A fast multiple decomposition of a discrete-time signal on bases in vector spaces by the polynomial time-frequency transformation

We investigate the relationships among the Pade table, continued fraction expansions and perfect reconstruction (PR) filter banks. We show how the Pade table can be utilized to develop a new lattice structure for general two-channel bi-orthogonal perfect reconstruction (PR) filter banks. This is achieved through characterization of all two-channel biorthogonal PR filter banks. The parameterization found using this method is unique for each filter bank. Similar to any other lattice structure, the PR property is achieved structurally and the quantization of the parameters of the lattice does not effect this property. Furthermore, we demonstrate that for a given filter, the set of all complementary filters can be uniquely specified by two parameters, namely, the end-to-end delay of the system and a scalar quantity. Finally, we investigate the convergence of the successive filters found through the proposed lattice structure and develop a sufficient condition for this convergence.

This study presents design of 2D nonseparable Perfect Reconstruction Filter Bank (PRFB) for two different sampling lattices: the quincuncial and rectangular. In quincunx case z-domain PR conditions are mapped into Bernstein-x domain. Desired power spectrum of 2D nonseparable filter is approximated by using Bernstein polynomial. Since we introduce mapping from complex periodic domain to real polynomial domain, PRFB design in Bernstein-x domain is much easier to handle. The parametric solution for 2D nonseparable design technique is obtained with desired regularity for quincunx sampling lattices. This technique allows us to design of 2D wavelet transform. For rectangular downsampling, the use of signed shuffling operations to obtain a PRFB from a low pass filter enables the reduction of PR conditions. This design technique leads us to efficient implementation structure since all the filters in the bank have the same coefficients with sign and position changes. This structure overcomes the high complexity problem that is the major shortcoming of 2D nonseparable filter banks. Designed filter banks are tested on 2D image models and real images in terms of compaction performance. It has been shown that nonseparable design can outperform separable ones in the application of data compression.

A systematic multi-step approach is described for optimizing the stopband response of the prototype filter for low-delay critically sampled cosine-modulated filter banks in the least-mean-square sense subject to the maximum allowable aliasing and amplitude errors. In this approach, filter banks having several channels are designed by starting with a filter bank with a small number of channels. Then, the number of channels is gradually increased and a new prototype filter is optimized using the modified version of the prototype filter of the previous step as a good start-up solution. Several examples are included illustrating the flexibility of the proposed approach for making compromises between the required filter orders, the required filter bank delays, and the aliasing and amplitude errors. These examples show that by allowing very small amplitude and aliasing errors, the stopband performance of the resulting filter bank is significantly improved compared to the corresponding perfect-reconstruction filter bank. Alternatively, the filter bank delay and the order of the prototype filter can be significantly reduced while still achieving practically the same filter bank performance.

Non-uniform filter banks (NUFBs) are being preferred because it divides the signal into unequal bands. Among perfect reconstruction (PR) and Nearly Perfect Reconstruction (NPR) filter banks, the later is preferred since the filter banks can be realized with minimal complexity. In this paper, a prototype filter is designed using the windowing techniques and the pass-band edge frequency (ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> ) is optimized using the linear optimization technique for the design of a near perfect reconstructed filter bank. Among the different windowing techniques, Kaiser and dolph-chebyshev windows have been considered. The performance of the filter banks for each of these windowing techniques was measured in terms of Peak Reconstruction Error (PRE). These techniques are being applied on an Electro Cardio Gram (ECG) signal and their respective results are also being discussed.

This paper presents an eigenfilter based method for the design of 4-band perfect reconstruction (PR) filter banks. All the filters of the filter bank exhibit linear phase and orthogonality property. The eigenfilter method has been used so far to impose linear constraints on the filter, however, in this paper, we discuss how to impose nonlinear constraints on the filter using the eigenfilter method. The analysis lowpass filter impulse response is obtained using an iterative approach in the eigenfilter method to minimize the error as given by the alias cancellation equations in the PR filter bank. Once the analysis lowpass filter impulse response h <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> [n] is obtained, the other filters are obtained using shuffling matrices. We present two design examples of a 4-channel perfect reconstruction filter bank. We also report a minor modification in the eigenfilter method for imposing non-homogeneous linear constraints on the filter.

This paper proposes a new design method for a class of two-channel 2D non-separable perfect reconstruction (PR) filter banks (FBs) using the multiplet FBs. 1D multiplet FBs are PR FBs that can be obtained by frequency transformation of a prototype PR FB in the conventional lifting structure so that a better frequency characteristics can be obtained and varied online to process different signals. By employing the 1D to 2D transformation of Phoong et al., new 2D PR multiplet FBs with quincunx, hourglass, and parallelogram spectral support are obtained. These nonseparable multiplet FBs can be cascaded to realize new PR directional FB for image processing and motion analysis. The design procedure is very general and it can be applied to both linear-phase and low-delay 2D FBs. Design examples are given to demonstrate the usefulness of the proposed method

Expansions of complex discrete-time signals onto a single basis in the given vector space provide an incomplete and in a way potentially misleading information about the signals in signal analysis applications. We use a simple example to illustrate the incompleteness of the information about the signal in the decompositions on a single basis. Then, we limit our attention to an oversampled decimated perfect reconstruction filter bank. It is illustrated that oversampled decimated filter banks can serve as a tool which provides more complete information about the signal and at the same time the filter banks can enjoy an efficient polyphase component implementation of maximally decimated, i.e. nonredundant, filter banks. In an oversampled decimated filter bank the relation between analysis filters given by H/sub k/(z)=H/sub 0/(zW/sub M//sup k/) does not in general imply that the filter bank formed by these filters is a uniform filter bank with center frequencies of the filters uniformly spaced on the frequency axis. A simple example which removes a possible surprise which accompanies this statement is provided. It is also shown that if the complete set of filters H/sub k/(z)=H/sub 0/(zW/sub M//sup k/) of the oversampled decimated filter bank can be derived from more than a single prototype then the efficient polyphase implementations of the oversampled decimated filter bank can be readily available. Relation of the considered oversampled decimated filter bank with a decomposition of the discrete-time signal onto several bases of the given vector space is emphasized throughout the paper.

Classically, the filter banks (FBs) used in source coding schemes have been chosen to possess the perfect reconstruction (PR) property or to be maximally selective quadrature mirror filters (QMFs). This paper puts this choice back into question and solves the problem of minimizing the reconstruction distortion, which, in the most general case, is the sum of two terms: a first one due to the non-PR property of the FB and the other being due to signal quantization in the subbands. The resulting filter banks are called minimum mean square error (MMSE) filter banks. Several quantization noise models are considered. First, under the classical white noise assumption, the optimal positive bit rate allocation in any filter bank (possibly nonorthogonal) is expressed analytically, and an efficient optimization method of the MMSE filter banks is derived. Then, it is shown that while in a PR FB, the improvement brought by an accurate noise model over the classical white noise one is noticeable, it is not the case for the MMSE FB. The optimization of the synthesis filters is also performed for two measures of the bit rate: the classical one, which is defined for uniform scalar quantization, and the order-one entropy measure. Finally, the comparison of rate-distortion curves (where the distortion is minimized for a given bit rate budget) enables us to quantify the SNR improvement brought by MMSE solutions.

This paper considers a cosine-modulated, two-dimensional (2-D) perfect reconstruction (PR) filter banks theory. First, a 2-D digital filter with half-pass-band obtained by the sampling matrix had to be designed. Next, 2-D analysis filter banks are realized by cosine-modulating this prototype 2-D digital filter so that 2-D analyzed signals become real. It is shown that the modulation in the 2-D frequency plane is equivalent to 1-D modulation. A necessary and sufficient condition for 2-D perfect reconstruction filter banks is derived. If the polyphase filter pairs of the prototype filter doubly complement, the resulting 2-D filter bank satisfies the condition of perfect reconstruction.

Considers the theory of cosine-modulated 2 dimensional (2-D) perfect reconstruction (PR) filter banks. First, a 2-D digital filter design with half passband, obtained by the sampling matrix, is discussed. Next, 2-D analysis filter banks are realized by cosine-modulating this prototype 2-D digital filter. It is shown that the modulation in the 2-D frequency plane is equivalent to the 1-D modulation. A necessary and sufficient condition for 2-D perfect reconstruction filter banks is derived. If the polyphase filter pairs of the prototype filter have a double-complement, the resulting 2-D filter bank satisfies the condition of perfect reconstruction. >

In this paper, the author considers the theory of modulated 2 dimensional (2-D) perfect reconstruction (PR) filter banks with permissible passbands. At first, the author designs a 2-D complex digital filter with half passband obtained by the sampling matrix. Next, 2-D analysis filter banks are realized by modulating this prototype 2-D complex digital filter and by taking the real part of the output. It is also shown that the modulation in the 2-D frequency plane is equivalent to a 1-D DFT. A necessary and sufficient condition for 2-D perfect reconstruction filter banks is derived. Finally, some examples are shown.

B-Mode ultrasound images are degraded by inherent noise called Speckle, which creates a considerable impact on image quality. This noise reduces the accuracy of image analysis and interpretation. Therefore, reduction of speckle noise is an essential task which improves the accuracy of the clinical diagnostics. In this paper, a Multi-directional perfect-reconstruction (PR) filter bank is proposed based on 2-D eigenfilter approach. The proposed method used for the design of two-dimensional (2-D) two-channel linear-phase FIR perfect-reconstruction filter bank. In this method, the fan shaped, diamond shaped and checkerboard shaped filters are designed. The quadratic measure of the error function between the passband and stopband of the filter has been used an objective function. First, the low-pass analysis filter is designed and then the PR condition has been expressed as a set of linear constraints on the corresponding synthesis low-pass filter. Subsequently, the corresponding synthesis filter is designed using the eigenfilter design method with linear constraints. The newly designed 2-D filters are used in translation invariant pyramidal directional filter bank (TIPDFB) for reduction of speckle noise in ultrasound images. The proposed 2-D filters give better symmetry, regularity and frequency selectivity of the filters in comparison to existing design methods. The proposed method is validated on synthetic and real ultrasound data which ensures improvement in the quality of ultrasound images and efficiently suppresses the speckle noise compared to existing methods.

In this paper, we present a design method of 2 dimensional (2-D) perfect reconstruction (PR) filter banks with permissible passbands by modulating a complex prototype 2-D FIR digital filter. At first, we design a 2-D complex digital filter with half of a passband obtained by a sampling matrix. Then, we show that the 2-D analysis filter banks can be realized by modulating this prototype 2-D complex digital filter and taking the real part of the output. It is demonstrated that the modulation in 2-D frequency plane is equivalent to one dimensional (1-D) discrete-Fourier-transform (DFT). We also derive a necessary and sufficient condition for the 2-D PR filter banks.

This paper proposes a new design method for a class of two-channel perfect reconstruction (PR) filter banks (FBs) called multi-plet FBs with very sharp cutoff using frequency- response masking (FRM) technique. The multi-plet FBs are PR FBs and their frequency characteristics are controlled by a single subfilter. By recognizing the close relationship between the subfilter and the FRM-based halfband filter, very sharp cutoff PR multi-plet FBs can be realized with reduced implementation complexity. The design procedure is very general and it can be applied to both linear-phase and low-delay PR FBs. Design examples are given to demonstrate the usefulness of the proposed method.

In essence, designing a perfect-reconstruction (PR) biorthogonal cosine-modulated filter bank (BCM) is a non-convex constrained optimization problem that can be solved in principle using general optimization solvers. However, when the number of channels is large and the order of the prototype filter (PF) is high, numerical difficulties in using those optimization solvers often occur, and the computational efficiency also becomes a concern. This paper proposes an algorithm that carries out the design in two stages. In the first stage, a convex Lagrangian relaxation technique is used to obtain a near PR (NPR) filter bank and, in the second stage, the coefficient vector of the PF obtained is alternately projected onto the null-spaces that are associated with the PR constraints, which turns the NPR filter bank into a PR filter bank. Simulation results are included to demonstrate the robustness of the proposed algorithm for designing BCM filter banks with a large number of channels and high-order PF as well as satisfactory design efficiency.