Perfect Reconstruction Condition Research Articles

A novel optimal wavelet filter banks for automated diagnosis of Alzheimer’s disease and mild cognitive impairment using Electroencephalogram signals

Electroencephalogram (EEG) of Alzheimer’s disease (AD) patients show a slowing effect and less synchronization. EEG signal’s transient and abrupt nature is captured from various mother wavelets. However, better performance can be obtained by balancing time–frequency localization in wavelet filters. We propose a new approach for designing filter banks based on optimal time–frequency localization of a four-step lifting structure (FSLS). First, we design a FSLS using Euler’s Frobenius half-band polynomial (EFHBP). The perfect reconstruction condition and vanishing moments are imposed on EFHBP to achieve maximum flat half-band filters (HBFs). HBFs are optimized using a balanced uncertainty metric to obtain a time–frequency spread balance. Afterward, these HBFs are used in the FSLS to obtain the synthesis and analysis filter banks. The proposed time–frequency optimized biorthogonal wavelet filter banks (TFOBWFBs) achieved a balance between time–frequency localization. Further, these TFOBWFBs are applied to decompose the EEG signals of AD patients. Twenty different features were extracted from decomposed EEG subbands, of which twelve significant features were selected using Kruskal Walli’s test. The machine learning models were trained and tested using obtained features with 10-fold cross-validation and leave-one-subject-out cross-validation. To validate this study, the proposed TFOBWFBs have been applied to two publicly available EEG datasets of mild cognitive impairment (MCI), AD, and healthy control (HC) subjects. The proposed TFOBWFBs achieved 98.90% accuracy for 2-way (AD vs. HC) and 96.50% for 3-way (AD vs. MCI vs. HC) classification using a support vector machine model with a 10-fold cross-validation. The proposed method outperforms the existing AD and MCI detection techniques. The performance of the proposed machine learning framework with optimization of wavelet filter banks is more significant and fast compared to previous studies. Also, the proposed framework can be applied to detect other neurodegenerative disorders.

B-spline quarklets and biorthogonal multiwavelets

In this paper, we show that B-spline quarks and the associated quarklets fit into the theory of biorthogonal multiwavelets. Quark vectors are used to define sequences of subspaces [Formula: see text] of [Formula: see text] which fulfill almost all conditions of a multiresolution analysis. Under some special conditions on the parameters, they even satisfy all those properties. Moreover, we prove that quarks and quarklets possess modulation matrices which fulfill the perfect reconstruction condition. Furthermore, we show the existence of generalized dual quarks and quarklets which are known to be at least compactly supported tempered distributions from [Formula: see text]. Finally, we also verify that quarks and quarklets can be used to define sequences of subspaces [Formula: see text] of [Formula: see text] that yield non-orthogonal decompositions of [Formula: see text].

A Flexible and Simple Lossless DWT Filter Bank Using a MAXFLAT FIR Half-Band Filter

This paper describes a simple, lossless and computationally efficient two-band single (s-) filter bank that creates an opposite band output by subtracting the primary filtered data from the original data. For computationally efficient and error-free s-filter bank achievement, a maximally flat (MAXFLAT) half-band filter with zero odd-order coefficients is characterized from a unique perfect reconstruction condition, and an explicit impulse–response formula (for non-zero integer coefficients of even order) is derived in a closed form of the filter. The examples are shown to provide a complete and accurate solution for the design of such s-filter banks. In addition, the effectiveness of the proposed s-filter banks is clearly verified by comparing the lossless 5/3 and lossy 9/7 filter banks (in the JPEG2000). The simulation results show that the s-filter banks lead to better performance than the JPEG2000 filter banks using two filters although allowing low computational complexity of less than 50%. This new approach is shown to provide significant advantages over existing lossless discrete wavelet transform (DWT) filter banks in both design flexibility and computational complexity.

A Novel Design of Symmetric Daub-4 Wavelet Filter Bank for Image Analysis

The orthogonal wavelet filter banks (FBs) consume large resources for hardware implementation due to their irrational coefficients. This results in reduction in speed of operation, huge memory requirement and increased power dissipation. In order to solve these issues, a low complex symmetric Daub-4 (8-tap) orthogonal wavelet FB is proposed in this brief. This is achieved by slightly altering the perfect reconstruction (PR) condition to make proposed FB symmetric and to obtain dyadic filter coefficients. The proposed wavelet FB is multiplier free and achieves considerably reduction in hardware and dynamic power consumption by reducing number of adders and shifters. This is verified by implementing the proposed wavelet FB on Kintex-7 field programmable gate array (FPGA) using vivado 2019.2 software from Xilinx. The proposed FB is tested for its suitability in applications such as image compression and medical image retrieval. Simulations results prove that the designed wavelet FB gives better performances in terms of PSNR for image compression and retrieval accuracy (ARP, ARR) for medical image retrieval on the benchmark images as compared to state-of-art methods.

Noise Reduction in CT Using Learned Wavelet-Frame Shrinkage Networks.

Encoding-decoding (ED) CNNs have demonstrated state-of-the-art performance for noise reduction over the past years. This has triggered the pursuit of better understanding the inner workings of such architectures, which has led to the theory of deep convolutional framelets (TDCF), revealing important links between signal processing and CNNs. Specifically, the TDCF demonstrates that ReLU CNNs induce low-rankness, since these models often do not satisfy the necessary redundancy to achieve perfect reconstruction (PR). In contrast, this paper explores CNNs that do meet the PR conditions. We demonstrate that in these type of CNNs soft shrinkage and PR can be assumed. Furthermore, based on our explorations we propose the learned wavelet-frame shrinkage network, or LWFSN and its residual counterpart, the rLWFSN. The ED path of the (r)LWFSN complies with the PR conditions, while the shrinkage stage is based on the linear expansion of thresholds proposed Blu and Luisier. In addition, the LWFSN has only a fraction of the training parameters (<1%) of conventional CNNs, very small inference times, low memory footprint, while still achieving performance close to state-of-the-art alternatives, such as the tight frame (TF) U-Net and FBPConvNet, in low-dose CT denoising.

A New Approach to the Design and Implementation of a Family of Multiplier Free Orthogonal Wavelet Filter Banks

This paper presents a generalized novel design of a family of multiplier free orthogonal wavelet filter banks (FBs) with low complexity. In the proposed method, maximum number of zeros at <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z=-1$ </tex-math></inline-formula> is ensured for a desired length of analysis and synthesis filters. A new class of technique is proposed to obtain dyadic filter coefficients based on double shifting orthogonality property with allowable deviation from original filter coefficients. This is achieved by slightly altering perfect reconstruction (PR) condition to obtain dyadic filter coefficient. A general closed form expression is formulated using double shifting orthogonality property to reduce the error between original filter coefficients and proposed dyadic filter coefficients. The designed wavelet FB improved time-frequency product, Sobolev regularity and frequency selectivity (low energy in error) over well known existing wavelet FBs. In addition, a fixed point VLSI architecture is proposed for the designed family of orthogonal wavelet FBs. The suggested architecture is designed and implemented on Kintex7 FPGA board. The proposed family of orthogonal wavelet FBs is multiplier less with significantly reduced adders, shifters and dynamic power dissipation. The performance of the proposed wavelet FB is validated on two different applications namely image compression and medical image retrieval. The proposed FBs are tested on well known datasets such as CLASSIC, EPFL, RAISE, and FiveK for image compression and on three different medical image databases such as NEMA, OASIS, and EXACT09 for image retrieval.

High-concentration time–frequency analysis for multi-component nonstationary signals based on combined multi-window Gabor transform

PurposeThis study aims to reveal the essential characteristics of nonstationary signals and explore the high-concentration representation in the joint time–frequency (TF) plane.Design/methodology/approachIn this paper, the authors consider the effective TF analysis for nonstationary signals consisting of multiple components.FindingsTo make it, the authors propose the combined multi-window Gabor transform (CMGT) under the scheme of multi-window Gabor transform by introducing the combination operator. The authors establish the completeness utilizing the discrete piecewise Zak transform and provide the perfect-reconstruction conditions with respect to combined TF coefficients. The high-concentration is achieved by optimization. The authors establish the optimization function with considerations of TF concentration and computational complexity. Based on Bergman formulation, the iteration process is further analyzed to obtain the optimal solution.Originality/valueWith numerical experiments, it is verified that the proposed CMGT performs better in TF analysis for multi-component nonstationary signals.

Some new discrete biorthogonal wavelets constructed with Laguerre polynomials

In this study, we present a new family of discrete wavelets which are constructed with the help of Laguerre polynomials and the Daubechies biorthogonal wavelets construction method. Our aim is to propose the discrete version of some previously constructed continuous Laguerre wavelets and also to present a method of discrete wavelets construction by several iterations. With this scheme, we use two different sets of finite impulse response filters for the signal decomposition and their duals for reconstruction. The quadruplet finite impulse response filters respect the anti-aliasing and the perfect reconstruction conditions, and at the same time, they resemble as much as possible the continuous Laguerre wavelets when using the cascade algorithm. We use the mean squared error, the maximum deviation, and the standard deviation to quantify the similarity between the continuous Laguerre wavelets and the constructed discrete Laguerre wavelets. The results show that, they are both the same wavelets due to the small nature of these parameters. Our method is important because, it can permit the determination of the finite impulse response filter coefficients corresponding to many other continuous wavelets.

Design of Transmultiplexer Filter Banks Using Ramanujan Sums

Ramanujan sums are sequences introduced by the famous mathematician S. Ramanujan in 1918. Ramanujan sum $$c_{q}(n)$$ is the sequence in n with periodicity q. Many arithmetic functions such as Euler’s totient function and Riemann zeta function can be expressed using these sequences. These sums have many attractive properties which make them suitable for signal processing domain. Ramanujan sums are orthogonal and symmetric. Implementation of Ramanujan sum-based transforms is faster compared to discrete Fourier transform, since their computation involves only co-resonant frequencies. These transforms are particularly useful in identifying periodicities in signals. Ramanujan sums are integer valued. In this letter, it is proposed to use Ramanujan sums for the design of transmultiplexer filter banks. Here, the analysis and synthesis filter coefficients are integers. Computation time is reduced considerably by replacing floating point arithmetic with integer coefficients. Rounding errors in hardware implementation are avoided, and perfect reconstruction condition in the transmultiplexer system is ensured by using these integer coefficients.

On the Design of Two-Dimensional Quincunx Filterbanks with Directional Vanishing Moment Based on Eigenfilter Approach

This paper presents a two-dimensional (2-D) eigenfilter technique to design 2-D perfect reconstruction (PR) filterbanks with directional vanishing moment (DVM). In this paper, we first introduce the DVM constraint for 2-D directional filterbank design. The proposed DVM constraint is imposed in 2-D eigenfilter formulation to obtain maximally flat 2-D low-pass analysis filter. Then, we use this designed analysis filter in 2-D time-domain formulation of PR condition to obtain complementary low-pass synthesis filters. The PR and DVM constraints are imposed to obtain 2-D low-pass synthesis filter. It is shown that the proposed 2-D filters satisfy the PR criteria with prescribed DVM. The performance of 2-D PR filters is evaluated in image denoising application. The performance of designed filterbank is compared with well-known existing methods in terms of peak signal-to-noise ratio to validate the results.

Optimal design of window functions for filter window bank

This paper considers the designs of the periodic window functions in the filter window banks. First, the filter window bank with the constant synthesis periodic window functions is considered. The total number of the nonzero coefficients in the impulse responses of the analysis periodic window functions is minimized subject to the near perfect reconstruction condition. This is an $ L_0 $ norm optimization problem. To find its solution, the $ L_0 $ norm optimization problem is approximated by the $ L_1 $ norm optimization problem. Then, the column of the constraint matrix corresponding to the element in the solution with the smallest magnitude is removed. Next, it is tested whether the feasible set corresponding to the new $ L_0 $ norm optimization problem is empty or not. By repeating the above procedures, a solution of the $ L_0 $ norm optimization problem is obtained. Second, the filter window bank with the time varying synthesis periodic window functions is considered. Likewise, the design of the periodic window functions in both the analysis periodic window functions and the synthesis periodic window functions is formulated as an $ L_0 $ optimization problem. However, this $ L_0 $ norm optimization problem is subject to a quadratic matrix inequality constraint. To find its solution, the set of the synthesis periodic window functions is initialized. Then, the set of the analysis periodic window functions is optimized based on the initialized set of the synthesis periodic window functions. Next, the set of the synthesis periodic window functions is optimized based on the found set of the analysis periodic window functions. Finally, these two procedures are iterated. It is shown that the iterative algorithm converges. A design example of a filter window bank with the constant synthesis periodic window functions and a design example of a filter window bank with the time varying synthesis periodic window functions are illustrated. It is shown that the near perfect reconstruction condition is satisfied, whereas this is not the cases for the nonuniform filter banks with the conventional samplers and the conventional block samplers.

Length Reduction of Singular Spectrum Analysis With Guarantee Exact Perfect Reconstruction via Block Sliding Approach

The conventional singular spectrum analysis is to divide a signal into segments where there is only one non-overlapping point between two consecutive segments. By putting these segments into the columns of a matrix and performing the singular value decomposition on the matrix, various one dimensional singular spectrum analysis vectors are obtained. Since different one dimensional singular spectrum analysis vectors represent different parts of the signal such as the trend part, the oscillation part and the noise part of the signal, the singular spectrum analysis plays a very important role in the nonlinear and adaptive signal analysis. However, as the length of each one dimensional singular spectrum analysis vector is the same as that of the original signal, there is a redundancy among these one dimensional singular spectrum analysis vectors. In order to reduce the required computational power and the required units for the memory storage for performing the singular spectrum analysis, this article proposes a method to reduce the total number of the elements of all the one dimensional singular spectrum analysis vectors. In particular, the length of the shift block for generating the trajectory matrix is increased from one to a positive integer greater than one under a certain criterion. In this case, the total number of the columns of the trajectory matrix is reduced. As a result, the total number of the off-diagonals of all the two dimensional singular spectrum analysis matrices is reduced. Hence, the total number of the elements of all the one dimensional singular spectrum analysis vectors is reduced. In order to guarantee exact perfect reconstruction, this article reformulates the de-Hankelization process. In particular, the first element of the off-diagonals of all the two dimensional singular spectrum analysis matrices is taken as the elements of the one dimensional singular spectrum analysis vectors. Exact perfect reconstruction condition is derived. Simulations show that exact perfect reconstruction can be achieved while the total number of the elements of all the one dimensional singular spectrum analysis vectors is significantly reduced.

M-band flexible wavelet transform and its application to the fault diagnosis of planetary gear transmission systems

Abstract The fault diagnosis of planetary gear transmission systems is crucial for the safety of machineries and equipment. To identify the underlying fault features in measured signals, a novel M-band flexible wavelet transform is constructed. This transform provides a denser sampling of the time–frequency plane and preserves tunable filter parameters and dilation factors. A perfect reconstruction condition of the proposed transform is established, and its corresponding wavelet filter bank is designed to satisfy the perfect reconstruction condition. A numerical implementation algorithm of M-band flexible wavelet transform is investigated using a multirate filter bank and fast Fourier transform. Denoising the simulation signals demonstrates that the proposed transform exhibits better performance than analytic flexible wavelet transform, orthogonal wavelet transform, and biorthogonal wavelet transform. A new fault diagnosis method for planetary gear transmission systems is proposed on the basis of M-band flexible wavelet transform and spectral negentropy. Experimental and comparative results show that the proposed method can be more effectively and accurately applied to the fault diagnosis of planetary gear transmission systems compared with typical fault diagnosis methods based on analytic flexible wavelet transform, Morlet wavelet transform, and infograms.

Design of orthogonal graph filter bank with known eigenvalues of Laplacian matrix

In the conventional designs of orthogonal graph filter bank, both eigenvalues and eigenvectors of Laplacian matrix are assumed to be unknown such that graph filter bank can be efficiently implemented by using polynomial graph filters; that is, the output signal at each vertex is computed only from the signal at that vertex and its K -hop neighbourhoods. However, the disadvantage of these methods is that the designed orthogonal filter bank does not have the exact perfect reconstruction (PR). In this study, the authors first assume that the eigenvalues of Laplacian matrix are known but eigenvectors are still unknown. Then, the orthogonal graph filter bank is designed by solving the simultaneous equation or Lagrange multiplier method such that the PR condition at all eigenvalues can be achieved for a given graph structure. Although the proposed filter bank is a graph-structure-dependent design, it can be efficiently implemented by using polynomial graph filters. Finally, experimental results are shown to demonstrate the effectiveness of the proposed design method.

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

In this paper, we consider the design of wavelet filters based on the Thiran's common-factor approach proposed in [13] . This approach aims at building finite impulse response filters of a Hilbert-pair of wavelets serving as real and imaginary part of a complex wavelet. Unfortunately it is not possible to construct wavelets which are both finitely supported and analytic. The wavelet filters constructed using the common-factor approach are then approximately analytic. Thus, it is of interest to control their analyticity. The purpose of this paper is to first provide precise and explicit expressions as well as easily exploitable bounds for quantifying the analytic approximation of this complex wavelet. Then, we prove the existence of such filters enjoying the classical perfect reconstruction conditions, with arbitrarily many vanishing moments.

$M$-Channel Critically Sampled Spectral Graph Filter Banks With Symmetric Structure

This paper proposes a class of $M$-channel spectral graph filter banks with a symmetric structure, that is, the transform has sampling operations and spectral graph filters on both the analysis and synthesis sides. The filter banks achieve maximum decimation, perfect recovery, and orthogonality. Conventional spectral graph transforms with decimation have significant limitations with regard to the number of channels, the structures of the underlying graph, and their filter design. The proposed transform uses sampling in the graph frequency domain. This enables us to use any variation operators and apply the transforms to arbitrary graphs even when the filter banks have symmetric structures. We clarify the perfect reconstruction conditions and show design examples. An experiment on graph signal denoising conducted to examine the performance of the proposed filter bank is described.

Properties of approximated empirical mode decomposition and optimal design of its system kernel matrix for signal decomposition

An approximated empirical mode decomposition generates a set of approximated intrinsic mode functions via a linear, nonadaptive but iterative approach. The decomposition was found to be very useful for a content-independent pattern recognition application. As the process is characterized by a system kernel matrix and performed iteratively, the approximated intrinsic mode functions can be understood as the original signals processed by a set of mask operations. Here, some properties of the decomposition are studied and an optimal design of the system kernel matrix is proposed. It is found that there is only one approximated intrinsic mode function if the exact perfect reconstruction condition is satisfied. Obviously, the approximated intrinsic mode function is the original signal. Therefore, the decomposition is practically not meaningful. To address this issue, the infinite number of iterations in the algorithm is truncated to a finite number of iterations. Also, the design of the system kernel matrix is formulated as an optimization problem. In particular, the exact perfect reconstruction error between the sum of the approximated intrinsic mode functions and the original signal is minimized and the total absolute sum of the difference between any two different eigenvalues of the iterative matrix is maximized subject to a stability condition. Here, the stability condition refers to the eigenvalues of the iterative matrix being between zero and one. Since the optimization problem is nonsmooth and nonconvex, a genetic algorithm is employed for finding its near global optimal solution. Compared to the conventional approximated empirical mode decomposition, computer numerical simulations show that our proposed approach can achieve more than one approximated intrinsic mode function with each approximated intrinsic mode function corresponding to an output of a more meaningful mask operation.

Design and implementation of a novel low complexity symmetric orthogonal wavelet filter‐bank

The design of wavelet filter-bank in real continuum domain demands infinite precision to achieve perfect reconstruction (PR). The computations with irrational coefficients in orthogonal filters require a large amount of hardware that result in increased power dissipation, huge memory requirements and reduction in speed of operation. This paper presents a new approach to obtain a complete dyadic (low complexity) 6-tap orthogonal symmetric wavelet filter-bank (FB) with near perfect reconstruction. This is achieved by slightly altering the PR conditions to make orthogonal filters symmetric and to obtain complete dyadic filter coefficients. The proposed wavelet FB has reduced dynamic power dissipation and significantly reduced adder and shifter count. The VLSI architecture of the proposed FB is designed and implemented on Kintex-7 FPGA. The architectural design is carried out using Verilog and functional simulation and synthesis are executed using Vivado 2016.4 software. The proposed FB dissipates 114mW dynamic power and requires only 5 adders and 3 shifters. The effectiveness of the proposed wavelet is verified in three different applications namely image compression, iris recognition system and orthogonal frequency division multiplexing (OFDM). The proposed wavelet FB gives comparable performance in image compression and attains superior performance in iris recognition system (feature extraction) and OFDM.

De-Hankelization of singular spectrum analysis matrices via L1 norm criterion

For the de-Hankelization process in the singular spectrum analysis (SSA), the elements in the one dimensional SSA vectors are obtained via the conventional diagonal averaging method. That is, the average value of each off-diagonal of each two dimensional SSA matrix is computed. However, there are many large valued elements in the error vectors defining as the absolute differences between the off-diagonal vectors and the vectors with all the elements being the represented values. In other words, many elements in the off-diagonal vectors are different from the represented values. To address this issue, this paper proposes to find the represented values using the L 1 norm criterion subject to the exact perfect reconstruction condition. Compared to the conventional diagonal averaging method, the computer numerical simulation results show that our proposed method could yield a smaller total number of the large valued elements in the error vectors.

Characteristic Vectors in p/q-Channel Orthonormal Wavelet

Wavelet analysis is a newly rapidly developing subject in the late twentieth century. As a time-frequency analysis tool, wavelet analysis has many advantages over other time-frequency tools, such as in signal processing, image processing, speech processing, pattern recognition, quantum physics and other fields. Multiresolution analysis (MRA for short) is an important method for studying wavelet orthonormal wavelet bases with rational dilation 2. However, p/q-band wavelet is known to have advantages over 2-band wavelet in some aspects such as in signal processing and attracted more and more interest in recent years. But there are relatively less results for the case of p/q-band. This paper studies the orthonormal wavelet bases with rational dilation factor p/q based on multiresolution analysis by a polyphase decomposition technique. First, we gave the concept of Multiresolution analysis with rational dilation p/q and deduced an identity of the masks matrix. Also, a perfect reconstruction condition in terms of masks was presented. Further, we gave the refinement and wavelet matrices respectively and derived the characteristic roots and the corresponding orthonormal characteristic vectors of the wavelet matrix, and then a method with characteristic vectors was reduced to achieve the orthonormal wavelet bases with rational dilation factor p/q. In the end, an example is offered to verify this theory.