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Compactly Supported Multivariate Pairs of Dual Wavelet Frames Obtained by Convolution Martin Ehler
∗
Philipps–University of Marburg Department of Mathematics and Computer Science Hans–Meerwein–Strasse 35032 Marburg / Germany October 26, 2006
Abstract In this paper, we present a construction of compactly supported multivariate pairs of dual wavelet frames. The approach is based on the convolution of two refinable distributions. We obtain smooth wavelets with any preassigned number of vanishing moments. Their underlying refinable function is fundamental. In the examples, we obtain symmetric wavelets with small support from optimal refinable functions, i.e., the refinable function has minimal mask size with respect to smoothness and approximation order of its generated multiresolution analysis. The wavelet system has maximal approximation order with respect to the underlying refinable function. AMS subject classification: 42C40, 42C15 Key words: wavelet frames; dual frame; fundamental refinable function; mixed extension principle.
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 ∗
The author acknowledges the financial support provided through the European Union’s Human Potential Programme, under contract HPRN–CT–2002–00285 (HASSIP)
1
treatment of operator equations. Classically, they are constructed from a refinable function ϕ, i.e., ϕ satisfies X ϕ(x) = ak ϕ(M x − k), k∈Zd
where M is an expanding integer matrix and (ak )k∈Zd is a finitely supported sequence. Fundamental refinable functions, i.e., ϕ is continuous, and it interpolates the integer grid, ϕ(k) = δ0,k , for all k ∈ Zd , are also used in computer aided geometric design. Moreover, if a function f can be represented by X ck ϕ(x − k), for all x ∈ Rd , f (x) = k∈Zd
with ϕ fundamental, then the coefficients are given explicitly by ck = f (k), for all k ∈ Zd . This is advantageous for the application of the wavelet algorithms. Smooth ϕ provides high approximation order of its generated multiresolution analysis, see [9, 10, 31]. In the frame setting, in contrast to orthogonal wavelets, the approximation order of the wavelet system can be less than the approximation order of the underlying multiresolution analysis. It is mainly influenced by their number of vanishing moments, where we say a wavelet ψ has L vanishing moments if Z xα ψ(x)dx = 0, for all α ∈ Nd , |α| < L, Rd
see [12] and Section 2.3 for details. Furthermore, vanishing moments lead to high compression rates, see e.g. [1, 11]. Symmetry and good localization of the wavelets are desired in many applications as for instance in image and signal analysis, see for example [32]. The number of wavelets should be small to bound the complexity of the wavelet algorithms. One–dimensional orthogonal dyadic wavelets have been sucessfully constructed in [11] by applying the Fejer–Riesz Lemma. The wavelets are not symmetric, and the lemma does not hold in higher dimensions. One overcomes these problems constructing biorthogonal wavelets, see [7, 14, 16, 20, 21, 23, 25]. However, the support of the wavelets rapidly grows with increasing smoothness, see for example [3, 21, 22, 23, 24]. The concept of wavelet frames allows for redundancy. It provides enough freedom to overcome all problems mentioned so far, see Section 2.1 for the concepts of tight wavelet frames and wavelet bi–frames. However, we have to pay attention to the number of vanishing moments as described above. Successful one–dimensional wavelet frame constructions are given in [6, 12, 34, 36]. Multivariate constructions are presented in [5, 15, 35], but the wavelets have only one vanishing moment, and their number increases with higher smoothness. In [28], the wavelets have only one vanishing moment, or they are no longer compactly supported. Arbitrary smooth 2
wavelets with an arbitrary number of vanishing moments are constructed in [18], but neither symmetry nor optimality conditions are treated. In this paper, we overcome all problems mentioned above by the construction of wavelet bi–frames. We suppose, we already have two refinable distributions satisfying the necessary conditions for biorthogonality. They can be chosen well localized because they do not need to be in L2 (Rd ). If they are carefully chosen, then their convolution is smooth and results in a fundamental refinable function. From this function, we obtain well localized and smooth wavelets with a high number of vanishing moments. For a dilation matrix M , the approach yields |det(M )|2 − 1 wavelets. Our construction is simple and painless. Surprisingly, it is often optimal. Here, optimality includes two criteria: first, the approximation order of the wavelet bi–frame reaches the approximation order of the underlying multiresolution analysis. This is the highest possible, see Section 2.3. Second, the underlying refinable function has minimal mask size with respect to smoothness and approximation order of its multiresolution analysis. The support of the refinable function can be estimated by the support of its mask, see [2]. If the start distributions are chosen optimal, then optimality is kept invariant under convolution in most of our examples. We obtain an optimal d–dimensional wavelet bi–frame with arbitrary smooth symmetric wavelets and an arbitrary high number of vanishing moments. It is generated by only three wavelets. Then we construct symmetric two–dimensional wavelet bi–frames for the quincunx dilation matrix. Finally, we treat a two–dimensional fundamental box spline. The box spline bi–frame is optimal, and it has significantly smaller support than biorthogonal wavelets. It should be mentioned that the number of wavelets can be reduced to 2|det(M )|−1 at the expense of the loss of a fundamental refinable function and optimality. The outline is as follows: in Section 2, we introduce wavelet frames, the multiresolution analysis, symbols, and the mixed extension principle. Moreover, we recall the concept of approximation order of a wavelet bi–frame. This yields an optimality criteria, and it establishes the importance of vanishing moments. Section 3 is devoted to setting up the main ideas of our construction. We state the convolution method, and we derive start symbols. In Section 4, we recall the treatment of symmetry in terms of symbols, and we consider optimality conditions in terms of the underlying refinable function. Finally, we present our examples of optimal symmetric wavelet bi–frames drawing comparisons to biorthogonal wavelets.
2
General Setting
In this section, we recall the concepts of wavelet bi–frames and their construction from refinable functions, see [4, 11]. Finally, we introduce the approximation order of a wavelet bi–frame as discussed in [12].
3
2.1
Wavelet Frames
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 C1 , C2 such that C1 kf kH ≤ k(hf, fi i)i∈I k`2 (I) ≤ C2 kf kH ,
for all f ∈ H,
(1)
where h·, ·i denotes the inner product in H. Given a frame {fi : i ∈ I}, we have two bounded operators F : H → `2 (I), f 7→ (hf, fi i)i∈I , X F ∗ : `2 (I) → H, (ci )i∈I 7→ c i fi . i∈I
Then S := F ∗ F is called the frame operator. It is boundedly invertible. The system {S −1 fi : i ∈ I} is again a frame, called the canonical dual frame. It gives rise to the expansion X ® f= f, S −1 fi fi , for all f ∈ H. (2) i∈I
If we can choose C1 = C2 in (1), then {fi : i ∈ I} is called a tight frame, and we have S = C11 IdH . Given a nontight frame, the canonical dual does not always have convenient properties, or its computation is problematic. The frame concept allows for redundancy. Thus, there might be further expansions similar to (2): two frames {fi : i ∈ I}, {fei : i ∈ I} in H are called a pair of dual frames (or a bi–frame) if the expansion E XD f= f, fei fi (3) i∈I
is valid, for each f ∈ H. Let us recall the concept of wavelet frames: a matrix M ∈ Zd×d is called a dilation matrix if all its eigenvalues are larger than one in modulus. Let M denote a dilation matrix with m := |det(M )| throughout. For ψ ∈ L2 (Rd ) and j ∈ Z, k ∈ Zd , let ¡ ¢ j ψj,k (x) := m 2 ψ M j x − k . Given a finite set of functions {ψ 1 , . . . , ψ n } ⊂ L2 (Rd ), we denote ¡© ª¢ © µ ª X ψ 1 , . . . , ψ n := ψj,k : j ∈ Z, k ∈ Zd , µ = 1, . . . , n .
(4)
If (4) is a frame in L2 (Rd ), then we call it a wavelet frame. However, the canonical dual might not always have the wavelet structure as well. But possibly, we can replace the canonical dual by a dual wavelet frame. Given ψ µ , ψeµ ∈ L2 (Rd ), µ = 1, . . . , n, then X({ψ 1 , . . . , ψ n }), X({ψe1 , . . . , ψen }) are called a wavelet bi–frame if they are a pair of dual frames in L2 (Rd ).
4
2.2
The Mixed Extension Principle
Let us recall the basics of classical wavelet constructions. An increasing sequence (Vj )j∈Z of closed subspaces in L2 (Rd ) is called a multiresolution analysis if the following holds: i) f (·) ∈ Vj iff f (M −j ·) ∈ V0 , S ii) j∈Z Vj is dense in L2 (Rd ), T iii) j∈Z Vj = {0}, iv) there exists ϕ ∈ V0 such that V0 is the closed linear span of its integer shifts. If ϕ is given, then the sequence (Vj )j∈Z is already completely determined by applying i), iv), and we say ϕ generates the multiresolution analysis. Then ϕ(M −1 ·) is contained in V0 . We are interested in the special case, when there exists a finitely supported sequence (ak )k∈Zd such that ϕ satisfies the refinement equation X ϕ(x) = ak ϕ(M x − k). (5) k∈Zd
Let us define the Fourier transform of f ∈ L1 (Rd ) ∩ L2 (Rd ) by Z f (x)e−2πix·ξ dx. fb(ξ) := Rd
We denote the extension to the class of tempered distributions in the same way. A finitely supported sequence (ak )k∈Zd is called a mask, and its trigonometric polynomial 1 X ak e−2πik·ξ a(ξ) := m d k∈Z
is called a symbol. We suppose a(0) = 1. Then applying the Fourier transform to both sides of (5) yields by iteration that ´ Y ³ −j ϕ(ξ) b = a M> ξ (6) j≥1
is the Fourier transform of a compactly supported solution of the refinement equation (5), see for example [11]. We say a generates ϕ if ϕ b is given by (6). If a generates d ϕ ∈ L2 (R ), then ϕ already generates a multiresolution analysis, see [13]. Remark 2.1. Throughout the paper, symbols are trigonometric polynomials. Therefore, their masks are finitely supported sequences. Thus, our refinable distributions always have compact support in this paper. It should be mentioned that the term symbol is often used in a more general way in literature.
5
Classical wavelet constructions are based on a multiresolution analysis. The closed linear span of the integer shifts of the wavelets is an algebraic complement of V0 in V1 . Thus, the wavelets are contained in V1 . Then they can be represented by linear combinations of ϕ(M x − k), for k ∈ Zd . In the wavelet frame setting, we still use this idea: we start with two refinable functions ϕ and ϕ e contained in L2 (Rd ). 0 0 They are given implicitly by two symbols a and b . The wavelets are defined by X µ X µ bk ϕ(M e x − k), (7) ak ϕ(M x − k) and ψeµ (x) := ψ µ (x) := k∈Zd
k∈Zd
where aµ , bµ are additional symbols, for µ = 1, . . . , n. We need the following conditions to assure that X({ψ 1 , . . . , ψ n }), X({ψe1 , . . . , ψen }) are a wavelet bi–frame: let us say the symbol family {(aµ , bµ ) : µ = 0, . . . , n} satisfies condition (I) if the following holds: (I–a) a0 (0) = b0 (0) = 1. (I–b) aµ (0) = bµ (0) = 0, for all µ = 1, . . . , n. (I–c) For all γ ∈ ΓM ,
n X
aµ (ξ + γ)bµ (ξ) = δ0,γ ,
(8)
µ=0
where ΓM is a complete set of representatives of M −> Zd /Zd . The following mixed extension principle (MEP) has been stated in [6, 12]. It has its roots in the fundamental work [34]. Theorem 2.2 (MEP). Let the symbol family {(aµ , bµ ) : µ = 0, . . . , n} satisfy condition (I), and let a0 , b0 generate refinable functions ϕ, ϕ e ∈ L2 (Rd ), respectively. For µ = 1, . . . , n, define ψ µ , ψeµ by (7). Then X({ψ 1 , . . . , ψ n }), X({ψe1 , . . . , ψen }) are a wavelet bi–frame.
2.3
Approximation Order
This section provides an optimality criteria for wavelet bi–frames. Due to [12], we recall the concept of approximation order. We suppose throughout the present section that M is isotropic, i.e., M can be diagonalized, and all its eigenvalues have the same modulus ρ. Definition 2.3. (i) We say a multiresolution analysis (Vj )j∈Z provides approximation order L if, for every f in the Sobolev space W L (L2 (Rd )), there exists a positive constant Cf such that ª © dist(f, VN ) := min kf − gkL2 (Rd ) : g ∈ VN ≤ Cf ρ−N L .
6
(ii) We say a wavelet bi–frame X({ψ 1 , . . . , ψ n }), X({ψe1 , . . . , ψen }) provides approximation order L if, for all f ∈ W L (L2 (Rd )), there exists a positive constant Cf such that kf − QN (f )kL2 (Rd ) ≤ Cf ρ−N L , (9) where the truncated representation QN is given by E X D µ µ e QN (f ) := . f, ψj,k ψj,k µ=1,...,n j ξ) , a ˇn2 +µ := aµ , Then the family
©¡
ˇbν (ξ) := b0 (ξ)dν (M > ξ) , ˇbn2 +µ := bµ .
(14)
¢ ª a ˇµ , ˇbµ : µ = 0, . . . , n1 + n2
satisfies condition (I). For aµ = bµ = cµ = dµ , (13) becomes Lemma 3.2 in [18] for the construction of tight wavelet frames. For aµ = bµ , and cν = dν , (14) simplifies to the inductive construction algorithm given in [15, 35]. By an ”inverse labelling” of primal and dual symbols in (13) and (14), we have generalized these ideas to pairs of symbols. Proof. The conditions (I–a) and (I–b) are obviously satisfied.
8
(a): Let γ ∈ ΓM . By applying (I–c), we obtain X X aµ (ξ + γ)cν (ξ + γ)bµ (ξ)dν (ξ) = aµ (ξ + γ)cν (ξ + γ)bµ (ξ)dν (ξ) µ,ν
µ,ν
=
X µ,ν
= δ0,γ
aµ (ξ + γ)bµ (ξ)cν (ξ + γ)dν (ξ) X
cν (ξ + γ)dν (ξ)
ν
= δ0,γ . (b): By M > γ ∈ Zd , the definitions (14) yield nX 1 +n2
a ˇµ (ξ
+ γ)ˇbµ (ξ) =
µ=0
n2 X
a0 (ξ + γ)cν (M > ξ)b0 (ξ)dν (M > ξ)
ν=0
+
n1 X
aµ (ξ + γ)bµ (ξ)
µ=1
=
a0 (ξ
0
+ γ)b (ξ)
n2 X
cν (M > ξ)dν (M > ξ)
ν=0
+
n1 X
aµ (ξ + γ)bµ (ξ)
µ=1
=
a0 (ξ
0
+ γ)b (ξ) +
n1 X
aµ (ξ + γ)bµ (ξ)
µ=1
=
n1 X
aµ (ξ + γ)bµ (ξ)
µ=0
= δ0,γ . In the examples of Section 4, we apply part (a) of Theorem 3.2 to two identical symbol families: Corollary 3.2. Let the symbol family {(aµ , bµ ) : µ = 0, . . . , n} satisfy condition (I). Then the family {(aµ bν , bµ aν ) : µ, ν = 0, . . . , n} (15) satisfies condition (I). In the following, we translate Corollary 3.2 into the function setting. Let ϕ = ψ 0 and ϕ e = ψe0 be the refinable distributions with respect to a0 and b0 . We denote the wavelets corresponding to aµ , bµ by ψ µ , ψeµ , respectively. Then ϕ ∗ ϕ e is refinable with respect to a0 b0 . The wavelets associated to aµ bν are ψ µ ∗ ψeν . If ϕ ∗ ϕ e ∈ L2 (Rd ), Theorem 2.2 implies, that X({ψ µ ∗ ψeν : µ, ν = 0, . . . , n, (µ, ν) 6= (0, 0)}) 9
is a frame in L2 (Rd ), and the reconstruction formula E X D µ ν e f= f, (ψ ∗ ψ )j,k (ψ ν ∗ ψeµ )j,k µ,ν=0,...,n (µ,ν)6=(0,0) j∈Z,k∈Zd
holds, for all f ∈ L2 (Rd ).
3.2
Finding Start Symbols
In this section, we treat the problem of finding appropriate start symbols for Theorem 3.1. Given symbols a0 and b0 , a symbol family {(aµ , bµ ) : µ = 0, . . . , n} satisfying condition (I) can be constructed. A symbol a is called interpolatory if X a(ξ + γ) = 1. (16) γ∈ΓM
Let ϕ be refinable with respect to a. If ϕ is fundamental, then a is interpolatory. The converse implication is valid under additional assumptions, see [29] for details. We choose n = m − 1, because it minimizes the number of wavelets for a given dilation matrix. Then (I–c) is equivalent to X aµ (ξ + γ)bµ0 (ξ + γ) = δµ,µ0 . γ∈ΓM
Thus, b0 has to satisfy
X
a0 (ξ + γ) b0 (ξ + γ) = 1 .
(17)
γ∈ΓM
Such a symbol is called dual to a0 . Thus, b0 is dual to a0 iff a0 b0 is interpolatory. Given an interpolatory symbol a0 , an explicit construction method for b0 is obtained in [14, 25], see also [16, 22]. Additionally, a representation of dual symbols is presented in the next section. Given a0 interpolatory and a dual symbol b0 , a construction algorithm for the family of start symbols {(aµ , bµ ) : µ = 0, . . . , m − 1} has been presented in [25]: we obtain, for µ = 1, . . . , m − 1, X ¡ ∗ ∗ ¢ aµ (ξ) = e−2πiγµ ·ξ a0 (ξ + γ) − a0 (ξ)e−2πiγµ ·γ b0 (ξ + γ),
(18)
γ∈ΓM \{0}
bµ (ξ) =
1 −2πiγµ∗ ·ξ e m
X
¡ ∗ ¢ 1 − e−2πiγµ ·γ a0 (ξ + γ),
γ∈ΓM \{0}
10
(19)
∗ where {0, γ1∗ , . . . , γm−1 } is a complete set of representatives of Zd /M Zd . Then the symbol family {(aµ , bµ ) : µ = 1, . . . , m − 1} satisfies condition (I). We still have to verify vanishing moments of the wavelets: let the start symbols a0 , b0 satisfy the sum rules of order L. Then, by applying (18),(19), all start wavelet symbols have L vanishing moments. Note that applying Corollary 3.2 preserves the number of vanishing moments.
3.3
A Representation of Dual Symbols
Let d be a dual symbol to a. It can easily be verified that, for all symbols c, X b(ξ) = d(ξ) + c(ξ) − d(ξ) (ca)(ξ + γ)
(20)
γ∈ΓM
is also dual to a. Eq. (20) is the analogon in terms of symbols of the nonorthogonal projection formula for pseudoframes obtained in Corollary 2 of [30]. We suppose a is interpolatory. Then d = 1 is a dual symbol, and (20) reduces to X (ca)(ξ + γ). b(ξ) = 1 + c(ξ) − (21) γ∈ΓM
The construction of b in (21) follows the spirit of the lifting scheme. From the trivial dual symbol d = 1, we construct another dual symbol b with hopefully better properties. Theorem 3.3. Let an interpolatory symbol a satisfy the sum rules of order L. Then: (a) If an interpolatory symbol c satisfies the sum rules of order L, then X (ca)(ξ + γ) b(ξ) = c(ξ) + 1 −
(22)
γ∈ΓM
is dual to a, and it satisfies the sum rules of order L. (b) Let b be dual to a, and let it satisfy the sum rules of order L. Then à ! X 1 c(ξ) := b(ξ) + 1− b(ξ + γ) m γ∈Γ
(23)
M
is interpolatory, it satisfies the sum rules of order L, and b has the representation (22). Remark 3.4. The CBC algorithm proposed in [16, 22] for the construction of dual symbols additionally deals with noninterpolatory symbols. The results are presented in terms of masks, and the sum rule order is allowed to differ between primal and dual symbols. Theorem 3.3 is a specific result of [16, 22] formulated in terms of symbols. Our restrictions provide the following structural observation in a simple 11
way: recall, if a refinable function has orthogonal integer shifts, then its generating symbol a is orthogonal, i.e., X |a(ξ + γ)|2 = 1. γ∈ΓM
In (22), the natural choice c = a means that b equals a up to the amount which a lacks in order to be orthogonal. Proof of Theorem 3.3. (a) As already mentioned, it can directly be verified that b is indeed dual to a. For interpolatory a and c, the sum rules imply ∂ α a(0) = δ0,α
and ∂ α c(0) = δ0,α ,
for all |α| < L.
(24)
Furthermore, sum rules and (24) imply the sum rules of order L for b. (b) A short calculation yields that c is interpolatory. Let us verify the sum rules for c: by applying the duality (17), we obtain X ¯ (ba)(· + γ)¯0 = δ0,α . ∂α γ∈ΓM
The sum rules and the Leibniz formula for differentiation yield ∂ α b(0) = δ0,α ,
for all |α| < L.
(25)
Thus, the sum rules for a and b together with (25) imply the sum rules for c. Let us verify the representation (22): by applying the duality (17), the interpolatory property, and X X b(ξ + γ e), for γ ∈ ΓM , b(ξ + γ + γ e) = γ e∈ΓM
γ e∈ΓM
we obtain X
(ca)(ξ + γ) =
γ∈ΓM
X γ∈ΓM
X b(ξ + γ) + 1 − 1 b(ξ + γ + γ e) a(ξ + γ) m m
1 1 X =1+ − b(ξ + γ). m m γ∈Γ
γ e∈ΓM
M
With (23), this yields c(ξ) + 1 −
X
(ca)(ξ + γ) = b(ξ).
γ∈ΓM
It should be mentioned that, for m = 2 and c = a, the mask of b in (22) coincides with the smallest nontrivial dual masks constructed in [14, 25]. 12
4
Symmetry, Optimality, and Examples
In this section, we present examples of our convolution method with a discussion about optimality. We explain the treatment of symmetry in terms of masks as proposed in [19]. Then we discuss optimality conditions in terms of the refinable function, i.e., minimal mask size with respect to smoothness and approximation order of the underlying multiresolution analysis. Due to Section 2.3, a wavelet bi– frame has maximal approximation order, if it reaches the approximation order of the underlying multiresolution analysis. Under these optimality criteria, we obtain several examples of optimal symmetric wavelet bi–frames. First, we recall smoothness criteria, see for ¡ ¢ example [21]: Let 1 ≤ p ≤ ∞. For 0 < η ≤ 1, the Lipschitz space Lip η, Lp (Rd ) consists of those functions f ∈ Lp (Rd ) such that kf − f (· − t)kLp ≤ Cf ktkη , for all t ∈ Rd , where Cf is a positive constant. For f ∈ Lp (Rd ), we call © ¡ ¢ ª νp (f ) := sup k + η : ∂ α f ∈ Lip η, Lp (Rd ) , for all |α| = k its Lp –critical exponent. Moreover, given symbols a and b in case m = 2, we choose ∗
a1 (ξ) := e−2πiξ·γ a(ξ + γ),
∗
b1 (ξ) := e−2πiξ·γ b(ξ + γ),
(26)
where ΓM = {0, γ}, and {0, γ ∗ } is a set of representatives of Zd /M Zd . To have a short notation for the mask supports, we define, for nonnegativ t, © ª ♦t := (k1 , . . . , kd )> ∈ Zd : |k1 | + . . . + |kd | ≤ t , © ª ¤t := (k1 , . . . , kd )> ∈ Zd : |k1 |, . . . , |kd | ≤ t . In the following, we additionally apply z–notation, i.e., zj := e−2πiξj , for j = 1, . . . , d.
4.1
Symmetry
This section is devoted to the treatment of wavelet symmetries in terms of their masks. It is fundamental examples in the © for the ª following sections. d×d A finite set G ⊂ U ∈ Z : |det(U )| = 1 is called a symmetry group with respect to M if G forms a group under matrix multiplication and M U M −1 ∈ G, for all U ∈ G. A symbol a is called G–symmetric if there exists p ∈ Rd such that aU (k−p)+p = ak ,
for all U ∈ G, k ∈ Zd .
Thus, we implicitly impose (Id − U )p ∈ Zd , for all U ∈ G. A function f is called G–symmetric if there exists q ∈ Rd such that f (U (x − q) + q) = f (x),
for all U ∈ G, x ∈ Rd .
The following has been shown in [19]. Lemma 4.1. If a is G–symmetric and a(0) = 1, then the generated refinable function is G–symmetric. If, in addition, the wavelet symbol is also G–symmetric, then the associated wavelet is G–symmetric. 13
4.2
Arbitrary Smooth Optimal Wavelet Bi–Frames
In this section, we obtain arbitrary smooth optimal d–dimensional wavelet bi–frames with only three wavelets. In between, we discuss optimality constraints for a large class of dilation matrices. These results are also applied in the following sections. Let M be a dilation matrix such that n o ΓM = 0, (1/2, . . . , 1/2)> . (27) ©¡ 0 0 ¢ ¡ 1 1 ¢ª Let a ˇ , ˇb , a ˇ , ˇb be a one–dimensional symbol family satisfying condition (I) for dyadic dilation. We define multivariate symbols by b0 (ξ1 , . . . , ξd ) := ˇb0 (ξ1 ), b1 (ξ1 , . . . , ξd ) := ˇb1 (ξ1 ).
a0 (ξ1 , . . . , ξd ) := a ˇ0 (ξ1 ), a1 (ξ1 , . . . , ξd ) := a ˇ1 (ξ1 ),
Due to ΓM in (27), {(a0 , b0 ) , (a1 , b1 )} satisfies condition (I) with respect to the dilation matrix M . So far, we have applied the ideas in [18] to dilation matrices satisfying (27). These families can act as start symbols for Corollary 3.2. Let us have a look at optimality of our construction. The following Theorem is borrowed from [20]. Theorem 4.2. Let cˇ be a one–dimensional interpolatory symbol satisfying the sum rules of order L with respect to dyadic dilation. If its mask is supported on [−S, R], for S, R ∈ N, then ¹ º ¹ º S+1 R+1 L≤ + . (28) 2 2 Furthermore, there exists a unique interpolatory mask, which is supported on [−2N + 1, 2N − 1] and satisfies the sum rules of order 2N . Let us apply the ideas in [21] of the two–dimensional quincunx dilation to our d–dimensional setting: for nonnegative S, R, we define ( ) d X > d [−S, R]Σ := (k1 , . . . , kd ) ∈ Z : −S ≤ ki ≤ R . i=1
Then we obtain the following generalization. Corollary 4.3. Assume that M satisfies (27). Let c be a d–dimensional interpolatory symbol satisfying the sum rules of order L with respect to M . For S, R ∈ N, let its mask be supported on [−S, R]Σ . Then, (28) holds. Proof. The one–dimensional symbol cˇ(ξ1 ) := c(ξ1 , . . . , ξ1 ) is interpolatory, and its mask is supported on [−S, R]. By applying the chain rule of differentiation, it satisfies the sum rules of order L with respect to dyadic dilation. Due to Theorem 4.2, we conclude the proof. 14
Let us choose matrices M mentioned in [17]: for d = 2, 3, let µ ¶ 0 2 1 −1 1 M= , M = −1 −1 0 . 1 1 1 1 1
(29)
For d > 3, we define
0 .. . . . M = . 0 −1 1 Then M d = 2Id , and
....... 1 1 0 . . . 0 . . . . . . . . . .. . . ....... 0 1 0 . . . . . . . . . . . . −1 0 ................ 1 2 ...
1
(
M Zd =
(k1 , . . . , kd )> ∈ Zd :
d X
(30)
) ki ∈ 2Z
i=1
is the checkerboard lattice. Moreover, ΓM is given by (27). Next, let us choose µ ¶N µ ¶N 1+z 1 + 1/z 0 a ˇ (z) := · , (31) 2 2 ¶N µ ¶N ¶ µ ¶¶ µ µ µ 1 + 1/z 1 + 1/z 1 + z 1 + z 0 ˇb (z) := · · P2N 1 − · , (32) 2 2 2 2 where PN (x) :=
N −1 µ X j=0
¶ N −1+j j x j
(33)
0ˇ0
is the Bezout polynomial. Then a ˇ b is interpolatory, and it satisfies the sum rules of order 4N , see [11]. Its mask is supported on [−4N + 1, 4N − 1]. Thus, it is the unique interpolatory mask of Theorem 4.2. Applying (27), the multivariate symbol a0 b0 is interpolatory, and it satisfies the sum rules of order 4N . Its mask is supported on a straight line in [−4N +1, 4N −1]Σ . Due to Corollary 4.3, a0 b0 is optimal. Moreover, a0 , b0 , and a0 b0 are G–symmetric with G := {±Id } . (34) Let us have a look at optimality with respect to smoothness: Lemma 4.4. Given M by (29), (30), let a symbol a satisfy the sum rules of order 2N , and let it generate a fundamental refinable function ϕ. If its mask is supported on ¯ ¯ ) ( d ¯ ¯X ¯ ¯ (35) ki ¯ ≤ N − 1, l = 2, . . . , d , [−2N + 1, 2N − 1]Σ ∩ k ∈ Zd : ¯ ¯ ¯ i=l
then νp (ϕ) ≤ νp (ϕ0 ∗ ϕ e0 ), where ϕ0 ∗ ϕ e0 is generated by a ˇ0ˇb0 . 15
For d = 2, Lemma 4.4 is stated in [21]. The concrete choice of M allows the generalization to higher dimensions. We only give a brief outline of the proof of Lemma 4.4. Proof. Observe that ϕ is also refinable with respect to the symbol −d+1
c(ξ) := a(ξ) · a(2M >
−1
ξ) · · · a(2M > ξ).
with dilation matrix 2Id . Let us define the one–dimensional symbol cˇ(ξ1 ) := c(ξ1 , . . . , ξ1 ). We only have to show cˇ = a ˇˇb. Then Lemma 4.4 follows directly by following the lines in [21]. First, we verify a(0, ξ1 , . . . , ξ1 ) = 1,
...,
a(0, . . . , 0, ξ1 ) = 1,
for all ξ1 ∈ R.
(36)
Due to [21], (36) holds in case d = 2. For arbitrary d, we reduce to the two– dimensional setting by e a(ξ1 , ξ2 ) := a(ξ1 , . . . , ξ1 , ξ2 , . . . , ξ2 ), where we can vary the numbers of appearing ξ1 , ξ2 (both should appear at least once). Then e a is interpolatory, and its mask is supported on (35) with d = 2. By applying the chain rule of differentiation, it satisfies the sum rules of order 2N . This concludes the proof of (36). Due to (29), (30), and (36), we have cˇ(ξ1 ) = a(ξ1 , . . . , ξ1 ). Thus, it is interpolatory, and it satisfies the sum rules of order 2N . By applying (35), the mask of cˇ is supported on [−2N + 1, 2N − 1]. Thus, Theorem 4.2 yields cˇ = a ˇˇb. This concludes the proof. Let ϕ, ϕ e be generated by a0 , b0 , respectively. Lemma 4.5. With all choices above, we have ϕ(ξ) b =ϕ b0 (ξd − ξd−1 ) · · · ϕ b0 (ξ2 − ξ1 ) · ϕ b0 (ξ1 ), be = ϕ be (ξd − ξd−1 ) · · · ϕ be (ξ2 − ξ1 ) · ϕ be (ξ1 ). ϕ(ξ) 0
0
0
(37) (38)
Proof. First, observe that M d = 2Id implies −jd+l
M>
l
= 2−j M > ,
for all j, l ∈ N.
(39)
Applying (6) and (39) yields Y
0
a (M
>−j
ξ) =
j≥1
d−1 YY
a0 (M >
−jd
l
M > ξ)
j≥1 l=0
=
d−1 Y Y
l
a0 (2−j M > ξ).
l=0 j≥1
By applying the special choices of M and a0 , we conclude the proof of (37). Following the lines above, we obtain (38). 16
N mask1 ν∞ sum rules vm2 approx.3
1 ♦3 2 4 2 4
2 ♦7 3.5 8 4 8
3 ♦11 4.7 12 6 12
4 ♦15 ≥ 5.8 16 8 16
Table 1: Bi–frame of Section 4.2 1
the mask of the refinable function is supported on... number of vanishing moments, which all wavelets inherit. 3 due to (9), the approximation order of the wavelet bi–frame. 2
Figure 1: ϕ ∗ ϕ e for d = 2, N = 1 of Section 4.2
Due to Corollary 3.2, the family ©¡ 0 0 0 0 ¢ ¡ 1 0 0 1 ¢ ¡ 0 1 1 0 ¢ ¡ 1 1 1 1 ¢ª a b ,a b , a b ,a b , a b ,a b , a b ,a b satisfies condition (I). By applying (37), (38), a direct calculation yields that the fundamental refinable function of a0 b0 is given by ϕ ∗ ϕ(x) e = ϕ0 ∗ ϕ e0 (x1 + . . . + xd ) · ϕ0 ∗ ϕ e0 (x2 + . . . + xd ) · · · ϕ0 ∗ ϕ e0 (xd ). Thus, it inherits the full regularity of the one–dimensional function ϕ0 ∗ ϕ e0 . Due to Corollary 4.3 and Lemma 4.4, ϕ∗ϕ e has minimal mask size with respect to smoothness and sum rule order. For increasing N , we obtain arbitrary smooth wavelet bi–frames with maximal approximation order with respect to the underlying multiresolution analysis, see Theorem 2.4. Notice that, due to Lemma 4.1, the three wavelets are G–symmetric with G as in (34). See Table 1 for the wavelet bi–frame properties and Figure 1 for ϕ ∗ ϕ e with N = 1.
4.3
The Quincunx Dilation
In this section, we treat the quincunx dilation matrix µ ¶ 1 −1 M= . 1 1 17
(40)
−1 128
−3 128
0 −3 128
−3 128
0 39 128
0 −3 128
−1 128
0 39 128
1 39 128
0 −1 128
−3 128
0 39 128
0 −3 128
−3 128
0 −3 128
−1 128
−1 16
−1 8
0 −1 8
−1 16
0 3 4
0 −1 16
−1 8
0 −1 8
−1 16
(b) ab1
(a) ab
1 1024
1 256
0 1 256
3 512
0 −1 16
1 256
0 −17 128
0
0 3 512
−17 128
0 1 256
1 1024
0 −1 16
0 185 256
0 −1 16
0 1 1024
1 256
0 −17 128
0 −17 128
0 1 256
3 512
0 −1 16
0 3 512
1 256
0 1 256
1 1024
1 128
3 128
0 3 128
3 128
0 −39 128
0 3 128
1 128
0 −39 128
1 −39 128
0 1 128
3 128
0 −39 128
0 3 128
3 128
0 3 128
1 128
(d) a1 b1
(c) a1 b
Figure 2: Masks of Laplace bi–frame for N = 2
Thus, m = 2. Our convolution method of Corollary 3.2 with n = 1 yields 3 wavelets. The following example provides optimal wavelet bi–frames. Example 4.6 (Laplace). Let 1 e a(z1 , z2 ) := 2
µ
1 11 1 11 1 + z1 + + z2 + 4 4 z1 4 4 z2
¶ (41)
be the Laplace symbol. It is interpolatory, and it satisfies the sum rules of order 2, see [7]. For N ∈ N, let N a := e ad 2 e ,
N b := e ab 2 c PN (1 − e a),
where PN is again the Bezout polynomial (33). Here b·c, d·e denote the floor function, ceiling function, respectively. Then a, b are real–valued, and ab is interpolatory, see [11, 14]. Hence, by (26), {(a, b) , (a1 , b1 )} satisfies condition (I). By applying Corollary 3.2, also the family {(ab, ab) , (ab1 , a1 b) , (a1 b, ab1 ) , (a1 b1 , a1 b1 )} satisfies condition (I). See Figure 2 for the masks. Observe that a, b, and ab are G–symmetric with ½ µ ¶ µ ¶ µ ¶¾ 1 0 0 −1 0 1 G := ±I2 , ± ,± ,± . (42) 0 −1 1 0 1 0 For N ≥ 2, ϕ ∗ ϕ e is fundamental. Due to Lemma 4.1, all wavelets are G–symmetric, see Figure 3 with N = 2. Then a comparison to biorthogonal wavelet bases from 18
(a) ϕ ∗ ϕ e
(b) ϕ ∗ ψe
(c) ψ ∗ ϕ e
(d) ψ ∗ ψe
Figure 3: Laplace bi–frame for N = 2
[21] is given in Tables 2 and 3. As far as we know, the chosen biorthogonal bases are the best known which are comparitive to N = 2, 4 with respect to smoothness and approximation order and where the primal refinable function is fundamental. To obtain the same approximation order, the supports of the biorthogonal dual masks are significantly larger. Due to Corollary 4.3, our refinable function has minimal mask size with respect the sum rule order. The wavelet bi–frame has maximal approximation order, see Theorem 2.4. In the following example, we obtain three times differentiable wavelets. Example 4.7 (Three times differentiable). Given the symbol a by the mask in Figure 4, the generated refinable function ϕ is fundamental. It satisfies the sum rules of order 4, and it is contained in the H¨older class C α (R2 ), for α = 1.5, see [8]. By Theorem 3.3, b(ξ) := a(ξ) + 1 − |a(ξ)|2 − |a(ξ + γ)|2 is dual to a. Then, a, b, ab are G–symmetric with G as in (42). The refinable function ϕ∗ϕ e is fundamental. Due to Lemma 4.1, all wavelets are G–symmetric, see Figure 5. We obtain better results than the smoothest biorthogonal basis given in [21], see Table 4. Due to Theorem 2.4, the wavelet bi–frame has maximal approximation 19
mask ν∞ sum rules vm approx.
biorth.(1)1 ϕ ϕ e ♦1 ¤4 0.6 0.04 2 6 6 2 2
bi–frame ϕ∗ϕ e ♦3 1.3 4 2 4
biorth.(2)2 ϕ ϕ e ¤2 ¤7 1.4 1.2 4 10 10 4 4
Table 2: Laplace N = 2 versus biorthogonal bases 1 2
best choice in [21] with ϕ Laplace refinable function with mask (41), ϕ e in L2 (R2 ). 1 2 best choice in [21] with ϕ, ϕ e ∈ C (R ).
mask ν∞ sum rules vm approx.
bi–frame ϕ∗ϕ e ♦7 2.5 8 4 8
biorth.(1)1 ϕ ϕ e ¤2 ¤5 1.4 0.09 4 6 6 4 4
biorth.(2)2 ϕ ϕ e ¤3 ¤10 2.2 2.2 6 14 14 6 6
biorth.(3)3 ϕ ϕ e ¤4 ¤11 2.9 2.2 8 14 14 8 8
Table 3: Laplace N = 4 versus biorthogonal bases 1
best choice in [21] with ϕ ∈ C 1 (R2 ), ϕ e in L2 (R2 ). 2 2 best choice in [21] with ϕ, ϕ e ∈ C (R ). 3 best choice in [21] with ϕ, ϕ e ∈ C 2 (R2 ) and approximation order 8. 2
1 256
−9 256
0 −9 256
−9 256
0 81 256
0 −9 256
1 256
0 81 256
1 81 256
0 1 256
−9 256
0 81 256
0 −9 256
−9 256
0 −9 256
1 256
Figure 4: The mask of a in Example 4.7
20
(a) ϕ ∗ ϕ e
(b) ϕ ∗ ψe
(c) ψ ∗ ϕ e
(d) ψ ∗ ψe
Figure 5: Bi–frame of Example 4.7
order. Additionally, our Laplace bi–frame in Example 4.6, for N = 5, is even 3 times differentiable.
4.4
Box Splines
In this section, we deal with three–direction box splines. We choose M = 2I2 . Let ¶N µ ¶N µ ¶N µ ¶N µ 1 + z1 1 + z2 1 + z1 z2 1 · · · (43) e a(N ) (z) := 2 2 2 z1 z2 be the symbol of the three–direction box spline of equal multiplicities N . In [33], for N = 2, 4, 6, and 8, a symbol eb(N ) has been constructed such that e a(N )eb(N ) is nonnegative and interpolatory. Its mask is G–symmetric with ½ µ ¶ µ ¶ µ ¶ µ ¶ µ ¶¾ 0 1 1 0 0 1 1 −1 −1 1 G := ±I2 , ± ,± ,± ,± ,± . 1 0 1 −1 −1 1 1 0 0 1 Fixing N = 2, we can split into a(z) := e a(1) (z),
b(z) := e a(1) (z)eb(2) (z). 21
(44)
mask ν∞ sum rules vm approx.
bi–frame ϕ∗ϕ e ♦9 2.9 8 4 8
Laplace N = 5 ϕ∗ϕ e ♦9 3.0 10 4 8
biorth. ϕ ϕ e ¤4 ¤12 2.9 2.8 8 16 16 8 8
Table 4: Bi–frame versus Laplace for N = 5 versus biorthogonal basis
mask ν∞ sum rules vm approx.
bi–frame ϕ∗ϕ e ¤3 2 4 2 4
biorth.(1)1 ϕ ϕ e ¤1 ¤6 1 0.63 2 6 6 2 2
biorth.(2)2 ϕ ϕ e ¤3 ¤6 1.43 0.073 4 4 4 4 4
Table 5: Box spline bi–frames versus biorthogonal bases 1
ϕ is the three direction box spline with multiplicity 1. ϕ e is constructed in [22]. ϕ (butterfly mask) and ϕ e are given in [3]. 3 obtained by Sobolev embedding. 2
Then a and ab are interpolatory, and they generate fundamental refinable functions. Applying Corollary 3.2 yields a wavelet bi–frame with 15 wavelets. Part (b) of Theorem 3.1 yields 6 wavelets, but the refinable function is no longer fundamental. See Figure 6 for the fundamental refinable box spline. A comparison to biorthogonal bases from [3, 22] is given in Table 5. Observe that our convolved masks have smaller support than the comparative biorthogonal ones, while we obtain much smoother functions. By the results in [20], an interpolatory symbol whose mask is supported on ¤3 satisfies the sum rules of order at most 4. If, in addition, the generated refinable function is fundamental, then it is not contained in C 2 (Rd ), see [16]. Thus, our refinable function has minimal mask size. The wavelet bi–frame has maximal approximation order, see Theorem 2.4. Due to Lemma 4.1, see the Figures 7, 8 for the mask symmetries, the wavelets are Gi –symmetric for different subgroups Gi ⊂ G: ½ µ ¶¾ 0 1 G1 := ±I2 , ± , ϕ ∗ ψe3 , ψ 3 ∗ ϕ, e ψ 3 ∗ ψe3 are G1 –symmetric. 1 0 ½ µ ¶¾ −1 1 G2 := ±I2 , ± , ϕ ∗ ψe1 , ψ 1 ∗ ϕ, e ψ 1 ∗ ψe1 are G2 –symmetric. 0 1 ½ µ ¶¾ 1 0 G3 := ±I2 , ± , ϕ ∗ ψe2 , ψ 2 ∗ ϕ, e ψ 2 ∗ ψe2 are G3 –symmetric. 1 −1 G4 := {±I2 } ,
all other wavelets are G4 –symmetric. 22
−1 −3 −3 −1 64 64 64 64 −3 3 0 32 0 −3 64 64 −3 3 33 33 3 −3 64 32 64 64 32 64 −1 0 33 1 33 0 −1 64 64 64 64 −3 3 33 33 3 −3 64 32 64 64 32 64 −3 3 0 32 0 −3 64 64 −1 −3 −3 −1 64 64 64 64
(a) a0 b0
(b) ϕ ∗ ϕ e
Figure 6: The fundamental box spline
Acknowledgement: The author would like to thank Stephan Dahlke for his helpful comments.
References [1] G. Beylkin, R. Coifman, and V. Rokhlin, Fast wavelet transforms and numerical algorithms, Comm. Pure Appl. Math. 44 (1991), 141–183. [2] C. Cabrelli, C. Heil, and U. Molter, Accuracy of several multidimensional refinable distributions, J. Fourier Anal. Appl. 6 (2000), no. 5, 483–502. [3] D. R. Chen, B. Han, and S. D. Riemenschneider, Construction of multivariate biorthogonal wavelets with arbitrary vanishing moments, Adv. Comput. Math. 13 (2000), no. 2, 131–165. [4] O. Christensen, An Introduction to Frames and Riesz Bases, Birkh¨auser, Boston, 2003. [5] C. K. Chui and W. He, Construction of multivariate tight frames via kronecker products, Appl. Comput. Harm. Anal. 11 (2001), 305–312.
23
−1 1 1 −1 64 64 64 64 −1 1 0 32 0 −1 64 64 −1 −1 3 3 1 1 64 16 64 64 16 64 3 3 0 −35 1 −35 0 64 64 64 64 −1 3 −1 1 1 3 64 16 64 64 16 64 −1 1 0 32 0 −1 64 64 −1 1 −1 1 64 64 64 64
3 3 −1 −1 64 64 64 64 −1 3 1 0 16 0 64 64 −1 −1 −35 1 1 1 64 16 64 64 32 64 −1 −1 1 1 0 64 1 64 0 64 64 1 1 1 −35 −1 −1 64 32 64 64 16 64 −1 1 3 0 16 0 64 64 −1 −1 3 3 64 64 64 64
(a) a1 b1
"
#
(d) a0 b1
−1 −1 −3 3 1 1 64 64 16 16 64 64 −1 5 3 0 64 0 −7 0 64 64 64 −1 1 −31 7 −31 1 −1 64 16 64 8 64 16 64 3 5 0 −7 0 64 0 −1 64 64 64 −1 3 −3 −1 1 1 64 64 16 16 64 64
(e) a2 b1
−1
(g) a0 b2
−1 −1 64 64 −1 1 0 64 64 3 1 5 1 64 16 64 16 −31 −1 3 0 64 0 16 64 −3 −7 −7 −3 7 64 64 8 64 64 −1 −31 3 0 64 0 64 16 1 5 1 3 16 64 16 64 1 0 −1 64 64 −1 −1 64 64
(c) a3 b3
−3 −1 3 1 1 −1 64 64 16 16 64 64 −7 3 5 0 64 0 64 0 −1 64 64 −1 1 −31 7 −31 1 −1 64 16 64 8 64 16 64 −1 5 3 0 64 0 −7 0 64 64 64 −1 1 −1 −3 1 3 64 64 16 16 64 64
−1 16 16 −1 1 0 16 16 1 1 1 16 8 16 −1 1 0 16 16 −1 −1 16 16
−1 −1 3 3 64 64 64 64 −1 1 3 0 16 0 64 64 1 1 1 −35 −1 −1 64 32 64 64 16 64 −1 1 1 0 64 1 64 0 −1 64 64 −1 −1 −35 1 1 1 64 16 64 64 32 64 3 1 0 −1 0 64 64 16 −1 −1 3 3 64 64 64 64
(b) a2 b2
−1 1 1 −1 16 16 16 16 −1 1 0 8 0 −1 16 16 −1 1 1 −1 16 16 16 16
(f) a3 b1
−1 3 3 64 64 64 −1 −3 1 0 16 0 64 64 1 5 −31 −7 −1 64 64 64 64 16 1 7 1 0 0 16 16 8 −1 −7 −31 5 1 16 64 64 64 64 −3 −1 1 0 16 0 64 64 −1 −1 3 3 64 64 64 64 −1 64
(i) a3 b2
1 2
(h) a b
−1
1 16
0
16 −1 −1 16 16
−1 −1 16 16 1 0 −1 16 16 1 1 8 16 1 16
(j) a0 b3
−1 −1 64 64 1 0 −1 64 64 1 5 1 3 16 64 16 64 −1 −31 3 0 64 0 64 16 −3 −7 −7 −3 7 64 64 8 64 64 −31 3 0 64 0 −1 64 16 3 1 5 1 64 16 64 16 −1 1 0 64 64 −1 −1 64 64
−1 64 −1 64
3 3 −1 −1 64 64 64 64 −3 1 0 16 0 −1 64 64 −1 −7 −31 5 1 16 64 64 64 64 1 7 1 0 0 16 16 8 1 5 −31 −7 −1 64 64 64 64 16 1 0 16 0 −3 64 −1 3 3 64 64 64
(l) a2 b3
1 3
(k) a b
Figure 7: Masks of the box spline bi–frame (part I)
24
−1 256
−3 −3 −1 256 256 256 −3 −3 3 0 0 256 256 128 45 45 15 15 256 256 256 256 −1 −1 1 1 0 0 32 32 32 32 −21 −149 105 105 −149 −21 9 9 256 256 256 256 256 256 256 256 −67 −67 3 131 3 0 0 0 0 256 128 128 128 256 −21 −149 105 105 −149 −21 9 9 256 256 256 256 256 256 256 256 −1 −1 1 1 0 0 32 32 32 32 15 45 45 15 256 256 256 256 −3 −3 3 0 0 256 128 256 −1 −3 −3 −1 256 256 256 256
−3 9 1 256 32 256 −21 9 15 0 0 0 256 256 256 −21 −67 −149 −1 45 1 3 32 256 128 256 32 256 128 −149 105 45 0 0 0 256 256 256 −3 15 −1 105 131 105 −1 15 −3 256 256 32 256 128 256 32 256 256 −1 45 0 256 0 105 0 −149 256 256 256 −3 3 45 −1 −149 −67 −21 1 256 128 256 32 256 128 256 32 −3 −21 15 9 0 0 0 256 256 256 256 −1 −3 1 9 3 256 256 32 256 256 3 256
(b) a2 b0
1 0
(a) a b
−1 256
−3 1 9 3 256 32 256 256 −3 −21 15 9 0 256 0 256 256 256 −3 −1 −149 −67 −21 3 45 1 256 128 256 32 256 128 256 32 −1 −149 45 105 0 0 256 0 256 256 256 −3 −1 −1 −3 15 105 131 105 15 256 256 32 256 128 256 32 256 256 −149 −1 105 45 0 256 0 0 256 256 256 −21 −67 −149 −1 −3 1 45 3 32 256 128 256 32 256 128 256 −3 9 15 0 −21 0 256 256 256 256 −3 −1 3 9 1 256 256 32 256 256
(c) a3 b0
Figure 8: Masks of the box spline bi–frame (part II)
25
−1 256 −3 256 −3 256 −1 256
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27
List of Figures 1 2 3 4 5 6 7 8
ϕ∗ϕ e for d = 2, N = 1 of Section 4.2 . . . Masks of Laplace bi–frame for N = 2 . . . Laplace bi–frame for N = 2 . . . . . . . . The mask of a in Example 4.7 . . . . . . . Bi–frame of Example 4.7 . . . . . . . . . . The fundamental box spline . . . . . . . . Masks of the box spline bi–frame (part I) . Masks of the box spline bi–frame (part II)
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
. . . . . . . .
17 18 19 20 21 23 24 25
Bi–frame of Section 4.2 . . . . . . . . . . . . . . . . . . . . . Laplace N = 2 versus biorthogonal bases . . . . . . . . . . . Laplace N = 4 versus biorthogonal bases . . . . . . . . . . . Bi–frame versus Laplace for N = 5 versus biorthogonal basis Box spline bi–frames versus biorthogonal bases . . . . . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
17 20 20 22 22
List of Tables 1 2 3 4 5
28
