Wavelet Filter Banks Research Articles

Design of Time–Frequency-Localized Two-Band Orthogonal Wavelet Filter Banks

In this paper, we design time–frequency-localized two-band orthogonal wavelet filter banks using convex semidefinite programming (SDP). The sum of the time variance and frequency variance of the filter is used to formulate a real symmetric positive definite matrix for joint time–frequency localization of filters. Time–frequency-localized orthogonal low-pass filter with specified length and regularity order is designed. For nonmaximally regular two-band filter banks of length twenty, it is found that, as we increase the regularity order, the solution of the SDP converges to the filters with time–frequency product (TFP) almost same as the Daubechies maximally regular filter of length twenty. Unlike the class of Daubechies maximally regular minimum phase wavelet filter banks, a rank minimization algorithm in a SDP is employed to obtain mixed-phase low-pass filters with TFP of the filters as well as the scaling and wavelet function better than the equivalent two-band Daubechies filter bank.

Design of High-Performance ECG Detector for Implantable Cardiac Pacemaker Systems using Biorthogonal Wavelet Transform

A digital electrocardiogram (ECG) detector with low power consumption and high performance based on biorthogonal 2.2 wavelet transform and applicable for the modern implantable cardiac pacemakers is proposed in the present work. Biorthogonal 2.2 wavelet transform is chosen due to its high SNR, less number of coefficients, resemblance of shape with ECG wave and ability to increase QRS complex detection performance. Architecture of the proposed ECG detector includes modified biorthogonal 2.2 wavelet filter bank and a modified soft threshold-based QRS complex detector. Three low-pass filters and one high-pass filter with pipelined architecture are used which are lesser than the earlier designed detectors. Various blocks of proposed detector are designed to denoise the input ECG signal and then to find the correct location of R-wave. Verilog hardware description language for design entry, Modelsim embedded in Xilinx ISE v.14.1 for simulation, Virtex-6 FPGAs for synthesis and Xilinx ISE tools are used to measure the performance, area and power of the proposed ECG detector and its constituent blocks. A low detection error rate of 0.13%, positive predictivity ( $$\hbox {P}^{+}$$ ) of 99.94% and sensitivity ( $$\hbox {S}_{\mathrm{e}}$$ ) of 99.92% are achieved for the proposed ECG detector which are better compared to the previous results. Also, it consumes only 20 mW of total power at 50 KHz and shows the overall delay of 18.924 ns which makes it useful for the low power and high-performance applications.

Online Wavelet Complementary velocity Estimator

In this paper, we have proposed a new online Wavelet Complementary velocity Estimator (WCE) over position and acceleration data gathered from an electro hydraulic servo shaking table. This is a batch estimator type that is based on the wavelet filter banks which extract the high and low resolution of data. The proposed complementary estimator combines these two resolutions of velocities which acquired from numerical differentiation and integration of the position and acceleration sensors by considering a fixed moving horizon window as input to wavelet filter. Because of using wavelet filters, it can be implemented in a parallel procedure. By this method the numerical velocity is estimated without having high noise of differentiators, integration drifting bias and with less delay which is suitable for active vibration control in high precision Mechatronics systems by Direct Velocity Feedback (DVF) methods. This method allows us to make velocity sensors with less mechanically moving parts which makes it suitable for fast miniature structures. We have compared this method with Kalman and Butterworth filters over stability, delay and benchmarked them by their long time velocity integration for getting back the initial position data.

Dual PHY Layer for Non-Orthogonal Multiple Access Transceiver in 5G Networks

Non-orthogonal multiple access (NOMA) is a promising multiple access technique, proposed in literature for the fifth generation (5G) mobile networks. The NOMA system model consists of the conventional orthogonal frequency division multiplexing (OFDM), as a pulse shaping technique in conjunction with a variable power domain for various users, allocated in proportion to each user’s channel gain. OFDM technique based on wavelet filter banks, namely wavelet OFDM (WOFDM) has been utilized in digital communication to improve the system robustness to noise and adjacent channel interference, and is therefore anticipated to be adopted for the NOMA technique. WOFDM in NOMA (WNOMA) outperforms OFDM-based conventional NOMA (CNOMA) with reference to interference mitigation, bandwidth efficiency, spectral confinement, and multi-user capacity. Most of the fourth generation (4G) networks are based on OFDM and its variants. Therefore, in this paper, keeping in view the interoperability with the 4G networks and the latency requirements in 5G, a dual physical layer based on conventional OFDM and WOFDM as pulse shaping methods, is proposed for the NOMA transceiver. Performance of WNOMA and CNOMA is analyzed for bit error rate in the presence of channel impairments including additive noise and IQ imbalance and multiuser capacity is also computed. Comparison of various parameters indicates the advantage of adopting WNOMA over its conventional counterpart for relatively poor channel conditions.

Fast statistical model-based classification of epileptic EEG signals

This paper presents a supervised classification method to accurately detect epileptic brain activity in real-time from electroencephalography (EEG) data. The proposed method has three main strengths: it has low computational cost, making it suitable for real-time implementation in EEG devices; it performs detection separately for each brain rhythm or EEG spectral band, following the current medical practices; and it can be trained with small datasets, which is key in clinical problems where there is limited annotated data available. This is in sharp contrast with modern approaches based on machine learning techniques, which achieve very high sensitivity and specificity but require large training sets with expert annotations that may not be available. The proposed method proceeds by first separating EEG signals into their five brain rhythms by using a wavelet filter bank. Each brain rhythm signal is then mapped to a low-dimensional manifold by using a generalized Gaussian statistical model; this dimensionality reduction step is computationally straightforward and greatly improves supervised classification performance in problems with little training data available. Finally, this is followed by parallel linear classifications on the statistical manifold to detect if the signals exhibit healthy or abnormal brain activity in each spectral band. The good performance of the proposed method is demonstrated with an application to paediatric neurology using 39 EEG recordings from the Children's Hospital Boston database, where it achieves an average sensitivity of 98%, specificity of 88%, and detection latency of 4s, performing similarly to the best approaches from the literature.

Biorthogonal Halfband Perfect Reconstruction Filterbank for Multimodal Image Fusion

Objectives: To design a wavelet filter bank which helps to detect more prominently continuous changes in tumor cells. Methods/Statistical Analysis: In this paper, a method of designing two-channel wavelet base FIR filter bank using factorization of a half band filter is presented. Here factorization is done considering maximum vanishing moments for construction of decomposition as well as reconstruction filters. The 14th order maximally flat halfband filter is proposed with factorization, leading to design of 8/8 orthogonal and 9/7 & 6/10 symmetric filters. Findings: The fusion performance of designed wavelet filter bank is evaluated using various performance metrics, like, cross entropy, standard deviation, mean square error and PSNR. The results are compared with the Daubechies filter bank where db4 is used for implementation. It is clear from the results that designed filter bank improves fusion performance. Further, proposed filters have maximum number of vanishing moments which gives smooth scaling and wavelet functions and consequently provides flat frequency response. Application/Improvements: The designed 8/8 orthogonal wavelet filters are implemented in fusion application for multimodal biomedical images of a subject for detection of an abnormality (cancerous growth). The novelty of this method is the adaptive design of the filterbank for the given multimodal images, so that fused images shall have more clarity, further, it will help to improve the results with enhancement in the entropy levels of fused image. Keywords: Filter Banks, Half Band Filter, Medical Image Processing, Multimodal Image Fusion, Spectral Factorization, Wavelet Transforms

Bipartite Approximation for Graph Wavelet Signal Decomposition

To compactly represent a graph signal in the frequency domain, critically sampled biorthogonal wavelet filterbanks have been proposed to decompose signals on bipartite graphs. However, in practice graph signals often reside on general graph structures that are not bipartite. Thus, an original nonbipartite graph must be decomposed into a sequence of bipartite graph approximations, so that the filterbanks can be applied successively for signal decomposition. In this paper, unlike previous proposals that are heuristic in nature, we design new bipartite approximation strategies for model-based and empirically derived probability distributions. In the first case, a signal prior assumes a Gaussian Markov Random Field (GMRF) model parametrized by the original graph. In the second case, beyond the GMRF signal prior, empirical signal observations are available to compute a posterior probability distribution. In both cases, we optimize for energy compaction in the bipartite subgraphs with two criteria: the Kullback–Leibler divergence metric, which encourages preserving the spectral characteristics of the original graph; and multiplicity of eigenvalue at frequency 1 for graph Laplacian, which is the frequency with minimal energy discrimination. The relative importance between the two criteria is determined by a proposed measure that evaluates the degree of mismatch between the signal prior and posterior. Given these criteria, we first design a global numerical optimization, and then propose a local heuristic approach for fast implementation with simplifications leading to local metric computation. Experimental results show that our proposed bipartite subgraph decomposition outperforms competing proposals in terms of energy compaction.

Splines and wavelets on circulant graphs

We present novel families of wavelets and associated filterbanks for the analysis and representation of functions defined on circulant graphs. In this work, we leverage the inherent vanishing moment property of the circulant graph Laplacian operator, and by extension, the e-graph Laplacian, which is established as a parameterization of the former with respect to the degree per node, for the design of vertex-localized and critically-sampled higher-order graph (e-)spline wavelet filterbanks, which can reproduce and annihilate classes of (exponential) polynomial signals on circulant graphs. In addition, we discuss similarities and analogies of the detected properties and resulting constructions with splines and spline wavelets in the Euclidean domain. Ultimately, we consider generalizations to arbitrary graphs in the form of graph approximations, with focus on graph product decompositions. In particular, we proceed to show how the use of graph products facilitates a multi-dimensional extension of the proposed constructions and properties.

Bipartite Graph Filter Banks: Polyphase Analysis and Generalization

The work by Narang and Ortega [“Perfect reconstruction two-channel wavelet filter banks for graph structured data,” IEEE Trans. Signal Process. , vol. 60, no. 6, pp. 2786–2799, Jun. 2012], [“Compact support biorthogonal wavelet filterbanks for arbitrary undirected graphs,” IEEE Trans. Signal Process. , vol. 61, no. 19, pp. 4673–4685, Oct. 2013] laid the foundations for the two-channel critically sampled perfect reconstruction filter bank for signals defined on undirected graphs. This basic filter bank is applicable only to bipartite graphs but using the notion of separable filtering, the basic filter bank can be applied to any arbitrary undirected graphs. In this paper, several new theoretical results are presented. In particular, the proposed polyphase analysis yields filtering structures in the downsampled domain that are equivalent to those before downsampling and, thus, can be exploited for efficient implementation. These theoretical results also provide new insights that can be exploited in the design of these systems. These insights allow us to generalize these filter banks to directed graphs and to using a variety of graph base matrices, while also providing a link to the $\text{DSP}_G$ framework of Sandryhaila and Moura [“Discrete signal processing on graphs,” IEEE Trans. Signal Process. , vol. 61, no. 7, pp. 1644–1636, Apr. 2013], [“Discrete signal processing on graphs: Frequency analysis,” IEEE Trans. Signal Process. , vol. 62, no. 12, pp. 3042–3054, Jun. 2014]. Experiments show evidence that better nonlinear approximation and denoising results may be obtained by a better selection of these base matrices.

Instantaneous brain dynamics mapped to a continuous state space

Measures of whole-brain activity, from techniques such as functional Magnetic Resonance Imaging, provide a means to observe the brain's dynamical operations. However, interpretation of whole-brain dynamics has been stymied by the inherently high-dimensional structure of brain activity. The present research addresses this challenge through a series of scale transformations in the spectral, spatial, and relational domains. Instantaneous multispectral dynamics are first developed from input data via a wavelet filter bank. Voxel-level signals are then projected onto a representative set of spatially independent components. The correlation distance over the instantaneous wavelet-ICA state vectors is a graph that may be embedded onto a lower-dimensional space to assist the interpretation of state-space dynamics. Applying this procedure to a large sample of resting-state and task-active data (acquired through the Human Connectome Project), we segment the empirical state space into a continuum of stimulus-dependent brain states. Upon observing the local neighborhood of brain-states adopted subsequent to each stimulus, we may conclude that resting brain activity includes brain states that are, at times, similar to those adopted during tasks, but that are at other times distinct from task-active brain states. As task-active brain states often populate a local neighborhood, back-projection of segments of the dynamical state space onto the brain’s surface reveals the patterns of brain activity that support many experimentally-defined states.

A novel three-band orthogonal wavelet filter bank method for an automated identification of alcoholic EEG signals

Alcoholism is a critical disorder related to the central nervous system, caused due to repeated and excessive consumption of alcohol. The electroencephalogram (EEG) signals are used to depict brain activities. It can also be employed for diagnosis of subjects consuming excessive alcohol. In this study, we have developed an automated system for the classification of alcoholic and normal EEG signals using a recently designed duration-bandwidth product (DBP), optimized three-band orthogonal wavelet filter bank (TBOWFB), and log-energy (LE). First, we obtain sub-bands (SBs) of EEG signals using the TBOWFB. Then, we use logarithms of the energies of the SBs as the discriminating features which are fed to the least square support vector machine (LS-SVM) for the discrimination of normal and alcoholic EEG signals. We have achieved a classification accuracy (CA) of 97.08%, with ten-fold cross validation strategy. The proposed model presents a promising performance, and therefore it can be used in a practical setup to assist the medical professionals in the diagnosis of alcoholism using EEG signals automatically.

A novel approach for time–frequency localization of scaling functions and design of three-band biorthogonal linear phase wavelet filter banks

Design of time–frequency localized filters and functions is a classical subject in the field of signal processing. Gabor's uncertainty principle states that a function cannot be localized in time and frequency domain simultaneously and there exists a nonzero lower bound of 0.25 on the product of its time variance and frequency variance called time–frequency product (TFP). Using arithmetic mean (AM)– geometric mean (GM) inequality, product of variances and sum of variances can be related and it can be shown that sum of variances has lower bound of one. In this paper, we compute the frequency variance of the filter from its discrete Fourier transform (DFT) and propose an equivalent summation based discrete-time uncertainty principle which has the lower bound of one. We evaluate the performance of the proposed discrete-time time–frequency uncertainty measure in multiresolution setting and show that the proposed DFT based concentration measure generate sequences which are even more localized in time and frequency domain than that obtained from the Slepian, Ishii and Furukawa's concentration measures. The proposed design approach provides the flexibility in which the TFP can be made arbitrarily close to the lowest possible lower bound of 0.25 by increasing the length of the filter. In the other proposed approach, the sum of the time variance and frequency variance is used to formulate a positive definite matrix to measure the time–frequency joint localization of a bandlimited function from its samples. We design the time–frequency localized bandlimited low pass scaling and band pass wavelet functions using the eigenvectors of the formulated positive definite matrix. The samples of the time–frequency localized bandlimited function are obtained from the eigenvector of the positive definite matrix corresponding to its minimum eigenvalue. The TFP of the designed bandlimited scaling and wavelet functions are close to the lowest possible lower bound of 0.25 and 2.25 respectively. We propose a design method for time–frequency localized three-band biorthogonal linear phase (BOLP) wavelet perfect reconstruction filter bank (PRFB) wherein the free parameters can be optimized for time–frequency localization of the synthesis basis functions for the specified frequency variance of the analysis scaling function. The performance of the designed filter bank is evaluated in classification of seizure and seizure-free electroencephalogram (EEG) signals. It is found that the proposed filter bank outperforms other existing methods for the classification of seizure and seizure-free EEG signals.

An analysis method for wearable electrocardiogram measurement based on non-orthogonal complex wavelet expansion.

An analysis method for wearable electrocardiogram (ECG) measurement is proposed, which is based on the non-orthogonal complex wavelet expansion. The standard continuous wavelet transform performs the signal filtering that corresponds to the orthogonal projection of the ECG signal onto each of the non-orthogonal wavelets. This causes feature leakage from the neighboring wavelet filterbanks and degrades the detection accuracy of the QRS complexes or T waves, even though the wavelets whose waveforms fit to the morphologies of the ECG signals are chosen. The proposed method involves a different signal filtering that corresponds to the non-orthogonal projection of the signal onto each wavelet so that the filter outputs can be the expansion coefficients of the wavelets to approximate the ECG signal. The concept of our proposed method is supported by simulation results.

Compactly Supported Biorthogonal Wavelet Bases on the Body Centered Cubic Lattice

AbstractIn this work, we present a family of compact, biorthogonal wavelet filter banks that are applicable to the Body Centered Cubic (BCC) lattice. While the BCC lattice has been shown to have superior approximation properties for volumetric data when compared to the Cartesian Cubic (CC) lattice, there has been little work in the way of designing wavelet filter banks that respect the geometry of the BCC lattice. Since wavelets have applications in signal de‐noising, compression, and sparse signal reconstruction, these filter banks are an important tool that addresses some of the scalability concerns presented by the BCC lattice. We use these filters in the context of volumetric data compression and reconstruction and qualitatively evaluate our results by rendering images of isosurfaces from compressed data.

Design of Two Channel Biorthogonal Graph Wavelet Filter Banks with Half-Band Kernels

Design of Two Channel Biorthogonal Graph Wavelet Filter Banks with Half-Band Kernels

A parametrization technique to design joint time–frequency optimized discrete-time biorthogonal wavelet bases

The accurate and efficient representation of a signal in terms of elementary atoms has been a challenge in many signal processing applications including harmonic analysis. The wavelet bases have been proved to be very efficient and flexible atoms. Towards the goal of obtaining optimal wavelet bases, we present a simple and efficient parametrization technique for constructing linear phase biorthogonal discrete-time wavelet bases that have joint time–frequency localization (JTFL) close to the lower bound of 0.25. In this paper, we first develop a parametrization technique to design biorthogonal filter banks (FBs). Then an optimization method is formulated to design jointly time–frequency localized discrete wavelet bases employing the designed FBs. Finally, the performance of the optimal wavelet bases is evaluated in image coding application. The proposed parametrization method presents a general and yet a very simple framework to construct a linear phase biorthogonal FB of desired order, with the prescribed number of vanishing moments (VMs) and free parameters. Several examples are presented to demonstrate the effectiveness and flexibility of the technique to design different classes of FB with various degrees of freedom. The performance of the designed FBs is compared with the other popular biorthogonal wavelet FBs.

Time–frequency localized three-band biorthogonal wavelet filter bank using semidefinite relaxation and nonlinear least squares with epileptic seizure EEG signal classification

In this paper, we design time–frequency localized three-band biorthogonal linear phase wavelet filter bank for epileptic seizure electroencephalograph (EEG) signal classification. Time–frequency localized analysis and synthesis low-pass filters (LPF) are designed using convex semidefinite programming (SDP) by transforming a nonconvex problem into a convex SDP using semidefinite relaxation technique. Three-band parameterized lattice biorthogonal linear phase perfect reconstruction filter bank (BOLPPRFB) is chosen and nonlinear least squares algorithm is used to determine its parameters values that generate the designed analysis and synthesis LPF such that the band-pass and high-pass filters are also well localized in time and frequency domain. The designed analysis and synthesis three-band wavelet filter banks are compared with the standard two-band filter banks like Daubechies maximally regular filter banks, Cohen–Daubechies–Feauveau (CDF) biorthogonal filter banks and orthogonal time–frequency localized filter banks. Kruskal–Wallis statistical test is employed to measure the statistical significance of the subband features obtained from the various two and three-band filter banks for epileptic seizure EEG signal classification. The results show that the designed three-band analysis and synthesis filter banks both outperform two-band filter banks in the classification of seizure and seizure-free EEG signals. The designed three-band filter banks and multi-layer perceptron neural network (MLPNN) are further used together to implement a signal classifier that provides classification accuracy better than the recently reported results for epileptic seizure EEG signal classification.

Hypercomplex polynomial wavelet-filter bank transform for color image

Starting from a link with the discrete polynomial decomposition, we propose an extension of the non-uniform filter bank framework to vector-valued signals to carry out a non marginal color wavelet transform. Recently quaternions or Clifford algebras have been used to perform color image processing. This paper uses the embedding of color information into the imaginary part of a Quaternion. From this coding strategy, discrete Hypercomplex 2D polynomial transform is presented as generalizing the real polynomial transform to the Hypercomplex domain. We illustrate how this numerical approach can be used to define Wavelet filter bank transform for color image, with piecewise Hypercomplex polynomial. From this, the connection between discrete polynomial representation and multiscale, filter bank algorithm already defined for Grayscale image is described and the proposed color representation is discussed. This new hypercomplex polynomial-wavelet transform has an extra parameter: an imaginary unit quaternion corresponding to a privileged color direction. This new color representation satisfies the conditions of orthogonality, perfect reconstruction and linear phase filter. Moreover, a link between hypercomplex polynomial projections and discrete multiscale differentiators is studied. We also give a practical study of this new Hypercomplex transform in the contexts of image denoising, which validates the interest of our construction.

An automatic detection of focal EEG signals using new class of time–frequency localized orthogonal wavelet filter banks

It is difficult to detect subtle and vital differences in electroencephalogram (EEG) signals simply by visual inspection. Further, the non-stationary nature of EEG signals makes the task more difficult. Determination of epileptic focus is essential for the treatment of pharmacoresistant focal epilepsy. This requires accurate separation of focal and non-focal groups of EEG signals. Hence, an intelligent system that can detect and discriminate focal–class (FC) and non–focal–class (NFC) of EEG signals automatically can aid the clinicians in their diagnosis. In order to facilitate accurate analysis of non-stationary signals, joint time–frequency localized bases are highly desirable. The performance of wavelet bases is found to be effective in analyzing transient and abrupt behavior of EEG signals. Hence, we employ a novel class of orthogonal wavelet filter banks which are localized in time–frequency domain to detect FC and NFC EEG signals automatically. We classify EEG signals as FC and NFC using the proposed wavelet based system. We compute various entropies from the wavelet coefficients of the signals. These entropies are used as discriminating features for the classification of FC and NFC of EEG signals. The features are ranked using Student’s t-test ranking algorithm and then fed to Least Squares-Support Vector Machine (LS–SVM) to classify the signals. Our proposed method achieved the highest classification accuracy of 94.25%. We have obtained 91.95% sensitivity and 96.56% specificity, respectively, using this method.The classification of FC and NFC of EEG signals helps in localization of the affected brain area which needs to undergo surgery.

Optimal duration-bandwidth localized antisymmetric biorthogonal wavelet filters

We present a design of a new class of compactly supported antisymmetric biorthogonal wavelet filter banks which have the analysis as well as the synthesis filters of even-length. Here, the analysis and the synthesis filters are designed to have minimum joint duration-bandwidth localization (JDBL). The design of filters has been formulated as a direct time-domain linearly constrained eigenvalue problem that does not involve any parametrization and iterations. The optimal analysis and synthesis filters have been obtained as the eigenvectors of the positive definite matrices. The closed form analytic expression for the objective function has been presented. The perfect reconstruction and regularity conditions have been incorporated in the design by employing time-domain matrix characterization. The method can control duration and bandwidth localizations of the analysis and synthesis filters, independently. A few design examples have been presented and compared with previous works. The performance of the optimal filter banks designed by employing the proposed method has been evaluated in image coding and signal denoising applications.