Characterization of oblique dual frame pairs

Given a frame for a subspace of a Hilbert space, we consider all possible families of oblique dual frame vectors on an appropriately chosen subspace. In place of the standard description, which involves computing the pseudoinverse of the frame operator, we develop an alternative characterization which in some cases can be computationally more efficient. We first treat the case of a general frame on an arbitrary Hilbert space, and then specialize the results to shift-invariant frames with multiple generators. In particular, we present explicit versions of our general conditions for the case of shift-invariant spaces with a single generator. The theory is also adapted to the standard frame setting in which the original and dual frames are defined on the same space.


INTRODUCTION
Frames are generalizations of bases which lead to redundant signal expansions [1][2][3][4]. A frame for a Hilbert space is a set of not necessarily linearly independent vectors that has the property that each vector in the space can be expanded in terms of these vectors. Frames were first introduced by Duffin and Schaeffer [1] in the context of nonharmonic Fourier series, and play an important role in the theory of nonuniform sampling [1,2,5,6]. Recent interest in frames has been motivated in part by their utility in analyzing wavelet expansions [7,8], and by their robustness properties [3,[8][9][10][11][12][13].
Frame-like expansions have been developed and used in a wide range of disciplines. Many connections between frame theory and various signal processing techniques have been recently discovered and developed. For example, the theory of frames has been used to study and design oversampled filter banks [14][15][16][17] and error correction codes [18]. Wavelet families have been used in quantum mechanics and many other areas of theoretical physics [8,19].
One of the prime applications of frames is that they lead to expansions of vectors (or signals) in the underlying Hilbert space in terms of the frame elements. Specifically, if H is a separable Hilbert space and { f k } ∞ k=1 is a frame for H , then any f in H can be expressed as for some dual frame {g k } ∞ k=1 for H . In order to use this representation in practice, we need to be able to calculate the coefficients f , g k . A popular choice of {g k } ∞ k=1 is the minimal-norm dual frame, that is, the canonical dual frame. However, computing the minimal-norm dual is highly nontrivial in general. Another issue is that the frame { f k } ∞ k=1 might have a certain structure which is not shared by the minimal-norm dual. This complication appears, for example, if { f k } ∞ k=1 is a wavelet frame: there are cases where the canonical dual of a wavelet frame does not have the wavelet structure (cf. [8]). One way to circumvent these types of problems is to search for more general choices of duals. Usually, one requires additional constraints on the choice of k=1 has a shift-invariant structure, it is natural to require that {g k } ∞ k=1 also share this structure.
More recently, the traditional concept of frames has been broadened to include frames on subspaces. Oblique frame decompositions, which were suggested in [10,20] and further developed in [21][22][23], allow for frame expansions in which (1) is restricted to signals f in a given closed subspace X of H . The vectors { f k } ∞ k=1 and {g k } ∞ k=1 are still required to be frames, but only for subspaces of H ; { f k } ∞ k=1 forms a frame for X and {g k } ∞ k=1 constitutes a frame for a possibly different subspace S such that H = X⊕S ⊥ , where S ⊥ denotes the orthogonal complement of S in H . By choosing S = X = H , we recover the conventional dual frames; however, oblique dual frames allow for more freedom in the design since the analysis space S is not restricted to be equal to the synthesis space X as in traditional frame expansions. A further 2 EURASIP Journal on Applied Signal Processing generalization of this concept leads to pseudoframes [24]. As in oblique dual frames, (1) is restricted to f ∈ X, but { f k } ∞ k=1 and {g k } ∞ k=1 are no longer constrained to be frame sequences. Since, in this paper, we are interested in frame decompositions, we focus our attention on oblique dual frames which provide a general setting for frame analysis.
Given a frame { f k } ∞ k=1 for a subspace X, a complete characterization of all possible oblique dual frames on a subspace S has been obtained in [22,24]. This characterization involves computing the pseudoinverse of the frame operator TT * , where T is the preframe operator associated with the frame { f k } ∞ k=1 . In many cases, computing this pseudoinverse is computationally demanding. An interesting question therefore is whether there is an alternative characterization for all oblique duals which does not necessarily involve the pseudoinverse of TT * . Our main result, derived in Section 3, shows that the oblique dual frames can be characterized in an alternative way in which the pseudoinverse of TT * is replaced by the pseudoinverse of HT * , where H is an appropriately chosen operator. The advantage of this characterization is that there is freedom in choosing the operator H so that it can be tailored such that the pseudoinverse of HT * is easier to compute than the pseudoinverse of TT * . Concrete examples demonstrating this computational advantage have recently been explored in [25][26][27] in the context of Gabor expansions.
An important class of frames in signal processing applications are shift-invariant frames, which are generated by translates of a set of generators [6]. The advantage of these frames is that the corresponding frame expansion can be implemented using linear time-invariant (LTI) filters. In Section 4, we specialize our results to the case of shiftinvariant frames. As we show, while the classical frame representation may involve ideal filters which cannot be implemented in practice, by using the proposed alternative representation, the ideal filters can often be replaced by non ideal realizable filters. Furthermore, our general conditions take a particular simple form in the case of a shift-invariant space generated by a single function.
Before proceeding to the detailed development, in the next section, we summarize the required mathematical preliminaries.

DEFINITIONS AND BASIC RESULTS
We now introduce some definitions and results that will be used throughout the paper.
Given a transformation T, we denote by N (T) and R(T) the null space and range space of T, respectively. The Moore-Penrose pseudoinverse of T is written as T † and the adjoint is denoted by T * . The inner product between vectors x, y ∈ H is denoted by x, y , and is linear in the first argument. We use R and Z to denote the reals and integers, respectively. The complex conjugate of a complex function f (x) is denoted by f (x). For a subspace W of a Hilbert space H , W ⊥ is the orthogonal complement of W in H . Given a sequence of vectors {g k } ∞ k=1 ⊂ H , we let span{g k } ∞ k=1 be the closure of the span of {g k } ∞ k=1 , that is, the smallest closed subspace containing {g k } ∞ k=1 (the span of a set of vectors consists by definition of all finite linear combinations of the vectors with complex coefficients). Projection operators play an important role in our development. Given closed subspaces W and V of a Hilbert space H such that H = W ⊕ V ⊥ (a direct sum, not necessarily orthogonal), the oblique projection E WV ⊥ onto W along V ⊥ is defined as the unique operator satisfying That is, there is a one-to-one correspondence between decompositions of H and projections on H . Thus, our results in this paper obtained via the splitting assumption H = W ⊕ V ⊥ could as well be formulated starting with a projection.
For f ∈ L 1 (R), we denote the Fourier transform by As usual, the Fourier transform is extended to a unitary operator on L 2 (R). For a sequence c = {c k } ∈ 2 , we define the discrete-time Fourier transform as the 1-periodic function in L 2 (0, 1) given by The discrete-time convolution a k = c k * d k between two sequences c, d ∈ 2 is defined by The continuous-time convolution between two functions φ, φ 1 ∈ L 2 (R) is given by A set of vectors { f k } ∞ k=1 forms a Bessel sequence for a Hilbert space H if there exists a constant B < ∞ such that for all x ∈ H [3]. The preframe operator associated with a Bessel sequence { f k } ∞ k=1 is given by and its adjoint is given by The assumption H = W ⊕ V ⊥ will play a crucial role throughout the paper. Lemma 1, proved by Tang (see [29,Theorem 2.3]), deals with this condition, and relies on the concept of the angle between two subspaces. The angle from V to W is defined as the unique number Lemma 1. Given closed subspaces V, W of a separable Hilbert space H , the following are equivalent: More information on the condition H = W ⊕ V ⊥ in general Hilbert spaces can be found in [22].

Oblique dual frames
The terminology oblique dual frame originates from the relation of these frames with oblique projections, as incorporated in the following lemma [22].
Then the following are equivalent. In case the equivalent conditions are satisfied,

Lemma 2 leads to a simple geometric interpretation of the oblique dual frames. Given a classical dual
to an oblique dual on V by constructing the sequence . This interpretation is illustrated in Figure 1. Denoting by T the preframe operator of the frame { f k } ∞ k=1 , it was shown in [22,24] that the oblique dual frames where {h k } ∞ k=1 ∈ V is a Bessel sequence. The characterization (14) involves computing the pseudoinverse of TT * which can be computationally demanding. An interesting question therefore is whether there is an alternative characterization for the duals which does not involve the pseudoinverse of TT * . Our main result, Theorem 1, shows that the oblique dual frames can be characterized in an alternative way in which the pseudoinverse (TT * ) † is replaced by (HT * ) † , where H is an appropriately chosen operator. The advantage of this characterization is that there is freedom in choosing the operator H so that it can be tailored such that (HT * ) † is easier to compute than (TT * ) † . Furthermore, in this representation, the infinite sum is no longer required.
In Section 4, we specialize the results to the case of shiftinvariant frames which are important in signal processing applications since frame expansions involving shift-invariant frames can be implemented using LTI filters.

Mathematical preliminaries
The proof of our main theorem is based on some general results from the theory of operators on Hilbert spaces. Therefore, before stating our result, we collect the needed facts in Lemma 4. We first present a well-known identity, which we will use in the sequel.
Proof. The lemma is proven in a straightforward manner by showing that under the conditions of the lemma, B † A † satisfies the Moore-Penrose conditions [30].
Then the following hold.
(iv) The operator is independent of the choice of the bounded operator U : For the proof, see the appendix. We note that Lemma 4(iii) provides an explicit method for computing the oblique projection E VW ⊥ ; it is especially convenient if we choose H 1 = 2 , in which case Y * U becomes an operator on 2 .

Oblique dual families
We now present our main result, which provides an alternative characterization of all oblique duals.
where {h k } ∞ k=1 is a frame sequence with preframe operator H, satisfying that N (H) ⊕ R(T * ) = 2 . Alternatively, where B : 2 → H is any bounded operator with R(B) = V, S is a closed subspace of 2 such that 2 = R(T * ) ⊕ S, and is the canonical orthonormal basis for 2 . Note that from Lemma 4(iv), it follows that the families defined by (19) differ only in the choice of S.
Proof. The proof of the theorem relies on the following lemma.
Lemma 5 (see [22]). Let { f k } ∞ k=1 be a frame for W , and let V be a closed subspace such that k=1 be the canonical orthonormal basis for 2 . The oblique dual frames By Lemmas 4 and 5, we can characterize the oblique dual frames on V along W ⊥ as all families of the form where H : 2 → H is a bounded operator with closed range, satisfying that 2 = N (H) ⊕ R(T * ). Such an operator has the form H{c ∈ H a frame sequence. By inserting this expression for H in (20), we get From Lemma 4(iii), we can write E VW ⊥ as where M = B(T * B) † . Substituting (22) into (18), we have that with S = N (H), thus completing the proof.
In the special case in which W = H , Theorem 1 implies that the classical dual frames of { f k } ∞ k=1 are the families where {h k } ∞ k=1 is a frame sequence, satisfying that N (H) ⊕ R(T * ) = 2 . This should be compared with the known characterization [31] Eldar and O. Christensen 5 and only if H is injective. However, if { f k } ∞ k=1 is overcomplete, then R(T * ) is a subspace of 2 ; the more redundant the frame is, the "smaller" R(T * ) is, that is, the larger the kernel of H is forced to be.
In [25][26][27], it is shown that using the characterization (24) in a finite-dimensional setting can lead to Gabor expansions that are computationally much more efficient than conventional Gabor expansions. Furthermore, by proper choice of H, one can improve the condition number of HT * . Specifically, consider the case in which we are given the Gabor expansion of a finite-length signal, and the goal is to reconstruct the signal from these samples. Instead of using the minimal-norm dual for reconstruction, corresponding to (TT * ) † T, it is suggested to use a nonminimal norm dual of the form (HT * ) † H, where H is chosen such that HT * is efficient to compute. For example, if T is a frame operator corresponding to a Gabor frame with a Gaussian window φ[k] = e −k 2 /σ 2 1 for some σ 2 1 > 0, then we can choose H as a frame operator corresponding to a Gabor frame with a Gaussian window h[k] = e −k 2 /σ 2 2 , where σ 2 is chosen such that the effective spread of h[k] is equal to a. If L/b is divisible by a, where L is the length of the signal and a and b are the shifts along the time and frequency axes, respectively, then the matrix HT * is invertible for any choice of σ 2 . Because of the limited spread of h[k], the matrix HT * can be computed very efficiently, resulting in an efficient method for reconstructing the signal from its Gabor coefficients.
One more advantage of the approach is that for large values of σ 1 , the matrix TT * can be poorly conditioned. By appropriately selecting the spread σ 2 of h[k], it is possible to improve the condition number of HT * , leading to a more stable reconstruction algorithm.

Minimal-norm duals
We now use the representation of Theorem 1 to develop alternative forms of the minimal-norm oblique duals.
Given a bounded operator B with R(B) = V, the minimal-norm oblique dual vectors of { f k } ∞ k=1 on V along W , that is, the oblique dual vectors leading to coefficients with minimal 2 norm, can be written as [10,20] The representation (26) follows from Theorem 1 if we choose (19) reduces to (26). Alternatively, it was shown in [22] that the minimal-norm oblique duals can be expressed as This characterization also follows from Theorem 1, with H = T. More generally, we can obtain this characterization by choosing H as an arbitrary operator with N (H) = N (T), as incorporated in the following theorem.
where {h k } ∞ k=1 is a frame sequence with preframe operator H, satisfying that N (H) = N (T). Alternatively, where B is a bounded operator with R(B) = V and {δ k } ∞ k=1 is the canonical orthonormal basis for 2 .
Proof. The proof of the theorem follows from the fact that if T : H 1 → H 2 is a bounded operator with closed range, then the operator is independent of the choice of the bounded operator U : From Lemma 3, it then follows that Therefore, thus completing the proof.
If V = W , then the vectors g k defined by Theorem 2 are the conventional minimal-norm dual frame vectors. Thus, Theorem 2 provides an alternative method for computing the conventional dual frame vectors, which are typically given by By using Theorem 2, we may choose B so that (T * B) † is easier to compute than (T * T) † ; alternatively, we may choose H such that (HT * ) † can be evaluated more efficiently than (TT * ) † .

FRAME SEQUENCES IN SHIFT-INVARIANT SPACES
We now consider frames of translates in shift-invariant spaces. The importance of this class of frames stems from the fact that the corresponding frame expansions can be implemented using LTI filters.

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EURASIP Journal on Applied Signal Processing

Shift-invariant frames
A shift-invariant frame with multiple generators is a frame { f k j } k∈Z, j∈J of the form where J is an index set, φ j ∈ L 2 (R) and we define the translation operator acting on functions in where h k j = T k h j and φ k j = T k φ j , can be implemented using a bank of LTI filters, as depicted in Figure 2. To see this, we first note that for fixed j, the coefficients can be expressed as samples at x = k of a convolution integral where g(x) = h j (−x). Thus, the sequence c k j can be viewed as samples at x = k of the output of an LTI filter with impulse response h j (−x), with f (x) as its input. Next, we note that the sum k∈Z c k j φ j (x − k) can be expressed as a convolution where p(x) is the modulated impulse train

Shift-invariant duals
Having defined shift-invariant frames, our goal now is to obtain shift-invariant oblique dual frames via Theorem 1.
For φ j , h j ∈ L 2 (R), j ∈ J, we let We further denote by T and H the preframe operators of the sequences {T k φ j } k∈Z, j∈J and {T k h j } k∈Z, j∈J , respectively. Throughout the section, we make the following assumptions: Note that if {T k φ j } k∈Z, j∈J is a frame sequence, then these conditions can be formulated entirely in terms of the operators T and H via This formulation shows that in general, the two conditions are unrelated. In fact, if {T k φ j } k∈Z, j∈J and {T k h j } k∈Z, j∈J are frames for L 2 (R), then the first condition holds; but if, for example, {T k φ j } k∈Z, j∈J is a Riesz basis and {T k h j } k∈Z, j∈J is overcomplete, then the second condition does not hold. On the other hand, if {T k φ j } k∈Z, j∈J and {T k h j } k∈Z, j∈J are Riesz sequences, then the second condition holds; but in case one of these sequences spans L 2 (R) and the other does not, then the first condition is not satisfied.
Theorem 3. Let φ j , h j ∈ L 2 (R), j ∈ J, and assume that {T k φ j } k∈Z, j∈J and {T k h j } k∈Z, j∈J are frame sequences. Then, under assumptions (i) and (ii), the sequence Proof. We first show that Indeed, for any f ∈ H , Y. C. Eldar and O. Christensen 7 Now, h j = Ha j for some a j . From assumption (ii), we can express a j as a j = a H j + a T j , where a H j ∈ N (H) and a T j ∈ R(T * ). Therefore, h j = Ha j = Ha T j . But since a T j ∈ R(T * ), we have that a T j = T * b j for some b j ∈ N (T * ) ⊥ = R(T) = W . We conclude that h j = HT * b j for some b j ∈ W , and Substituting (44) into (46), we have that where P is an orthogonal projection onto N (HT * ) ⊥ . But, by assumption (ii), N (HT * ) = N (T * ) = R(T) ⊥ = W ⊥ , so that P = P W . Since E VW ⊥ P W = E VW ⊥ , (47) reduces to Now, it was shown in [22,Corollary 4.2] that if W and V are shift-invariant, then E VW ⊥ T k = T k E VW ⊥ , which from (47) implies that where

Single generator
An important special case of a shift-invariant frame is a frame of the form {T k φ} k∈Z , with a single generator φ. These frames are especially easy to analyze. In particular, as the following proposition shows, one can immediately characterize the generators that create a frame for their closed linear span ({T k φ} k∈Z cannot be a frame for all of L 2 (R), cf. [32]).

Then {T k φ} k∈Z is a frame sequence with bounds A, B if and only if
It turns out that for single-generated systems, the conditions L 2 (R) = W ⊕ V ⊥ and 2 = R(T * ) ⊕ N (H) of the previous section are also easy to verify. Suppose that {T k φ} k∈Z and {T k h} k∈Z are frame sequences, and let The following proposition, proved in [22], provides an easily verifiable condition on the generators φ and h such that

Ψ) and there exists a constant A > 0 such that
We now show that the second condition 2 = R(T * ) ⊕ N (H) is actually contained in the first condition L 2 (R) = W ⊕ V ⊥ . Thus, only the first condition needs to be verified, which can be done in a straightforward way by using Proposition 2. Proof. It was shown in [22,Lemma 4.7] that the range of the adjoint of the preframe operator associated to any singlegenerated shift-invariant frame is Thus, if R(T * ) + N (H) = 2 , then (56) implies that for any c ∈ 2 , its discrete-time Fourier transform satisfies C(e 2πiω ) = 0 on N (Φ) ∩ N (Ψ) c , from which we conclude that N (Φ) ∩ N (Ψ) c = ∅.
Combining our results leads to the following characterization of all oblique duals in the single-generated shiftinvariant case.
and let Suppose that {T k φ} k∈Z is a frame sequence so that for some A > 0. Then, the sequence

LTI representation of minimal-norm duals
We now develop an LTI representation of the minimal-norm duals of a single-generated shift-invariant frame. We have seen in Theorem 2 that the minimal-norm oblique duals can be characterized as g k = B(T * B) † δ k , where B : 2 → H is a bounded operator with range V such that H = W ⊕ V ⊥ . Suppose now that we let T be the preframe operator of a shift-invariant frame {T k φ} k∈Z for W and choose B as the preframe operator of a shift-invariant frame {T k b} k∈Z . Proposition 2 provides necessary and sufficient conditions onb(ω) such that H = W ⊕ V ⊥ . Given a generator b(x) satisfying these conditions, we now show how to implement the operator B(T * B) † using LTI filters. , Φ e 2πiω = 0, 0, Φ e 2πiω = 0. (63) Proof. We first show that if c = (T * B) † d, then the sequence c k can be obtained by filtering the sequence d k with the filter A(e j2πω ). To this end, we note that if d = T * Bg, then d can be obtained by filtering the sequence g k with a filter Indeed, where It then follows that h k are the samples at the points x = k of the function f (x) whose Fourier transform is given bŷ f (ω) =φ(ω)b(ω). Therefore, H e j2πω = k∈Zf (ω + k) = k∈Zφ (ω + k)b(ω + k). (67) Thus, (T * B) † is equivalent to filtering the input sequence with the filter A(e j2πω ). To conclude the proof, we note that if f = Bg, then f (x) = k∈Z g k b(x − k), which is equivalent to modulating the sequence g k by an impulse train, and filtering the modulated sequence with a filter with impulse response b(x).
Lemma 6 can be used to develop an efficient method for reconstructing a signal g(x) in W from coefficients c = T * g. Specifically, the reconstruction is obtained as g = B(T * B) † c which is the output of the block diagram in Figure 3 with the sequence c as its input. Now, the kth coefficient c k can be written as and thus can be obtained by filtering the input signal g(x) with a filter with impulse response f (−x) and frequency responsef (ω), and then sampling the output at x = k.
The advantage of this reconstruction is that given the samples c, we have freedom in choosing the filterb(ω) so that it can be tailored such that the filtersb(ω) and A(e 2πiω ) are easy to implement.
Note that if the signal g(x) does not lie in the space W spanned by the signals { f (x − k)}, then the output of the block diagram of Figure 3 will be equal to P W g(x). This follows immediately from the fact that B(T * B) † T * = T(T * T) † T * = P R(T) = P W . A similar idea was first introduced in [34] in the context of consistent sampling. In that setting, it was suggested to choose a filterb(ω) that spans a space V, different from the sampling space W , that is easy to implement, and then use a discrete-time correction filter in order to compensate for the mismatch between the sampling filter and the reconstruction filter. Here we use a similar idea where the essential difference is that in the scheme of Figure 3, the overall reconstruction is equivalent to an orthogonal projection onto the reconstruction space, while the scheme of [34] is equivalent to an oblique projection.

CONCLUSION
We have obtained a complete characterization of the oblique dual frames associated with a frame for a subspace of a Hilbert space. Compared to the use of the classical dual frame, this leads to considerable freedom in the design. In [25,26], we demonstrated that these results can lead to much more efficient representations in the case of finitedimensional spaces; we believe that the results presented here will lead to similar gains in the general case. As an important special case, we considered frame expansions in shiftinvariant spaces. For the case of a single generator, our general conditions take a particular simple form.