Bounds on the dimension of splines spaces

Nelly Villamizar1 joint work with Bernard Mourrain2

1Centre of Mathematics for Applications, University of Oslonellyyv@cma.uio.no

2GALAAD, INRIA Mediterranee, Sophia AntipolisBernard.Mourrain@inria.fr

MEGA 2011: Effective Methods in Algebraic GeometryJune 3, 2011

Introduction

A typical surface model has a large percentage of superfluous control points.One method to eliminate them is using T-spline local refinements as in[Sederberg et al.04].

1

1[Sederberg et al.04]

Smooth parametric surfaces interpolating triangular meshes are very useful for

modeling surfaces of arbitrary topology. In [Yvart et al.05] it is introduced a

hierarchical triangular surface model.

2

2Local surface refinement [Yvart et al.05].

A base mesh is built first, representing a coarse approximation of the object.Finner details are then added wherever needed with local refinement.

This model allows us to easily deform objects while preserving local details byediting only a few points.

3

3A dog’s head model [Yvart et al.05].

Triangular spline space

Let ∆ be a connected simplicial complex embeded in R2.

A spline on ∆ is a piecewise polynomial function such that thepolynomials meet with some order of smoothness.

The set of Cr splines on ∆ of degree at most k is a vector space, whichwill be denoted as Crk(∆).

Previous related workThe earliest paper where the problem of finding dimR C

rk(∆) is explicitly

formulated is due to Gilbert Strang.

I Strang’s conjecture ([Str73] and [Str74]).I Morgan and Scott computed the dimension for k ≥ 5, [M-S74].

Strang’s conjecture is not valid for general triangulations.I In [Sch79] Schumaker presented a lower and upper bounds for

arbitrary triangulations.I Alfeld [Alf-S87] obtained a formula for the dimension for k ≥ 4r+ 1.

The results were extended to k ≥ 3r + 2 by Hong [Hon91].I Billera (in [Bil88]) introduced the use of homological algebra in the

study of splines. Generic dimension of C13 (∆).

I Geramita and Schenk (in [G-S98]) applying homological techniquesderived a formula which gives the number of planar splines insufficiently high degree.

Following their ideas, these results can also be applied to splinespaces of small degree k. A lower and an upper bound fordimCrk(∆) can be found as follows.

A lower bound on the dimension of Crk(∆)

We embed ∆ in the hyperplane z = 1 ⊆ R3 and form the cone ∆ over∆ with vertex at the origin.

b

b b bb∆

∆

(0, 0, 0)

If Crk(∆) is the set of splines on ∆ of degree exactly k, then there is avector space isomorphism

Crk(∆) ∼= Crk(∆).

The abelian group⊕

k≥0 Crk(∆) = Cr(∆) can be viewed as a graded

module over the polynomial ring R := R[x, y, z].

Thus dimR Crk(∆) = dimR C

r(∆)k.

For an interior edge τ , let lτ denote the homogeneous linear formvanishing on τ .

f

g

τ

lτ

γ

lτ1

lτ2

If f, g are two polynomials supported in two triangles which share theedge τ , then the algebraic formulation of Cr smoothness is that lr+1

τ

divides f − g.J (σ) = 0 for σ ∈ ∆2 (triangles),

J (τ) = (lr+1τ ) for τ ∈ ∆0

1 (interior edges),

J (γ) =∑γ⊆τi∈∆0

1(lr+1τi ) for γ ∈ ∆0

0 (interior vertices).

Let f0i = |∆0

i | the number of i-faces.

Let ∂i be the relative (modulo ∂∆) boundary map and R = R[x, y, z],

R : 0 −→⊕σ∈∆2

Rσ∂2−→

⊕τ∈∆0

1

Rτ∂1−→

⊕γ∈∆0

0

Rγ −→ 0.

The homology of the chain complex R is the relative homology withcoefficients in R. From this chain complex, we construct R/J :

0 −→⊕σ∈∆2

R/J (σ)∂2−→

⊕τ∈∆0

1

R/J (τ)∂1−→

⊕γ∈∆0

0

R/J (γ) −→ 0

Using Euler characteristic equation and the properties [Bil88]:

I H2(R/J ) = Crk(∆),I H1(R/J ) = H0(J ),I H0(R/J ) = 0,

dimR Crk(∆) =

2∑i=0

(−1)i∑

β∈∆02−i

dimRR/J (β)k + dimRH1(R/J k).

Since dimH1(R/J ) ≥ 0 for any value of k ≥ 1,

dimR Crk(∆) ≥

2∑i=0

(−1)i∑

β∈∆02−i

dimRR/J (β)k.

•∑σ∈∆0

2

dimRRk = f02

(k + 2

2

)

•∑τ∈∆0

1

dimRR/J (τ)k =

f01∑

i=1

[(k + 2

2

)−(k + 2− r − 1

2

)]•∑γ∈∆0

0

dimRR/J (γ)k = ?

Being lr+11 , . . . , lr+1

ti a minimal generating set for J (γi), we have theresolution for R/J (γi) given by:

0→ R(−Ωi − 1)ai ⊕R(−Ωi)bi → ⊕tij=1R(−r − 1)→R→ R/J (γi)→ 0,

where Ωi − 1 is the socle degree of R/J (γi), given by the formula

Ωi =

⌊ti r

ti − 1

⌋+ 1, ai = ti (r + 1) + (1− ti) Ωi and bi = ti − 1− ai.

Thus we have,

dimR⊕γi∈∆0

0

R/J (γi)k =

f00∑

i=1

[(k + 2

2

)− ti

(k + 2− (r + 1)

2

)+

bi

(k + 2− Ωi

2

)+ ai

(k + 2− Ωi − 1

2

)].

Theorem (Lower bound)The dimension of Crk(∆) is bounded below by

dimRCrk(∆)≥

(k + 2

2

)+

f01∑i=1

(k + 2− r − 1

2

)

−f00∑i=1

[ti

(k + 2− (r + 1)

2

)− bi

(k + 2− Ωi

2

)− ai

(k + 2− Ωi − 1

2

)],

where ti is the number of edges with distinct slopes at the interior vertices γi,

Ωi =

⌊ti r

ti − 1

⌋+ 1, ai = ti (r + 1) + (1− ti) Ωi and bi = ti − 1− ai.

For example, for this polygon we have f02 = 14, f0

1 = 18 and f00 = 5.

Giving a numbering to the interior vertices, we find ti for each.

1

3

4

5

2

t1 = 3

1

3

4

5

2

t2 = 5

and counting in the same way we get t3 = 6, t4 = 3 and t5 = 5.

Applying the formula in the theorem we get,

dimR C12 (∆) ≥ 9.

An upper bound on the dimension of Crk(∆)

Let γ1, . . . , γf00

be an ordering of the interior vertices. For i = 1, . . . , f00 ,

let ti be the number of edges with different slopes join the vertex γi witheither a vertex of lower index or a vertex on the boundary.

With the numbering above we would have,

3

4

5

1

t1 = 2

2

3

4

5

1

t2 = 2

2

Theorem (Upper bound)The dimension of Crk(∆) is bounded by

dimRCrk(∆)≤

(k + 2

2

)+

f01∑

i=1

(k + 2− r − 1

2

)−

f00∑

i, ti=1

(k + 2− r − 1

2

)

−f00∑

i=1,ti≥2

[ti

(k + 2− (r + 1)

2

)− bi

(k + 2− Ωi

2

)− ai

(k + 2− Ωi − 1

2

)],

where ti is the number of edges with different slopes attaching the vertexγi to vetices on the boundary or of lower index, for some order of the

interior edges γ1, . . . , γf00, Ωi =

⌊ti rti−1

⌋+ 1, ai = ti (r+ 1) + (1− ti) Ωi

and bi = ti − 1− ai.

The problem is not purely combinatorialIn the Morgan-Scott triangulation we mentioned above, we have

t1 = t2 = t3 = 4.

And no matter how we number the interior vertices, we get

t1 = 2, t2 = 3 and t3 = 4.

1

3

2

t1 = 2

1

3

2

t2 = 3

1

3

2

t3 = 4

With the formulas above we get:

6 ≤ dimR C12 (∆) ≤ 7.

Effect of ordering on the upper bound

Let us consider the triangulated polygon from last example,numberings ofthe interior vertices in three different ways.

1

3

4

5

2

(1)

1

3

4

5

2

(2)

1

3

4

5

2

(3)

As we said before, we have f02 = 14, f0

1 = 18 and f00 = 5.

For numbering (1) we get:

3

4

5

1

t1 = 2

2

3

4

5

1

t2 = 2

2 2

4

5

1

t3 = 4

3

2

3

5

1

t4 = 2

4

3

4

5

1

t5 = 5

2

Hence dimR C12 (∆) ≤ 12.

For numbering (2) we have:

3

4

5

2

t1 = 2

1

3

4

5

1

t2 = 4

2 2

4

5

1

t3 = 3

3

2

3

5

1

t4 = 4

4

2

3

4

1

t5 = 3

5

Thus dimR C12 (∆) ≤ 10.

For numbering (3):

3

4

1

5

t1 = 3

2

3

2

1

5

t2 = 3

4

3

4

5

1

t3 = 3

2

3

2

5

1

t4 = 3

4

3

2

1

5

t5 = 3

4

And we get that dimR C12 (∆) ≤ 9. But we already know 9 is a lower

bound of the dimension hence dimR C12 (∆) = 9.

Theorem (Schumaker [Sch84])Numbering the interior vertices in such a way that each pair ofconsecutive vertices in the list are corners of a common triangle.For each i = 1, 2, . . . , V , let ei =number of edges with different slopesattached to the i-th vertex but not attached to any of the first i− 1vertices in the list,and let

σi =

k−r∑j=1

(r + j + 1− j · ei)+.

ThendimCrk(∆) ≤ α+ βE − γV +

V∑i=1

σi,

where E and V denote the number of interior edges and interior vertices

respectively, α = (k+2)(k+1)2 , β = (k−r)(k−r+1)

2 andγ = 1

2 [(k + 1)(k + 2)− (r + 1)(r + 2)].

In the example, the first two numberings (1) and (2), but not (3).

1

3

4

5

2

(1)

1

3

4

5

2

(2)

1

3

4

5

2

(3)

In fact, 10 is the best upper bound we can get.Since we can use any ordering,

UpperBoundHOM(dimCrk(∆)) ≤ UpperBoundSCH(dimCrk(∆)),

where UpperBoundSCH(dimCrk(∆)) and UpperBoundHOM(dimCrk(∆))denotes the best upper bound we get for dimCrk(∆) by usingShchumaker’s theorem and Theorem 1 respectively.

Sketch of proofThe formula for the upper bound given in the theorem is proved usingthe fact that

H1(R/J ) = H0(J )

and “half-edge” representations [γ|τ ] used by Schenck and Stillman.From the resolution

0→ R(−Ωi − 1)ai ⊕R(−Ωi)bi → ⊕tij=1R(−r − 1)→ J(γi)→ 0

we have:

H0(J )=⊕γ∈∆0

0

⊕τ3γ

[γ|τ ]R(−r− 1)/

∑τ=(γ,γ′)∈∆0

1

([γ|τ ]− [γ′|τ ])R(−r − 1) +

f00∑i=1

K(γi)

where K(γi) is the syzygy module of J (γi), isomorphic toR(−Ωi − 1)ai ⊕R(−Ωi)

bi .

A bound on the dimension of H0(J ) is obtained by triangulating therelations [γ|τ ]− [γ′|τ ] and pruning the syzygy modules K(γi).

Since H1(R/J ) is a finite dimensional graded vector space, it vanishes in

sufficiently high degree. Thus for k 0, [G-S98] we have that

dimR Crk(∆) =

2∑i=0

(−1)i∑

β∈∆02−i

dimRR/J (β)k.

TheoremThe formula for the lower bound above gives the exact dimension for allk ≥ 3r + 1.

Lemma ([SchS97])If ∆ is a triangulated region in R2, then there exits a total order on ∆0

such that for every γ ∈ ∆00, there exist vertices v′, v′′ adjacent to γ with

γ > v′, v′′, and such that γv′, γv′′ have distinct slopes.

b

b

b

b

τγ

γ0

v′

v′′

τγ

τb

This allows to proceed by induction on the interior vertices, starting witha vertex connected to the boundary by at least two different edges.

bγ

ω2

ω1

ω3

ω4

ω5

l1

l2

l3l4

l5 τ′

τ′′

L23

L34

L45

L41

LemmaLet l, l1, l3 three different lines with no common point. Then

(lr+1, lr+11 , lr+1

2 )N = (xr+1, yr+1, zr+1)N = (x, y, z)N ,

for N ≥ 3r + 1.

Some examples - Hierarchical subdivisions

Suppose we have some triangulated domain in the plane.

We would like to find recursively a refinement of this triangulation forwhich the lower and the upper bound give us the exact dimension andthe spline space is not trivial.

Let us consider the following hierarchical subdivision.

First we add a vertex in the interior of each triangle.

b

b

b

b

b

b

b

b

b

b

b

b

b

bb

b

b

b

b

Each time two triangles share an edge we join the two vertices inside thetwo triangles.

b

b

b

b

b

b

b

b

b

b

b

b

b

bb

b

b

b

b

Now we join each new vertex to the three vertices of the triangle itbelongs to.

b

b

b

b

b

b

b

b

b

b

b

b

b

bb

b

b

b

b

And adding edges in the triangles in the boundary we get a Powell-Sabinrefinement of the initial triangulation.Each initial triangle is now triangulated into six triangles.

b

b

b

b

b

b

b

b

b

b

b

b

b

bb

b

b

b

b

∆

Numbering the interior vertices in the following way.

b

b

b

b

b

b

b

b1

2

3

4

bb

b

b

bb

b

5

6

7

8

9

10

1112

bb

b

b

**

*13

14

15

16b

bb

By the formulas for the bounds we get that for k ≥ 3:

dimR C1k(∆) =

(k + 2

2

)+ E0

(k − 2

2

)+ 2V0

(k − 1

2

)+

(2k − 1

2

)(V − 2).

And as Powell and Sabin proved [P-S77] for k = 2

dimR C12 (∆) = 3V,

where E0, V0 and V denote respectively the number of interior edges, interior

vertices and total number of vertices in the initial triangulation.

The results about the bounds on the dimension can also be applied toany rectilinear subdivision of a polygonal domain (not onlytriangulations), and to mixed splines, which are splines where the order ofsmoothness may differ on the various edges.

Thanks for your attention!

Principal References

Anthony V. Geramita and Henry K. Schenck. Fat Points, InverseSystems, and Piecewise polynomial Functions. Journal of Algebra204, 116-128. 1998.

Ming-Jun Lai and Larry L. Schumaker. Spline Functions onTriangulations. Encyclopedia of Mathematics, Cambridge UniversityPress, 2007.

Larry L. Schumaker. Bounds on the dimension of spaces ofmultivariate piecewise polynomials. Rocky Mountain J. Math. 14,251-264, 1984.

Hal Schenck and Mike Stillman. Local cohomology of bivariatesplines. Journal of Pure and Applied Algebra, 117-118: 535-548,1997.

Some more references

Peter Alfeld and Larry L. Schumaker. The dimension of BivariateSpline Spaces of Smoothness r for degree d ≥ 4r + 1. Constr.Approx. 3, 189-197, 1987.

Bernard Mourrain. On the dimension of spline spaces on planarT-subdivisions. 2010.

Louis J. Billera. Homology of smooth splines: generic triangulationsand a conjecture of Strang. Transactions of the AmericanMathematical Society 310, 325-340, 1988.

Louis J. Billera and Lauren L. Rose. A dimension series formultivariate splines. Discrete and Computational Geometry 6,107-128, 1991.

J. Austin Cottrell, Thomas J. R. Hughes, and Yuri Bazilevs.Isogeometric analysis: toward integration of CAD and FEA. JohnWiley& sons, Ltd, 2009.

G. Farin. Curves and Surfaces for Computer Aided GeometricDesign: A Practical Guide. Academic Press, 1993.

Hong Dong. Spaces of bivariate spline functions over triangulation.Journal of Approximation Theory and its Applications v. 7, n. 1,56-75, 1991.

John Morgan and Ridgway Scott. A Nodal Basis for C1 PiecewisePolynomials of Degree n ≥ 5. Mathematics of Computation, v.29,n.131, 736-740, 1974.

M.J.D. Powell and Malcom A. Sabin. Piecewise quadraticapproximations on triangles. ACM Trans. Math. Softw. 3, 316-325.1977.

Henry K. Schenck. A spectral sequence for splines. Advances inApplied Mathematics 19, 283-199, 1997.

Larry L. Schumaker. On the dimension of spaces of piecewisepolynomials in two variables. Multivariate Approximation Theory,W. Schempp and K. Zeller (eds.), Basel, Birkhauser, 396-412, 1979.

Thomas Sederberg, David Cardon, G. Thomas Finnigan, NicholasNorth, Jianmin Zheng and Tom Lyche. T-splines Simplification andLocal Refinement. 2004.

Edwin H. Spanier. Algebraic Topology. McGraw Hill, New York,1966.

Gilbert Strang. Piecewise Polynomials and the Finite ElementMethod. Bulletin of the American Mathematical Society, Volume 79,Number 6, November 1973.

Gilbert Strang. The Dimension of Piecewise Polynomial Spaces, AndOne–Sided Approximation. Lecture Notes in Mathematics 363,Conference on the numerical solution of differential equations,Springer-Verlag 1974.

Alex Yvart, Stefanie Hahmann and Georges-Pierre Bonneau.Hierarchical Triangular Splines. ACM Transactions on Graphics, Vol.24, No. 4, 13741391, October 2005.