Lecture IV Bivariate subdivision schemes for quad meshes with arbitrary connectivity

Bivariate subdivision schemes for quad meshes with arbitrary connectivity

Ulrich Reif

Technische Universit¨at Darmstadt

Bertinoro, May 19, 2010

The reason for success in Computer Graphics:

Subdivision for cubic B-splines uses two different rules:

Subdivision for univariate cubic B-splines

Subdivision for cubic B-splines uses two different rules:

Subdivision for cubic B-splines uses two different rules:

1 6

1

Subdivision for univariate cubic B-splines

Subdivision for cubic B-splines uses two different rules:

1 6

1 1

1

Subdivision for bicubic tensor product B-splines uses three different rules:

Subdivision for bivariate cubic B-splines

Subdivision for bicubic tensor product B-splines uses three different rules:

1 6 1

1 6 1

1 6 1

6 36

6

1 6 1

new vertex point

Subdivision for bicubic tensor product B-splines uses three different rules:

1 6 1

1 1

1 6 1

1 6 1

new edge point

Subdivision for bivariate cubic B-splines

Subdivision for bicubic tensor product B-splines uses three different rules:

1

1

1 1

1 1

1 1

new face point

Topological rules:

Each old n-gonis split into n quadrilaterals.

After the first step,allfaces are quadrilaterals.

After a few steps, all extraordinary vertices are sufficiently well separated, and the vast majority of and vertices is regular.

Catmull-Clark subdivision for general meshes

Geometric rules:

Use standard stencils, wherever it is possible.

Geometric rules:

Use standard stencils, wherever it is possible.

Catmull-Clark subdivision for general meshes

Geometric rules:

Use standard stencils, wherever it is possible.

Geometric rules:

Use standard stencils, wherever it is possible.

Catmull-Clark subdivision for general meshes

Geometric rules:

Use standard stencils, wherever it is possible.

Geometric rules:

Use standard stencils, wherever it is possible.

Only at the extraordinary vertex itself, a new stencil is needed.

β ββ γ

γ

ββ γ

ββ

ββ γ

ββ ββ γ

γ

β

α

Catmull and Clark suggest:

α= 1− 7 4n

β = 3

2n²

γ = 1

4n²

Known parts of the limit surface

Always 4×4 control points define one patch.

Always 4×4 control points define one patch.

Known parts of the limit surface

Always 4×4 control points define one patch.

Always 4×4 control points define one patch.

Known parts of the limit surface

At a given level, an n-sided regionaround the extraordinary vertex is unknown.

As subdivision proceeds, the known parts grow and the unknown parts shrink.

Eventually, only a single point at the center is not covered.

Parts which are covered at different levels, coincide.

Subdivision surface as union of spline rings

Subdivision surface as union of spline rings

The subdivision surface can be regarded as the union of those ring-shaped parts which are newly added at every step of refinement.

Subdivision surface as union of spline rings

Subdivision surface as union of spline rings

The subdivision surface can be regarded as the union of those ring-shaped parts which are newly added at every step of refinement.

Locally, a subdivision surface can be represented as the union of spline rings, and a limit point, called thecentral point,

x= [

m∈N

x^m∪x^c, x^m : Σ0×Zn3(s,t,j)7→R^d.

All spline rings have a similar structure. They consist of a fixed numbern of L-shaped patches,

x^m = [

j∈Zn

x^m_j , x^m_j : Σ₀ 3(s,t)7→R^d.

General setup

Locally, a subdivision surface can be represented as the union of spline rings, and a limit point, called thecentral point,

x= [

m∈N

x^m∪x^c, x^m : Σ0×Zn3(s,t,j)7→R^d.

All spline rings have a similar structure. They consist of a fixed numbern of L-shaped patches,

x^m = [

j∈Zn

x^m_j , x^m_j : Σ0 3(s,t)7→R^d.

Being part of aregular subdivision surface, the spline rings can be parametrized with the help of the basic limit functions of theregular subdivision scheme and the control points at level m,

x^m(s,t,j) =

L

X

`=0

f_{`</sub>(s,t,j)p<sup>m</sup><sub>`} =F(s,t,j)P^m,

where

F = [f0,f1, . . . ,fL], P^m =



 p^m₀

... p^m_L



.

The control points at level m are obtained from the previous level by application of squaresubdivision matrix A,

P^m =AP^m−1, P^m =A^mP,

General setup

The sequence of spline rings to be analyzed is

x^m =FP^m=FA^mP.

F is built from the basic limit functions of the regular rules.

I F is mapping control points to the corresponding spline ring.

I F forms a partition of unity,P

ifi= 1.

I F is assumed to be at leastC¹.

I F is linearly independent.

Arepresents the special rules.

I Ais mapping control points from one level to the next finer one.

I The rows ofA(the stencils) sum to 1.

Pcontains the user-given initial set of control points.

Defining equation

Av_i =λ_iv_i.

Eigenvalues, ordered by modulus,

|λ₀| ≥ |λ₁| ≥ · · · ≥ |λ_L|, D=

"

λ₀ 0

0 . .. λ_L

# .

Right eigenvectors, existence assumed, V = [v0, . . . ,vL], AV =VD.

Left eigenvectors

W =V⁻¹ =



 w0

...



, WA=DW.

Eigendecomposition

WithA^m =VD^mV⁻¹ =VD^mW, the spline rings are

x^m =FA^mP=FVD^mWP.

The row-vectorG =FV contains the eigen-functions g_`,

G = [g₀, . . . ,g_L], g_i =Fv_i.

The column-vector Q=WPcontains the eigen-coefficientsq_`,

Q=

" q₀

... q_L

#

, q_{`</sub> =w<sub>`}P.

Finally,

x^m =GD^mQ=X

`

λ^m<sub>`</sub> g`q`.

WithA^m =VD^mV⁻¹ =VD^mW, the spline rings are

x^m =FA^mP= (FV)D^m(WP) =GD^mQ.

The row-vectorG =FV contains the eigen-functions g_`,

G = [g₀, . . . ,g_L], g_i =Fv_i.

The column-vector Q=WPcontains the eigen-coefficientsq_`,

Q=

"

q₀ ... q_L

#

, q_{`</sub> =w<sub>`}P.

Finally,

x^m =GD^mQ=X

λ^mg q .

Convergence and the dominant eigenvalue

If|λ₀|>1, then the sequence

x^m =X

i

λ^m_i g_iq_i

is typically divergent. This case is excluded.

Since all rows of Asum to 1,

A



 1... 1



=



 1... 1



 ⇒ λ₀ = 1, v₀ =



 1... 1



.

The eigen-function to λ₀ = 0 is

g0 =Fv0=X

i

fi = 1.

The eigen-coefficient toλ₀= 0 is

q₀ =w₀P, wherew₀A=w₀.

Asymptotic expansion:

x^m =λ^m₀g0q0+X

i≥1

λ^m_i g_iq_i =q0+O(λ^m₁).

If 1 =λ0>|λ₁|, then the sequence of spline rings converges to the central point

x^c := lim

m→∞x^m =q0.

In other words: Ifλ0= 1 is the strictly dominanteigenvalue of the subdivision matrix, then the subdivision surfacex iscontinuous.

Ineffective eigenvectors

What happens if the generating systemG is not linearly independent?

Convergence of

x^m =GA^mP

is possible even if %(A)>1.

There might exist ineffective eigenvectorsofA, i.e.,

Av =λv, λ6= 0, Gv = 0.

If so, spectral properties of Acannot be related to smoothness properties of the subdivision scheme.

For any given matrixA there exists anequivalent matrix A∗ without ineffective eigenvectors, i.e.,

GA^mP=GA^m_∗P for all mand P.

The eigenfunctions of A∗ corresponding to equal eigenvalues are linearly independent.

Construction: For an ineffective eigenvectorv ofA, computew such that

w^tv =λ

w^tv⁰ = 0 for all other eigenvectors v⁰ of A.

Set ˜A:=A−vw^t and repeat.

Popular schemes for quad meshes

Catmull-Clark (generalizing cubic B-splines) Doo-Sabin (generalizing quadratic B-splines)

Cashman’s NURBS subdivision (generalizing B-splines of any order) Simplest subdivision (generalizing C¹ four-direction splines)

Velho’s 4-8 scheme (generalizing C⁴ four-direction box splines) Kobbelt’s interpolatory scheme (generalizing the four-point scheme)

Geri’s game – An Oscar for subdivision