The biorthogonal wavelets that are redundant-free and nearly shift-insensitive
© Shi and Luo; licensee Springer. 2012
Received: 20 June 2012
Accepted: 23 August 2012
Published: 7 September 2012
It is well known that discrete wavelet transform (DWT) is sensitive to shift, which means a slight shift of feature in the original signal may cause unpredictable changes in the analysis subbands. Some modified versions of DWT can reduce the shift sensitivity, however, they are all redundant. In this article, we shows the shift sensitivity is caused by the aliasing terms formed in the downsampling operation during analysis process. A novel scheme for the design of wavelet is proposed to reduce the effect of aliasing terms as much as possible in the general framework of DWT. A few of biorthogonal wavelets have been designed and applied in the simulation examples. The results of examples demonstrate the efficiency of the designed wavelets in the term of shift insensitivity and nonredundancy.
Discrete wavelet transform (DWT) has been applied in many fields as a tool of signal processing, e.g., signal de-noising, feature extraction, pattern recognition and image registration[1–3]. However, DWT is shift-sensitive. A slight shift of feature in the original signals or images may generate unpredictable changes in its DWT analysis subbands. For example, for level-d low-low (LL) subbands of a two-dimensional figure, only the features that consist of more than 22d pixels in the original images can be insensitive to shift in image registration. Some new wavelets and modified calculation frameworks of DWT have been presented to reduce the shift sensitivity. However, all these DWTs become redundant, i.e., they are no longer critically-sampled[6–10]. For example, Kingsbury’s dual tree complex wavelet transform (DTCWT) and Selesnick’s double-density wavelet transform (DDWT) are all redundant. A pair of filter banks is employed in Kingsbury’s DTCWT, which leads to a constant redundancy rate of 2:1 for 1-D signals and 2 m :1 for m-dimensional signals. The other modified DWTs, such as DDWT, also cannot be critically-sampled. The lack of directionality is other main drawback of DWTs. Many modified transformations have been presented to improve directional selectivity, such as curvelet and contourlet transformations[11–13]. These multiresolution representations are much more redundant.
The shift sensitivity of DWT
The filter bank ( Q ( z ) and are 9- and 3-tap,)
The filter bank ( Q ( z ) and are 19- and 3-tap, )
The nearly shift-invariant and critically-sampled DWT
In order to avoid redundancy, the framework of general DWT, Figure1, is remained in the proposed DWT except the wavelet filters are designed to satisfy some extra requirements.
The extra requirements on wavelets
Namely, the effect of aliasing terms in the low-pass subband has been approximately removed. The design scheme of wavelets that satisfy these extra requirements will be given in the next. First of all, it is necessary to review the design of biorthogonal wavelet.
The design of biorthogonal wavelet
y = sin2(ω/2), (it means the vanishing moments of H0(z) and must be either both odd or both even); R(·) is an odd polynomial, which is chosen such that for all ω∈[0,Π.
We can obtain constraints about the coefficients of Q(z), and R(·)(Though there are equations on Q(z) and, only of them are independent because of symmetry). It has been shown even when R(·) = 0, Q(z) and may be not unique for a identical. Furthermore, when R(·) ≠ 0, there are more choices in designing Q(z) and, so are H0(z) and. Therefore, it is possible to construct biorthogonal wavelet filters that satisfy the PR requirement and approximately satisfy (4) and (5). The design scheme is presented in the following section.
The design of biorthogonal wavelets for the proposed DWT
The design and application examples
In this section, two design examples are presented to illustrate the design process, and three simulation examples are employed to demonstrate the performances of designed wavelets in the applications.
The design examples
Two biorthogonal that approximately satisfy (5) are designed. In the first bank, F and G are 5- and 4-tap, respectively, (Q(z) = F(z) + z−1G(z) becomes 9-tap); is 3-tap; (it means R(·) = 0 in Equation (7)). H0and become 20- and 4-tap, respectively, which are identical with “bior3.9” wavelet filter banks in structure. The filter coefficients obtained are given in Table1. It can be verified that H0and satisfies the PR requirement (3) (Let and). In the second filter bank, H0 and are also 20- and 4-tap, however, we choose R(·) ≠ 0 to improve the approximation performance. The filter coefficients are shown in Table2.
The application examples
In order to show the intensity difference after 2D wavelet analysis, the pixels whose intensities are smaller than a certain percent of Imax have been set as Imin, where Imax and Imin are the maximum and minimum of the LL subband coefficients, respectively. This operation is similar to the feature extraction according to the pixel intensities automatically. When the pixels in Figure5c,d whose intensities are smaller than 0.4Imax are set as Imin, Figure5c,d become Figure5e,f. It also illustrates the effect of shift sensitivity has been reduced efficiently using the designed wavelets.
For the general wavelets, the aliasing terms formed in analysis process of DWT can be eliminated in the synthesis process by the anti-aliasing properties of filter banks. However, the aliasing terms remain in the analysis outputs. In this article, it shows the aliasing terms cause the shift sensitivity of DWT. A novel scheme is proposed to reduce the effect of aliasing terms. Some extra requirements on the design of wavelets are proposed. The design scheme are presented and two biorthogonal wavelets that approximately satisfy the extra requirements have been designed. The shift sensitivity of the wavelet analysis has been reduced effectively by the new wavelets, which is very favorable for many signal processing applications, such as image registration, feature extraction and pattern recognition (the process is usually achieved using the analysis outputs rather than the synthesis results in these cases). The other superiority of the proposed wavelet analysis is that the wavelet representation remains critically-sampled and does not bring out any redundancy.
This study had 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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