Optimal Wavelet Filter Research Articles

Optimised DWT using cooperative particle swarm optimiser for hybrid domain based medical and natural image denoising

The quest for productive image denoising systems still is a valid challenge, at the intersection of practical investigation and measurements. In spite of the sophistication of the recently proposed systems, most calculations have not yet achieved an attractive level of applicability. In this research, an optimal wavelet filter coefficient design-based methodology is proposed for image denoising. The method utilises new wavelet filter whose coefficients are derived by discrete wavelet (Haar) transform using CPSO optimisation and bilateral filter. The optimal wavelet coefficient based denoising methods minimise the noise, while bilateral filter further decreases the noise and increases the PSNR without any loss of relevant image information. Overall, the proposed approach consists of two stages namely, (i) design of optimal wavelet filter, (ii) image denoising using a bilateral filter. At first, wavelet optimal coefficients are selected using cooperative particle swarm optimiser (CPSO). After that, the hybrid domain based algorithm (wavelet with bilateral filter) is applied to the noisy image which is helpful to obtain the denoised image. A comparative study of the performance of different existing approaches and the proposed denoised approach is made in terms of PSNR, SDME, SSIM and GP. When compared, the proposed algorithm gives better PSNR compared to the existing methods.

Automated heartbeat classification and detection of arrhythmia using optimal orthogonal wavelet filters

Abstract Electrocardiogram (ECG) is commonly used to analyze heartbeats. Arrhythmia is a disturbance of the heartbeat that can potentially be lethal. Arrhythmia can be detected by identifying an individual abnormal heartbeat, which can occur in isolation or sequentially. Heartbeats can be classified into five types: non-ectopic (N), supra ventricular ectopic (S), ventricular ectopic (V), fusion (F) and unknown beats (Q). Automated classification of the incidence and pattern of abnormal heartbeats enables arrhythmia detection, triage and life-threatening arrhythmia. Therefore, it is vital to develop a tool for the classification of these classes. ECG signals are contaminated by noise rendering difficulty and challenging to discriminate between different types of heartbeats. In this work, we used noisy as well as denoised (clean) ECG signals to classify heartbeats. Using stop-band energy (SBE) minimized dyadic orthogonal filter bank, wavelet decomposition of the ECG signals was performed. After this, fuzzy entropy, Renyi entropy and fractal dimension features were extracted for easier and accurate classification into the five classes. These features were then fed to classifiers, achieving maximum accuracy (MAAC) of 98%, maximum sensitivity (MASE) of 85.33% and maximum specificity (MASP) of 98.22% for noisy data, and MAAC of 98.1%, MASE of 85.63% and MASP of 98.27% for clean data, with KNN classifier. Our developed system can be used in intensive care units to assist the clinicians to aid in their diagnosis.

A new approach to identify obstructive sleep apnea using an optimal orthogonal wavelet filter bank with ECG signals

Abstract Obstructive sleep apnea (OSA) is a severe sleep ailment. It manifests when a person's breathing is interrupted while sleeping due to an inadequate supply of oxygen to the brain and physical body. The detection of OSA at an early stage can provide better mitigation from severe health impairments. An accurate method to detect OSA can improve the quality of life significantly. The computer-aided diagnosis (CAD) system of OSA developed using single-channel, single-signal physiological signals can efficiently cut costs, and can be used even at home. Therefore, a simple and portable OSA-CAD system using single-channel electrocardiogram (ECG) is introduced in this paper. This study proposes the automated identification of ECG signal affected from OSA employing an optimal two-band filter bank (FB) technique. While designing the filter bank, our central objective is to minimize the spectral localization, subjected to exact regeneration and regularity criteria. The identification of OSA using ECG signals is duly based on the newly designed FB. Using the proposed FB, ECG signals were split into wavelet frequency-bands (WFBs). The fuzzy-entropy (FUEN) and the log of signal-energy (LOEN) of WFBs have been employed as the distinguishing features. The 35-fold cross-validation technique was exercised using various classifiers, namely K nearest neighbor (KNN), decision tree (DT), linear discriminant, logistic regression, and support vector machine (SVM) to separate into normal and OSA affected subjects. The proposed model has attained respective highest average accuracy (AVAC), average sensitivity (AVSE), average specificity (AVSP) and F1-score of 90.87%, 92.43% 88.33% and 92.61%. Our proposed model outperformed the existing systems developed using the same database and was found to be more efficient, robust, and easy to use.

Multi-objective iterative optimization algorithm based optimal wavelet filter selection for multi-fault diagnosis of rolling element bearings

Rolling element bearings (REBs) play an essential role in modern machinery and their condition monitoring is significant in predictive maintenance. Due to the harsh operating conditions, multi-fault may co-exist in one bearing and vibration signal always exhibits low signal-to-noise ratio (SNR), which causes difficulties in detecting fault. In the previous studies, maximum correlated kurtosis deconvolution (MCKD) has been validated as an efficient method to extract fault feature in the fault signals. Nonetheless, there are still some challenges when MCKD is applied to fault detection owing to the rigorous requirements of multiple input parameters. To overcome limitation, a multi-objective iterative optimization algorithm (MOIOA) for multi-fault diagnosis is proposed. In this method, correlated kurtosis (CK) is taken as a criterion to select optimal Morlet wavelet filter using the whale optimization algorithm (WOA). Meanwhile, to further eliminate the effect of the inaccurate period on CK, the update process of period is incorporated. After that, the simulated and experimental signals are utilized to testify the validity and superiority of the MOIOA for multiple faults detection by the comparison with MCKD. The results indicate that MOIOA is efficient to extract weak fault features even with heavy noise and harmonic interferences.

MMSFL-OWFB: A novel class of orthogonal wavelet filters for epileptic seizure detection

The optimal filters with minimal bandwidth are highly desirable in many applications such as communication and biomedical signal processing. In this study, we design optimally frequency localized orthogonal wavelet filters and evaluate their performance using electroencephalogram (EEG) signals for automated detection of the epileptic seizure. The paper presents a novel method for designing optimal orthogonal wavelet filter banks (OWFB) with the objective of minimizing their frequency spreads. The designed wavelet filter also possesses the desired degree of regularity. The regularity condition has been imposed analytically so as to satisfy the constraint accurately. We propose a novel semi-definite programming (SDP) formulation which does not involve any parametrization. The solution of the SDP yields optimal orthogonal wavelet filter for the given length of the filter. We have developed an automated diagnosis system that identifies epileptic seizure EEG signals using the features obtained from the designed minimally mean squared frequency localized (MMSFL) OWFB. We have tested the performance of the proposed model using two independent EEG databases in order to ensure the consistency and robustness of the model. Interestingly, the proposed MMSFL-OWFB feature-based model exhibits ceiling level of performance, with classification accuracy ≥ 99% in classifying seizure (ictal) and seizure-free (non-ictal) EEG signals for both databases. Our developed system can be employed in hospitals and community cares to aid the epileptologists in the accurate diagnosis of seizures.

Application of an optimal class of antisymmetric wavelet filter banks for obstructive sleep apnea diagnosis using ECG signals

Obstructive sleep apnea (OSA) is a sleep disorder caused due to interruption of breathing resulting in insufficient oxygen to the human body and brain. If the OSA is detected and treated at an early stage the possibility of severe health impairment can be mitigated. Therefore, an accurate automated OSA detection system is indispensable. Generally, OSA based computer-aided diagnosis (CAD) system employs multi-channel, multi-signal physiological signals. However, there is a great need for single-channel bio-signal based low-power, a portable OSA-CAD system which can be used at home. In this study, we propose single-channel electrocardiogram (ECG) based OSA-CAD system using a new class of optimal biorthogonal antisymmetric wavelet filter bank (BAWFB). In this class of filter bank, all filters are of even length. The filter bank design problem is transformed into a constrained optimization problem wherein the objective is to minimize either frequency-spread for the given time-spread or time-spread for the given frequency-spread. The optimization problem is formulated as a semi-definite programming (SDP) problem. In the SDP problem, the objective function (time-spread or frequency-spread), constraints of perfect reconstruction (PR) and zero moment (ZM) are incorporated in their time domain matrix formulations. The global solution for SDP is obtained using interior point algorithm. The newly designed BAWFB is used for the classification of OSA using ECG signals taken from the physionet's Apnea-ECG database. The ECG segments of 1 min duration are decomposed into six wavelet subbands (WSBs) by employing the proposed BAWFB. Then, the fuzzy entropy (FE) and log-energy (LE) features are computed from all six WSBs. The FE and LE features are classified into normal and OSA groups using least squares support vector machine (LS-SVM) with 35-fold cross-validation strategy. The proposed OSA detection model achieved the average classification accuracy, sensitivity, specificity and F-score of 90.11%, 90.87% 88.88% and 0.92, respectively. The performance of the model is found to be better than the existing works in detecting OSA using the same database. Thus, the proposed automated OSA detection system is accurate, cost-effective and ready to be tested with a huge database.

Compound faults detection of the rolling element bearing based on the optimal complex Morlet wavelet filter

Wavelet filter is widely used in extracting fault features embedded in the noisy vibration signal, especially the complex Morlet wavelet. In most occasions, the filter parameters are optimized adaptively with a suitable objective function. And then, with the Hilbert transform demodulation analysis, the single localized fault in rolling element bearings can be detected. To extend it for compound faults detection, a novel index deduced from the different intervals of the prominent bearing fault frequencies and subsequent harmonics in the envelope spectrum is proposed. By maximizing the ratio of correlated kurtosis to kurtosis of the envelope spectrum amplitudes of the filtered signal, the optimal complex Morlet wavelet filters corresponding to the different faults are designed by the particle filtering method, respectively. Two cases of real signals are analyzed to evaluate the performance of the proposed method, which include one case of experiment signal with artificial outer race fault coupled with roller fault, as well as one case of engineering data with outer race fault coupled with inner race fault. Furthermore, some comparisons with a previous method are also conducted. The results demonstrate the effectiveness and robustness of the method in compound faults diagnosis of the rolling element bearings.

Bearing condition monitoring approaches - envelope and cepstrum analyses

Envelope and cepstrum analyses are useful tools to identify the fault bearing characteristic frequencies (BCFs) and distinguish them from other frequencies. However, the traditional envelope analysis has disadvantages such as the choice of central frequency of filter needs prior experience. To overcome the traditional envelope shortage, an optimal wavelet filter is used as a band-pass filter whose parameters are optimised based on the maximum kurtosis value. To further reduce the residual in-band noise and highlight the periodic impulsive feature, an envelope enhancement is applied to the filtered signal. The article aims to compare these two condition monitoring tools in a pragmatic way and demonstrate their properties by using a practical example of a rolling bearing fault. The results prove that cepstrum analysis provides a clear indication of fault for machine parts such as vehicle gearbox, in addition to evaluating the efficiency of proposed band-pass filter in the envelope analysis.

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.

Heuristic feature extraction method for BCI with harmony search and discrete wavelet transform

For the brain-computer interface system (BCI), pre-processing has an important role to ensure system performance. However, the speech recognition system using electroencephalogram (EEG) is weak against temporal effects. Therefore, in general cases, wavelet transform has been used to cope with the temporal effects and non-stationary characteristic of EEG. The discrete version of wavelet transform, called DWT, requires a filter of the system for use in downsampling the signal. In other words, it is important to determine the suitable type of filter. In many cases, it is difficult to find an adequate filter for DWT because of differences in the characteristics of the input signal. In this paper, we proposed a heuristic approach to finding the optimal filter of the system for EEG signals. The harmony search algorithm (HSA) was used for finding of the optimal filter. In the learning process with the EEG system, the optimal wavelet filter could be found, which is automatically designed for subject personality.

Design of Time–Frequency Optimal Three-Band Wavelet Filter Banks with Unit Sobolev Regularity Using Frequency Domain Sampling

In this paper, we design three-band time---frequency-localized orthogonal wavelet filter banks having single vanishing moment. We propose new expressions to compute mean and variances in time and frequency from the samples of the Fourier transform of the asymmetric band-pass compactly supported wavelet functions. We determine discrete-time filter of length eight that generates the time---frequency optimal time-limited scaling and wavelet functions using cascade algorithm. Time---frequency product (TFP) of a function is defined as the product of its time variance and frequency variance. The TFP of the designed functions is close to 0.25 with unit Sobolev regularity. Three-band filter banks are designed by minimizing a weighted combination of TFPs of wavelets and scaling functions. Interestingly, empirical results show that time---frequency optimal, filter banks of length nine, designed with the proposed methodology, have unit Sobolev regularity, which is maximum achievable with single vanishing moment. Design examples for length six and length nine filter banks are given to demonstrate the effectiveness of the proposed design methodology.

Design of Time–Frequency Localized Filter Banks: Transforming Non-convex Problem into Convex Via Semidefinite Relaxation Technique

We present a method for designing optimal biorthogonal wavelet filter banks (FBs). Joint time---frequency localization of the filters has been chosen as the optimality criterion. The design of filter banks has been cast as a constrained optimization problem. We design the filter either with the objective of minimizing its frequency spread (variance) subject to the constraint of prescribed time spread or with the objective of minimizing the time spread subject to the fixed frequency spread. The optimization problems considered are inherently non-convex quadratic constrained optimization problems. The non-convex optimization problems have been transformed into convex semidefinite programs (SDPs) employing the semidefinite relaxation technique. The regularity constraints have also been incorporated along with perfect reconstruction constraints in the optimization problem. In certain cases, the relaxed SDPs are found to be tight. The zero duality gap leads to the global optimal solutions. The design examples demonstrate that reasonably smooth wavelets can be designed from the proposed filter banks. The optimal filter banks have been compared with popular filter banks such as Cohen---Daubechies---Feauveau biorthogonal wavelet FBs, time---frequency optimized half-band pair FBs and maximally flat half-band pair FBs. The performance of optimal filter banks has been found better in terms of joint time---frequency localization.

Novel Gauss–Hermite integration based Bayesian inference on optimal wavelet parameters for bearing fault diagnosis

Rolling element bearings are commonly used in machines to provide support for rotating shafts. Bearing failures may cause unexpected machine breakdowns and increase economic cost. To prevent machine breakdowns and reduce unnecessary economic loss, bearing faults should be detected as early as possible. Because wavelet transform can be used to highlight impulses caused by localized bearing faults, wavelet transform has been widely investigated and proven to be one of the most effective and efficient methods for bearing fault diagnosis. In this paper, a new Gauss–Hermite integration based Bayesian inference method is proposed to estimate the posterior distribution of wavelet parameters. The innovations of this paper are illustrated as follows. Firstly, a non-linear state space model of wavelet parameters is constructed to describe the relationship between wavelet parameters and hypothetical measurements. Secondly, the joint posterior probability density function of wavelet parameters and hypothetical measurements is assumed to follow a joint Gaussian distribution so as to generate Gaussian perturbations for the state space model. Thirdly, Gauss–Hermite integration is introduced to analytically predict and update moments of the joint Gaussian distribution, from which optimal wavelet parameters are derived. At last, an optimal wavelet filtering is conducted to extract bearing fault features and thus identify localized bearing faults. Two instances are investigated to illustrate how the proposed method works. Two comparisons with the fast kurtogram are used to demonstrate that the proposed method can achieve better visual inspection performances than the fast kurtogram.

Smoothness index-guided Bayesian inference for determining joint posterior probability distributions of anti-symmetric real Laplace wavelet parameters for identification of different bearing faults

Rolling element bearings are one of the most common components used in machines and they are used to support rotating shafts. Unexpected bearing failures may cause machine breakdown which results in economic loss. Detection of bearing faults is crucial to prevent bearing failures. Vibration based signal processing methods have been proven to be effective in identifying different bearing faults. Among different signal processing methods, wavelet analysis is widely investigated because it is able to highlight the similarity between wavelet functions with different wavelet parameters and impulses caused by bearing faults. In wavelet analysis, two topics are of great concern. The first is how to choose a suitable wavelet mother function for bearing fault diagnosis. In recent studies, an anti-symmetric real Laplace wavelet or impulse response wavelet has been experimentally proven to have a high similarity with impulses caused by bearing faults. Therefore, anti-symmetric real Laplace wavelet is chosen as the wavelet mother function in this paper. The second is how to determine the optimal wavelet parameters so as to enhance the ability of wavelet analysis. Based on the anti-symmetric real Laplace wavelet, smoothness index based Bayesian inference is proposed in this paper to establish joint posterior probability density functions of wavelet parameters, which reflect graphical relationships between wavelet parameters. The smoothness index is chosen because it is not only able to quantify bearing fault signals, but also has upper and lower bounds, compared with other metrics, such as wavelet entropy, Shannon entropy, kurtosis, sparsity measurement, etc. For Bayesian inference, a general particle filter is adopted to iteratively calculate and update joint posterior probability density functions of wavelet parameters. Once the joint posterior probability density functions of wavelet parameters are available, the optimal wavelet parameters are determined and an optimal wavelet filtering is conducted to extract bearing fault signatures. Real case studies are investigated to illustrate how the proposed method works. The results show that the proposed method can determine joint posterior probability density functions of wavelet parameters and is effective in identifying different bearing faults.

A general sequential Monte Carlo method based optimal wavelet filter: A Bayesian approach for extracting bearing fault features

A general sequential Monte Carlo method, particularly a general particle filter, attracts much attention in prognostics recently because it is able to on-line estimate posterior probability density functions of the state functions used in a state space model without making restrictive assumptions. In this paper, the general particle filter is introduced to optimize a wavelet filter for extracting bearing fault features. The major innovation of this paper is that a joint posterior probability density function of wavelet parameters is represented by a set of random particles with their associated weights, which is seldom reported. Once the joint posterior probability density function of wavelet parameters is derived, the approximately optimal center frequency and bandwidth can be determined and be used to perform an optimal wavelet filtering for extracting bearing fault features. Two case studies are investigated to illustrate the effectiveness of the proposed method. The results show that the proposed method provides a Bayesian approach to extract bearing fault features. Additionally, the proposed method can be generalized by using different wavelet functions and metrics and be applied more widely to any other situation in which the optimal wavelet filtering is required.

On the application of optimal wavelet filter banks for ECG signal classification

This paper discusses ECG signal classification after parametrizing the ECG waveforms in the wavelet domain. Signal decomposition using perfect reconstruction quadrature mirror filter banks can provide a very parsimonious representation of ECG signals. In the current work, the filter parameters are adjusted by a numerical optimization algorithm in order to minimize a cost function associated to the filter cut-off sharpness. The goal consists of achieving a better compromise between frequency selectivity and time resolution at each decomposition level than standard orthogonal filter banks such as those of the Daubechies and Coiflet families. Our aim is to optimally decompose the signals in the wavelet domain so that they can be subsequently used as inputs for training to a neural network classifier.

The automatic selection of an optimal wavelet filter and its enhancement by the new sparsogram for bearing fault detection: Part 2 of the two related manuscripts that have a joint title as “Two automatic vibration-based fault diagnostic methods using the novel sparsity measurement—Parts 1 and 2”

Rolling element bearings are the most important components used in machinery. Bearing faults, once they have developed, quickly become severe and can result in fatal breakdowns. Envelope spectrum analysis is one effective approach to detect early bearing faults through the identification of bearing fault characteristic frequencies (BFCFs). To achieve this, it is necessary to find a band-pass filter to retain a resonant frequency band for the enhancement of weak bearing fault signatures. In Part 1 paper, the wavelet packet filters with fixed center frequencies and bandwidths used in a sparsogram may not cover a whole bearing resonant frequency band. Besides, a bearing resonant frequency band may be split into two adjacent imperfect orthogonal frequency bands, which reduce the bearing fault features. Considering the above two reasons, a sparsity measurement based optimal wavelet filter is required to be designed for providing more flexible center frequency and bandwidth for covering a bearing resonant frequency band. Part 2 paper presents an automatic selection process for finding the optimal complex Morlet wavelet filter with the help of genetic algorithm that maximizes the sparsity measurement value. Then, the modulus of the wavelet coefficients obtained by the optimal wavelet filter is used to extract the envelope. Finally, a non-linear function is introduced to enhance the visual inspection ability of BFCFs. The convergence of the optimal filter is fastened by the center frequencies and bandwidths of the optimal wavelet packet nodes established by the new sparsogram. Previous case studies including a simulated bearing fault signal and real bearing fault signals were used to show that the effectiveness of the optimal wavelet filtering method in detecting bearing faults. Finally, the results obtained from comparison studies are presented to verify that the proposed method is superior to the other three popular methods.

A joint sparse wavelet coefficient extraction and adaptive noise reduction method in recovery of weak bearing fault features from a multi-component signal mixture

Rolling element bearings are widely used to support rotating components of a machine. Due to close space locations of components in the machine, a vibration signal caused by bearing localized defects is easily overwhelmed by other strong vibration signals. Extracting the bearing fault signal from a multi-component signal mixture is thus significant to detect early bearing fault features and prevent machine breakdown. In this paper, a bearing fault diagnosis method, named cyclic spike detection method, is proposed to extract the weak bearing fault features from a multi-component signal mixture. Firstly, the optimal center frequency and bandwidth of a complex Morlet wavelet filter are determined by a simplex-simulated annealing algorithm along with a maximum sparsity objective function. The filtered signal is then obtained by applying the optimal wavelet filter to the multi-component signal mixture. After that, a new adaptive local maximum selection method is proposed to make the filtered signal succinct. Only a few spikes are retained to reveal potential cyclic intervals caused by bearing localized defects. Two multi-component signal mixtures, including a simulated signal and a real vibration signal collected from an industrial machine, are used to validate the effectiveness of the proposed cyclic spike detection method. The results demonstrate that the proposed method can extract the weak bearing fault features from other strong masking vibration signals and noise.

The Optimized Wavelet Filters and Real-Time Implementation of Speech Codec Base on DWT on TMS320C64xx

This paper presents, optimized wavelet filters using windowing techniques and linear optimization, which are used as mother wavelets for speech compression based on wavelets. Proposed design and real-time implementation of the optimized speech codec based on discret wavelet transform (DWT), was performed. The comparison tests between the proposed optimized wavelet filters and others wavelets based on CR, SNR, PSNR and NRMSE computation, confirm the efficiency of the developed wavelet filters. The real-time implementation using DSP/BIOS tool, on the TMS320C6416 fixed-processor shows that the encoding and decoding algorithms satisfy the real-time processing.

The optimal Mexican hat wavelet filter de-noising method based on cross-validation method

A new de-noising method based on parameter optimized Mexican hat wavelet was put forward in this paper. For the similar shape to the mechanical shock vibration signal, the Mexican hat wavelet is chosen as the mother wavelet and improved by the shape parameters optimization. The noise jamming in the raw vibration signals can be filtered by the continue wavelet transform (CWT) using the improved Mexican hat wavelet as the mother wavelet. The shape parameters of the Mexican hat wavelet are optimized by the cross-validation method (CVM). In the CWT process, the optimal scale factor is also obtained by the circle CVM. The useful components can be extracted by the CWT with the optimal shape parameters and scale factor. The experimental result shows that the proposed method can not only de-noise the useless noise effectively but also extract the fault feature availably.