Compactly Supported Multivariate Wavelet Frames Obtained by Convolution Martin Ehler ∗ Philipps–University of Marburg Department of Mathematics and Computer Science Hans–Meerwein–Strasse 35032 Marburg / Germany
[email protected]
August 12, 2005
Abstract In this paper, we construct pairs of compactly supported dual wavelet frames for general dilations. We obtain smooth wavelets with small supports by convolution. Our approach leads to wavelets that have a high number of vanishing moments. Moreover, the wavelet system reaches the approximation order of the corresponding multiresolution analysis. The refinable function is interpolating and the underlying filter bank provides perfect reconstruction. The wavelets for the quincunx dilation matrix in our examples additionally have very high symmetries. AMS subject classification: 65T60, 42C40, 42C15, 41A63 Key words: wavelet frames; vanishing moments; interpolating refinable functions; multiresolution analysis.
1
Introduction
Wavelet analysis and fast wavelet algorithms are widely used in applied mathematics. Wavelets are applied in image and signal analysis as well as for the numerical treatment of operator equations. In general, the construction of compactly supported wavelets is based on the construction ∗
The author acknowledges the financial support provided through the European Union’s Human Potential Programme, under contract HPRN–CT–2002–00285 (HASSIP)
1
of a refinable function φ. That means that φ satisfies the refinement equation X φ(·) = ak φ(M · −k) k∈Zd
where M is an expanding integer matrix and (ak )k∈Zd is a finite sequence. The wavelets are defined by finite linear combinations of φ(M · −k) for k ∈ Zd . In many applications one is interested in refinable functions φ that in addition are interpolating. That means that φ is continuous and satisfies φ(k) = δ0,k , for all k ∈ Zd . Interpolating refinable functions are, for example, used in subdivision schemes for computer aided geometric design. Moreover, if φ is interpolating and a function f can be represented by X f (x) = ck φ(x − k) , for all x ∈ Rd , k∈Zd
then the coefficients are given explicitly by ck = f (k) for all k ∈ Zd . The smoothness of the refinable function is another desired property because it leads to high approximation order of the corresponding multiresolution analysis, see [15, 16, 33] for details. The construction by finite linear combinations leads to smooth wavelets that are, for instance, required for the characterization of smoothness spaces. In the frame setting, in contrast to orthogonal wavelets, the approximation order of the wavelet system can lag behind the approximation order of the multiresolution analysis. Let us say a wavelet ψ has L vanishing moments if Z xα ψ(x)dx = 0 , for all α ∈ Nd with |α| < L . Rd
Vanishing moments influence the approximation order of the wavelet system. See [19] for a detailed discussion of the connection of vanishing moments and approximation order. Moreover, vanishing moments lead to high compression rates, see for example [2, 17]. Small supports of the wavelets are very important in many applications. Another desired feature is the perfect reconstruction of the underlying filter bank. In that case, the wavelet algorithms can be applied without any inconvenient deconvolutions at the end of the reconstruction process. Moreover, for applications as for instance in image and signal analysis, symmetric wavelets are often preferable, see for example [34]. In this paper, we construct wavelets that simultaneously have all of the above mentioned properties. Arbitrarily smooth, compactly supported orthogonal wavelet bases in one dimension were constructed by I. Daubechies, see [17] for details. A general multivariate construction method for arbitrarily smooth orthogonal wavelets with compact support for arbitrary expanding integer matrices is still not known. The construction already seems to be very complicated in R2 for the quincunx dilation matrix. In [1], arbitrarily smooth orthogonal wavelets for R2 have been constructed. However, the quincunx matrix is not treated there and neither it is in [25]. In [12, 29], orthogonal wavelets for the quincunx matrix are constructed, but differentiability is not achieved. The wavelets in [5, 21] do not have compact support. It should be mentioned that it is not possible to construct orthogonal wavelets with compact support that satisfy 2
certain symmetry conditions. Moreover, already in dimension one, no orthogonal wavelets that have compact support and whose refinable function is interpolating can be constructed. In recent years, one could overcome that restriction with the concept of multi–wavelets. Smooth orthogonal multi–wavelets whose refinable function vector is interpolating have successfully been constructed in dimension one, see [40] for example. A general biorthogonal construction principle that leads to symmetric wavelets is known, see [7, 26] for example. However, as a result of constructing smoother biorthogonal wavelets, their support increases dramatically. This is problematic in many applications. One can overcome these difficulties with the concept of wavelet frames. By skipping the geometrical orthogonality or biorthogonality conditions, we obtain more freedom for the construction of the wavelets. In contrast to bases, frames do not necessarily lead to unique expansions. On the one hand, these redundancies increase the amount of data for expansions but, on the other hand, besides the additional freedom, redundancy is advantageous in many applications, as for instance in denoising strategies or pattern recognition. In general, the construction of wavelet frames is based on extension principles, see [36, 38] and [19]. The unitary extension principle (UEP) leads to tight wavelet frames. The mixed extension principle (MEP) generalizes the UEP and it leads to pairs of dual wavelet frames. In [19], pairs of dual wavelet frames are called bi–framelets. They are called sibling frames in [10] if the corresponding primal and dual refinable functions coincide. In [10, 19], it is shown that, by using the UEP with a b–spline refinable function, one always obtains one wavelet that has only one vanishing moment which means that the approximation order of the wavelet system is at most equal to 2. In the oblique extension principle (OEP) as well as in the mixed oblique extension principle (MOEP), a function Θ is introduced that is called fundamental function in [19] and vanishing moment recovery function in [10]. For Θ = 1, they become the UEP and the MEP, respectively. These oblique extension principles are used in [10, 19] for the construction of tight wavelet frames as well as pairs of dual wavelet frames from a given b–spline and from pseudo–splines to obtain wavelets with a high number of vanishing moments. They only need two wavelets to generate the frame. The construction uses the factorization of trigonometric polynomials in one dimension. In [18], by applying the MOEP, one–dimensional symmetric pairs of dual wavelet frames are constructed from two arbitrary refinable functions with finite masks. In [11], the construction in [10] is generalized to arbitrary integer dilations in one dimension. Symmetric or anti–symmetric tight wavelet frames in one dimension are constructed in [39] by using the UEP. In [38], the box–spline matrix is treated by the UEP for the construction of two–dimensional wavelets. The symbols themselves consist of tensor products while the wavelets are nonseparable. They obtain arbitrarily smooth wavelets with a high number of vanishing moments but the number of wavelets increases with growing smoothness. Moreover, a general inductive algorithm is presented for the construction of tight wavelet frames. This algorithm increases the regularity but there is always one wavelet that still has only the same number of vanishing moments as the starting wavelets. The inductive algorithm in [38] is used in [22] for the construction of arbitrarily smooth tight wavelet frames for all dimensions and arbitrary dilations. Again, the number of wavelets increases with growing smoothness. The special choice of the starting wavelets in [22] implies that there is always one wavelet with only one vanishing moment. Furthermore, it can be shown that the corresponding refinable function cannot have stable integer shifts by using their construction. In [24], it has been shown that 3
one can obtain arbitrarily smooth tight wavelet frames in any dimension. Starting with one– dimensional wavelet frames, multivariate wavelet frames are obtained by tensor products and a linear transform. This approach leads to tight wavelet frames with a high number of vanishing moments. Unfortunately, one obtains separable wavelets for the box–spline dilation matrix. In case of the quincunx dilation matrix, high smoothness leads to a large number of wavelets or to large supports. This approach does not lead to symmetric wavelets in general. While the constructions in [10, 11, 18, 19, 39] are one–dimensional, we construct multivariate pairs of dual wavelet frames for arbitrary dilations. In contrast to the UEP and the MEP, the oblique extension principles do not lead to an underlying filter bank that provides perfect reconstruction. In our inductive approach, we use the MEP. Therefore, on the one hand, we obtain a filter bank with perfect reconstruction. On the other hand, we can start with symbols that satisfy the necessary conditions for the construction of biorthogonal wavelet bases instead of the more restrictive orthogonal conditions needed in [38]. Thus, with our approach, obtaining symmetric starting symbols with a high number of vanishing moments is well–known and painless, see for example [26]. It is worth mentioning that we do not assume any smoothness of the corresponding starting wavelets. They can be tempered distributions that are not contained in L2 (Rd ). Therefore, we are able to start with filters that have small supports. Furthermore, in contrast to the necessary factorization in [10, 19], we only need to multiply these symbols by themselves. Thus, we can avoid the problem of factorizing multivariate trigonometric polynomials that would arise by generalizing the one–dimensional constructions in [10, 19] to the multivariate setting. This multiplication process does not lead to a tight wavelet frame but it leads to a pair of dual wavelet frames. Nevertheless, we are very close to tight wavelets because our primal and dual functions coincide. They only differ by their labelling. In other words, our dual frame is a permutation of the primal one. The corresponding refinable function is interpolating and we achieve high regularity as well as a high number of vanishing moments for all wavelets with small supports. Contrary to [22], the approximation order of our wavelet system reaches the highest possible order that is offered by the underlying multiresolution analysis. It should be mentioned that we do not obtain interorthogonality as they do in [10]. Overcoming the disadvantages of [22, 38], with our approach, the number of wavelets does not depend on the smoothness. It only depends on the modulus m of the determinant of the dilation matrix. The number of wavelets is m2 − 1. Thus, in dimension one by dilating with 2, we obtain three wavelets. That means, we have only one wavelet more than in [10, 19]. In our examples, the whole construction is applied to the case of the quincunx dilation matrix. Additionally, in contrast to [24], we obtain very symmetric wavelets because our construction preserves the symmetries of our starting functions. As far as we know, no existing approach leads to all these properties. The construction in this paper leads to a general construction algorithm. The only input is a compactly supported interpolating refinable function. Finally, the output is a pair of dual wavelet frames. Furthermore, we give another construction method in this paper to reduce the number of the wavelets. We obtain a pair of dual wavelet frames that is generated by 2m − 2 primal as well as dual wavelets. In that case, primal and dual functions differ and the corresponding refinable functions do not have stable integer shifts. In dimension two with dilation matrix 2I2 , we obtain only six wavelets. Thus, we have two wavelets less than by applying the tensor product construction to the wavelet frames in [10, 19]. In the last example, we apply the construction 4
for the reduced number of wavelets to a three–direction box–spline. The outline of this paper is as follows: in Section 2, we briefly recall the definition of wavelet frames and the multiresolution analysis. Moreover, the mixed extension principle is presented, see [37]. In Section 3, we state our construction principle that is based on the MEP and we present a construction method for these starting symbols, see [26]. Then, we verify the approximation order of the constructed wavelet system. In Section 4, we apply our wavelet frame construction by using the starting symbols from Section 3. We give concrete examples for the quincunx dilation matrix and the dilation matrix 2I2 .
2
General Setting
In this section, we recall wavelet frames, the multiresolution analysis and the refinement equation. Furthermore, we state the MEP as derived in [37], that we shall apply to the construction of pairs of dual wavelet frames in the following sections.
2.1
Wavelet Frames
A wavelet basis consists of a finite number of functions whose dilates and translates form a basis of L2 (Rd ). Let us specify: an integer matrix M is called expanding if all its eigenvalues are larger than one in modulus. Let M be an expanding integer matrix throughout the paper. We call it the dilation matrix and set m := |det(M )|. Let ΓM with 0 ∈ ΓM and Γ∗M with −1 0 ∈ Γ∗M be a complete set of representatives of M t Zd /Zd and Zd /M Zd , respectively. For a function ψ ∈ L2 (Rd ) and j ∈ Z, k ∈ Zd , we set j
ψj,k (·) := m 2 ψ(M j · −k) .
(1)
Moreover, we say that {ψ 1 , . . . , ψ r } ⊂ L2 (Rd ) generates a wavelet basis if © µ ª ψj,k | 1 ≤ µ ≤ r, j ∈ Z, k ∈ Zd is a basis of L2 (Rd ). The concept of wavelet bases seems to be very restrictive for the construction of wavelets. Thus, in the following, we weaken the classical basis concept to obtain more freedom in the construction methods. Let I be a countable index set. A set {fi | i ∈ I} in a Hilbert space H is called a frame in H if there exist two positive constants A and B so that X |hf |fi iH |2 ≤ Bkf k2H , for all f ∈ H . Akf k2H ≤ i∈I
If in addition A = B = 1, we call {fi | i ∈ I} a (normalized) tight frame. In this case, every f ∈ H has the expansion X f= hf |fi iH fi . i∈I
5
To obtain expansions for the case A 6= B, we need the concept of dual frames. The two sets {fi | i ∈ I} and {fei | i ∈ I} in H are called a pair of dual frames if both are frames in H and the expansion X f= hf |fei iH fi (2) i∈I
is valid for every f ∈ H. At this point, let ψ µ , ψeµ ∈ L2 (Rd ), for 1 ≤ µ ≤ r. We say that {(ψ µ , ψeµ ) | 1 ≤ µ ≤ r}
(3)
generates a pair of dual wavelet frames if µ {ψj,k | 1 ≤ µ ≤ r, j ∈ Z, k ∈ Zd }
and
µ {ψej,k | 1 ≤ µ ≤ r, j ∈ Z, k ∈ Zd }
are a pair of dual frames in the Hilbert space L2 (Rd ). In this case, by applying (2), one has the wavelet expansion r X X D ¯ E X ¯ eµ µ f= f ¯ψj,k ψj,k , L2 (Rd )
µ=1 j∈Z k∈Zd
for all f ∈ L2 (Rd ). We call {ψ µ | 1 ≤ µ ≤ r} the set of primal and {ψeµ | 1 ≤ µ ≤ r} the set of dual wavelets. Notation (3) is used because the labelling of primal and dual wavelets is very important in our construction methods in the following sections.
2.2
Multiresolution Analysis and the Mixed Extension Principle
At this point, we recall the concept of multiresolution analysis followed by the MEP for the construction of wavelet frames. An increasing sequence (Vj )j∈Z of closed subspaces in L2 (Rd ) is called a multiresolution analysis if the following holds: (i) f (·) ∈ Vj if and only if f (M −j ·) ∈ V0 , T (ii) j∈Z Vj = {0} , S (iii) j∈Z Vj = L2 (Rd ). (iv) There exists a function φ ∈ V0 so that V0 is the closed linear span of its integer shifts. For a given multiresolution analysis, we say that the function φ in (iv) generates it. By (i) and (iv), the sequence (Vj )j∈Z is determined by φ. The multiresolution analysis provides us the structure for the construction of wavelet frames. Our aim is to find functions ψ 1 , . . . , ψ r ∈ V1 so that the closed linear spans of their integer shifts W01 , . . . , W0r in L2 (Rd ) satisfy r X V1 = V0 + W0µ . µ=1
6
Defining the wavelet spaces Wjµ by µ Wjµ := span{ψj,k | k ∈ Zd } ,
for j ∈ Z and 1 ≤ µ ≤ r, leads to the not necessarily direct decompositions Vj+1 = Vj +
r X
Wjµ .
µ=1
By (ii) and (iii), the wavelet system µ {ψj,k | 1 ≤ µ ≤ r, j ∈ Z, k ∈ Zd }
(4)
spans L2 (Rd ) and, under additional assumptions, (4) is also a wavelet frame. Let us have a closer look at the construction of a multiresolution analysis. Very often, one requires that the generator φ has stable integer shifts. That means that there are positive constants A and B so that X Akck`2 (Zd ) ≤ k ck φ(· − k)kL2 (Rd ) ≤ Bkck`2 (Zd ) , (5) k∈Zd
for all c = (ck )k∈Zd ∈ `2 (Zd ). By using φ ∈ V0 ⊂ V1 , (5) and (i), one can show that there exists a sequence (ak )k∈Zd ∈ `2 (Zd ) so that X ak φ(M · −k) . (6) φ(·) = k∈Zd
Eq. (6) is called refinement equation. We will use the refinement equation as a starting point for the construction of a multiresolution analysis and wavelet frames. Some preparations are necessary. As usual, let the Fourier transform of a function f ∈ L1 (Rd ) be defined by Z b f (ξ) = f (x) e−ξ (x) dx , for ξ ∈ Rd , Rd
where ew (·) := e2πihw|·iRd ,
for w ∈ Rd .
Let us assume that (ak )k∈Zd is a finite sequence. By applying the Fourier transform to both sides of the refinement equation (6), one obtains b t −1 ·) b = a(M t −1 ·) φ(M φ(·) where a :=
(7)
1 X ak e−k . m d k∈Z
We call the trigonometric polynomial a a symbol and the finite sequence (ak )k∈Zd the mask b = 1. The or the filter. Let a be a symbol with a(0) = 1 and let φb be continuous in 0 with φ(0) iteration of (7) leads to ∞ Y −k b φ(·) = a(M t ·) . (8) k=1
7
With (8), we are able to construct the opposite way around. We start with a symbol a that satisfies a(0) = 1. Afterwards, we define φ by Eq. (8). The infinite product converges uniformly on compact sets. Therefore, φb is holomorphic. The function φb given by (8) is polynomially bounded and therefore, by applying the Paley–Wiener Theorem, φ is a tempered distribution with compact support, see [17] for details. In the following, we want to use the refinement equation (6) also for tempered distributions. A short overview of operations on tempered distributions can be found in [42]. We will write the translation and the dilation of a tempered distribution in the same way as for functions although pointwise evaluation does not make any sense. Now, for a symbol a, we read the refinement equation (6) more generally in terms of tempered distributions. We call a tempered distribution φ refinable with respect to the trigonometric polynomial a if it satisfies the refinement equation with corresponding mask (ak )k∈Zd . Let a be a symbol with a(0) = 1. It has been shown in [6] b = 1 that is that there exists a unique tempered distribution φ with compact support and φ(0) refinable with respect to a. It is given by (8). In the following, the term refinable distribution, or refinable function in case φ ∈ L2 (Rd ), corresponding to a symbol a with a(0) = 1 always refers to this unique solution in (8). Remark 2.1. Throughout the paper, symbols are trigonometric polynomials. Therefore, their masks are finite sequences. Thus, in this paper, refinable distributions always have compact support. It should be mentioned that the term symbol is often used in a more general way in literature. Let us have a closer look at interpolating refinable functions. We say a symbol a satisfies the interpolation condition if X a(· + γ) = 1 . (9) γ∈ΓM
The following theorem is proved in [31]. Theorem 2.2. A continuous refinable function φ corresponding to a symbol a with a(0) = 1 is interpolating if and only if φ has stable integer shifts and a satisfies the interpolation condition (9). Let us mention a necessary property for smooth refinable functions. We say the refinable function φ satisfies the Strang Fix conditions of order L if b ∂ α φ(k) = 0,
for all k ∈ Zd \{0} and α ∈ Nd , |α| < L.
With increasing smoothness, the refinable function φ satisfies a higher order of Strang Fix conditions, see [28]. Let a be the corresponding symbol. The order of the Strang Fix conditions is equivalent to the reproduction of polynomials of the corresponding shift invariant space. Therefore, it influences the approximation order of the corresponding multiresolution analysis, see [28, 33] for a detailed discussion of smoothness, Strang Fix conditions and approximation order. Now, let us come to the construction of wavelet frames. Given symbols a0 , b0 , . . . , ar , br , we say that {(aµ , bµ ) | 0 ≤ µ ≤ r} satisfies the perfect reconstruction conditions (PR 8
conditions) if
r X
aµ (·) bµ (· + γ) = δ0,γ ,
for all γ ∈ ΓM .
(10)
µ=0
Let {γ0 , . . . , γm−1 } = ΓM with γ0 = 0. We define the modulation matrices for the symbols aµ and bµ by a := ( aµ (· + γν ) )µ=0,...,r
ν=0,...,m−1
and
b := ( bµ (· + γν ) )µ=0,...,r
ν=0,...,m−1
.
It can easily be verified that the PR conditions (10) are equivalent to at b = Im . For r = m − 1, the modulation matrices a and b are quadratic. Therefore, b is the inverse of at and we obtain abt = Im . (11) Eq. (11) is equivalent to
X
aµ (· + γ) bν (· + γ) = δµ,ν ,
(12)
γ∈ΓM
for 0 ≤ µ, ν ≤ m − 1. We say that {(aµ , bµ ) | 0 ≤ µ ≤ r} satisfies condition (I) if the following conditions hold: (a) a0 (0) = b0 (0) = 1. (b) aµ (0) = bµ (0) = 0, for all 1 ≤ µ ≤ r. (c) {(aµ , bµ ) | 0 ≤ µ ≤ r} satisfies the PR conditions. In [37], the next theorem has been shown under an additional Bessel assumption and a e With the results in [24], the Bessel assumption is always certain decay condition on φ and φ. satisfied in our setting because we only work with symbols that are trigonometric polynomials and, thus, they have finite masks. The decay condition could be removed in [4, 9]. Theorem 2.3 (MEP). Let the symbol family {(aµ , bµ ) | 0 ≤ µ ≤ r} satisfy condition (I) and let φ be refinable with respect to a0 and φe be refinable with respect to b0 . In addition, let φ and φe be contained in L2 (Rd ). For µ = 1, . . . , r, we define X µ X µ e ψ µ (·) := ak φ(M · −k) and ψeµ (·) := bk φ(M · −k) . (13) k∈Zd
Then, the set
k∈Zd
{(ψ µ , ψeµ ) | 1 ≤ µ ≤ r}
generates a pair of dual wavelet frames.
9
In Theorem 2.3, we call φ the primal and φe the dual refinable function. By applying the Fourier transform to (13), we obtain the equivalent equations c b t −1 cµ (·) = aµ (M t −1 ·) φ(M b t −1 ·) and ψ eµ (·) = bµ (M t −1 ·) φ(M e ψ ·) .
(14)
Under the assumptions of Theorem 2.3, the refinable functions φ and φe generate two multiresolution analyses (Vj )j∈Z and (Vej )j∈Z , respectively, see [3] for details. Furthermore, for 1 ≤ µ ≤ r and j ∈ Z, we define µ f µ := span{ψeµ | k ∈ Zd } . Wjµ := span{ψj,k | k ∈ Zd } and W j j,k
By applying the PR conditions, one obtains Vj+1 = Vj +
r X
Wjµ
and Vej+1 = Vej +
µ=1
r X
fµ , W j
(15)
µ=1
see [8] for details. In our setting, these decompositions are not necessarily direct.
3
Compactly Supported Wavelet Frames
First, we present our construction method that leads to a pair of dual wavelet frames which is based on multiplying a given family of starting symbols by itself. Next, we describe the construction in [26] that can be applied to the construction of our starting symbols. Furthermore, for a given symbol that satisfies the interpolation condition (9), we characterize all dual symbols. Subsequently, we show that, by applying our wavelet frame construction, the approximation order of the wavelet system reaches the approximation order of the underlying multiresolution analysis.
3.1
Pairs of Dual Wavelet Frames
Our construction of pairs of dual wavelet frames is based on the multiplication of a given family of symbols by itself. Thus, this approach leads to wavelets that are convolutions of tempered distributions. The main results of this section are Theorem 3.2 and Theorem 3.5. First, we need the following lemma. Part (a) is a generalization of Lemma 3.2 in [24] and part (b) generalizes the idea of the inductive construction algorithm given in [22, 38] to dual pairs. Lemma 3.1. Let the symbol family {(aµ , bµ ) | 0 ≤ µ ≤ r} satisfy the PR conditions (10) and let another symbol family {(cν , dν ) | 0 ≤ ν ≤ s} satisfy s X
cν (·) dν (·) = 1 .
ν=0
10
(16)
(a) The family {(aµ cν , dν bµ ) | 0 ≤ µ ≤ r and 0 ≤ ν ≤ s} also satisfies the PR conditions. (b) For 0 ≤ ν ≤ s and 1 ≤ µ ≤ r, we define b0 (·) dν (M t ·) =: ˇbν (·) , bµ =: ˇbs+µ .
a ˇν (·) := a0 (·) cν (M t ·) , a ˇs+µ := aµ , Then, the family
{(ˇ aµ , ˇbµ ) | 0 ≤ µ ≤ r + s}
satisfies the PR conditions. Proof.
(a) Let γ ∈ ΓM . By applying the PR conditions (10) and Eq. (16), we obtain X X X aµ (·) cν (·) dν (· + γ) bµ (· + γ) = cν (·) dν (· + γ) aµ (·) bµ (· + γ) µ,ν
ν
=
X
µ ν
c (·) dν (· + γ) δ0,γ
ν
= δ0,γ . (b) For γ ∈ ΓM , we have
dν (M t (· + γ)) = dν (M t ·)
because dν is Zd –periodic and M t γ is contained in Zd . This leads to r+s X
a ˇ (·) ˇbµ (· + γ) = µ
µ=0
s X
0
ν
a (·) c (M
t
·) b0 (·
+
γ) dν (M t (·
+ γ)) +
aµ (·) bµ (· + γ)
µ=1
ν=0
= a0 (·) b0 (· + γ)
r X
s X
cν (M t ·) dν (M t ·) +
ν=0
r X
aµ (·) bµ (· + γ) .
µ=1
By applying (16) and the PR conditions, we obtain r+s X
a ˇ (·) ˇbµ (· + γ) = a0 (·) b0 (· + γ) + µ
µ=0
r X
aµ (·) bµ (· + γ)
µ=1
= δ0,γ .
Now, let us state our theorem for the construction of pairs of dual wavelet frames that is based on the MEP 2.3. Theorem 3.2. Let the symbol family {(aµ , bµ ) | 0 ≤ µ ≤ r} satisfy condition (I) and let φ ∗ φe be contained in L2 (Rd ). 11
(a) The family {(aµ bν , aν bµ ) | 0 ≤ µ, ν ≤ r}
(17)
satisfies the PR conditions. e the wavelet family (b) With the notation in (13) and ψ 0 = φ and ψe0 = φ, {(ψ µ ∗ ψeν , ψ ν ∗ ψeµ ) | 0 ≤ µ, ν ≤ r ,
µ + ν > 0}
(18)
generates a pair of dual wavelet frames. It is worth mentioning that we do not have to assume φ, φe ∈ L2 (Rd ) in Theorem 3.2. Therefore, we can start with small masks (aµk )k∈Zd and (bµk )k∈Zd . The constructed primal and dual functions coincide but they are differently labelled. That means the dual wavelet frame is a permutation of the primal one. Thus, we are very close to the case of tight wavelet frames. Furthermore, the number of vanishing moments of the starting symbols is preserved. Under the assumptions of Theorem 3.2, for 0 ≤ µ, ν ≤ r, we define Wjµ,ν := span{(ψ µ ∗ ψeν )j,k | k ∈ Zd } , f µ,ν := span{(ψ ν ∗ ψeµ )j,k | k ∈ Zd } . W j Thus, for all 0 ≤ µ, ν ≤ r and j ∈ Z, we have f ν,µ . Wjµ,ν = W j b e The refinable function φ ∗ φe has compact support and we know (φb φ)(0) = 1. With the results 0,0 e in [3], we obtain that φ ∗ φ generates the multiresolution analysis (Wj )j∈Z . Moreover, the PR conditions imply the decompositions X 0,0 Wj+1 = Wjµ,ν , for j ∈ Z, 0≤µ,ν≤r
see [8] for details. Proof of Theorem 3.2. and
(a) This part follows by applying part (a) of Lemma 3.1. With s := r (cν , dν ) := (bν , aν ) ,
for all 0 ≤ ν ≤ r.
Eq. (10) for γ = 0 implies the assumption (16). (b) By applying (14), we obtain b t −1 c e b t −1 ·) φ(M cµ ψ eν )(·) = aµ (M t −1 ·) bν (M t −1 ·) φ(M ·) . (ψ
(19)
Therefore, φ ∗ φe is the primal and dual refinable function with corresponding symbol a0 b0 . We have (a0 b0 )(0) = 1 and (aµ bν )(0) = 0 , for all 0 ≤ µ, ν ≤ r with µ + ν > 0. Further, we have assumed that φ ∗ φe ∈ L2 (Rd ). With (a) and (19), part (b) follows by applying Theorem 2.3.
12
Let us have a look at the constructed wavelet spaces. The following theorem shows the structure of Wjµ,ν . We apply notation (1) also to tempered distributions. Theorem 3.3. Under the notation of Theorem 3.2, let the symbol family {(aµ , bµ ) | 0 ≤ µ ≤ r} satisfy condition (I) and let φ ∗ φe be contained in L2 (Rd ). b (a) Let us suppose that φ is contained in L2 (Rd ) and that φe is bounded. For 0 ≤ µ ≤ r and j ∈ Z, we have ν Wjµ ∗ ψej,0 = Wjµ,ν . b (b) Let r = m − 1 and suppose that φb and φe have no real Zd –periodical zeros. Then, the functions φ ∗ φe and ψ µ ∗ ψeµ , for 1 ≤ µ ≤ m − 1, have stable integer shifts. Under the assumptions of part (b) in Theorem 3.3, let additionally φ ∗ φe be continuous and b be real–valued. Then, (12) implies that a0 b0 satisfies the interpolation condition (9). By applying Theorem 2.2, the refinable function φ ∗ φe is interpolating. 0
For the proof of Theorem 3.3, we need some auxiliary results. For f, g ∈ L1 (Rd ), by a simple substitution, one obtains j
fj,k ∗ gj,0 = m− 2 (f ∗ g)j,k ,
for all j ∈ Z, k ∈ Zd .
(20)
The convolution of a function in L2 (Rd ) with a Schwartz function is again a Schwartz function. Thus, if g is a tempered distribution and f ∈ L2 (Rd ), then the convolution f ∗ g is again a tempered distribution. It can easily be verified that (20) still holds. In order to prove Theorem 3.3, we will also apply the following characterization of functions with stable integer shifts as given in [27]. Theorem 3.4. A compactly supported function φ ∈ L2 (Rd ) has stable integer shifts if and only if φb has no real Zd –periodical zeros. Now, let us prove Theorem 3.3. Proof of Theorem 3.3. map
b (a) The function φe is bounded. Therefore, it can be verified that the ν ν : f 7→ f ∗ ψej,0 · ∗ ψej,0
is a bounded linear map from L2 (Rd ) to L2 (Rd ). This implies ν Wjµ ∗ ψej,0 ⊂ L2 (Rd ) .
By applying Eq. (20), it follows that ν span{(ψ µ ∗ ψeν )j,k | k ∈ Zd } ⊂ Wjµ ∗ ψej,0 .
This leads to
ν Wjµ,ν ⊂ Wjµ ∗ ψej,0 .
13
(21)
ν Now, let f ∈ Wjµ ∗ ψej,0 . Thus, there exists g ∈ Wjµ so that ν . f = g ∗ ψej,0 µ Moreover, there are gn ∈ span{ψj,k | k ∈ Zd } converging to g in L2 (Rd ). By the continuity ν ν of the mapping (21), the functions gn ∗ ψej,0 converge to f = g ∗ ψej,0 . By (20), we have ν gn ∗ ψej,0 ∈ Wjµ,ν .
Now, f is contained in Wjµ,ν because the space Wjµ,ν is closed. Finally, we have also shown ν Wjµ ∗ ψej,0 ⊂ Wjµ,ν .
(b) It follows by Theorem 3.4 that φ ∗ φe has stable integer shifts. Let 1 ≤ µ ≤ m − 1. Again, c cµ ψ eµ by applying Theorem 3.4, the function ψ µ ∗ ψeµ has stable integer shifts if and only if ψ c eµ . cµ ψ has no real Zd –periodical zeros. We assume that ξ ∈ Rd is a Zd –periodical zero of ψ Eq. (12) shows that there exists γ ∈ ΓM with −1
(aµ bµ )(M t ξ + γ) 6= 0 . For all k ∈ Zd , Eq. (14) yields c cµ ψ eµ )(ξ + M t γ + M t k) 0 = (ψ −1 b t −1 e = (aµ bµ )(M t ξ + γ) (φb φ)(M ξ + γ + k) . b −1 It follows that φb φe has a Zd –periodical zero in M t ξ + γ. This is a contradiction to the e Therefore, we have shown that ψ µ ∗ ψeµ has stable integer shifts. stability of φ ∗ φ.
The construction by Theorem 3.2 leads to (r + 1)2 − 1 wavelets. For increasing r, this might be a problem for numerical applications. In the following theorem, we present a construction method that leads to fewer wavelets. It is a generalization of the construction for tight wavelet frames in [22, 38]. Theorem 3.5. Let the symbol family {(aµ , bµ ) | 0 ≤ µ ≤ r} satisfy condition (I) and let φ ∗ φe e we define be contained in L2 (Rd ). With the notation in (13) and ψ 0 = φ and ψe0 = φ, := .. .
1 m
ψ 0 ∗ (ψe0 (M −1 ·)) ,
1 m
Ψr := r+1 Ψ := .. .
1 m 1 m
ψ 0 ∗ (ψer (M −1 ·)) , ψ 1 ∗ ψe0 ,
1 m 1 m
Ψ2r
1 m
ψ r ∗ ψe0 ,
1 m
Ψ0
:=
Then, the wavelet family
ψe0 ∗ (ψ 0 (M −1 ·)) =: .. . 0 r −1 e ψ ∗ (ψ (M ·)) =: ψe1 ∗ ψ 0 =: .. . r 0 e ψ ∗ψ =:
e µ ) | 1 ≤ µ ≤ 2r} {(Ψµ , Ψ
generates a pair of dual wavelet frames. 14
e0 , Ψ er , Ψ e r+1 , Ψ e 2r . Ψ
Proof. For 0 ≤ ν ≤ r and 1 ≤ µ ≤ r, let us define b0 (·) aν (M t ·) =: ˇbν (·) , bµ =: ˇbr+µ .
a ˇν (·) := a0 (·) bν (M t ·) , a ˇr+µ := aµ ,
(22)
e 0 are refinable with respect to a It can easily be shown that Ψ0 and Ψ ˇ0 and ˇb0 and that Ψµ and e µ are primal and dual wavelets corresponding to the wavelet symbols a Ψ ˇµ and ˇbµ , respectively. d The assumption φ ∗ φe ∈ L2 (R ) together with the refinement equation (6) implies that also Ψ0 e 0 are contained in L2 (Rd ). By applying part (b) of Lemma 3.1 with s := r and and Ψ (cν , dν ) := (bν , aν ) , we obtain that
for all 0 ≤ ν ≤ r,
{(ˇ aµ , ˇbµ ) | 0 ≤ µ ≤ 2r}
satisfies the PR conditions. Moreover, it also satisfies condition (I). Thus, the assertion follows by the MEP 2.3. e 0 generates a multiresolution analysis. FurtherIt should be mentioned that Ψ0 as well as Ψ more, we also obtain the decompositions (15). Theorem 3.5 leads to 2r wavelets. Our construction principle in Theorem 3.2 yields (r + 1) − 1 wavelets. However, we pay for this smaller number of wavelets with the loss of stability. e 0 do not have stable integer shifts. As we show in the following lemma, in general, Ψ0 and Ψ e e 0 . But, by applying the refinement equation to φ and to φ, For a0 6= b0 , we also have Ψ0 6= Ψ µ µ d e e the functions Ψ and Ψ , for 0 ≤ µ ≤ r, are linear combinations of (φ ∗ φ)(· − k) for k ∈ Z . 2
Lemma 3.6. Let φ be refinable with respect to the symbol a with a(0) = 1. Moreover, let φ satisfy the Strang Fix conditions of order 1. Then, ( m1 φ(M −1 ·))b has real Zd –periodical zeros. Proof. First, we have
1 b t ·) . φ(M −1 ·))b = φ(M m Therefore, m1 φ(M −1 ·) also satisfies the Strang Fix conditions of order 1. Moreover, it is refinable with respect to a(M t ·). We have that (
a(M t γ) = 1 ,
for all γ ∈ ΓM .
Thus, a(M t ·) does not satisfy the Strang Fix conditions of order 1. In [28], it is shown that if b t ·) has real φ satisfies Strang Fix conditions and the corresponding symbol does not, then φ(M Zd –periodical zeros. More explicitly, by applying the representation (8), it can easily be shown that ΓM \{0} is a set of Zd –periodical zeros of ( m1 φ(M −1 ·))b. We should recall that, in this paper, we only speak of a refinable function φ if it is compactly supported. Thus, φ ∈ L2 (Rd ) already implies φ ∈ L1 (Rd ). In [27], it has been shown that a refinable function φ ∈ L1 (Rd ) always satisfies the Strang Fix conditions of order 1. Thus, by applying Lemma 3.6 and Theorem 3.4, it cannot have stable integer shifts. In general, Theorem 15
3.5 is applied with φ ∈ L2 (Rd ) and φe a tempered distribution. Both functions satisfy the Strang e cannot have stable integer shifts. Fix conditions of order at least 1. Thus, Ψ and Ψ By applying Theorem 3.2, we obtain three wavelets by using a dilation matrix with m = 2. With Theorem 3.5, we could reduce the number of the wavelets to two primal and two dual wavelets. In two dimensions with the dilation matrix 2I2 , one would prefer to apply Theorem 3.5 that leads to six wavelets instead of fifteen by applying Theorem 3.2.
3.2
The Construction of Starting Symbols
By applying Theorem 3.2 to r = m − 1 and under the assumptions of part (b) in Theorem 3.3, we obtain an interpolating refinable function. Furthermore, the choice r = m − 1 minimizes the number of wavelets for a given dilation matrix. Therefore, in this section, we describe the construction of starting symbols {(aµ , bµ ) | 0 ≤ µ ≤ m − 1}. It is based on [26]. We need some notations and auxiliary results. For a given symbol a, we call X Aγ ∗ := aM k+γ ∗ e−k k∈Zd
the subsymbol of a for γ ∗ ∈ Γ∗M . It is well known by character sums that, for k ∈ Zd , we have ( X m , if k ∈ M Zd , ek (γ) = (23) 0, otherwise . γ∈Γ M
Applying (23), one obtains the following relations X Aγ ∗ (M t ·) = eγ ∗ (· + γ) a(· + γ)
(24)
γ∈ΓM
and a(·) =
1 X Aγ ∗ (M t ·) e−γ ∗ (·) . m γ ∗ ∈Γ∗
(25)
M
For a family of symbols {(aµ , bµ | 0 ≤ µ ≤ m − 1}, let Aµγ∗ and Bγµ∗ be the subsymbols of aµ ∗ and bµ , respectively. Let {γ0∗ , . . . , γm−1 } = Γ∗M with γ0∗ = 0. The matrices ¡ ¢ ¡ ¢ A = Aµγν∗ µ=0,...,m−1 and B = Bγµν∗ µ=0,...,m−1 ν=0,...,m−1
ν=0,...,m−1
are called the primal and dual polyphase matrices for the symbol family {(aµ , bµ | 0 ≤ µ ≤ m − 1}. By applying (23), it can be verified that the symbol family {(aµ , bµ ) | 0 ≤ µ ≤ m − 1} satisfies the PR conditions (10) if and only if the corresponding polyphase matrices satisfy ABt = mIm ,
(26)
see [26] for details. For r = m − 1, the PR conditions are equivalent to (12). Therefore, b0 has to satisfy X a0 (· + γ) b0 (· + γ) = 1 . γ∈ΓM
16
We call such a symbol b0 dual to a0 . Thus, b0 is dual to a0 if and only if a0 b0 satisfies the interpolation condition (9). For a symbol a0 that satisfies the interpolation condition, an explicit construction method for b0 is given in [20, 26]. Moreover, in the next section, we give a complete characterization. Therefore, we focus on symbols a0 that satisfy the interpolation condition. Let us assume we are given a symbol a0 that satisfies the interpolation condition (9) and a dual symbol b0 with corresponding subsymbols A0γi∗ and Bγ0i∗ , respectively. It can be shown that a symbol a0 satisfies the interpolation condition (9) if and only if A00 = 1 ,
(27)
see for example [13]. In this case, a construction algorithm for the whole family of starting symbols {(aµ , bµ ) | 0 ≤ µ ≤ m − 1} is given in [26]: the matrix . . . A0γm−1 1 A0γ1∗ A0γ2∗ ∗ 0 . e . (28) A= mIm−1 . .. . 0 is invertible. By defining
m 0 ... ... 0 −A0 ∗ γ1 .. e , . Im−1 B= .. . 0 −Aγm−1 ∗
we obtain
e t = mIm . eB A
These polyphase matrices correspond to a symbol family {(e aµ , ebµ ) | 0 ≤ µ ≤ m − 1} that satisfies the PR conditions with e a0 = a0 and the trivial dual symbol eb0 = 1. By orthogonalizing e to the vector of subsymbols of b0 , see [26] for details, we obtain the following the rows of A primal and dual polyphase matrices 1 A0γ1∗ A0γ2∗ ... A0γm−1 ∗ −Bγ01∗ A0γm−1 ∗ −Bγ01∗ m − Bγ01∗ A0γ1∗ −Bγ01∗ A0γ2∗ . . . .. . . 0 0 . . . (29) A= . −Bγ2∗ Aγ1∗ . . . . . .. .. .. .. −Bγ0m−1 A0γ1∗ ... . . . m − Bγ0m−1 A0γm−1 −Bγ0m−1 ∗ ∗ ∗ ∗ and
B00
−A0γ1∗ B= .. . −A0γm−1 ∗
Bγ01∗
... Im−1
17
Bγ0m−1 ∗
.
(30)
It can easily be verified that ABt = mIm . Now, the symbol family {(aµ , bµ ) | 0 ≤ µ ≤ m − 1} can be reconstructed by (25). For 1 ≤ µ ≤ m − 1, we obtain ´ X ³ aµ (·) = e−γµ∗ (·) − a0 (·) e−γµ∗ b0 (· + γ) (31) γ∈ΓM
and 1 bµ (·) = m
à e−γµ∗ (·) −
X ³
´
!
e−γµ∗ a0 (· + γ)
.
(32)
γ∈ΓM
It should be mentioned that the procedure described above inherits the idea of the lifting scheme in [41]. The symbol e a0 = a0 and the wavelet symbols eb1 , . . . , ebm−1 remain unchanged. We only modify eb0 as well as the wavelet symbols e a1 , . . . , e am−1 . Indeed, the polyphase matrices in [30], which are constructed by means of the lifting scheme, are very similar to (29) and (30). Let us mention that, given a symbol a0 and a dual symbol b0 , it is only necessary to find an e whose first row consists of the subsymbols of a0 . Then, one can apply the invertible matrix A e in (28) is not needed. orthogonalization process. The special structure of A
3.3
A Characterization of Dual Symbols
For the construction of the starting symbols in Section 3.2, we have to find a symbol a0 that satisfies the interpolation condition (9) and a symbol b0 that is dual to a0 . The next theorem gives a characterization of all such dual symbols b0 . Let us say a symbol a satisfies the Strang Fix conditions of order L if ∂ α a(γ) = 0 ,
for all γ ∈ ΓM \{0} and α ∈ Nd , |α| < L .
If the symbol a with a(0) = 1 satisfies the Strang Fix conditions of order L, then the corresponding refinable function also satisfies them, see [28] for details. Let us be given a symbol a and a symbol d that is dual to a. It can easily be verified that, for all symbols c, the symbol X b(·) = d(·) + c(·) − d(·) (ac)(· + γ) (33) γ∈ΓM
is also dual to a. Eq. (33) is the analogon in terms of symbols of the non–orthogonal projection formula for pseudoframes obtained in Corollary 2 of [32]. Now, let us suppose that a satisfies the interpolation condition (9). Then, d = 1 is a dual symbol and (33) reduces to X (34) b(·) = 1 + c(·) − (ac)(· + γ) . γ∈ΓM
The construction of b in (34) follows the spirit of the lifting scheme. Starting with the trivial dual symbol d = 1, we construct another dual symbol b with hopefully better properties. We obtain the following characterization by involving the Strang Fix conditions. 18
Theorem 3.7. Let a and c be symbols satisfying the interpolation condition (9) and the Strang Fix conditions of order L. (a) The symbol b(·) = c(·) + 1 −
X
(ca)(· + γ)
(35)
γ∈ΓM
is dual to a and satisfies the Strang Fix conditions of order L. (b) Every symbol b that is dual to a and satisfies the Strang Fix conditions of order L is of the form (35). It should be mentioned that, for m = 2 and c = a, the mask of b in (35) coincides with the smallest nontrivial dual masks constructed in [20, 26]. To prove Theorem 3.7, we need some auxiliary results. Let Aγ ∗ and Bγ ∗ be the subsymbols of a and b, respectively. By applying (23), (24) and (25), it can be verified that b is dual to a if and only if X Aγ ∗ Bγ ∗ = m . (36) γ ∗ ∈Γ∗M
Let a satisfy the interpolation condition. Thus, by applying (27), the duality condition (36) reduces to X Bγ ∗ Aγ ∗ . (37) B0 = m − γ ∗ ∈Γ∗M \{0}
Our characterization is based on the next theorem that has been shown in [28]. In the following, let ΠL the space of all polynomials of total degree less or equal to L. Theorem 3.8 (Sum Rules). A symbol a satisfies the Strang Fix conditions of order L if and only if X X aM k p(M k) = aM k+γ ∗ p(M k + γ ∗ ) , for all p ∈ ΠL−1 and γ ∗ ∈ Γ∗M . (38) k∈Zd
k∈Zd
The conditions (38) are called sum rules. In the following, we restate these sum rules in terms of subsymbols and incorporate the conditions for interpolation and duality. P We define the semi–convolution of a trigonometric polynomial Q = k∈Zd fk e−k with an arbitrary sequence (hk )k∈Zd by Q ? (hk )k∈Zd := (fk )k∈Zd ∗ (hk )k∈Zd . For a polynomial p and γ ∗ ∈ Zd , we set ¡ ¢ pγ ∗ := p(k + M −1 γ ∗ ) k∈Zd .
(39)
The following corollary shows that the Strang Fix conditions of a symbol can be expressed in terms of convolutions of the corresponding subsymbols with polynomial sequences. 19
Corollary 3.9. A symbol a satisfies the Strang Fix conditions of order L if and only if the corresponding subsymbols satisfy Aγ ∗ ? p0 = A0 ? pγ ∗ ,
for all p ∈ ΠL−1 and γ ∗ ∈ Γ∗M .
(40)
Proof. 1. Let a satisfy the Strang Fix conditions of order L and let p ∈ ΠL−1 . Furthermore, let l ∈ Zd be given. With the notation q l (·) := p(l − M −1 · +M −1 γ ∗ ) ∈ ΠL−1 , we obtain (Aγ ∗ ? p0 ) (l) =
X
aM k+γ ∗ p(l − k)
k∈Zd
=
X
aM k+γ ∗ q l (M k + γ ∗ ) .
k∈Zd
By applying (38), it follows that (Aγ ∗ ? p0 ) (l) =
X
aM k q l (M k)
k∈Zd
=
X
aM k p(l − k + M −1 γ ∗ )
k∈Zd
= (A0 ? pγ ∗ ) (l). 2. Let us assume that (40) is valid and let p ∈ ΠL−1 be given. We define q(·) := p(−M · +γ ∗ ) ∈ ΠL−1 . Thus, we obtain X
aM k p(M k) =
k∈Zd
X
aM k q(−k + M −1 γ ∗ )
k∈Zd
= (A0 ? qγ ∗ ) (0) . By applying (40), it follows that X
aM k p(M k) = (Aγ ∗ ? q0 ) (0)
k∈Zd
=
X
aM k+γ ∗ q(−k)
k∈Zd
=
X
k∈Zd
Now, Theorem 3.8 implies the assertion.
20
aM k+γ ∗ p(M k + γ ∗ ).
In the next corollary, we show the consequences of Corollary 3.9 for symbols that satisfy the interpolation condition and their duals. It is already implicitly used in [30]. We present a proof here because we obtain it directly from Corollary 3.9. Corollary 3.10. Let a be a symbol that satisfies the interpolation condition (9). (a) The symbol a satisfies the Strang Fix conditions of order L if and only if A γ ∗ ? p0 = p γ ∗ ,
for all p ∈ ΠL−1 and γ ∗ ∈ Γ∗M .
(41)
(b) Let a satisfy the Strang Fix conditions of order L and let the symbol b be dual to a. Then, b satisfies the Strang Fix conditions of order L if and only if Bγ ∗ ? p0 = pγ ∗ ,
for all p ∈ ΠL−1 and γ ∗ ∈ Γ∗M .
(42)
Proof. (a) The subsymbol A0 of a symbol a that satisfies the interpolation condition is given by A0 = 1. Thus, (41) follows from Corollary 3.9. (b) Let p ∈ ΠL−1 and γ ∗ ∈ Γ∗M . It has been shown in [30] that Aγ ∗ ? p0 = pγ ∗
if and only if Aγ ∗ ? p0 = p−γ ∗ .
(43)
P Let (40) be valid. By multiplying two trigonometric polynomials Q = k∈Zd fk e−k and e = P d fek e−k , we obtain again a trigonometric polynomial QQ e whose coefficient Q k∈Z sequence is given by (fk )k∈Zd ∗ (fek )k∈Zd . The classical convolution is associative. Thus, for the semi–convolution, it follows that ¢ ¢ ¡ ¡ Bγe∗ Aγe∗ ? pγ ∗ = Bγe∗ ? Aγe∗ ? pγ ∗ . By applying (37), we obtain X
B0 ? p γ ∗ = m ? p γ ∗ −
¡
¢ Bγe∗ Aγe∗ ? pγ ∗ .
γ e∗ ∈Γ∗M \{0}
Moreover, (41), (43) and Corollary 3.9 lead to X
B0 ? pγ ∗ = mpγ ∗ −
Bγe∗ ? pγ ∗ −eγ ∗
γ e∗ ∈Γ∗M \{0}
X
= mpγ ∗ −
B0 ? p γ ∗
γ e∗ ∈Γ∗M \{0}
= mpγ ∗ − (m − 1)B0 ? pγ ∗ . It follows that B0 ? p γ ∗ = p γ ∗ . By applying Corollary 3.9, we obtain (42). 21
Now, let (42) be valid. Again, we obtain B0 ? pγ ∗ = mpγ ∗ −
X
Bγe∗ ? pγ ∗ −eγ ∗
γ e∗ ∈Γ∗M \{0}
= mpγ ∗ − = pγ ∗ .
X
pγ ∗
γ e∗ ∈Γ∗M \{0}
Thus, it follows (40).
Now, we have collected all the ingredients for the proof of Theorem 3.7. Proof of Theorem 3.7. (a) For γ ∗ ∈ Γ∗M , let Aγ ∗ and Cγ ∗ be the subsymbols of a and c, respectively. We define a symbol d with corresponding subsymbols Dγ ∗ by X D0 := m − Aγ ∗ Cγ ∗ γ ∗ ∈Γ∗M \{0}
and for γ ∗ ∈ Γ∗M \{0} .
Dγ ∗ := Cγ ∗ ,
By (36), the symbol d is dual to a. Now, Corollary 3.10 yields that d satisfies the Strang Fix conditions of order L. Next, we will show that d = b. Therefore, by applying (25), we reconstruct d from its subsymbols. This leads to X X 1 m− Aγ ∗ (M t ξ) Cγ ∗ (M t ξ) + d(ξ) = Cγ ∗ (M t ξ) e−γ ∗ (ξ) . m ∗ ∗ ∗ ∗ γ ∈ΓM \{0}
γ ∈ΓM \{0}
We apply (24) and C0 = 1: d(ξ) = 1 −
1 m
X
X
e−γ ∗ (ξ + γ e) a(ξ + γ e) eγ ∗ (ξ + γ) c(ξ + γ) + c(ξ) −
γ ∈ΓM γ ∗ ∈Γ∗M \{0} γ,e
= 1 + c(ξ) −
1 1 X − c(ξ + γ) a(ξ + γ e) m m γ,e γ ∈ΓM
Eq. (23) yields
X
X
eγ ∗ (γ − γ e) .
γ ∗ ∈Γ∗M \{0}
eγ ∗ (γ − γ e) = mδγ,eγ − 1 .
γ ∗ ∈Γ∗M \{0}
Therefore, it follows that 1 1 X − e) (mδγ,eγ − 1) c(ξ + γ) a(ξ + γ m m γ,e γ ∈ΓM X 1 X 1 − c(ξ + γ) a(ξ + γ) + e) . = 1 + c(ξ) − c(ξ + γ) a(ξ + γ m γ∈Γ m
d(ξ) = 1 + c(ξ) −
γ,e γ ∈ΓM
M
22
1 m
The interpolation condition (9) for the symbols c and a yields X c(ξ + γ) a(ξ + γ e) = 1 . γ,e γ ∈ΓM
Thus, we obtain d(ξ) = 1 + c(ξ) −
X
c(ξ + γ) a(ξ + γ) .
γ∈ΓM
(b) For the subsymbols Cγ ∗ of a symbol c, we set C0 := 1 and Cγ ∗ := Bγ ∗ ,
for γ ∗ ∈ Γ∗M \{0}.
Therefore, c satisfies the interpolation condition. Corollary 3.10 implies that c also satisfies the Strang Fix conditions of order L. Following the lines of part (a), we obtain the representation (35).
3.4
Approximation Order
In this section, we will show that our wavelet frame construction leads to the highest possible approximation order. That means it reaches the approximation order of the underlying multiresolution analysis. Let us suppose that the dilation matrix M is isotropic and let λ be the modulus of the eigenvalues. We say that a multiresolution analysis (Vj )j∈Z provides approximation order L if, for every f in the Sobolev space W L (L2 (Rd )), dist(f, Vn ) := min{kf − gkL2 (Rd ) | g ∈ Vn } = O(λ−nL ) . While {(ψ µ , ψeµ ) | 1 ≤ µ ≤ r} generates a pair of dual wavelet frames, we define the truncated representation Qn by X D ¯¯ µ E µ Qn (f ) := f ¯ψej,k ψj,k , for all f ∈ L2 (Rd ). 1≤µ≤r k∈Zd , j 0. If M is isotropic, then, by applying Theorem 3.12, the constructed wavelet system {(ψ µ ∗ ψeν , ψ ν ∗ ψeµ ) | 0 ≤ µ, ν ≤ m − 1, µ + ν > 0} provides approximation order 2L. If we apply Theorem 3.5 instead of Theorem 3.2, then the constructed refinable functions e 0 satisfy the Strang Fix conditions of order 2L. It should be mentioned that the Ψ and Ψ corresponding symbols do not. However, the products of primal and dual wavelet symbols a ˇµˇbµ , for 1 ≤ µ ≤ 2m − 2, have vanishing moments of order 2L. For isotropic M , Theorem 3.12 implies that the constructed wavelet system provides approximation order 2L. 0
Thus, the approximation order of the wavelet system reaches the approximation order of the corresponding multiresolution analysis in both cases. Moreover, by applying our construction, the role of primal and dual wavelets can be interchanged in the definition of approximation order.
4
Explicit Constructions
In this section, we explicitly apply Theorem 3.2 and Theorem 3.5. We construct examples of pairs of dual wavelet frames in two dimensions. To obtain wavelet frames which can be used in applications, the number of generating wavelets should be small. Therefore, we use a dilation matrix with a small determinant. Let M be a dilation matrix with m = 2 and let us be given a symbol a0 that satisfies the interpolation condition and the Strang Fix conditions of order L. We choose A = a0 in Theorem 3.7. For simplicity, we denote ΓM = {0, γ} and Γ∗M = {0, γ ∗ }. Thus, we obtain b0 (·) = a0 (·) + 1 − |a0 (·)|2 − |a0 (· + γ)|2 .
(45)
The symbol b0 is dual to a0 and satisfies the Strang Fix conditions of order L. Moreover, we define for t ∈ C 1 a1t (·) := b0 (· + γ) e−γ ∗ (·) and b1t (·) := ta0 (· + γ) e−γ ∗ (·) . t
(46)
It can be shown that the primal and dual polyphase matrices (29) and (30) correspond to the symbol family {(a0 , b0 ), (a1t , b1t )} (47) with t = 1. Therefore, it satisfies the PR conditions. With this result, it can easily be shown that (47) also satisfies the PR conditions for arbitrary t ∈ R. Since a0 and b0 satisfy the Strang Fix conditions of order L, the symbols a1t and b1t have L vanishing moments. 25
Now, we apply Theorem 3.2 to the starting symbols (47). With the notation of Theorem 3.2, the functions φ and φe are refinable with respect to a0 and b0 . We start with a smooth refinable function φ. High regularity for the constructed pair of dual wavelet frames can be achieved by applying Theorem 3.2 although we convolute φ with a tempered distribution. Starting with a real–valued symbol a0 that satisfies the interpolation condition (9), Eq. (45) yields b0 = a0 (3 − 2a0 ) . Therefore, we obtain φe = φ ∗ η with a tempered distribution η that is refinable with respect to 3 − 2a0 . This leads to φ ∗ φe = φ ∗ φ ∗ η . By the convolution product φ ∗ φ, we increase the smoothness and, by the convolution with the tempered distribution η, we do not decrease the regularity too much. At this point, we are able to apply Theorem 3.2 to construct concrete examples of smooth pairs of compactly supported dual wavelet frames. The constructed wavelets have a high number of vanishing moments and the corresponding refinable function is even interpolating. Moreover, the wavelets as well as the refinable function satisfy several symmetry conditions. We choose the quincunx dilation matrix µ ¶ 1 −1 M= , 1 1 which satisfies m = 2. In the next two examples, we start with a symbol a0 that satisfies the interpolation condition. Next, we define the starting symbol b0 by (45). By denoting γ ∗ = (1, 0)t , the wavelet symbols a1 = a1t and b1 = b1t are defined by (46) for t = 2. We observed that the choice t = 2 reduces the L2 (Rd ) distance of φ ∗ ψe and ψ ∗ φe in the following examples. We apply Theorem 3.2 to these starting symbols. The smoothness of the refinable function in the following examples was estimated by an implementation of the Villemoes algorithm, cf. [43, 44]. Example 4.1. As starting symbol a0 , we choose the Laplace symbol µ ¶ 1 1 1 1 1 0 a = 1 + e−(1,0)t + e(1,0)t + e−(0,1)t + e(0,1)t . 2 4 4 4 4 It has been shown in [12] that a0 satisfies the Strang Fix conditions of order 2 and the interpolation condition (9). The corresponding refinable function φ is interpolating and contained in the H¨older class C α (R2 ) for α = 0.6. The refinable distribution φe is not contained in L2 (R2 ), but the convolution product φ ∗ φe is contained in the H¨older class C α (R2 ) for α = 1.3. It is interpolating and satisfies the Strang Fix conditions of order 4. By applying (18), we obtain wavelets φ ∗ ψe and ψ ∗ φe that have 2 vanishing moments. The wavelet ψ ∗ ψe has 4 vanishing moments. Thus, the wavelet system provides approximation order 4. Due to the symmetry properties of the corresponding mask, the refinable function is symmetric to the four lines x1 = 0, x2 = 0, x1 = x2 and x1 = −x2 , see [23] for details. The wavelets satisfy the same symmetry conditions e + (−1, 1)t ) is interpolating. See modulo translation. Moreover, it can be verified that (ψ ∗ ψ)(· Figure 1 for the corresponding masks and Figure 2 for the plotted functions. 26
1 − 128
1 − 128
1 − 16
3 − 128
0
3 − 128
0
39 128
0
3 − 128
0
39 128
1
39 128
0
0
39 128
0
3 − 128
3 − 128
0
3 − 128
3 − 128
3 − 128
1 − 128
−1 8
0
1 −8
0
3 4
0
0
1 −8
1 − 16
−1 8
1 − 16
1 − 16
1 − 128
(b) a0 b1
(a) a0 b0 1 1024
1 1024
1 256
0
1 256
1 128
3 512
0
1 − 16
0
1 256
0
17 − 128
0
17 − 128
0
1 256
0
1 − 16
0
185 256
0
1 − 16
0
1 256
0
17 − 128
0
17 − 128
0
1 256
0
1 − 16
0
3 512
1 256
0
1 256
3 512
3 512
1 1024
1 128
3 128
0
3 128
3 128
0
39 − 128
0
3 128
0
39 − 128
1
39 − 128
0
3 128
0
39 − 128
0
3 128
0
3 128
3 128
1 128
1 128
1 1024
(d) a1 b1
(c) a1 b0
Figure 1: The masks of the symbols in Example 4.1 In the second example, we start with a symbol a0 that leads to a smoother refinable function and satisfies a higher order of the Strang Fix conditions. Therefore, the smoothness and the number of vanishing moments of the constructed pair of dual wavelet frames increase. Example 4.2. Let the symbol a0 be given by the mask in Figure 3. It satisfies the interpolation condition (9). In [14], it has been shown that the corresponding refinable function φ is interpolating. The Strang Fix conditions of order 4 are satisfied and φ is contained in the H¨older class C α (R2 ) for α = 1.5. Like in Example 4.1, the refinable distribution φe is not contained in L2 (R2 ) but φ ∗ φe is very smooth. The convolution product is contained in the H¨older class C α (R2 ) for α = 2.9. Furthermore, it is interpolating and satisfies the Strang Fix conditions of order 8. The corresponding wavelets φ ∗ ψe and ψ ∗ φe have 4 vanishing moments. The wavelet ψ ∗ ψe has 8 vanishing moments. Thus, the approximation order of the wavelet system is 8. Like in Example 4.1, the refinable function φ ∗ φe is symmetric to the four lines x1 = 0, x2 = 0, x1 = x2 and x1 = −x2 . All wavelets possess similar symmetry properties and e + (−1, 1)t ) is interpolating. For s ≥ 0, we define (ψ ∗ ψ)(· ♦s := {k ∈ Z2 | |k1 | + |k2 | ≤ s} . The masks of a0 b0 , a0 b1 , a1 b0 and a1 b1 are supported in ♦9 , ♦6 + (1, 0)t , ♦12 + (1, 0)t and ♦9 + (2, 0)t , respectively. We have plotted the corresponding functions in Figure 4. Let us make a short comparison to the biorthogonal construction given in [26]. 27
(a) φ ∗ φe
(b) φ ∗ ψe
(c) ψ ∗ φe
(d) ψ ∗ ψe
Figure 2: Refinable function and wavelets in Example 4.1 28
1 256 9 − 256
0
9 − 256
0
81 256
0
9 − 256
0
81 256
1
81 256
0
9 − 256
0
81 256
0
9 − 256
0
9 − 256
1 256
9 − 256
9 − 256
1 256
1 256
Figure 3: The mask of a0 in Example 4.2 Example 4.3. Let φ be refinable with respect to the symbol a that satisfies the interpolation condition (9). For simplicity, let a be real–valued. The dual symbol b(N ) is defined by b(N ) := aN −1 PN (1 − a) where PN (x) :=
N −1 µ X j=0
(48)
¶ N −1+j j x j
is the B´ezout polynomial. (Here aN −1 means a to the power of N −1. It is not the label.) For all 0 < N ∈ N, the symbol b(N ) is dual to a, see [26] for details. Let us denote the corresponding refinable function by φe(N ) . For the biorthogonal construction, one defines primal and dual wavelets by linear combinations of φ(M · −k)
and
φe(N ) (M · −k) ,
respectively. In our construction, all wavelets are defined by linear combinations of the convolution product (φ ∗ φe(N ) )(M · −k) , which is much smoother than φe(N ) itself. We split ab(N ) into two factors and use them as starting symbols in Theorem 3.2. Let a be the Laplace symbol. It is well known that φe(2) is not contained in L2 (R2 ). To obtain biorthogonal wavelets, one has to choose N at least equal to 5. By choosing N = 4, we split ab(4) into a0 := aa
b0 := aaP4 (1 − a) .
and
While φe(4) is not contained in L2 (R2 ), the refinable function φ ∗ φe(4) is two times differentiable. Moreover, it is interpolating. The symbols a1 and b1 can be defined by (46). Now, a0 and b0 satisfy the Strang Fix conditions of order 4 and a1 as well as b1 has 4 vanishing moments. The constructed symbol family {(a0 , b0 ), (a1 , b1 )} satisfies condition (I). Applying Theorem 3.2, we obtain a pair of dual wavelet frames where all wavelets have 4 vanishing moments. Thus, by applying Theorem 3.12, the constructed wavelet system provides approximation order 8. The wavelets have the same symmetry properties as those in the Examples 4.1 and 4.2. Table 1 shows a comparison of the biorthogonal construction 29
(a) φ ∗ φe
(b) φ ∗ ψe
(c) ψ ∗ φe
(d) ψ ∗ ψe
Figure 4: Refinable function and wavelets in Example 4.2 30
and our wavelet frame construction. The dual refinable function of the biorthogonal construction φe(5) is not differentiable and the mask of its symbol b(5) is supported in ♦8 . With our approach, we used the refinable function φ ∗ φe(4) that is contained in the H¨older class C α (R2 ) for α = 2.5. Its corresponding mask is smaller than the mask of φe(5) . N 2 3 4 5
b(N ) φˆ˜ φe(N ) φˆ˜ φˆ˜ ♦2 φˆ˜ – φ˜ˆ ♦4 φ˜ˆ – ˆ ˆ φ˜ ♦6 φ˜ – ˆ ˆ 0.3 φ˜ ♦ φ˜ C (R2 ) 8
a b(N ) φˆ˜ φ ∗ φe(N ) φˆ˜ φˆ˜ ♦3 φˆ˜ C 1.3 (R2 ) ♦5
C 1.9 (R2 )
♦7
C 2.5 (R2 )
♦9
C 3.0 (R2 )
Table 1: Biorthogonality versus frames
The choice
1 a := (1 + e−(1,0)t )(1 + e(1,0)t ) 4 instead of the Laplace symbol in Example 4.3 leads to less smooth and non–symmetric wavelets but also to smaller filters. Vanishing moments and approximation order remain unchanged. Remark 4.4. Due to the symmetry properties of the masks, we would have obtained the same refinable functions and wavelets in our examples by using the box–spline dilation matrix ¶ µ 1 1 Mb = 1 −1 instead of the quincunx dilation. The examples show that we can construct pairs of compactly supported dual wavelet frames with high smoothness. All wavelets have a high number of vanishing moments and the corresponding refinable function is interpolating. The wavelets as well as the refinable function are very symmetric. Furthermore, the dual wavelet frame is only a permutation of the primal frame. As far as we know, no existing approach leads to all these properties. Finally, let us compare our construction with two–dimensional tensor product wavelets. In [10, 19], tight wavelet frames in one dimension are constructed with 2 wavelets. By tensor products of these wavelets, one can construct a tight wavelet frame in two dimensions with the dilation matrix M = 2I2 . This way, one obtains 8 wavelets. By applying Theorem 3.2, we would obtain 15 wavelets because we have m = 4. To reduce the number of the wavelets, we apply Theorem 3.5. Then, we obtain 6 primal and 6 dual wavelets. Thus, in contrast to 8 high–pass channels for the tensor product wavelets, by applying our construction, we only have 6 high–pass channels in the underlying filter bank. The following example deals with starting symbols that are constructed in [35].
31
Example 4.5. Let M = 2I2 and let n be a natural number. The symbol (49) c(n) = 2−3n (1 + e−(1,0)t )n (1 + e−(0,1)t )n (1 + e−(1,1)t )n ¡1¢ ¡0¢ ¡1¢ leads to the refinable box–spline with the direction vectors 0 , 1 , 1 , where each direction has multiplicity n. In [35], for 2 ≤ n ≤ 8, a symbol d(n) is constructed so that c(n) d(n) satisfies the interpolation condition and the corresponding refinable function has high symmetries and smoothness. Now, let us define the new starting symbol a0(n) := c(n) d(n) . By Theorem 3.7, a dual symbol b0(n) can be constructed that satisfies the Strang Fix conditions of order n. While a0(n) satisfies the interpolation condition, we obtain a family of starting symbols that satisfies the PR conditions. By applying Theorem 3.5, we obtain 6 wavelets. The refinable function satisfies Strang Fix conditions of order 2n and the wavelet system provides approximation order 2n. Acknowledgement: The author would like to thank Stephan Dahlke and Karsten Koch for their helpful comments and accurate review.
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[43] L. Villemoes, Wavelet analysis of refinement equations, SIAM J. Math. Anal. 25 (1994) 1433–1460. [44] http://www.mathematik.uni–marburg.de/˜dahlke/ag–numerik/research/software/.
List of Tables 1
Biorthogonality versus frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
List of Figures 1
The masks of the symbols in Example 4.1 . . . . . . . . . . . . . . . . . . . . . 27
2
Refinable function and wavelets in Example 4.1 . . . . . . . . . . . . . . . . . . 28
3
The mask of a0 in Example 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4
Refinable function and wavelets in Example 4.2 . . . . . . . . . . . . . . . . . . 30
35
