3.3 Spline Wavelets
3.3.1 A Biorthogonal B-Spline Multiresolution on the Interval
We begin with the description of a B-spline multiresolution on the real line. A biorthogonal multires-olution on the unit interval (0, 1) is then derived by restriction and subsequent modifications at the boundary [49].
Biorthogonal B-Spline Multiresolution of L2(R)
The construction of B-spline wavelets which we describe here is based on the concept of refinable functions.
A function φ ∈ L2(R) is called refinable with mask a = {ak}k∈Z if φ(x) =X
k∈Z
akφ(2x − k) . (3.3.1)
We say that two refinable functions φ, and ˜φ with mask ˜a = {˜ak}k∈Z form a dual pair if φ, ˜φ(· − k)
L2 = δ0,k, k ∈ Z . (3.3.2)
We assume in the following that φ and ˜φ are normalised, Z
R
φ(x) dx = Z
R
φ(x) dx = 1 .˜ (3.3.3)
The refinable function φ is used to generate the family of functions
φ[j,k]:= 2j/2φ(2j· −k) , j, k ∈ Z . (3.3.4) Defining
Φj:= {φ[j,k]: k ∈ Z} and Φ˜j:= { ˜φ[j,k]: k ∈ Z} , (3.3.5) 42
3.3. Spline Wavelets
we obtain a multiresolution basis for L2(R) as introduced in Section 2.2.1. If φ and ˜φ have compact support, it follows that the bases Φj and ˜Φj are uniformly stable, and also that the number of nonzero entries of the masks a and ˜a is finite.
The approximation power of the spaces Sj: = S(Φj) and ˜Sj: = S( ˜Φj) depends on their polynomial exactness. We say that φ is exact of order d if all polynomials of degree of at most d − 1 can be expressed as a linear combination of the integer translates φ(· − k). Likewise, the dual order of exactness is denoted by ˜d. These properties are equivalent to the existence of the following representations of the monomials xr,
with primal and dual expansion coefficients ˜αk,r and αk,r.
We will choose the cardinal B-spline of order d as generator for the primal multiresolution. To this end, we shortly review the associated definitions. Let [t0, . . . , td]f denote the d-th order divided difference of f ∈ Cd(R) at the nodes t0≤ . . . ≤ td. With the setting xd+:= (max{0, x})d, the cardinal B-spline φd of
and its support is given by
supp φd= [`1, `2] with `1:= −jd 2
k, `2:=ld 2
m. (3.3.9)
Thus, it is centred around x = 0 for even orders d and around x = 12 for odd d. Note the identities
d = `2− `1 and µ(d) = `1+ `2. (3.3.10)
The B-spline φd is an example of a refinable function according to (3.3.1) with mask
ak= 21−d
d
k + bd2c
, k = `1, . . . , `2. (3.3.11)
Furthermore, φd is exact of order d. Thus, the generator for the primal multiresolution can be specified using standard B-spline theory. However, the construction of a dual generator which satisfies (3.3.2) is non-trivial. Such a result has been first obtained in [42] based on Fourier decompositions.
Theorem 3.6. For each d, and ˜d ≥ d, ˜d ∈ N, with d + ˜d even, there exists a function ˜φd, ˜d∈ L2(R) with the following properties.
• The support is given by
supp ˜φd, ˜d= [˜`1, ˜`2] with `˜1:= `1− ( ˜d − 1) , `˜2:= `2+ ( ˜d − 1) . (3.3.12)
• ˜φd, ˜d is refinable with a finitely supported mask ˜a = {˜ak}`k=˜˜2 `1.
43
Chapter 3. Two Constructions on the Interval polynomial exactness d and ˜d do exist. To explicitly construct a wavelet basis for L2(R), it remains to calculate the coefficients ˜αk,rand αk,r, and to specify the complement bases Ψj and ˜Ψj.
We infer from the biorthogonality condition (3.3.2) that the expansion coefficients from (3.3.6) have the explicit form
˜
αy,r= (·)r, ˜φ(· − y)
L2, r = 0, . . . , d − 1 . (3.3.13) Using the normalisation (3.3.3), we derive the coefficients for the case r = 0 as
˜
αy,0= 1 . (3.3.14)
The translated coefficients of higher order r > 0 for arbitrary y can be reduced to y = 0 via
˜
Finally, ˜α0,r can be determined recursively. We refer to [50,132] for the full derivation and only state the result, generator functions ψ and ˜ψ,
Ψj:= {ψ[j,k]: k ∈ Z} and Ψ˜j:= { ˜ψ[j,k]: k ∈ Z} . (3.3.17) If the generators have the form
ψ(x) :=X it follows that the biorthogonality conditions are satisfied,
φ, ˜ψ(· − k)
L2= ˜φ, ψ(· − k)
L2 = 0 , ψ, ˜ψ(· − k)
L2 = δ0,k, k ∈ Z . (3.3.20) Due to the refinability of φ and ˜φ, the full wavelet bases Ψ and ˜Ψ as assembled in (2.2.17) are also biorthogonal. Since ˜φ is exact of order ˜d, we infer that ψ and therefore all wavelets ψ[j,k]have ˜d vanishing moments,
Z
R
xrψ[j,k](x) dx = 0 , r = 0, . . . , ˜d − 1 . (3.3.21) Finally, the biorthogonality condition (2.2.44) and the finite masks a, ˜a and b, ˜b entail that Ψ and ˜Ψ are Riesz bases for L2(R).
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3.3. Spline Wavelets
Restriction to the Interval
The restriction of the collections Φj, ˜Φj to the interval (0, 1) poses two essential difficulties.
• Basis functions which contain x = 0 or x = 1 in their support are truncated, with the result that the respective scalar products change their values.
• As the generators φ and ˜φ generally have supports of different lengths, the cardinalities of Φj and Φ˜j are now finite but not equal.
As a result, biorthogonality is destroyed. In [49], this issue has been resolved by taking linear combinations of the truncated original functions in such a way that the new functions are identical to monomials near the boundary. We will describe this ansatz in the following.
Since the support of φ is bounded according to (3.3.9), the nonzero part of the mask is given by a = {ak}`k=`2 1. For an arbitrary but fixed parameter ` ≥ −`1, we introduce the following function on the left and then declines to zero. It is known from [43] that this function is refinable. The precise form of the refinement relation reads
At the right end of the interval, we define φRj,`−d+ranalogously to (3.3.22) and obtain a similar refinement relation. On the dual side, we introduce the parameter ˜` ≥ −˜`1and define the boundary functions ˜φLj,˜`− ˜d+r
Chapter 3. Two Constructions on the Interval
The indexing of these functions was carefully chosen in [49] to allow for continuous numbering of all functions left, middle and right. Assigning the indices as follows,
k ∈ ∆Lj for φLj,k, k ∈ ∆0j for φ[j,k], k ∈ ∆Rj for φRj,k, (3.3.28a) and likewise for the dual side with parameters ˜d and ˜`, we can set
∆j:= ∆Lj ∪ ∆0j∪ ∆Rj , ∆˜j:= ˜∆Lj ∪ ˜∆0j∪ ˜∆Rj , (3.3.30) and a first version of the single-scale bases for the spaces Sj and ˜Sj on the interval is given by
Φ(0)j := {φLj,k} ∪ {φ[j,k]} ∪ {φRj,k} , Φ˜(0)j := { ˜φLj,k} ∪ { ˜φ[j,k]} ∪ { ˜φRj,k} . (3.3.31) Note in particular that the functions adapted to the left and right borders are refinable, cf. (3.3.23).
Defining βm,r analogously to ˜βm,r (3.3.24), and setting
β˜j,m,rL := ˜βm,r, β˜j,m,rR := ˜β2j+1−m−µ(d),r, r = 0, . . . , d − 1 , (3.3.32a)
The dual relations are defined analogously. Writing these relations in matrix form produces the boundary blocks of the refinement matrices M(0)j,0 and ˜M(0)j,0, while their interior blocks are given by the masks a and ˜a, see Table 3.11.
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3.3. Spline Wavelets
Remark 3.7. The number of boundary functions on the primal side is always d, independent of the length of their support which is controlled by `. The number of boundary functions on the dual side is ˜d, independent of ˜`. It follows that a choice of
` = ` + ( ˜˜ d − d) (3.3.34)
leads to equal cardinality of the primal and dual bases on the interval, which is a prerequisite for biorthog-onality.
Remark 3.8. Homogeneous boundary conditions on either the primal or the dual side, or on both sides, and either on the right or the left end, or on both ends, in any combination, can be achieved by deleting the monomial of degree 0 (corresponding to the constant x0 = 1) from the appropriate index set(s). If different boundary conditions are to be fulfilled on the left and the right end, the parameters ` and/or ˜` might additionally need to be chosen differently on the left and the right end. Equal cardinality of the primal and dual bases can always be reestablished by a suitable modification of (3.3.34).
Reestablishing Biorthogonality
In the previous paragraph, we have constructed primal and dual multiresolutions restricted to the interval (0, 1). We still need to confirm that the collections of functions Φ(0)j and ˜Φ(0)j are respectively linearly independent. Secondly, biorthogonality has to be reestablished, which has been lost due to the restriction.
The route proposed in [49] is to introduce local transformations on the left and right boundaries which lead again to a biorthogonal basis. Linear independence is then a trivial consequence.
As we have d ≤ ˜d, the primal index set ∆Lj for the left boundary functions is generally smaller than the dual set ˜∆Lj. By enlarging ∆Lj with the ˜d − d leftmost primal functions from the inner set ∆0j, and repeating the procedure analogously for the right end of the interval, we can define the square matrices
ΓLj:= We cite from [49] the following central
Theorem 3.9. The matrices ΓXj, X ∈ {L, R}, are independent of j and symmetric with respect to the left and right sides,
ΓLj = ΓL= Γ , ΓRj = ΓR= Γl. (3.3.36) In the situation of Theorem 3.6, Γ is always nonsingular.
It follows that the bases defined by
Φj:= Φ(0)j , Φ˜j:= Γ−Tj Φ˜(0)j :=
are biorthogonal. The primal refinement relation is not affected by this transformation, while the dual refinement matrix changes,
Mj,0 = M(0)j,0, M˜j,0= Γj+1M˜(0)j,0Γ−1j . (3.3.38) However, an exact calculation of the entries of Γ requires some non-trivial calculations [49]. From (3.3.26) and (3.3.27), we infer for r = 0, . . . , d − 1 and s = 0, . . . , ˜d − 1
Chapter 3. Two Constructions on the Interval Obviously, these expressions can be reduced to a calculation of
I(µ, ν) :=
Z ∞
0 φ(x − µ) ˜φ(x − ν) dx . (3.3.41)
From the length of the support of φ and ˜φ, we deduce that
I(µ, ν) =( 0 for µ ≤ −`2 or ν ≤ −˜`2,
δµ,ν for −`1≤ µ or −˜`1≤ ν . (3.3.42)
It remains to determine the values for I(µ, ν) in the remaining range of µ, ν. We reformulate
I(µ, ν) =
where we have used the definition z(µ, ν) := Inserting the refinement relations for χ, φ and ˜φ in the form of (3.3.1), we obtain the identity
z(β) =X where β, γ and η are two-dimensional indices. Due to the finite support of φ and ˜φ, the index sets are also finite. This leads to an eigenvector problem for z with eigenvalue 1, whose solution is uniquely determined by the normalisation
X
β
z(β) = 1 . (3.3.47)
We solve this eigenvector problem for z and insert the values into the equation for I(µ, ν) (3.3.43), which in turn can be used to compute the entries of Γ. After these steps, the biorthogonalisation is complete [50].