New results on approximate Hilbert pairs of wavelet filters with common factors

10/25/2017 ∙ by Sophie Achard, et al. ∙ 0

In this paper, we consider the design of wavelet filters based on the Thiran common-factor approach proposed in Selesnick [2001]. This approach aims at building finite impulseresponse filters of a Hilbert-pair of wavelets serving as real and imaginary part of a complexwavelet. Unfortunately it is not possible to construct wavelets which are both finitelysupported and analytic. The wavelet filters constructed using the common-factor approachare then approximately analytic. Thus, it is of interest to control their analyticity. Thepurpose of this paper is to first provide precise and explicit expressions as well as easilyexploitable bounds for quantifying the analytic approximation of this complex wavelet.Then, we prove the existence of such filters enjoying the classical perfect reconstructionconditions, with arbitrarily many vanishing moments.

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1 Introduction

Wavelet transforms provide efficient representations for a wide class of signals. In particular signals with singularities may have a sparser representation compared to the representation in Fourier basis. Yet, an advantage of Fourier transform is its analyticity, which enables to exploit both the magnitude and the phase in signal analysis. In order to combine both advantages of Fourier and real wavelet transform, one possibility is to use a complex wavelet transform. The analyticity can be obtained by choosing properly the wavelet filters. This may offer a true enhancement of real wavelet transform for example in singularity extraction purposes. We refer to

Selesnick et al. (2005); Tay (2007) and references therein for an overview of the motivations for analytic wavelet transforms. A wide range of applications can be addressed using such wavelets as image analysis (Chaux et al., 2006), signal processing (Wang et al., 2010), molecular biology (Murugesan et al., 2015), neuroscience (Whitcher et al., 2005).

Several approaches have been proposed to design a pair of wavelet filters where one wavelet is (approximately) the Hilbert transform of the other. Using this pair as real and imaginary part of a complex wavelet allows the design of (approximately) analytic wavelets. The simplest complex analytic wavelets are the generalized Morse wavelets, which are used in continuous wavelet transforms in Lilly and Olhede (2010). The approximately analytic Morlet wavelets can also be used for the same purpose, see Selesnick et al. (2005). However, for practical or theoretical reasons, it is interesting to use discrete wavelet transforms with finite filters, in which case it is not possible to design a perfectly analytic wavelets. In addition to the finite support property, one often requires the wavelet to enjoy sufficiently many vanishing moments, perfect reconstruction, and smoothness properties. Among others linear-phase biorthogonal filters were proposed in Kingsbury (1998a, b) or q-shift filters in Kingsbury (2000). We will focus here on the common-factor approach, developed in Selesnick (2001, 2002). In Selesnick (2002) a numerical algorithm is proposed to compute the FIR filters associated to an approximate Hilbert pair of orthogonal wavelet bases. Improvements of this method have been proposed recently in Tay (2010); Murugesan and Tay (2014). The approach of Selesnick (2001) is particularly attractive as it builds upon the usual orthogonal wavelet base construction by solving a Bezout polynomial equation. Nevertheless, to the best of our knowledge, the validity of this specific construction have not been proved. Moreover the quality of the analytic approximation have not been thoroughly assessed. The main goal of this paper is to fill these gaps. We also provide a short simulation study to numerically evaluate the quality of analyticity approximation for specific common-factor wavelets.

After recalling the definition of Hilbert pair wavelet filters, the construction of the Thiran’s common-factor wavelets following Thiran (1971); Selesnick (2002) is summarized in Section 2. Theoretical results are then developed to evaluate the impact of the Thiran’s common-factor degree on the analytic property of the derived complex wavelet. In Section 3, an explicit formula to quantify the analytic approximation is derived. In addition, we provide a bound demonstrating the improvement of the analytic property as increases. These results apply to all wavelets obtained from FIR filters with Thiran’s common-factor. Of particular interest are the orthogonal wavelet bases with perfect reconstruction. Section 4 is devoted to proving the existence of such wavelets arising from filters with Thiran’s common-factor, which correspond to the wavelets introduced in Selesnick (2001, 2002). Finally, in Section 5, some numerical examples illustrate our findings. All proofs are given in the Appendices.

2 Approximate Hilbert pair wavelets

2.1 Wavelet filters of a Hilbert pair

Let and be two real-valued wavelet functions. Denote by and their Fourier transform,

We say that forms a Hilbert pair if

where denotes the sign function taking values and 1 for , and , respectively. Then the complex-valued wavelet is analytic since its Fourier transform is only supported on the positive frequency semi-axis.

Suppose now that the two above wavelets are obtained from the (real-valued) low-pass filters and , using the usual multi-resolution scheme (see Daubechies (1992)

). We denote their z-transforms by

and , respectively. In Selesnick (2001) and Ozkaramanli and Yu (2003), it is established that a necessary and sufficient condition for to form a Hilbert pair is to satisfy, for all ,

(1)

Since takes different values at and , we see that this formula cannot hold if both and are continuous on the unit circle, which indicates that the construction of Hilbert pairs cannot be obtained with usual convolution filters and in particular with finite impulse response (FIR) filters. Hence a strict analytic property for the wavelet is not achievable for a compactly supported wavelet, which is also a direct consequence of the Paley-Wiener theorem.

However, for obvious practical reasons, the compact support property of the wavelet and the corresponding FIR property of the filters must be preserved. Thus the strict analytic condition (1) has to be relaxed into an approximation around the zero frequency,

(2)

Several constructions have then been proposed to define approximate Hilbert pair wavelets, that is, pairs of wavelet functions satisfying the quasi analytic condition (2) (Tay, 2007). The common-factor procedure proposed in Selesnick (2002), is giving one solution to the construction of approximate Hilbert pair wavelets. This is the focus of the following developments.

2.2 The common-factor procedure

The common-factor procedure (Selesnick, 2002) is designed to provide approximate Hilbert pair wavelets driven by an integer and additional properties relying on a common factor transfer function . Namely, the solution reads

(3)
(4)

where is the transform of a causal FIR filter of length , , such that

(5)

In Thiran (1971), a causal FIR filter satisfying this constraint is defined, the so-called maximally flat solution given by (see also (Selesnick, 2002, Eq (2))):

(6)

The cornerstone of our subsequent results is the following simple expression for , which appears to be new, up to our best knowledge.

Proposition 1.

Let be a positive integer and where the coefficients are defined by (6). Then, for all , we have

(7)

where denotes any of the two complex numbers whose squares are equal to .

Here denotes the set of all non-zero complex numbers.

Remark 1.

In spite of the ambiguity in the definition of , the right-hand side in (7

) is unambiguous because, when developing the two factors in the expression between square brackets, all the odd powers of

cancel out.

Remark 2.

It is interesting to note that the closed form expression (7) of directly implies the approximation (5). Indeed, the right-hand side of (7) yields

It is then straightforward to obtain (5).

Proof.

See Section A. ∎

To summarize the common-factor approach, we use the following definition.

Definition (Common-factor wavelet filters).

For any positive integer and FIR filter with transfer function , a pair of wavelet filters is called an -approximate Hilbert wavelet filter pair with common factor if it satisfies (3) and (4) with .

Condition is equivalent to

(8)

A remarkable feature in the choice of the common filter is that it can be used to ensure additional properties such as an arbitrary number of vanishing moments, perfect reconstruction or smoothness properties.

First an arbitrary number of vanishing moments is set by writing

(9)

with the -transform of a causal FIR filter (hence a real polynomial of ).

An additional condition required for the wavelet decomposition is perfect reconstruction. It is acquired when the filters satisfy the following conditions (see Vetterli (1986)):

(PR-G)
(PR-H)

This condition is classically used for deriving wavelet bases and , , which are orthonormal bases of . This will be investigated in Section 4.

3 Quasi-analyticity of common-factor wavelets

We now investigate the quasi-analyticity properties of the complex wavelet obtained from Hilbert pairs wavelet filters with the common-factor procedure.

Let be respectively the father and the mother wavelets associated with the (low-pass) wavelet filter . The transfer function is normalized so that (this is implied by (3) and (8)). The father and mother wavelets can be defined through their Fourier transforms as

(10)
(11)

where is the corresponding high-pass filter transfer function defined by (see e.g. Selesnick (2001)). We also denote by the father and the mother wavelets associated with the wavelet filter . Equations similar to (10) and (11) hold for , and using and in place of and (see e.g. Selesnick (2001)).

We first give an explicit expression of and of with respect to and .

Theorem 2.

Let be a positive integer. Let be an -approximate Hilbert wavelet filter pair. Let denote the father and mother wavelets defined by (10) and (11) and denote the wavelets defined similarly from the filter . Then, we have, for all ,

(12)
(13)

where

(14)
(15)
(16)

In (14), we use the convention so that is well defined on .

Proof.

See Section A. ∎

Following Theorem 2, we can write, for all ,

(17)

This formula shows that the quasi-analytic property and the Fourier localization of the complex wavelet can be respectively described by

  1. [label=()]

  2. how close the function is to the step function (or to the sign function);

  3. how localized the (real) wavelet is in the Fourier domain.

Property 2 is a well known feature of wavelets usually described by the behavior of the wavelet at frequency 0 (e.g. vanishing moments implies a behavior in ) and by the polynomial decay at high frequencies. This behavior depends on the wavelet filter (see Villemoes (1992); Eirola (1992); Ojanen (2001)) and a numerical study of property 2 is provided in Section 5.

Note that, remarkably, property 1, only depends on . Figure 1 displays the function for various values of . It illustrates the fact that as grows, indeed gets closer and closer to the step function . We can actually prove the following result which bounds how close the Fourier transform of the wavelet is to .

Denote, for all and , the distance of to by

(18)
Theorem 3.

Under the same assumptions as Theorem 2, we have, for all ,

where is a function satisfying, for all ,

(19)
Proof.

See Section A. ∎

This result provides a control over the difference between the Fourier transform of the complex wavelet and the Fourier transform of the analytic signal associated to . In particular, as , the relative difference converges to zero exponentially fast on any compact subsets that do not intersect .

Figure 1: Plots of the function for , 4, 8, 16.

4 Solutions with perfect reconstruction

Let us now follow the path paved by Selesnick (2002) to select appearing in the factorization (9) of the common factor to impose vanishing moments. First observe that, under (3), (4) and (9), the perfect reconstruction conditions (PR-G) and (PR-H) both follow from

(20)

where we have set and .

To achieve (20), the following procedure is proposed in Selesnick (2002), which follows the approach in Daubechies (1992) adapted to the common factor constraint in (3).

  1. [label=Step 0]

  2. Find with finite, real and symmetric impulse response satisfying (20).

  3. Find a real polynomial satisfying the factorization .

However, in Selesnick (2002), the existence of solutions and is not proven, although numerical procedures indicate that solutions can be exhibited. We shall now fill this gap and show the existence of such solutions for any integers .

We first establish the set of solutions for .

Proposition 4.

Let and be two positive integers. Let be defined as in Proposition 1 and let . Then the two following assertions hold.

  1. [label=()]

  2. There exists a unique real polynomial of degree at most such that satisfies (20) for all .

  3. For any real polynomial , the function satisfies (20) on if and only if it satisfies

    (21)

    where

    (22)

    and is any real polynomial satisfying .

Proof.

See Section B. ∎

Proposition 4 provides a justification of 1. In particular, a natural candidate for 1 is . Now, by the Riesz Lemma (see e.g. (Daubechies, 1992, Lemma 6.1.3)), the factorization of 2 holds if and only if takes its values in on the unit circle , or equivalently for all . Although easily verifiable in practice (using a numerical computation of the roots of ), checking this property theoretically for all integers is not yet achieved.

Nevertheless we next prove that 2 can always be carried out for any , at least by modifying into a polynomial of the form (21) with a conveniently chosen .

Theorem 5.

Let and be two positive integers and let and be the polynomials defined as in Proposition 4. Then there exists a real polynomial such that is a solution of (20) and satisfies the factorization where , real polynomial of , does not vanish on the unit circle.

Proof.

See Section B. ∎

Proposition 4 and Theorem 5 allows one to carry out the usual program to the construction of compactly supported orthonormal wavelet bases, as described in Daubechies (1992). Hence we get the following.

Corollary 6.

Let and be two positive integers. Let be as in Theorem 5. Define as in (9) and let be the -approximate Hilbert wavelet filter pair associated to . Then the wavelet bases and are orthonormal bases of .

Observe that Theorem 5 states the existence of the polynomial but does not define it in a unique way. We explain why in the following remark.

Remark 3.

Since in Proposition 4 is defined uniquely, it follows that, if we require that all the roots of are inside the unit circle, there is at most one solution for with degree at most , which correspond to the case . This solution, when it exists, is usually called the minimal phase, minimal degree solution. However we were not able to prove that does not vanish on , which is a necessary and sufficient condition to obtain such a minimal degree solution for . Hence we instead prove the existence of solutions for by allowing to be non-zero.

5 Numerical computation of approximate Hilbert wavelet filters

5.1 State of the art

Let and be positive integers. Then, by Theorem 5, we can define the polynomial and derive from its coefficients the impulse response of the corresponding -approximate Hilbert wavelet filter pair with vanishing moments and perfect reconstruction.

We now discuss the numerical computation of the coefficients of in the case where the polynomial defined by Proposition 4 does not vanish on . Indeed suppose that one can obtain a numerical computation of this polynomial . Then the roots of can also be computed by a numerical solver and, as explained in Remark 3, if they do not belong on (which has to be checked taking into account the possible numerical errors), it only remains to factorize into by separating the roots conveniently. Taking all roots of modulus inferior to 1 leads to “mid-phase” wavelets. There are other ways of factorizing , namely “min-phase” wavelets, see Selesnick (2002), leading to wavelets with Fourier transform of the same magnitude but with different phases. This difference can be useful in some multidimensional applications where the phase is essential.

Hence the computation of the wavelet filters boils down to the numerical computation of the polynomial defined by Proposition 4. In Selesnick (2002), this computation is achieved by using the following algorithm.

  • Let and , where denotes the convolution for sequences. Then with . The filter has length .

  • The filter is such that is half-band. Let denote the Toeplitz matrix associated with

    , vector of length

    , that is, if and else. We introduce the matrix obtained by keeping only the even rows of , which has size . Then is the solution of the equation

    (23)

    with a -vector with a at the middle (i.e. at -th position).

We implemented this linear inversion method but it turned out that the corresponding linear equation is ill posed for too high values of and (for instance ). For smaller values of and , we recover the wavelet filters of the hilbert.filter program of the R-package waveslim computed only for equal to (3,3), (3,5), (4,2) and (4,4), see Whitcher (2015).

5.2 A recursive approach to the computation of the Bezout minimal degree solution

We propose now a new method for computing the -approximate common-factor wavelet pairs with vanishing moments under the perfect reconstruction constraint. As explained previously, this computation reduces to determining the coefficients of the polynomial defined in Proposition 4. Our approach is intended as an alternative to the linear system resolution step of the approach proposed in Selesnick (2002). Since our algorithm is recursive, to avoid any ambiguity, we add the subscripts for denoting the polynomials and appearing in 4. That is, we set

and is the unique polynomial of degree at most satisfying the Bezout equation

  1. [label=,labelsep=1.4cm,align=left]

We propose to compute for all , by using the following result.

Proposition 7.

Let . Define

(24)

Then the solution of the Bezout equation is given by

(25)

Moreover, for all , we have the following relation between the solution of and that of :

(26)
Proof.

See Appendix C. ∎

This result provides a recursive way to compute by starting with

using the interpolation formula (

25) and then using the recursive formula (26) to compute up to . In contrast to the method of Selesnick (2002) which consists in solving a (possibly ill posed) linear system, this method is only based on product and composition of polynomials.

5.3 Some numerical result on smoothness and analyticity

We now provide some numerical results on the quality of the analyticity of the -approximated Hilbert wavelet. All the numerical computations have been carried out by the method of Selesnick (2002) which seems to be the one used by practitioners (as in the software of Whitcher (2015)). Recall that as established in Theorem 3, for all ,

where is displayed in Figure 2. Thus the quality of analyticity relies on the behavior of but also of . First, goes to when thanks to the property of vanishing moments given by (9). Secondly, decays to zero as

goes to infinity. This last point is verified numerically, by the estimation of the Sobolev exponents of

using Ojanen (2001)’s algorithm. Values are given in Table 1. For Sobolev exponents are greater than 1. Notice that “min-phase” and “mid-phase” factorizations of have the same exponents since the methods do not change the magnitude of .

M L 1 2 3 4 5 6 7 8
1 0.60 0.72 0.81 0.89 0.94 0.98 0.99 1.00
2 1.11 1.23 1.34 1.44 1.54 1.63 1.73 1.82
3 1.52 1.64 1.74 1.83 1.92 2.01 2.09 2.17
4 1.87 1.98 2.07 2.16 2.24 2.32 2.40 2.48
5 2.19 2.29 2.37 2.45 2.53 2.60 2.68
6 2.48 2.57 2.65 2.72 2.80 2.87
7 2.74 2.83 2.91 2.98 3.05 3.12
8 3.00 3.09 3.16 3.23 3.29
Table 1: Sobolev exponent estimated for functions. Dots correspond to configurations where numerical instability occurs in the numerical inversion of (23).

Figure 2 displays the overall shapes of the Fourier transforms of orthonormal wavelets with common-factor for various values of and . Their quasi-analytic counterparts are plotted below in the same scales. It illustrates the satisfactory quality of analityc approximation.

Figure 2: Top row: Plots of for (left), 3 (center), 4 (right) and (black), 4 (red), 8 (green). Bottom row: same for .

Tay et al. (2006) propose two objective measures of quality based on the spectrum,

Numerical values of and are computed using numerical evaluations of on a grid, and, concerning , using Riemann sum approximations of the integrals. Such numerical computations of and are displayed in Figure 3 for various values of and . The functions and are decreasing with respect to (which corresponds to the behaviour of ). They are also decreasing with respect to (through the faster decay of around zero and infinity). Moreover, the values illustrate the good analyticity quality of common-factor wavelets. For example, values appear to be lower than those of approximate analytic wavelets based on Bernstein polynomials given in Tay et al. (2006).

Figure 3: Plot of and with respect to for different values of .

6 Conclusion

Approximate Hilbert pairs of wavelets are built using the common-factor approach. Specific filters are obtained under perfect reconstruction conditions. They depend on two integer parameters and which correspond respectively to the order of the analytic approximation and the number of null moments. We demonstrate that the construction of such wavelets is valid by proving their existence for any parameters . Our main contribution in this paper is to provide an exact formula of the relation between the Fourier transforms of the two real wavelets associated to the filters. This expression allows us to evaluate the analyticity approximation of the wavelets, i.e. to control the presence of energy at the negative frequency. This result may be useful for applications, where the approximated analytic properties of the wavelet have to be optimized, in addition to the usual localization in time and frequency. Numerical simulations show that these wavelets are easy to compute for not too large values of and , and confirm our theoretical findings, namely, that the analytic approximation quickly sharpens as increases.

7 Acknowledgements

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Appendix A Proofs of Section 3

a.1 Proof of Proposition 1

Proof of (7).

Notice that and that for all ,

It is then easy to check that

The fact that concludes the proof. ∎

a.2 Technical results on

We first establish the following result, which will be useful to handle ratios with .

Lemma 8.

Let be a positive integer. Define as in Proposition 1. Then does not vanish on the unit circle () and

Proof.

Since is a real polynomial of , we have for all such that , . Moreover, as shown in the proof of Proposition 4, if with , then reads as in (43), which is minimal and maximal for and , respectively. ∎

We now study on the circle.

Lemma 9.

For all with , we have

(27)

where is the function defined on by (14).

Proof.

Observe that, for all , denoting by any of the two roots of ,

Set now . We deduce that

The result then follows from the classical result with here . ∎

a.3 Proof of Theorem 2

Proof of equality (12).

Equation (10) provides the relation between and . The same relation holds between and . It follows with Lemma 8, (3) and (4), that, for all ,

Applying Lemma 9, we get that, for all ,

We thus obtain (12) using the definition of given by (15). ∎

Proof of equality (13).

First observe that the relation between the high-pass filters and follows from that between the low-pass filter and , namely

The relationship between and is given by (11) (exchanging and ), yielding, for all ,

We now replace by the expression obtained in (12) and thanks to (11),

Since has a real impulse response and , Lemma 9 gives that, for all ,

Hence, we finally get that, for all ,

(13) is proved. ∎

a.4 Proof of Theorem 3

Approximation of

We first state a simple result on the function .

Lemma 10.

Let be a positive integer. The function defined by (14) is -periodic. Moreover is continous on and we have, for all ,

(28)

where

(29)

and

(30)
Proof.

By definition (14), is -periodic and continuous on . Moreover, at any of its discontinuity points in , jumps have height . Hence