangles control S0 Small triangles control W0 Local edit

Resolution level 1

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

The hierarchical basis

{Vi ∈ ∆ℓ} ⊂ {Vi ∈ ∆ℓ+1} ∆PS

ℓ

⊂ ∆PS

ℓ+1

Sℓ := S1

2(∆PS ℓ ),

Sℓ ⊂ Sℓ+1 S2 = S0 ⊕ W0 ⊕ W1 Largest triangles control S0 Medium triangles control W0 Smallest triangles control W1

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

The hierarchical basis

Basis functions: Sℓ = span{φℓ,k : k = 1, . . . , 3Nℓ} sℓ(x, y) = φℓcℓ =

Nℓ

• i=1

3

• j=1

Bijℓ(x, y)cijℓ φℓ+1 = [φo

ℓ+1 φn ℓ+1],

φo

ℓ+1 correspond to old reused vertices of level ℓ

φn

ℓ+1 correspond to the new vertices of level ℓ + 1

The set of splines φ0 ∪

m

• ℓ=1

φn

ℓ

forms a hierarchical basis for Sm.

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

Wavelets via the lifting scheme

φℓ = φℓ+1Pℓ φℓ+1 = φo

ℓ+1

φn

ℓ+1

• φℓ

ψℓ

• = φℓ+1
• Pℓ

Qℓ

• (Wim Sweldens, 1994)

Lifting ψℓ = φn

ℓ+1 − φℓUℓ

with Uℓ the update matrix. We find a relation of the form

• φℓ

ψℓ

• = φℓ+1
• Pℓ

0ℓ Iℓ

• − PℓUℓ

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

The update step

Problems Semi-orthogonality ⇒ Uℓ not sparse Fix Uℓ sparse ⇒ ψℓ local support Want stability ⇒ need 1 vanishing moment for ψℓ Remaining orthogonality conditions approximated by least squares

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

The update step

Problems Semi-orthogonality ⇒ Uℓ not sparse Fix Uℓ sparse ⇒ ψℓ local support Want stability ⇒ need 1 vanishing moment for ψℓ Remaining orthogonality conditions approximated by least squares

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

The update step

Problems Uℓ not sparse ⇒ ψℓ no local support Fix Uℓ sparse ⇒ ψℓ local support Want stability ⇒ need 1 vanishing moment for ψℓ Remaining orthogonality conditions approximated by least squares

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

The update step

Problems Want local support ⇒ Uℓ sparse Fix Uℓ sparse ⇒ ψℓ local support Want stability ⇒ need 1 vanishing moment for ψℓ Remaining orthogonality conditions approximated by least squares

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

The update step

Problems Want local support ⇒ Uℓ sparse Fix Uℓ sparse ⇒ ψℓ local support Want stability ⇒ need 1 vanishing moment for ψℓ Remaining orthogonality conditions approximated by least squares

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

The update step

Problems Want local support ⇒ Uℓ sparse Orthogonalize w.r.t. scaling functions in the update stencil Want stability ⇒ need 1 vanishing moment for ψℓ Remaining orthogonality conditions approximated by least squares

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

The update step

Problems Want local support ⇒ Uℓ sparse Orthogonalize w.r.t. scaling functions in the update stencil Want stability ⇒ need 1 vanishing moment for ψℓ Remaining orthogonality conditions approximated by least squares

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

The update step

Problems Want local support ⇒ Uℓ sparse Orthogonalize w.r.t. scaling functions in the update stencil i.e. ˜ φℓ has to reproduce constants Remaining orthogonality conditions approximated by least squares

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

The update step

Problems Want local support ⇒ Uℓ sparse Orthogonalize w.r.t. scaling functions in the update stencil An extra linear constraint Remaining orthogonality conditions approximated by least squares

Powell–Sabin splines Spherical Powell–Sabin splines Multiresolution analysis

The update step

Problems Want local support ⇒ Uℓ sparse Orthogonalize w.r.t. scaling functions in the update stencil An extra linear constraint Remaining orthogonality conditions approximated by least squares

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 scattered data Spherical PS spline surface with 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–Bézier polynomials on spheres and sphere-like surfaces. Comput. Aided

• Geom. 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 spherical

Powell–Sabin B-splines. Comput. Aided Geom. Design, to appear.

• M. Neamtu and L. L. Schumaker. On the approximation order of splines
• n spherical triangulations. Adv. in Comp. Math., 21:3–20, 2004.
• W. Sweldens. The lifting scheme: A construction of second generation
• wavelets. SIAM J. Math. Anal., 29(2):511–546, 1997.