Complex Eigenvalues for Binary Subdivision Schemes

0.15 MB·English

by  Christian Kuehn

Complex Eigenvalues for Binary Subdivision Schemes
Christian Ku¨hn
Cornell University,Center for Applied Mathematics, 14853 Ithaca NY,United States
e-mail: [email protected]
Abstract
8
Convergencepropertiesofbinarystationarysubdivisionschemesforcurveshavebeenanalyzedusing
0
the techniques of z-transforms and eigenanalysis. Eigenanalysis provides a way to determine derivative
0
2 continuityatspecificpointsbasedontheeigenvaluesofafinitematrix. Noneofthewell-knownsubdivi-
sionschemesforcurveshavecomplexeigenvalues. Weprovewhenaconvergentschemewithpalindromic
n
mask can have complex eigenvalues and that a lower limit for the size of the mask exists in this case.
a
Wefindaschemewith complexeigenvaluesachievingthislowerbound. Furthermoreweinvestigatethis
J
scheme numerically and explain from a geometric viewpoint why such a scheme has not yet been used
1
in computer-aided geometric design.
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Keywords: subdivision; z-transform; eigenanalysis; computer-aided geometric design
]
R
G 1 Introduction and Notation
.
s
c Subdivision schemes have been used successfully in computer-aided geometric design for more than two
[ decades. The basic idea is to consider a mesh of points in either one or two dimensions and to construct a

1 curve or a surface out of this mesh by an iterative procedure. This allows the representation of geometric
v objectsbyarelativelysmallnumberofpoints. Inthispaperweonlyconsidersubdivisionschemesforcurves.
9 Inthefollowingparagraph,wereviewtherequiredbackgroundverybrieflyandreferto[5]and[6]foramore
4
detailed introduction.
2
3
. We start with an inital mesh of points Nk = 2−kZ and associate with it a sequence of control points
1 Pk = Pk . As an example consider the case k = 1; then we can view the previous construction as a
0 { i }
bi-infinite sequence of points
8
0
...,( 1,Pk ),( 1/2,Pk ),(0,Pk),(1/2,Pk),...
v: { − −2 − −1 0 1 }
i A binary subdivision scheme doubles the number of points at each step, obtaining all new points only
X
using the points of the last step. The mesh N is refined to N for primal schemes and to a mesh
k k+1
ar 2−(k−1)Z 2−(k−2)Z for dual schemes (see Figure 1).
{ − }
To simplify notation we simply suppress this underlying refinement of the mesh and focus on the con-
struction of the sequence of control points Pk. The main data defining a binary stationary subdivision
scheme S is given by a so-called mask a= a Ri σ(a) Z and refinement rules
a i
{ ∈ | ∈ ⊂ }
Pk+1 = a Pk and Pk+1 = a Pk (1)
2i 2j i−j 2i+1 2j+1 i−j
2j∈Xσ(A) 2j+X1∈σ(A)
where σ(a) is called the support of S . The iteration of a subdivision scheme is denoted by Pk+1 = S Pk.
a a
Furthermore we consider only palindromic masks with finite width, i.e. we suppose σ(a) = n, n+
{− −
1
primal
dual
Figure 1: Difference between primal and dual refinement of the mesh
1, ,n 1,n and a =a if n is odd (primal schemes) and σ(a)= n+1, ,n 1,n and a =a
i −i i 1−i
··· − } {− ··· − }
if n is even (dual schemes).
The next natural step is to analyze the limiting properties of a scheme as k . Let pk(t) denote the
function obtained from the control points Pk (attached to the underlying mesh)→by∞linear interpolation. A
subdivision scheme S is termed Cm-convergentif there exists a Cm function f :R R such that
a
→
lim max f(t) pk(t) =0
k→∞A≤t≤Bk − k
for every fixed A,B R and if f(t) 0 for some initial sequence of control points. Two paradigms of
∈ 6≡
analysishavebeen developedto answerwhether a scheme S is convergent. First, we define the z-transform
a
of a subdivision scheme by
S s (z)= a zi (2)
a a i
←→
i∈Xσ(a)
giving a one-to-onecorrespondence between the mask a and a formal Laurent series. We have the following
well-known results (see [1] or [2]):
Theorem 1.1. S is not convergent if s (1)=2 or s ( 1)=0.
a a a
6 − 6
Hence convergent schemes have s (z)=(1+z)s (z) where the scheme S relates to S via ∆(S Pk)=
a b b a a
S ∆Pk where ∆Pk = Pk Pk−1 i Z . We call S the difference scheme associated to S .
b { i − i | ∈ } b a
Theorem 1.2. S is C0-convergent if S is contractive, i.e. S maps any initial sequence of control points
a b b
to 0 upon iteration.
Note that contractivity of S can be checked by computing if S <1 where
b b
k k
Sb =max b2j , b2j+1 
k k Xj | | Xj | |
 
This reduces the question of convergence to a direct computation. Theorem 1.2 can be improved to an “if
andonly if” statementin the caseof asymptotic equivalenceofthe operatorsS and S (see [2]for details).
a b
The easiest way to improve the level of differentiability is given in the next theorem.
Theorem 1.3. Let s (z)= (1+z)s (z). If S is Ck-convergent then S is Ck+1-convergent.
a 2 q q a
Hence we can try to divide out (1+z) factors from the z-transform and check the resulting scheme for
2
convergence. Conversely we can improve the level of smoothness of a convergent scheme by multiplying
the z-transform by (1+z) factors. Note that the correspondence (2) provides the corresponding mask a.
2
Expositions and proofs of the previous results are found in [1], [2] and [3].
2
2 Complex Eigenvalues: An Example and Minimum Width
An alternative way of analyzing convergence is provided by setting up a matrix A associated to S . Note
a
that the refinement rules (1) require an infinite matrix but to investigate smoothness near a given point
it suffices to perform eigenanalysis on a finite matrix A called the local subdivision matrix. Eigenanalysis
provides an upper bound on the level of derivative continuity. If the width of S is n then A Rn×n. For a
a
∈
more detailed dsicussion of eigenanalysis consider [3] or [5].
WeaskthequestionifthematrixAcanhavecomplexeigenvaluesforaconvergentschemeS . Reviewing
a
manywell-knownsubdivisionschemesfor curvesthe surprisingresultis thatallclassicalschemessuchasB-
spline schemesorthe four-pointscheme (see [3]or[5])do nothavecomplex eigenvalues. A priorithere is no
reasonfor a matrix with real entries not to have complex eigenvalues. We remark that complex eigenvalues
havebeenobservedinthe analysisofsomesubdivisionschemes forsurfacessuchas the √3-scheme(see [4]).
For curves no complex eigenvalue scheme seems to be known. Therefore we ask three questions:
1. Doesthereexistaconvergentbinarysubdivisionschemewithpalindromicmaskhavingcomplexeigen-
values? (Answer: Yes!)
2. What is the minimum width of such a scheme?
3. Why did complex eigenvalue schemes not appear “naturally” in computer-aided geometric design for
curves?
Proposition 2.1. There does not exist a binary subdivision scheme S with width less than or equal to 5
a
and complex eigenvalues in the local subdivision matrix A.
Proof. We begin with the case n = 5. First note that since s (z) = a z2+a z +a +a z−1+a z−2 we
a 2 1 0 1 2
get using Theorem 1.1 that 2 = s (1) = 2a +2a +a and 0 = s ( 1) = 2a 2a +a for a convergent
a 2 1 0 a 2 1 0
− −
scheme. This implies the conditions:
1
a =1 2a and a =
0 2 1
− 2
Therefore the local subdivision matrix A is given by
a 1 2a a 0 0
−
 0 1/2 1/2 0 0 
A= 0 a 1 2a a 0
 − 
 0 0 1/2 1/2 0 
 
 0 0 a 1 2a a 
 − 
which has eigenvalues 1,1,1 2a,a,a and since a R we conclude that no convergent scheme with a
{ 2 2 − } ∈
mask of width 5 can have complex eigenvalues. The cases n 4 are dealt with in the same way and all
≤
matrices have real eigenvalues in these cases.
Note that the previous analysis fails for the case n = 6. We show now that it is possible to find a
convergentscheme of width 6 with “simple” coefficients having a small common denominator.
Proposition 2.2. The mask a = a = 1 ,a = 3 ,a = 4,a = 4,a = 3 ,a = 1 defines a
{ −2 −10 −1 10 0 5 1 5 2 10 2 −10}
C0-convergent binary scheme S with complex eigenvalues in the local subdivision matrix A.
a
3
Proof. Since s (z) = (1+z) z2 + 2z + 2 + 2 1 := (1+z)s (z) we find S (z) = 4 < 1 so that
a (cid:16)−10 5 5 5z − 10z2(cid:17) b k b k 5
S is C0-convergentby Theorem 1.2. The local subdivision matrix is given by
a
1/10 4/5 3/10 0 0 0
−
 0 3/10 4/5 1/10 0 0 
−
0 1/10 4/5 3/10 0 0
A= − 
 0 0 3/10 4/5 1/10 0 
 − 
 0 0 1/10 4/5 3/10 0 
 − 
 0 0 0 3/10 4/5 1/10 
 − 
which has eigenvalues 1 , 1 ,2,1,1 2 i√2 ,1 2+i√2 so that the result is proved.
{−10 −10 5 5 − 5 }
(cid:0) (cid:1) (cid:0) (cid:1)
Thismeansthatcomplexeigenvaluesimposeconditionsonthewidthofamask. Itisingeneraldesirable
to have masks with relatively small size to guarantee locality, i.e. moving a control point does not affect
partsofthecurvefarawayfromthispoint. Manyclassicalschemeshavewidthbetween6and10. Therefore
we can conclude that there is no algebraic reasonwhy we have not yet seen a complex eigenvalue scheme in
practice.
We remark that we cannot use Theorem 1.3 to find a C1-convergent scheme with width 6. Since
s (z) = (1+z) 2az2+2(b a)z+2 4b+ 2(b−a) + 2a = (1+z)s (z) using Theorem 1.1 we get the dif-
a 2 (cid:16) − − z z2(cid:17) 2 q
ference scheme S . Using Theorem 1.1 on S gives
q q
1
0=s ( 1)=2(1+4a 4b) b= +a
q
− − ⇒ 4
This condition implies that the eigenvalues of the localsubdivision matrix A are real. But obviouslywe can
use Theorem 1.1 to construct a scheme with higher smoothness out of the complex eigenvalue scheme we
presented by multiplying the z-transform with (1+z) factors. For example we get from the scheme given in
2
Proposition 2.2 the C1 scheme
1 1 11 4 11 1 1
a = ,a = ,a = ,a = ,a = ,a = ,a =
−3 −2 −1 0 1 2 3
(cid:26) −20 10 20 5 20 10 −20(cid:27)
It again has a pair of complex conjugate eigenvalues.
3 Numerical Comparison: Complex vs. Real Eigenvalues
Ourgoalistoanswerthelastofourthreequestionsandwestartbycomparingtheresultsofseveralschemes:
Mask Smoothness
(a) a = 1/10,a =3/10,a =4/5,a =4/5,a =3/10,a = 1/10 C0
−2 −1 0 1 2 2
{ − − }
(b) a = 1/20,a =1/10,a =11/20,a =4/5,a =11/20,a =1/10,a = 1/20 C1
−3 −2 −1 0 1 2 3
{ − − }
(c) a =1/2,a =1,a =1/2 C0
−1 0 1
{ }
(d) a =1/8,a =4/8,a =6/8,a =4/8,a =1/8 C2
−2 −1 0 1 2
{ }
Scheme(d)isderivedfromcubicB-splinesandscheme(c)isknownas“simplest”or“two-point”scheme;
itiseasytocheckthat(c)-(d) haveonlyrealeigenvalues. We comparetheschemeswiththe mostimportant
test sequence of control points given by
P0 = 0,0,0,0,1,0,0,0,0
{ }
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on the integer mesh N0 on [ 4,4]. The reason for its importance is that all convergence properties can be
−
derivedby investigatingthis sequence andusing linearityofthe scheme (see [3]). The schemes wereiterated
10 steps and linear interpolation was used to display the function constructed from P10. The results are
shown in Figure 2.
(a) complex eigenvalues (b) C1 complex eigenvalues
1 0.8
0.6
0.5
0.4
0.2
0
0
−0.5 −0.2
−4 −2 0 2 4 −4 −2 0 2 4
(c) − simplest scheme (d) − cubic B−spline scheme
1 0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0 0
−4 −2 0 2 4 −4 −2 0 2 4
Figure 2: (a)-(b) complex eigenvalue schemes; (c)-(d) two “standard” schemes.
ThecubicB-splineschemeisC2 andproducestheexpectedsmoothcurve. Thesimplestschemeproduces
the classical C0 tent function. For the complex eigenvalue scheme (a) we find a C0 curve, which is neither
smooth nor contains clear connecting points between several parts of the curve like displayed for the tent
function. Instead we see small oscillations or “ripples”. The complex eigenvalue scheme (b) gives a C1
curve with two small negative bumps outside [ 2,2]. In computer-aided geometric design undesired ripples
−
or bumps of the limiting curve are usually called artefacts. In the last two decades the focus has been to
find schemes which produce smooth curves or curves with well-defined sharp transitions without artefacts.
This is one possible explanation why practioners might have overlookedor dismissed schemes with complex
eigenvalues. In the next section we show why artefacts are expected to appear.
4 A Geometric Explanation using Dynamical Systems
Weaimtoexplainwhyweexpectartefactsineverysubdivisionschemeforcurveswithcomplexeigenvalues.
Fix a convergent scheme S with one pair of complex conjugate eigenvalues. Fix a mesh point x Nk0.
Consider the n control poinats in Pk0 attached to mesh points closest to x . By shifting the indicke0s∈we can
k0
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assume without loss of generality that these points are given by Pk0 for i 1,2,...,n . Define a vector
i ∈{ }
v =(Pk0,Pk0,...,Pk0) (3)
k0 1 2 n
This process can be continued for the n-points closest to x in Pk0+1, Pk0+2, etc. Note that the local
k0
subdivision matrix A associated to S produces this sequence of vectors under iteration:
a
v =Av for k k and v given by (3)
k+1 k ≥ 0 k0
This means we are dealing with a particularly simple discrete dynamical system defined by the map A :
Rn Rn. It is known from eigenanalysisthat the eigenvalues µ of A must obey the restriction µ =1 and
i 1
→
µ <1 for i 2,3,...,n (see [5]). Furthermore it is immediate by the convergence of S that
i a
| | ∈{ }
Av =v v¯ as k for some vector vector v¯ with identical components.
k k+1
→ →∞
In fact we find that Av¯=v¯ so that v¯ is a fixed point. Since we have at least one pair of complex conjugate
eigenvalues there exists a change of basis so that A has a two-dimensional eigenspace R in which A is a
rotation-scalingwith scalingfactor λ<1. The scalingfactor is givenby the realpartof the pairof complex
conjugate eigenvalues. Due to the rotational component the convergence to v¯ is not monotone. This lack
of monotonicity is reflected in the different behaviour of the complex eigenvalue schemes in comparison to
some of the classical schemes.
In ourexamples ofcomplex eigenvalues there is anotherreasonfor non-monotoneconvergencecausedby
thetworealnegativeeigenvalues(seeProposition2.2). Thisgivesalternatingconvergenceoftwocomponents
of v to v¯. This alternating convergence is unavoidable for complex eigenvalue schemes with width 6 as the
k
next proposition shows.
Proposition 4.1. Every binary complex eigenvalue subvision scheme with palindromic mask of width 6 has
at least 2 eigenvalues with negative real part.
Proof. The local subdivision matrix A is given by
a c b 0 0 0
 0 b c a 0 0 
0 a c b 0 0
A= 
 0 0 b c a 0 
 
 0 0 a c b 0 
 
 0 0 0 b c a 
 
Using the condition a+b+c=1 we first eliminate c and then compute the eigenvalues explicitly. They are
given by:
µ =1, µ =a, µ =b a, µ =1 a b 1+2a 7a2 6b+2ab+9b2
1 2,3 4 5,6
− − − ± − −
p
For the eigenvalues to be complex we get that the terms inside the square-root for µ must be negative.
5,6
The inequality to be satisfied is:
1+b 2 2(1 5b+8b2)
a< − −
p 7
But we can calculate directly that
1+b 2 2(1 5b+8b2) 0 for b R
− − ≤ ∈
p
Therefore we can conclude that µ =a<0.
2,3
We leave it as an open question whether any binary complex eigenvalue scheme with palindromic mask
must have eigenvalues with negative real part. Hence it might be possible to avoid alternating convergence
for a complex eigenvalue scheme but non-monotonicity can obviously not be avoided. Hence we expect
artefacts to appear.
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5 Conclusion
We have demonstrated that subdivision schemes for curves can have complex eigenvalues. We have shown
that the minimum width of a scheme with complex eigenvalues is 6. Furthermore we defined a scheme
achievingthis minimum withsimple coefficients. Numericalcomparisonofcomplexeigenvalueschemeswith
twowell-knownschemesshowedthatthemainreasonwhycomplexeigenvalueschemeshavebeenoverlooked
or disregardedis the appearance of ripples or artefacts. They appear due to the lack of monotonicity of the
dynamical system defined by iteration of the local subdivision matrix A.
We recommend to consider the use ofcomplex eigenvalueschemes in computer-aidedgeometric designif
thereisaneedforcurvesorobjectswithroughsurfaces. Ifthescalingfactorsofcomplexconjugateeigenval-
ues and the magnitude of real negative eigenvalues is small we expect a limiting curve with small artefacts.
This canbe usedto scalehowpronouncedthe artefactsshouldappear. Note thatwecanalwaysconstructa
surfacesubdivisionschemefromacomplexeigenvaluecurveschemebyusingthetensorproductofthemask.
Therefore we can hope to use these schemes to define geometric objects which do not appear completely
smoothorwithsharptransitionsandcloseagapinthecurrentpracticalapplicationsofsubdivisionschemes.
References
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453., 1991
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School 2005, http://www.subdivision-summerschool.uni-kl.de/programme.html
[4] L. Kobbelt, √3 subdivision, SIGGRAPH proceedings, 2000
[5] P. Schr¨oder and D. Zorin, Subdivision for Modeling and Animation. Course Notes ACM SIGGRAPH,
2000
[6] J. Warren and H. Weimer, Subdivision Methods for Geometric Design: A Constructive Approach, Mor-
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