Energy optimized orthonormal wavelet filter bank with prescribed sharpness
Introduction
The Discrete-Wavelet-Transform (DWT) is an indispensable tool in many applications requiring the processing of numerical data [1], [2], [3]. The power of the DWT lies in its ability to give a versatile multiresolution decomposition of the data it is analyzing [4], [5]. The DWT is related to a two-channel multirate filter bank and is implemented using a tree-structured cascade of the basic two-channel system [6], [7]. Wavelet and filter bank theories provide different ways of looking at and interpreting the signal decomposition that is being performed. With the traditional filter bank theory the decomposition is viewed as a frequency partitioning of a fullband signal into subbands. Wavelet theory views the decomposition as a partitioning into nested function spaces within a multiresolution framework. Both views are related and together provide a better understanding of such systems and this has spurred great interest in the theory, design and applications of such systems. Wavelet theory emphasizes the importance of vanishing moments (VM) and regularity of the wavelet functions which is something not considered in traditional filter bank theory. Wavelet filters can be classified as either biorthogonal or orthogonal. This paper will consider only orthogonal filters which have the advantages of noise decorrelation in denoising, simple bit-allocation in compression and more generally the norm (energy) preserving property.
The design of wavelet filter bank can be viewed as a constrained optimization problem. The first essential constraint is the perfect reconstruction (PR) or orthogonality constraint (some filter banks known as QMF [8] technically only have approximate PR). The second constraint is the VM constraint to ensure regularity in the equivalent wavelet functions. After imposing these constraints the remaining degrees of freedom can be optimized with respect to some chosen criterion which can be application specific. One criterion that has received at lot of attention from the community is the maximization of the compaction energy of a specific signal the wavelet is analyzing, i.e. signal adapted filter banks [9], [10], [11], [12]. This compaction energy optimized filter has the potential of improved performance in applications such as signal analysis, coding and communications [11]. The frequency selectivity property of a traditional low-pass filter is however not ensured using this criterion. Good frequency selectivity is important to ensure the effects of aliasing or energy leakage is minimized in the subband signals and this is important in some applications such as system monitoring [13] and system identification in subbands [14]. Note that frequency selectivity only affects aliasing in the subbands and not the reconstructed signal (assuming no processing in the subbands). A perfect reconstruction filter bank has no aliasing in the reconstructed signal irrespective of the frequency selectivity as the aliasing in the subbands is canceled during reconstruction.
The challenge is therefore to design filters with good compaction energy while still having good frequency selectivity. This paper presents a method to design energy optimized filters that have a prescribed degree of transition band sharpness to ensure a desired degree of frequency selectivity. The method is based on the Zero-Pinning (ZP) technique on the Bernstein polynomial which is a simple and versatile technique for orthogonal wavelet filters. The ZP technique was first proposed in [15] where all the degrees of freedom were used to shape the frequency response of the filter by strategic pinning of the zeros of the polynomial. The ZP technique was then extended where only some of the degrees of freedom were used for pinning and the remaining degrees were used to optimize the filter with respect to the analytic quality (a criterion for the design of Hilbert-pairs) [16]. An exhaustive search based technique is required in the optimization in [16] because the objective function cannot be expressed as a convex function. The search technique is computationally inefficient and is only practical when the number of remaining degrees is small (less than two). In this paper only two degrees of freedom will be used for zero pinning and the remaining degrees will be used to optimize the filter with respect to some energy function which is convex. The optimization can be cast as a Semidefinite Programming (SDP) problem for which efficient algorithms and freeware which are widely available. With this approach both the frequency selectivity criterion and the compaction energy criterion can be simultaneously addressed in the filter design. The novelties of this paper are:
- 1.
The design of wavelet filters using two criteria, i.e. compaction energy and frequency selectivity, at the same time.
- 2.
Combining the ZP technique on the Bernstein polynomial with Semidefinite Programming in the design problem formulation.
- 3.
Derivation of the objective function in terms of the Bernstein parameters, and the linear inequality bound on the filter response to ensure good frequency selectivity.
In Section 2 a review of the fundamentals of wavelets, filter banks and Bernstein polynomial as it relates to the work of this paper is presented. Section 3 describes the method for optimizing the filter with respect to an energy measure but with a prescribed degree of sharpness in the transition band. Relevant constraints to the problem are formulated here and it is shown how to cast the design problem as a semidefinite programming (SDP) problem. Design examples and discussions are presented in Section 4. Section 5 presents an application in image denoising and the paper concludes in Section 6.
Section snippets
Preliminaries and background
A two-channel multirate filter bank is made up of the following filters: (low-pass analysis), (high-pass analysis), (low-pass synthesis) and (high-pass synthesis). The following conditions must be satisfied to ensure perfect reconstruction (PR) [17]: and where the product filter is defined as For an orthogonal filter bank the low-pass filters are time-reverse versions of each other . Define
Energy optimized filter with prescribed sharpness
Two commonly used energy measures will be considered in this work.
Design examples and discussions
The CQF coefficients of all examples are shown in Table 1. The number of zeros, including multiple ones, is equal to N (length minus one).
Example 1 The objective function is the stopband energy defined in (10) and the problem is one of minimization. , , giving a length 18 CQF with 5 VMs. Pinned zero which leaves 2 remaining degrees of freedom for optimization. The value in the stopband energy formula in (10). The optimized free-parameter values are:
Image denoising application
In this section we compare the performance of filters in image denoising application. We designed a length 12 filter (FB12) having 2 VMs with the pinned zero and optimized w.r.t (11). The Gaussian noise of variance was added to the image and a soft thresholding is applied after five levels of wavelet decomposition. The threshold is set to . The denoised image is then reconstructed and the performance in terms of PSNR is shown in Table 2 and Table 3 for Lena and Barbara images
Conclusion
The design of wavelet filters which are optimized with respect to some energy measure has been presented here. The filters have a prescribed degree of sharpness in the transition band and this is achieved by zero-pinning the Bernstein polynomial. The filters have the traditional frequency selectivity property of a traditional low-pass filter and this ensured regularity in the equivalent wavelet function. It is shown that two competing criteria can be simultaneously addressed with the approach
David B.H. Tay received the B.Eng. (Electrical & Electronic) and B.Sc. (Mathematics) degrees from the University of Melbourne and the Ph.D. (Signal Processing) degree from Cambridge University. He was a lecturer and then assistant professor in the School of Electrical and Electronic Engineering, Nanyang Technological University from 1995 till 1999. Since July 1999 he is with the Department of Electronic Engineering, La Trobe University where he is currently a Reader and Associate Professor.
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Cited by (0)
David B.H. Tay received the B.Eng. (Electrical & Electronic) and B.Sc. (Mathematics) degrees from the University of Melbourne and the Ph.D. (Signal Processing) degree from Cambridge University. He was a lecturer and then assistant professor in the School of Electrical and Electronic Engineering, Nanyang Technological University from 1995 till 1999. Since July 1999 he is with the Department of Electronic Engineering, La Trobe University where he is currently a Reader and Associate Professor. Presently he is an Associate Editor for the International Journal of Multidimensional Systems and Signal Processing. He also serves as a member of DSP Technical Committee in the IEEE Circuits and Systems Society. His main research interest is in the area of wavelets and filter banks and he also has interest in the areas biomedical engineering and machine learning.
Selvaraaju Murugesan received the B.Eng. degree in mechatronics engineering in 2006 from Kumaraguru College of Technology (affiliated to Anna University), Tamil Nadu, India, and the M.Eng. degree in 2008 from La Trobe University, Bundoora, Australia. He has just completed the Ph.D. degree in the Department of Electronic Engineering at La Trobe University. His research interests are in the areas of multirate filter banks and wavelets.