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A New Scheme for the Design of Hilbert Transform Pairs of Biorthogonal Wavelet Bases
EURASIP Journal on Advances in Signal Processing volume 2010, Article number: 712105 (2010)
In designing the Hilbert transform pairs of biorthogonal wavelet bases, it has been shown that the requirements of the equal-magnitude responses and the half-sample phase offset on the lowpass filters are the necessary and sufficient condition. In this paper, the relationship between the phase offset and the vanishing moment difference of biorthogonal scaling filters is derived, which implies a simple way to choose the vanishing moments so that the phase response requirement can be satisfied structurally. The magnitude response requirement is approximately achieved by a constrained optimization procedure, where the objective function and constraints are all expressed in terms of the auxiliary filters of scaling filters rather than the scaling filters directly. Generally, the calculation burden in the design implementation will be less than that of the current schemes. The integral of magnitude response difference between the primal and dual scaling filters has been chosen as the objective function, which expresses the magnitude response requirements in the whole frequency range. Two design examples illustrate that the biorthogonal wavelet bases designed by the proposed scheme are very close to Hilbert transform pairs.
The dual-tree complex wavelet transform (DTCWT) had become an attractive signal processing tool since it was proposed by Kingsbury [1, 2]. It overcomes the main drawbacks of real discrete wavelet transform (DWT), such as shift sensitivity and poor directional selectivity (in the case of multidimensional DWT). A pair of filter banks is used in DTCWT, with which the wavelet bases associated form (approximate) Hilbert transform pairs. This property is critical since it is vital to reduce shift sensitivity and improve directionality .
Selesnick , and Ozkaramanli and Yu [5, 6] had studied the conditions under which the corresponding orthogonal wavelet bases form Hilbert transform pairs. It shows that if and only if one orthogonal low-pass filter is half-sample delayed from the other, the corresponding wavelets form a Hilbert transform pair. The compactly supported real orthogonal wavelets (except Haar wavelet) cannot be of linear phase (symmetry), however, linear phase is essential in many signal processing applications. As a result, the biorthogonal wavelets are usually suggested to be used, for example, JPEG2000 standard. Yu and Ozkaramanli  had further proven that the requirements of half-sample offset are also the necessary and sufficient condition for the design of Hilbert transform pairs of biorthogonal wavelet bases.
In , Selesnick proposed a new design scheme and designed two filter banks that the corresponding wavelets forming approximate Hilbert transform pairs. In , the design is formulated as a sampled-data optimization problem (suppose the primal filters are known) and then solved by standard approximation theory. The design scheme usually gives infinite impulse response (IIR) filters, which have to be truncated to form FIR filters. As a result, the dual filter bank is not symmetric and of perfect reconstruction (PR). In , the design is achieved by a constrained optimization procedure, which can ensure the filter bank to be symmetrical and PR. In the design, the objective function and constraints are all expressed in terms of the low-pass filters, that is, , , , and (see Section 2 for the notations). The objective function in  is equivalent to a measure, which involves a constrained minimax procedure. A series of objective functions are formed by selecting certain frequencies in , which express the magnitude response requirements at these discrete frequencies.
In this paper, a new design scheme is presented and the design implementation is described in detail. We have derived a relationship between the phase offset and the vanishing moment difference of two biorthogonal low-pass filters, which can be used to choose the vanishing moments so that the phase requirement is satisfied structurally. The design is also achieved by a constrained optimization procedure, in which the objective function and constraints are all expressed in terms of the auxiliary filters (, , , and ) rather than the low-pass filters directly. When the low-pass filters are of high vanishing moments in total, the variable number can be reduced. The objective function chosen is equivalent to a measure, which expresses the magnitude response requirements in the whole frequency range. Therefore, a better approximation quality to Hilbert transform pair is possible.
2.1. Biorthogonal Wavelet
Suppose that and represent the biorthogonal low-pass filters in the decomposition and reconstruction sides in Figure 1, respectively. and represent the associated highpass filters. and denote the corresponding wavelet bases. By choosing and , the no-distortion and no-aliasing conditions require that 
Equation (1) expresses the PR requirement of biorthogonal filter bank. In terms of the filter coefficients, it becomes
In order to obtain smooth biorthogonal wavelet bases, certain numbers of zeros at for and , are always imposed; that is, the filters have certain numbers of vanishing moments . Suppose that , , where and each denote real coefficient symmetric filters, , , , and . and are known as the auxiliary filters of and , respectively. and are either both of odd length or both even length, (1) can be written as 
where (it means the vanishing moments of and must be either both odd or both even) and and represent the discrete time Fourier transform (DTFT) of and , respectively. and all can be expressed as real polynomials in terms of by introducing suitable integer translations. Cohen et al.  had given the solution of (3), that is,
where is an odd polynomial, and
Equation (4) is equivalent to (1) or (2), which is the foundation in designing biorthogonal wavelet.
2.2. Hilbert Transform Pairs of Biorthogonal Wavelets
where , , , and denote the Fourier transforms of , , , and , respectively. The filter bank in Figure 2 is named as the dual filter bank of that in Figure 1 . and are known as the primal low-pass filters, and and are known as the dual low-pass filters.
Yu and Ozkaramanli had proven that if and only if the primal and dual low-pass filters satisfy
the corresponding biorthogonal wavelets satisfy (6), where , , , and denote DTFTs of , , , and , respectively. The phase requirement in (8) can be satisfied by choosing the vanishing moments of the filters in a simple way, which will be described in the next section. However, the requirement of magnitude responses has to be approximately achieved by an optimization procedure. The optimization routine in  is simplified as follows:
(In the design example, Antonini filters are chosen as and , and the constraintis are no longer necessary).
The optimization routine (9) involves variables, where denotes the greatest integer not exceeding and , , , and denote the lengths of , , , and , respectively. The design scheme (9) involves a constrained minimax routine. In the design, a series of objective functions are formed by choosing certain frequencies, which express the magnitude response requirements at discrete frequencies. Since the low-pass filters can be represented by their auxiliary filters uniquely, the optimization routine can be expressed by the auxiliary filters uniquely. If the filters have high vanishing moments in total, the variable number of objective function and constraints can be reduced. As a result, the calculation burden in optimization routine can be reduced. Furthermore, the objective function can be chosen to express the magnitude response requirement in whole frequency range.
3. A New Design Scheme
The new design scheme is also based on the sufficient and necessary conditions that give Hilbert transform pairs of biorthogonal wavelet base, that is, (8). First of all, the biorthogonal filter banks should be designed to satisfy PR requirement.
A biorthogonal low-pass filter, , is either of odd length or even length, and it can be always expressed as with -length symmetric filter, , (if is a symmetric filter of -tap, it can be expressed as with being a symmetric filter of -tap). Let denote the primal low-pass filter with a -tap symmetric filter. By introducing a suitable integer translation, is expressed as . Similarly, with a -tap being symmetric filter, . Since , , according to (4), we have
where ; is an odd polynomial. Similarly,
where ; is an odd polynomial. Since is symmetric and has taps, we can derive constraints on the filter and and the polynomial coefficients of according to (10). Similarly, we can derive constraints on and and from (11).
Equations (10) and (11) ensure that the filters and , , and satisfy PR requirement, respectively. If , that is, the polynomial , is not unique, and so are and . Similarly, if , and are not unique. Therefore, it is possible to construct biorthogonal filter banks that satisfy the PR requirement in addition to approximately satisfying (8) when and/or .
For the requirement of half-sample offset, first consider the requirement of phase group delay/advance. The relationship between the phase offset and the vanishing moment difference of two biorthogonal low-pass filters can be characterized by the following proposition.
Supposing two biorthogonal low-pass filters, and , where and each indicate symmetric FIR filters, , , then
where denotes the phase angle; .
Since and each are symmetric and of odd length, they can be expressed as
where and each are real polynomials in terms of ; , . Since , it results in
This proposition implies a simple and unique way for choosing the vanishing moments of primal and dual low-pass filters so that the requirement of phase half-sample offset can be satisfied. Supposing the vanishing moments of , , , and are of , , , and , respectively, it is demanded that
Second, consider the requirements of identical magnitude responses, which have to be approximately achieved by an optimization procedure. Our strategy is to minimize the integral of magnitude response difference between the primary and dual low-pass filters in whole frequency range
As a result, the design becomes a constrained optimization question as follows:
In this way, the objective function and constraints are all expressed by the auxiliary filters. The constraints in (16) are equivalent to these in (9) essentially. If the low-pass filters have high vanishing moments in total, the variable number of objective function and constraints in (17) is less than that in (9). For example, the variables in (17) corresponding to the primal filters consist of the coefficients of and , whose numbers are and , respectively. The total number in (9) is . Therefor, if , the variable number in (17) is less than that in(9). In Example 1, and are all of taps with , and and are, respectively, tap and of taps with . The variable number for the design scheme (17) is 20 however, it is 25 for the design scheme (9). Therefore, the calculation burden in optimization process can be reduced efficiently because of less variables. In fact, the constraints in (9) can be expressed by the auxiliary filters only (without the coefficients of and ) therefore, the variable number can be further reduced, and only the expressions become quite complicated.
4. Design Examples
Two biorthogonal filter banks are designed, whose corresponding wavelet bases , , , and form Hilbert transform pairs, that is, they satisfy (6).
In this example, the design process can be described by the following steps.
(1) Selection Vanishing Moments
suppose the vanishing moments of the primal and dual filter banks are , , , and , respectively. and ( and ) are either both odd or even; it means that , and . According to the proposition in the previous section, it is required that . In order to make the two filters have equivalent vanishing moments, we select , . We select , to make the two filter banks have the equivalent lengths and , .
(2) Forming Constraints
let the auxiliary filters , , . From(17), is an odd polynomial of first-degree. Suppose that , from (17), we have
Similarly, the constraints on and can be formed.
(3) Forming Objective function the objective function becomes
The objective function (19) is then simplified using the integral function int in Symbolic Math Toolbox in Matlab. The coefficients of auxiliary filters are obtained using the fmincon routine in Matlab. Finally, the filters , , , and are obtained, which are listed in Table 1. The plots of magnitude spectra and , and are depicted in Figure 3. The magnitude spectra and each are almost one sided, which show that the wavelets have very good approximation quality to Hilbert transform pairs.
In , Tay et al. presented two measures, and , to evaluate the approximation quality to Hilbert transform pair
We employ these measures as criterions in evaluating the design schemes. For the wavelet bases in reconstruction side, and are defined as the reciprocal of that in decomposition side. The two evaluation measures are given in Table 2.
A biorthogonal filter bank (dual filter bank) is designed whose wavelet bases and the wavelet bases of Antonini biorthogonal filter bank (primal filter bank) form Hilbert transform pairs. The lengths of dual filter bank are chosen as with , which are identical with the choices in . Therefore, the auxiliary filter is of 7 taps with four unknown variables, and so is . The primal auxiliary filter is of 5 taps, whose first three coefficients are , , and is of 3 taps with the first two coefficients , . The filter coefficients we obtained are listed in Table 3. The plots of magnitude spectra and are depicted in Figure 4(a); and are depicted in Figure 4(b). and are all almost one sided, which implies that and . The measures and of decomposition and reconstruction wavelets together with their average values are given in Table 2, which shows that the approximation quality of the designed biorthogonal wavelets to Hilbert transform pair has improved to some degree in terms of the average values.
This paper describes a new scheme for the design of Hilbert transform pairs of biorthogonal wavelets in detail. The relationship between the phase offset and the difference of vanishing moments is useful in choosing the vanishing moments so that the phase requirement on the corresponding low-pass filters can be satisfied strictly. The design is also simplified as a constrained optimization procedure. The objective function and constraints are all expressed in terms of the auxiliary filters, which can reduce the total variable number when the low-pass filters are of high vanishing moments. Consequently, the computational complexity can be reduced efficiently. The objective function expresses the magnitude response requirements in the whole frequency range properly, which is useful in improving the approximation quality to Hilbert transform pairs.
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This work has been supported by National Natural Science Foundation of China (NSFC) under Grant no. 60972156 and Beijing Natural Science Foundation under Grant no. 4102017.
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Shi, H., Luo, S. A New Scheme for the Design of Hilbert Transform Pairs of Biorthogonal Wavelet Bases. EURASIP J. Adv. Signal Process. 2010, 712105 (2010). https://doi.org/10.1155/2010/712105
- Discrete Wavelet Transform
- Filter Bank
- Wavelet Base
- Magnitude Response
- Optimization Routine