Lecture VII Shape analysis and higher regularity of bivariate subdivision schemes

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Lecture VII

Shape analysis and higher regularity of bivariate subdivision schemes

Ulrich Reif

Technische Universit¨at Darmstadt

Bertinoro, May 21, 2010

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Assessment fo subdivision surfaces, today

Designer 1 (Nintendo):

Subdivision surfaces are sufficiently smooth, by far.

Designer 2 (Pixar):

Subdivision surfaces are sufficiently smooth, from afar. Designer 3 (Mercedes):

Subdivision surfaces arefar from sufficiently smooth.

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Assessment fo subdivision surfaces, today

Designer 2 (Pixar):

Subdivision surfaces are sufficiently smooth, from afar.

Designer 3 (Mercedes):

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Assessment fo subdivision surfaces, today

Designer 2 (Pixar):

Subdivision surfaces are sufficiently smooth, from afar.

Designer 3 (Mercedes):

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Setup

A subdivision surfacex is the union ofspline rings, x= [

m∈N0

x^m.

Each spline ring is a linear combination of generating functions and control points,

x^m =X

i

g_ip^m_i =GP^m.

The sequence of control points is obtained by repeated application of thesubdivision matrix,

P^m =A^mP⁰.

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Setup

Eigenvalues

|λ₀| ≥ |λ₁| ≥ · · · ≥ |λ_L|,

Left and righteigenvectors Av` =λ`v`, w`A=λ`w`, Eigenfunctionsandeigencoefficients

f_`=Gv_`, q_`=w_`P.

Eigen-expansion

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Generic assumptions

Thesub-dominant eigenvalue is double λ:=λ₁ =λ₂>|λ₃|

Thecharacteristic map is regular and injective, ΨΨΨ := [f1,f2] =G[v1,v2], detDΨΨΨ6= 0.

Thesubsub-dominant eigenvalue is denoted byµ, 1> λ1 =λ2

| {z }

λ

> λ3 =· · ·=λN

| {z }

µ

>|λ_N+1|.

Curvature near central point determined by third order expansion

x^m .

=q₀+ ΨΨΨ[q₁;q₂] +µ^m

N

X

`=3

f_`q_`.

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Curvature and the subsub-dominant eigenvalue

The principal curvatures converge to 0, if µ < λ², are bounded, if µ=λ², diverge, if µ > λ². are in L^p for

p < 2 lnλ 2 lnλ−lnµ. C¹ always implies H^2,2.

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C

²

-conditions

A subdivision schemes generates C²-surfaces if and only if µ=λ²

and if the subsub-dominant eigenfunctions satisfy f₃, . . . ,f_N ∈span{f₁²,f₁f₂,f₂²}.

Degree estimate: If, on the regular part of the grid, the scheme generates polynomial patches of degree d joiningC^k, then non-trivial curvature continuity is possible only if

d ≥2k+ 2.

This rules out schemes generalizing uniform B-spline subdivision and box splines. The lowest order candidate is of bi-degree 6 with 4-fold knots.

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Shape analysis

To achieve curvature continuity, convergence of the principal curvatures is not sufficient.

principal directions is not necessary.

Weingarten map is necessary and sufficient, but . . .

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The Weingarten map revisited

The Weingarten mapW is a linear map in the tangent spaceTx, defined by

∇n=−W∇x.

Its eigenvalues and eigenvectors are the principal curvatures and directions, respectively.

With respect to basis x_u,x_v ofTx,

Dn=−W Dx ⇒ −DnDx^t=W DxDx^t ⇒ W =H G⁻¹, where

D:=

∂u

∂_v

, G :=DxDx^t, H :=−DnDx^t.

Problem: For spline surfaces,Dx and henceW isdiscontinuous.

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The Weingarten map revisited

Trick: Instead of Dn=−WDx, consider the dual equation, Dn^t=−E Dx^t.

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The Weingarten map revisited

Trick: Instead of Dn=−WDx, consider theextendeddual equation, [Dn^t, 0] =−E[Dx^t, n^t].

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The Weingarten map revisited

Trick: Instead of Dn=−WDx, consider theextendeddual equation, [Dn^t, 0] =−E[Dx^t, n^t].

With

Dx⁺=Dx^tG⁻¹

denoting the pseudo-inverse ofDx, E =−Dx⁺Dn=Dx⁺H(Dx⁺)^t

is a symmetric map acting onR³. By duality,

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The Weingarten map revisited

Properties:

E is a second order geometric invariant.

The principal directions are eigenvectors with respect to the principal curvatures.

E refers to coordinates of the embedding space.

Continuity of E is necessary and sufficientfor x to be aC²-manifold, i.e., in the subdivision setup, the limit

E^c:= lim

m→∞E^m, E^m: Σ0× {1, . . . ,n} →R^3×3 has to exist and to be constant.

The integrability conditions are simple, nuE_v⁺=nvE_u⁺ ⇒ Dx=DnE⁺.

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The Weingarten map revisited

Properties:

E is a second order geometric invariant.

The principal directions are eigenvectors with respect to the principal curvatures.

E refers to coordinates of the embedding space.

Continuity of E is necessary and sufficientfor x to be aC²-manifold, i.e., in the subdivision setup, the limit

E^c:= lim

m→∞E^m, E^m: Σ0× {1, . . . ,n} →R^3×3 has to exist and to be constant.

The integrability conditions are simple,

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The central surface

For simplicity, let q₁

q2

=L e₁

e2

.

The third order asymptotic expansion of the rings is

x^m .

=q₀+ [λ^mΨΨΨL, µ^mϕ], ϕ:=

N

X

i=3

f_ihq_i,n^ci.

Definition: Thecentral surface is a spatial ring defined by

˜x:= (ΨΨΨL, ϕ).

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Asymptotic expansions

WithJ :=DΨΨΨL, thefirst fundamental form ofx^m is G^m .

=λ^2mG, G :=J J^T.

With ˜G and ˜H the fundamental forms of the central surface ˜x, the second fundamental formof x^m is

H^m .

=µ^mH, H:=

s det ˜G detG H,˜ . Theembedded Weingarten map ofx^m is

m . m

E 0

t −t −1 µ

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Asymptotic expansions

The embedded Weingarten map ofx^m is

E^m .

=%^m E 0

0 0

, E :=L^tJ^−tH J⁻¹L, %:= µ λ².

The Gausian curvature of x^m is κ^m_G .

=%^2m detE.

The mean curvature of x^m is κ^m_M .

=%^m traceE.

The principal directions of x^m are R^m .

= [R, 0], RE =KR.

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Consequences

The deviation of E from a constant is a reliable indicator for the quality of a subdivision algorithm.

An algorithm cannot generate elliptic shapeunless 0∈ F(µ).

An algorithm cannot generate hyperbolic shapeunless 1,n−1∈ F(µ).

Optimal spectrum

simple 1, F(1) ={0}

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C

²

-schemes

TURBS (R.’ 95)

Freeform splines (Prautzsch ’96)

Guided subdivision (Peters, Karciauskas ’06)

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General framework for C

²

-subdivision

Denote byC_d²(Rⁿ) the space of allC²-rings in Rⁿ composed of patches of coordinate degree d.

A ring ΨΨΨ∈C₃²(R²) is called aconcentric tesselation map with scale factor λ∈(0,1), if it is injective and regular, i.e., detDΨΨΨ6= 0, and if ΨΨΨ and λΨΨΨ join C² when regarded as consecutive rings.

The image of ΨΨΨ and itsextensionare denoted Ω := ΨΨΨ(Σ), Ω_e:= Ω∪λΩ.

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General framework for C

²

-subdivision

The reparametrization operator R is mapping rings x^m∈C₆²(Rⁿ) to functions on Ω⊂Rⁿ by

R[x^m] : Ω3ξξξ 7→x^m(ΨΨΨ⁻¹(ξξξ)).

× {1,...,n}

IRⁿ

x^m R[x^m]

Ψ⁻¹

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General framework for C

²

-subdivision

The extended reparametrization operator R_e maps a pair

x^m,x^m+1 ∈C₆²(Rⁿ) of consecutive rings to a single function acting on Ωe

according to

Re[x^m,x^m+1] : Ωe3ξξξ7→

(R[x^m](ξξξ) ifξξξ∈Ω R[x^m+1](ξξξ/λ) ifξξξ∈λΩ.

× {1,...,n}

IRⁿ

x^m, x^m+1 R_e[x^m, x^m+1]

Ψ⁻¹, Ψ⁻¹/λ

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General framework for C

²

-subdivision

The subdivision matrixA hasquadratic precision, if for consecutive rings x^m =B6Q^m,x^m+1 =B6AQ^m,

R[x^m]∈P2(Ω) implies R_e[x^m,x^m+1]∈P2(Ω_e).

If ΨΨΨ has scale factorλandA has quadratic precision, then there exist eigenvalues λ_i, eigenvectors v_i and eigenfunctionsf_i =B₆v_i satisfying

λ₀ = 1, λ₁=λ₂ =λ, λ₃=λ₄ =λ₅=λ² f0= 1, [f1,f2] = ΨΨΨ, f3=f₁², f4 =f1f2, f5=f₂².

Consequence: LetA be a subdivision matrix with quadratic precision and eigenvalues λ0, λ1, . . . , λ`¯, whereλ0, . . . , λ5 are given above. If ΨΨΨ has scale factor λand|λ_i|< λ² for all i >5, then Adefines a

C²-subdivision algorithm.

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General framework for C

²

-subdivision

Compute x^m+1 =S(x^m) in four steps:

Reparam

Extension

Projection S = PTER

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General framework for C

²

-subdivision

Extension: Choose a linearextension operator E mapping the

functionr=R[x^m] defined on Ω to the function r_e=E[r] defined on Ω_e such that

r∈P2 ⇒ r_e∈P2.

The subdivision matrixA corresponding to the schemeS =PTER is given by

GA=PTER[G].

The algorithm isC², if λ_i < λ², i >6.

Acan be precomputed once and for all.

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Summary

Linear, univariate subdivision well understood.

Linear, multivariate subdivision well understood in the regular setting.

C¹-schemes for arbitrary topology available.

C²-schemes for arbitrary topology available, but not well established.

Nonlinear schemes not well understood.

Schemes for perfect shape sought.

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