Orthogonal Wavelet Filter Banks Research Articles

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.

Analysis of knee-joint vibroarthographic signals using bandwidth-duration localized three-channel filter bank

A vibrarthographic (VAG) signal is the recorded vibration signal produced by human knee-joints during movements of the leg. These signals are useful in quantifying lubrication and roughness of articular cartilage layers of the knee-joint. The early detection and treatment of an abnormal knee-joint will help in providing a better quality of life for the suffering patients. The VAG signals are nonlinear and non-stationary. Hence, in the proposed study, we have employed an optimal bandwidth-duration localized three-band (OBDLTB) orthogonal wavelet filter banks (OWFB) to automatically identify the health of knee-joints using VAG signals. We have used the database created at the University of Calgary containing a total of 89 VAG signals (51 normal and 38 pathological). The log-energy (LOEN) features are obtained from various sub-bands using OBDLTB-OWFB. The extracted LOEN features are separated using supervised machine learning algorithms into normal and abnormal VAG signals. Our automated system obtained an average classification accuracy of 89.89% and F1-score of 87.67% with 10-fold cross-validation. The performance of the model indicates that the proposed technique is able to identify the abnormal knee-joint reliably and thus this technique can aid the orthopedic surgeons in the early diagnosis of knee-joint abnormalities.

An automated diagnosis of depression using three-channel bandwidth-duration localized wavelet filter bank with EEG signals

Depression is a mental illness. If not diagnosed and treated quickly, it can affect one’s mood and quality of life. Modern life is stressful and fast paced, owing to which depression has emerged as a major source of mental health disorder. The electroencephalogram (EEG) signals, which are used to diagnose depression, are non-stationary, non-linear and complex. Their visual interpretation is difficult and takes time. This makes computer-aided depression diagnosis systems highly desirable for the early detection of the depression. This study aims towards the development of depression detection system using EEG based measures. We propose a computer aided depression diagnosis system using newly designed bandwidth-duration localized (BDL) three-channel orthogonal wavelet filter bank (TCOWFB) and EEG signal for the detection of depression. The EEG signal is decomposed into seven wavelet sub-bands (WSBs) using a optimal six-length TCOWFB. The logarithm of L2 norm (LL2N) of six detailed WSBs and one approximate WSB are used as discriminating features.These features are used in the classification of normal or depression EEG signals by applying them to the least square support vector machine (LS-SVM). The proposed system attained the perfect value of 1 for area under the curve (AUC) of receiver’s operating characteristics (ROC) using seven features. The proposed system with ten-fold cross validation (CV) strategy attained an average classification accuracy (ACA) of 99.58%. The proposed model obtained better ACA than the existing automated depression diagnosis systems (ADDS) and perfect AUC-ROC. Hence, it can be used in a clinical setup to diagnose the depression disorder accurately in lesser time, without any subjectivity due to human intervention.

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.

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.

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.

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.

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.

Orthogonal Wavelet Filters with Minimum RMS Bandwidth

For a time-limited sequence, the Root-Mean-Square (RMS) bandwidth is the normalized second moment of the spectrum. The RMS bandwidth is a useful and analytically tractable measure of the frequency localization of a discrete-time filter. In this work the design of orthogonal wavelet filter banks with a prescribed number of Vanishing Moment (VM) and having a minimum RMS bandwidth is considered. It is shown that the design problem can be cast as a convex optimization problem for which efficient algorithms and software for its solution exist.

Tensor Product Complex Tight Framelets with Increasing Directionality

Tensor product real-valued wavelets have been employed in many applications such as image processing with impressive performance. Edge singularities are ubiquitous and play a fundamental role in image processing and many other two-dimensional problems. Tensor product real-valued wavelets are known to be only suboptimal since they can only capture edges well along the coordinate axis directions (that is, the horizontal and vertical directions in dimension two). Among several approaches in the literature to enhance the performance of tensor product real-valued wavelets, the dual tree complex wavelet transform ($DT-\mathbb{C}WT$), proposed by Kingsbury [Phil. Trans. R. Soc. Lond. A, 357 (1999), pp. 2543--2560] and further developed by Selesnick, Baraniuk, and Kingsbury [IEEE Signal Process. Mag., 22 (2005), pp. 123--151], is one of the most popular and successful enhancements of the classical tensor product real-valued wavelets by employing a correlated pair of orthogonal wavelet filter banks. The two-dimensional $DT-\mathbb{C}WT$ is obtained essentially via tensor product and offers improved directionality with six directions. In this paper we shall further enhance the performance of the $DT-\mathbb{C}WT$ for the problem of image denoising. Using a framelet-based approach and the notion of discrete affine systems, we shall propose a family of tensor product complex tight framelets $TP-\mathbb{C}TF_n$ for all integers $n\ge 3$ with increasing directionality, where $n$ refers to the number of filters in the underlying one-dimensional complex tight framelet filter bank. For dimension two, such a tensor product complex tight framelet $TP-\mathbb{C}TF_n$ offers $\frac{1}{2}(n-1)(n-3)+4$ directions when $n$ is odd and $\frac{1}{2}(n-4)(n+2)+6$ directions when $n$ is even. In particular, we shall show that $TP-\mathbb{C}TF_4$, which is different from the $DT-\mathbb{C}WT$ in both nature and design, provides an alternative to the $DT-\mathbb{C}WT$. Indeed, we shall see that $TP-\mathbb{C}TF_4$ behaves quite similar to the $DT-\mathbb{C}WT$ by offering six directions in dimension two, employing the tensor product structure, and enjoying slightly less redundancy than the $DT-\mathbb{C}WT$. Then we shall apply $TP-\mathbb{C}TF_n$ to the problem of image denoising. We shall see that the performance of $TP-\mathbb{C}TF_4$ for image denoising is comparable to that of the $DT-\mathbb{C}WT$. Better results on image denoising can be obtained by using other $TP-\mathbb{C}TF_n$, for example, n=6, which has 14 directions in dimension two. Moreover, $TP-\mathbb{C}TF_n$ allows us to further improve the $DT-\mathbb{C}WT$ by using $TP-\mathbb{C}TF_n$ as the first stage filter bank in the $DT-\mathbb{C}WT$. We shall also provide discussion and comparison of $TP-\mathbb{C}TF_n$ with several generalizations of the $DT-\mathbb{C}WT$, shearlets, and directional nonseparable tight framelets.

Algorithms for matrix extension and orthogonal wavelet filter banks over algebraic number fields

Algorithms for matrix extension and orthogonal wavelet filter banks over algebraic number fields

Design of IIR orthogonal wavelet filter banks using lifting scheme

The lifting scheme is well known to be an efficient tool for constructing second generation wavelets and is often used to design a class of biorthogonal wavelet filter banks. For its efficiency, the lifting implementation has been adopted in the international standard JPEG2000. It is known that the orthogonality of wavelets is an important property for many applications. This paper presents how to implement a class of infinite-impulse-response (IIR) orthogonal wavelet filter banks by using the lifting scheme with two lifting steps. It is shown that a class of IIR orthogonal wavelet filter banks can be realized by using allpass filters in the lifting steps. Then, the design of the proposed IIR orthogonal wavelet filter banks is discussed. The designed IIR orthogonal wavelet filter banks have approximately linear phase responses. Finally, the proposed IIR orthogonal wavelet filter banks are applied to the image compression, and then the coding performance of the proposed IIR filter banks is evaluated and compared with the conventional wavelet transforms

Fast fractal coding of subbands using a non-iterative block clustering

We propose a new image coding scheme which uses a fast non-iterative algorithm for block clustering in fractal coding of wavelet transform coefficients. The original image is first decomposed into subbands containing information in different spatial directions and different scales, using an orthogonal wavelet filter bank. Subbands are encoded using local iterated function systems (LIFS) with range and domain blocks presenting horizontal or vertical directionalities. Their sizes are adapted to the correlation lengths and the resolution of each subband. This hybrid compression scheme allows to use the natural classification obtained by using a subband decomposition and reduce the computational complexity by a 12:1 factor. In order to increase this speedup, we propose a fast non-iterative algorithm for block clustering which classifies the directional domain and range blocks and leads to an accelerated LIFS generation in each subband. The proposed approach reduces the computational complexity of the basic fractal compression algorithm by a 50:1 factor and gives better PSNR/bit rate results, especially for images presenting directionalities.

New technique for designing nearly orthogonal wavelet filter banks with linear phase

A new technique based on nonlinear optimisation to design nearly orthogonal wavelet filter banks with linear phase is proposed. The main idea is to impose a certain number of zeros at z=−1 for a symmetric filter and make it satisfy the power complementary condition as accurately as possible. From this filter, a semi-orthogonal wavelet filter bank which is nearly orthogonal can be constructed. This semi-orthogonal filter bank can be approximately implemented using a filter bank consisting of only one prototype filter. The frequency selectivity can also be designed at the same time by using a weighted cost function.

Fast M-band orthogonal wavelet transform algorithm when base length equals 2M

The fast implementation algorithm of M-band orthogonal wavelet filter bank (OWFB) with length L equal to 2M is proposed by using the lossless matrix factorisation method. The computational complexity of the proposed algorithm becomes lower than that of the direct filtering method (DFM):(M/sup 2/+M+1) multiplications and M/sup 2/+2M-1) additions against 2M/sup 2/ and (2M/sup 2/-M), respectively.

Complex, linear-phase filters for efficient image coding

With the exception of the Haar basis, real-valued orthogonal wavelet filter banks with compact support lack symmetry and therefore do not possess linear phase. This has led to the use of biorthogonal filters for coding of images and other multidimensional data. There are, however, complex solutions permitting the construction of compactly supported, orthogonal linear phase QMF filter banks. By explicitly seeking solutions in which the imaginary part of the filter coefficients is small enough to be approximated to zero, real symmetric filters can be obtained that achieve excellent compression performance.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

New regular M-band orthogonal wavelet filter bank design using zero-insertion method

A new regular M-band orthogonal wavelet filter bank (OWFB) is designed by inserting zeros into the filter transfer functions, which improves the characteristics of the frequency magnitude response. In an application of the sub-band coding of a still image, the new OWFB improves the PSNR by approximately 1 dB in the reconstructed image over the existing OWFB at the same bit rate. >