Adv Syst Sci Appl 2017; 3:49–57
Published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/500

Haar wavelets and subdivision algorithms on the plane

Tatyana Zaitseva1∗

1Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia

Abstract: This paper presents a classification of all two-digits Haar systems and tiles on the two-
dimension plane up to affine similarity. We obtain three cases, only two of them (rectangular
and Dragon tile) are well known. In all of them we compute the Hölder regularity in L2 of
corresponding Haar functions. The technique of calculating the Hölder regularity is well known
for univariate wavelets and it was recently expended into multivariate wavelets. These values also
give us the information about the rate of convergence of the corresponding subdivision algorithms
and the rate of convergence of the cascade algorithm of the corresponding Haar decomposition.

Keywords: self-similar tilings, Hölder regularity, Haar wavelets, subdivision schemes

1. INTRODUCTION

The subdivision schemes have been studied in the literature in great detail due to many
applications in the approximation algorithms and in the curve and surface design [6, 7,
11, 13]. These algorithms possess several remarkable properties: linearity, shift-invariance,
computational simplicity. A subdivision algorithm extrapolates a function by its values on a
regular grid. In some sense they are development of de Rham cutting corner algorithm. In
many numerical problems, they are much more effective than spline algorithms, etc. [5,8]. In
this paper, we address the problem of construction of multivariate subdivision algorithms
based on Haar functions in Rd. The procedure of construction of Haar systems is well
elaborated [2, 10]. Every Haar basis is generated by a characteristic function of a tile, which
is a self-similar attractor in Rd, whose translates form a partition of the entire space. Such
an attractor is defined by an expanding integer matrix M and a system of integer points
(”digits”) d0, . . . , dm−1, where m = | detM |. We classify all two-digits Haar systems and
tiles on the two-dimension plane. It turns out, that there are exactly three types of two-digit
flate tiles up to affine similarity. Then we compute the Hölder regularity in L2 for all those
Haar functions. The exponents of Hölder regularity are responsible for the rate of convergence
of the corresponding subdivision algorithms.

In the next section we introduce all notation and formulate basic properties of tiles
and of Haar functions in Rd. Then, in Section 3, we formulate and prove our main result,
the classification theorem. Finally, in Section 4, we compute the Hölder regularity of Haar
functions and make conclusions on the rate of convergence of the corresponding subdivision
algorithms.

∗Corresponding author: zaitsevatanja@gmail.com

http://ijassa.ipu.ru/ojs/ijassa/article/view/500


50 T. ZAITSEVA

2. ATTRACTORS, TILINGS, AND HAAR FUNCTIONS

Let M ∈ Zd×d be an integer dilation matrix all of whose eigenvalues are larger than one
in the absolute value. Let m = | detM |. The matrix M splits the integer lattice Zd into m
equivalence (quotient) classes defined by relation x ∼ y ⇔ y − x ∈MZd. Choosing one
representative di ∈ Zd from each equivalence class, we obtain a set of digits D(M) =
{di : i = 0, . . . ,m− 1}. We always assume that 0 ∈ D(M).

We use the notation 0.d1d2 . . . =
∞∑
i=1

M−idi, di ∈ D(M). Consider the following set

G =

{
∞∑
i=1

M−idi : di ∈ D(M)

}
.

We can construct this set even if D(M) is not a correct set of digits and is just a set of m
integer vectors including the zero vector (we will call it attractor).
By [1, 2], for every expansive integer matrix M and for an arbitrary set of digits D(M), the
set G is a compact set with a nonempty interior and possesses the properties:

1. the Lebesgue measure µ(G) ∈ N;
2. G =

⋃
d∈D(M)

M−1(G+ d), the sets M−1(G+ d) have intersections of zero measure;

3. the indicator function χ = χG(x) of G satisfies the refinement equation

χ(x) =
∑

d∈D(M)

χ(Mx− d), x ∈ Rd;

4.
∑
k∈Zd

χ(x+ k) ≡ µ(G), i.e. integer shifts of χ cover Rd with µ(G) layers;

5. µ(G) = 1 if and only if the function system {χ(·+ k)}k∈Zd is orthonormal.
If µ(G) = 1, thenG is called a tile. The integer shifts of a tile define a tiling. This is a partition
of Rd to integer translates of the tile. This partition is with disjoint interia.

Every tile defines a system of Haar functions, ψ(s)
jk (x) = m

j
2ψ(s)(Mj − k), j, k ∈ Zd,

where s = 1, . . .m− 1; ψ(s), s = 1, . . .m− 1, are generating function of the Haar system.
Each of them is defined as ψ(s)(x) =

√
m(χ(MG)− χ(MG+ ds)). This is a complete

orthonormal system inL2(Rd) [2]. Thus, to classify Haar wavelets on Rd one needs to classify
all attractors. We will do that for the case m = 2, d = 2 in the next section, and then compute
the smoothness of those functions. Let us remark, that the smoothness is equal to the rate of
convergence of the corresponding subdivision algorithms, therefore computing the regularity
give us tight bounds for the rate of convergence.

3. THE CLASSIFICATION THEOREM

Theorem 1:
For m = 2, d = 2, there exist three types of attractors up to affine similarity, those types are:

1) M =

(
0 −2
1 0

)
, D =

{(
0
0

)
,

(
1
0

)}
2) M =

(
1 −2
1 0

)
, D =

{(
0
0

)
,

(
1
0

)}
3) M =

(
1 1
−1 1

)
, D =

{(
0
0

)
,

(
1
0

)}
Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



HAAR WAVELETS AND SUBDIVISION ALGORITHMS ON THE PLANE 51

Thus, every two-digit attractor on the plain is affinely similar to one of those three. We
illustrate these three tiles in Fig. 1, 2, 3 (the bounding square in all figures has vertices
(−1,−1), (−1, 1), (1, 1), (1,−1)). In Fig. 4, 5, 6, we see the plain tilings by those attractors
(the black attractor is initial). This theorem classifies not only all possible attractors but
also all possible types of plain two-digit Haar wavelets and the corresponding subdivision
schemes.

Fig.1: rectangular tile Fig.2: third type of tile Fig.3: Dragon tile

Fig.4: rectangular tile Fig.5: third type of tile Fig.6: Dragon tile

The proof of this theorem will be splitted to several lemmas.
Lemma 1:
If m = 2, d = 2, attractors G1 and G2 with the same dilation matrix M and different sets of
digits D1(M) and D2(M) are affinely similar.

Proof
Since m = 2, we can consider an attractor G0 with matrix M and set of digits D0(M) ={(

0
0

)
,

(
1
0

)}
. It is sufficient to proof that G1 and G0 are affinely similar, then analogously

G2 and G0 are affinely similar and then G1 and G2 are affinely similar.

Let M =

(
u v
w z

)
, D1(M) =

{(
0
0

)
,

(
a
b

)}
. Let X =

(
a x1

b x2

)
, x1 ∈ R, x2 ∈ R.

We will proof that we can choose such x1 ∈ R and x2 ∈ R that XM = MX .
It is equivalent to (

a x1

b x2

)(
u v
w z

)
=

(
u v
w z

)(
a x1

b x2

)
.(

au+ x1w av + x1z
bu+ x2w bv + x2z

)
=

(
au+ bv x1u+ x2v
aw + bz x1w + x2z

)
Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



52 T. ZAITSEVA

{
x1w = bv

bu+ x2w = aw + bz
av + x1z = x1u+ x2v

(3.1)

Suppose that w = 0. Then | detM | = uz = 2.

Case 1: detM = 2. Since u ∈ Z and z ∈ Z, we have
[
u = 2
z = 1

or
[
u = −2
z = −1

or
[
u = 1
z = 2

or[
u = −1
z = −2

. Then trM = ±3, characteristic polynomial is λ2 ± 3λ+ 2 = 0 and it has a root

∓1, then it is not a dilation matrix, the contradiction concludes the proof for this case.

Case 2: detM = −2. Since u ∈ Z and z ∈ Z, we have
[
u = −2
z = 1

or
[
u = 2
z = −1

or[
u = −1
z = 2

or
[
u = 1
z = −2

. Then trM = ±1, the characteristic polynomial is λ2 ± λ− 2 = 0

and it has root ±1, then it is not a dilation matrix.
So w 6= 0 and x1 = bv

w
. From the second equality we have x2 = 1

w
(aw + bz − bu). It

remains to verify the third equality if x1, x2 are chosen as above. Since av + x1z = av +
bvz
w

= bvu
w

+ awv+bzv−buv
w

= x1u+ x2v, we arrive at the third equality in (3.1).
Thus, we find a proper matrix X . Since XM = MX , it follows that XM−i = M−iX

∀i ∈ Z. By construction X
(

1
0

)
=

(
a
b

)
.

G1 =

{∑
i∈I
M−i

(
a
b

)
, I ⊂ N

}
=

{∑
i∈I
M−iX

(
1
0

)
, I ⊂ N

}
=

=

{∑
i∈I
XM−i

(
1
0

)
, I ⊂ N

}
= X

{∑
i∈I
M−i

(
1
0

)
, I ⊂ N

}
= XG0.

Lemma 2:
Let M1 and M2 be integer dilation 2× 2 matrices such that trM1 = trM2, detM1 =
detM2 = ±2 and let D1 and D2 be arbitrary two-digit sets from Z2; then the attractors
G1 and G2 produced by pairs (M1, D1) and (M2, D2) are affinely similar.

Proof
It is easy to prove that eigenvalues ofM1 are different (otherwise the discriminantD = t2 ± 8
of the characteristic polynomial (x2 − tx± 2) has to be zero, which is impossible for integer
t). Then we have a basis of eigenvectors for M1 and M2. Since the assumptions of the lemma
are satisfied, there is X such that XM1 = M2X . This system with four integer coefficients
is degenerate because it has a non-trivial solution X . Then it has a rational solution X0, and
therefore an integer solution X∗. M1 = X−1

∗ M2X∗.

Using Lemma 1 we can choose any sets of digits. Let D(M1) =

{(
0
0

)
,

(
1
0

)}
,
(
a
b

)
=

X∗

(
1
0

)
, a, b ∈ Z, D(M2) =

{(
0
0

)
,

(
a
b

)}
.

G1 =

{∑
i∈I
M−i

1

(
1
0

)
, I ⊂ N

}
=

{∑
i∈I
X−1
∗ M−i

2 X∗

(
1
0

)
, I ⊂ N

}
=

= X−1
∗

{∑
i∈I
M−i

2

(
a
b

)
, I ⊂ N

}
= X−1

∗ G2.

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



HAAR WAVELETS AND SUBDIVISION ALGORITHMS ON THE PLANE 53

Lemma 3:
If m = 2, d = 2, M is a dilation integer matrix, there are six possible pairs of (trM , detM).

Proof
Let t = trM . The characteristic polynomial is x2 − tx± 2 = 0.
Let x1, x2 be eigenvalues of M . Then |x1| > 1, |x2| > 1, |x1 · x2| = 2, and hence |x1| <
2, |x2| < 2. |t| ≤ |x1|+ |x2| < 4. t is an integer.

Case 1: detM = 2: t 6= ±3, because x2 ± 3x+ 2 has root∓1, but M is a dilation matrix,
a contradiction. In this case t = 0,±1,±2, it is easy to construct examples for all these cases.

Case 2: detM = −2: Since x1 · x2 = −2, let x1 < 0, x2 > 0, then −2 < x1 < −1, 2 >
x2 > 1, then −1 < t = x1 + x2 < 1, t is an integer, then only t = 0 is possible, there are
easy examples of dilation M with t = 0.

Now we are ready to prove Theorem 1.

Proof
As we can see from these lemmas, there are six types of tiles with different characteristic
polynomials. But it is easy to see that cases

1) tiles with tr = 0, det = 2 and tr = 0, det = −2 are affinely similar, for example,(
0 −2
1 0

)
,

(
0 2
1 0

)
are both rectangles.

2) tiles with tr = 1, det = 2 and tr = −1, det = 2 are affinely similar, for example,(
1 −2
1 0

)
,

(
−1 −2
1 0

)
.

3) tiles with tr = 2, det = 2 and tr = −2, det = 2 are affinely similar, for example,(
1 1
−1 1

)
,

(
−1 1
−1 −1

)
.

So we can see that there are only three types. We can obtain the fact, that they are affinely
different from their different smoothness, it is calculated in the next section.

4. REGULARITY OF TWO-DIGIT ATTRACTORS

In this section we compute the exponent of regularity of the three types of the Haar wavelets,
classified in Theorem 1. The regularity is important for estimating the rate of convergence
of the cascade algorithm of the Haar decomposition and of the subdivision algorithms in
approximation theory and surface design.

We fix again an expanding matrix M and the set of digits D. They, as we know, define
a unique attractor set K. The characteristic function of this set can also be defined in the
framework of refinement equations. A refinement equation is the linear functional difference
equation with a contraction of the argument by the matrix M . Thus, general refinement
equation has the form

ϕ(x) =
∑
k∈Zd

ckϕ(Mx− k), (4.2)

where ck are prescribed coefficients. Refinement equations have been studied in the literature
in great detail due to countless applications in wavelets, approximation theory, etc. (see
[1, 2, 5, 7]). We are interested in a special case of refinement equations, where all the
coefficients ck are zeros and ones. This case will be referred to as Haar case. Refinement
equations of the Haar case generate scaling function for Haar wavelets. In the univariate case,

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



54 T. ZAITSEVA

there is only one Haar scaling function ϕ(x) = χ[0,1]. In the multivariate case, an arbitrary tile
generate Haar system by its characteristic function used as a scaling function.

Thus, we consider refinement equation (4.2) and suppose that ck = 1 ∀k ∈ D and ck = 0
if k /∈ D, then all solutions of refinement equation are ϕ = λχK [2].

Next part is devoted to calculating the exact Hölder exponent (regularity)

αϕ = sup{α ≥ 0 : ‖ϕ(·+ h)− ϕ‖ ≤ C|h|α,∀h ∈ Rd}
of these solutions ϕ in all three cases. The Hölder exponent is responsible for the rate of
convergence of wavelet expansions as well as of subdivision approximation. It is one of the
most important characteristics of wavelet systems.
For univariate wavelets, it is well known that Hölder exponent can be expressed with spectral
characteristic of special matrices. Recently, this technique was expended into multivariate
wavelets. To attack the computation of Hölder regularity, we first construct these matrices.
They will be matrices build by coefficients of refinement equation.

Let D0 =

{(
0
0

)
,

(
1
0

)}
. We construct the attractor G using M and D0, it is a tile in

these 3 cases.
Let Ω ⊂ Zd be a minimal subset of Zd with the property K ⊂ Ω +G =

⋃
k∈Ω

(k +G). We

denote N = |Ω|.
We denote by Td the transition N ×N matrices (Td)ab = cMa−b+d, a, b,∈ Ω ∀d ∈ D0. In

our case these matrices are Boolean (consist in 0 and 1). There are two matrices, we will call
them T0, T1.

It is easy to see that we can have at most one 1 in column: otherwise, for some
a1, a2, b ∈ Ω, d ∈ D0, d1, d2 ∈ D Ma1 − b+ d = d1,Ma2 − b+ d = d2, then d2 − d1 =
M(a2 − a1), which contradicts the fact that digits are from different classes of equivalence.

We consider the following affine subset of the space RN :

V =

{
ω = {ω1, ω2, . . . , ωN} ∈ RN :

N∑
j=1

ωj = 1

}
.

We denote the linear part of subspace V by

W =

{
ω = {ω1, ω2, . . . , ωN} ∈ RN :

N∑
j=1

ωj = 0

}
.

As we have seen above, all of Td are invariant respectively to V and W .
Let v(x) = (ϕ(x+ k1), . . . , ϕ(x+ kN), ki ∈ Ω) ∈ RN .

Then the refinement equation is equal to v(x) = Tdv(Mx− d), ∀x ∈M−1(G+ d), d ∈ D0.
Let U = span{v(x1)− v(x2) | x1, x2 ∈ G}, n = dimU . Since v(x) ∈ V ∀x ∈ G, it

follows that U ⊂ W and n ≤ N − 1.
Since all of Td are invariant respectively to V and W , we have they are invariant respectively
to U . Then the restrictions Ad = Td|U of the operators Td are well-defined. We have only two
operators in our case, we will call them A0, A1.

From [3, Theorem 7] it follows that αϕ = − log2 (ρ2(A0, A1)), where ρ2(A0, A1) =

lim
m→∞

(2−m ·
∑
σ

‖Aσ(1) . . . Aσ(m)‖2)
1

2m , σ : {1, 2, . . . ,m} → {0, 1}. ρ2 is called L2 joint

spectral radius. We will use the short notation L2 spectral radius.

Let us see an example. Let M =
(

1 −2
1 0

)
, it is the second type from Theorem 1. Let

D =

{(
0
0

)
,

(
2
1

)}
, it is a correct set of digits.

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



HAAR WAVELETS AND SUBDIVISION ALGORITHMS ON THE PLANE 55

In this case we obtain that

Ω =

{(
−1
−1

)
,

(
−1
0

)
,

(
0
−1

)
,

(
0
0

)
,

(
0
1

)
,

(
1
−1

)
,

(
1
0

)
,

(
1
1

)
,

(
2
0

)}
.

As we can see, N = 9.
Further there are illustrations of the process (the bounding square has vertices (−1,−1),

(−1, 1), (1, 1), (1,−1)):

Fig.1: Attractor K Fig.2: Tilings of G Fig.3: How to find Ω

Let T0 be the matrix which corresponds to d =

(
0
0

)
, let T1 correspond to d =

(
1
0

)
.

T0 =



0 0 0 0 0 1 0 0 0
1 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 1
0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 1 0
0 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0


T1 =



0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0
1 0 0 0 0 0 1 0 0
0 1 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 1
0 0 0 1 0 0 0 0 0
0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 1 0


Let v be the eigenvector of T0+T1

2
which corresponds to the value 1 (such an eigenvector

exists because both T0 and T1 have a common left eigenvector (1, 1, . . . , 1)). Then we have
v ∈ V and T0v ∈ V , then it follows that u0 := v − T0v ∈ U .
Using the algorithm from [3, section 3.2], we regard u0 as the first basis vector in U . Then
for all vectors v1 in the current basis we try to complement to basis vectors T0v and T1v
(evidently these vectors are in U ). If we complement vector T0v and vectors in the basis are
still linear independent, then we keep this vector in the basis. The same is done with vector
T1v. We keep doing this for all vectors v1 until our system stays the same.
In our case, T0|U and T1|U are 5× 5 matrices (the coefficients are rounded to three decimal
digits after the point):

A0 = T0|U =


0.0 0.0 −1.044 −0.159 −0.23
0.0 0.0 0.693 0.106 0.153

1.088 0.0 0.214 −0.167 0.256
0.0 0.0 −0.291 −0.044 −0.064
0.0 0.618 −0.29 −0.449 −0.17


Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



56 T. ZAITSEVA

A1 = T1|U =


0.0 0.0 −0.174 −0.425 −0.843

0.817 0.0 −0.354 −0.888 0.56
0.0 0.0 0.282 0.142 0.938
0.0 1.0 0.463 0.34 −0.235
0.0 0.0 −0.36 0.558 −0.622


Now we have A0, A1 and we can find the coefficient of smoothness. There are different

ways to calculate ρ2(A0, A1).
Firstly, we can use formula ρ2 = λmax(

1
2
(A0 ⊗ A0 + A1 ⊗ A1)). Secondly, we can

consider operator AX = 1
2
(AT0XA0 + AT1XA1) in the space of symmetric matrices and then

find its spectral radius: ρ2 = λmax(A ) (these formulas can be found in [4, 12, 14].
But both ways led us to the same result. In our case ρ2 ≈ 0.7607 and αϕ ≈ 0.3946.
We can calculate Hölder regularity of other types of tiles using the same method.

Let M =
(

1 1
−1 1

)
, it is the third type from Theorem 1. We suppose that in this case

D =

{(
0
0

)
,

(
2
1

)}
too.

Here N = 21 and A0,A1 are 5× 5 matrices. We obtain that ρ2 ≈ 0.8478 and αϕ ≈ 0.2382.

In the first case from Theorem 1, we suppose M =
(

0 −2
1 0

)
, D =

{(
0
0

)
,

(
1
2

)}
.

Here N = 18 and A0,A1 are 9× 9 matrices. We obtain that ρ2 ≈ 0.7071 and αϕ = 0.5.
This answer is natural because the Hölder L2 regularity of a charasteristic function of a
polygon is always 0.5.
Remark 1:
We can note, that in the case of Dragon type Hölder regularity is higher than in the case
with tr = 1, this means that the convergence of subdivision schemes in Dragon case is worse
although this case is better known.

5. CONCLUSION

We have classified all two-digit tiles on the plain up to affine similarity. It turned out that
there only three types, two of them (rectangular and Dragon tiles) are well known, the
third one is much less studied. For all the types, we computed the Hölder regularity of the
corresponding characteristic functions. The Hölder exponents are different, which proved that
these classes are also different (0.5 for rectangular, ≈ 0.2382 for Dragon type, ≈ 0.3946 for
third type). We find not only the smoothness of two-digit Haar functions, but also the rate
of convergence of the corresponding subdivision algorithms. Thus, for two-digit plain Haar
system the problem is completely solved. The generalization for bigger dimensions (d ≥ 3)
is possible for two-digit tiles, this is a subject of future research. On the other hand, the
generalization to a bigger number of digits seems to be a hard problem, because already for
three digits there are infinitely many types of flat tiles (but in case, when our digits are d0 = 0,
d1, d2 and det(d1, d2) = ±1, there are 10 types of tiles). The method for computing the L2

Hölder exponent (and correspondingly, the rate of convergence of the subdivision schemes)
is universal and can be applied for every tile, with an arbitrary number of digits and arbitrary
dimension.

ACKNOWLEDGEMENTS

The author is grateful to her supervisor V. Yu. Protasov for posing the problem and
permanent support.

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



HAAR WAVELETS AND SUBDIVISION ALGORITHMS ON THE PLANE 57

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Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)

https://arxiv.org/abs/1702.00269

	Introduction
	Attractors, tilings, and Haar functions
	The classification theorem
	Regularity of two-digit attractors
	Conclusion