Quadratic Spline Wavelet Research Articles

Wavelet-based real-time calculation of multiple physiological parameters on an embedded platform

Objective: This paper aims to present how physiological signals can be processed based on wavelet decomposition to calculate multiple physiological parameters in real-time on an embedded platform. Approach: An ECG and PPG are decomposed to the appropriate scale based on a quadratic spline wavelet base in order to obtain high and narrow pulse peaks at the location of the mutation points. Based on the decomposed waveforms, feature points are positioned to calculate physiological parameters in real-time, including heart rate, pulse rate, blood oxygen, and blood pressure. The proposed algorithm has been implemented on a Texas Instruments’ CC2640R2F. Main results: The misdetection rate of feature point location based on the square wavelet decomposition waveform is only 0.57% in the acquired ECG and 0.23% in the acquired PPG. Heart rate and pulse rate are both highly correlated with the reference, both having correlation coefficients of 0.99. The pulse rate and heart rate are 3.85% (51/1326) and 2.94% (39/1326) outside the 95% consistency limit, respectively. The systolic and diastolic blood pressures are significantly associated with standard equipment measurements, with correlation coefficients of 0.87 and 0.83. The systolic and diastolic blood pressures were 5.88% (21/357) and 5.32% (19/357) outside the 95% consistency limit, respectively. Significance: The real-time calculation of multiple physiological parameters based on wavelet decomposition on an embedded platform presented here shows outstanding accuracy.

Quadratic Spline Wavelets for Sparse Discretization of Jump–Diffusion Models

This paper is concerned with a construction of new quadratic spline wavelets on a bounded interval satisfying homogeneous Dirichlet boundary conditions. The inner wavelets are translations and dilations of four generators. Two of them are symmetrical and two anti-symmetrical. The wavelets have three vanishing moments and the basis is well-conditioned. Furthermore, wavelets at levels i and j where i - j > 2 are orthogonal. Thus, matrices arising from discretization by the Galerkin method with this basis have O 1 nonzero entries in each column for various types of differential equations, which is not the case for most other wavelet bases. To illustrate applicability, the constructed bases are used for option pricing under jump–diffusion models, which are represented by partial integro-differential equations. Due to the orthogonality property and decay of entries of matrices corresponding to the integral term, the Crank–Nicolson method with Richardson extrapolation combined with the wavelet–Galerkin method also leads to matrices that can be approximated by matrices with O 1 nonzero entries in each column. Numerical experiments are provided for European options under the Merton model.

Galerkin method with new quadratic spline wavelets for integral and integro-differential equations

The paper is concerned with the wavelet-Galerkin method for the numerical solution of Fredholm linear integral equations and second-order integro-differential equations. We propose a construction of a quadratic spline-wavelet basis on the unit interval, such that the wavelets have three vanishing moments and the shortest support among such wavelets. We prove that this basis is a Riesz basis in the space L 2 0 , 1 . We adapt the basis to homogeneous Dirichlet boundary conditions, and using a tensor product we construct a wavelet basis on the hyperrectangle. We use the wavelet-Galerkin method with the constructed bases for solving integral and integro-differential equations, and we show that the matrices arising from discretization have uniformly bounded condition numbers and that they can be approximated by sparse matrices. We present numerical examples and compare the results with the Galerkin method using other quadratic spline wavelet bases and other methods.

Postprocessing Galerkin method using quadratic spline wavelets and its efficiency

The wavelet-Galerkin method is a useful tool for solving differential equations mainly because the condition number of the stiffness matrix is independent of the matrix size and thus the number of iterations for solving the discrete problem by the conjugate gradient method is small. We have recently proposed a quadratic spline wavelet basis that has a small condition number and a short support. In this paper we use this basis in the Galerkin method for solving the second-order elliptic problems with Dirichlet boundary conditions in one and two dimensions and by an appropriate post-processing we achieve the L2-error of order Oh4 and the H1-error of order Oh3, where h is the step size. The rate of convergence is the same as the rate of convergence for the Galerkin method with cubic spline wavelets. We show theoretically as well as numerically that the presented method outperforms the Galerkin method with other quadratic or cubic spline wavelets. Furthermore, we present local post-processing for example of the equation with Dirac measure on the right-hand side.

Quadratic spline wavelets with short support satisfying homogeneous boundary conditions

In this paper, we construct a new quadratic spline-wavelet basis on the interval and on the unit square satisfying homogeneous Dirichlet boundary conditions of the first order. The wavelets have one vanishing moment and the shortest support among quadratic spline wavelets with at least one vanishing moment adapted to the same type of boundary conditions. The stiffness matrices arising from the discretization of the second-order elliptic problems using the constructed wavelet basis have uniformly bounded condition numbers, and the condition numbers are small. We present some quantitative properties of the constructed basis. We provide numerical examples to show that the Galerkin method and the adaptive wavelet method using our wavelet basis require fewer iterations than methods with other quadratic spline wavelet bases. Moreover, due to the small support of the wavelets, when using these methods with the new wavelet basis, the system matrix is sparser, and thus one iteration requires a smaller number of floating point operations than for other quadratic spline wavelet bases.

A Hybrid Wavelet-Based Method for the Peak Detection of Photoplethysmography Signals.

The noninvasive peripheral oxygen saturation (SpO2) and the pulse rate can be extracted from photoplethysmography (PPG) signals. However, the accuracy of the extraction is directly affected by the quality of the signal obtained and the peak of the signal identified; therefore, a hybrid wavelet-based method is proposed in this study. Firstly, we suppressed the partial motion artifacts and corrected the baseline drift by using a wavelet method based on the principle of wavelet multiresolution. And then, we designed a quadratic spline wavelet modulus maximum algorithm to identify the PPG peaks automatically. To evaluate this hybrid method, a reflective pulse oximeter was used to acquire ten subjects' PPG signals under sitting, raising hand, and gently walking postures, and the peak recognition results on the raw signal and on the corrected signal were compared, respectively. The results showed that the hybrid method not only corrected the morphologies of the signal well but also optimized the peaks identification quality, subsequently elevating the measurement accuracy of SpO2 and the pulse rate. As a result, our hybrid wavelet-based method profoundly optimized the evaluation of respiratory function and heart rate variability analysis.

Optimal designs for spline wavelet regression models

In this article we investigate the optimal design problem for some wavelet regression models. Wavelets are very flexible in modeling complex relations, and optimal designs are appealing as a means of increasing the experimental precision. In contrast to the designs for the Haar wavelet regression model (Herzberg and Traves 1994; Oyet and Wiens 2000), the I-optimal designs we construct are different from the D-optimal designs. We also obtain c-optimal designs. Optimal (D- and I-) quadratic spline wavelet designs are constructed, both analytically and numerically. A case study shows that a significant saving of resources may be realized by employing an optimal design. We also construct model robust designs, to address response misspecification arising from fitting an incomplete set of wavelets.

Medical Monitoring Model of Internet of Things Based on the Adaptive Threshold Difference Algorithm

There are still problems such as low detection accuracy and poor noise immunity in the application of the standard threshold difference algorithm in the signal detection of electrocardiosignal (ECG), in this paper, a medical monitoring model based on the adaptive threshold difference is proposed. First we use a nonlinear filter to filter the P wave and T wave which are low frequency in ECG signal. Then complex wave QRS will be tested. Then the algorithm will be more accuracy through the detection of the R-R interval length and the adjustment of threshold. Finally, the ECG signal will be test with quadratic spline wavelet twice, and the error judgment will be known through adaptive threshold difference. In the simulation experiments, after judging error by wavelet transformation and making the standard threshold difference algorithm optimize adaptively, algorithm showed excellent detection accuracy with and without noise.

Quadratic Spline Wavelets with Short Support for Fourth-Order Problems

In the paper, we propose constructions of new quadratic spline-wavelet bases on the interval and the unit square satisfying homogeneous Dirichlet boundary conditions of the second order. The basis functions have small supports and wavelets have one vanishing moment. We show that stiffness matrices arising from discretization of the biharmonic problem using a constructed wavelet basis have uniformly bounded condition numbers and these condition numbers are very small.

Characterising an ECG signal using statistical modelling: a feasibility study

For clinical use, in electrocardiogram (ECG) signal analysis it is important to detect not only the centre of the P wave, the QRS complex and the T wave, but also the time intervals, such as the ST segment. Much research focused entirely on qrs complex detection, via methods such as wavelet transforms, spline fitting and neural networks. However, drawbacks include the false classification of a severe noise spike as a QRS complex, possibly requiring manual editing, or the omission of information contained in other regions of the ECG signal. While some attempts were made to develop algorithms to detect additional signal characteristics, such as P and T waves, the reported success rates are subject to change from person-to-person and beat-to-beat. To address this variability we propose the use of Markov-chain Monte Carlo statistical modelling to extract the key features of an ECG signal and we report on a feasibility study to investigate the utility of the approach. The modelling approach is examined with reference to a realistic computer generated ECG signal, where details such as wave morphology and noise levels are variable. References S. Edla, N. Kovvali, and A. Papandreou-Suppappola. Sequential Markov chain Monte Carlo filter with simultaneous model selection for electrocardiogram signal modeling. In 34th Annual International Conference of the IEEE EMBS, San Diego, California USA, 2012. doi:10.1109/EMBC.2012.6346915. A. Gelman, J. Carlin, H. Stern, and D. Rubin. Bayesian Data Analysis: Second Edition. Chapman and Hall/CRC, 2003. J. Hampton. The ECG Made Easy. Elsevier/Churchill Livingstone, seventh edition, 2008. C. Lin, C. Mailhes, and J.-Y. Tourneret. P- and T-wave delineation in ECG signals using a Bayesian approach and a partially collapsed Gibbs sampler. IEEE Transactions on Biomedical Engineering, 57(12):2840–2849, 2010. doi:10.1109/TBME.2010.2076809. P. E. McSharry, G. D. Clifford, L. Tarassenko, and L. A. Smith. A dynamical model for generating synthetic electrocardiogram signals. IEEE Transactions on Biomedical Engineering, 50(3):289–294, 2003. doi:10.1109/TBME.2003.808805. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller. Equation of state calculations by fast computing machines. The Journal of Chemical Physics, 21(6):1087–1092, 1953. doi:10.1063/1.1699114. G. B. Moody, R. G. Mark, and A. L. Goldberger. PhysioNet: a web-based resource for the study of physiologic signals. IEEE Engineering in Medicine and Biology, 20(3):70–75, 2001. doi:10.1109/51.932728. M. Niknazar, B. V. Vahdat, and S. R. Mousavi. Detection of characteristic points of ECG using quadratic spline wavelet transform. In Interrnational Conference on Signals, Circuits and Systems. IEEE, 2009. doi:10.1109/ICSCS.2009.5412588. J. Pan and W. J. Tompkins. A real-time QRS detection algorithm. IEEE Transactions on Biomedical Engineering, BME-32(3):230–236, 1985. doi:10.1109/TBME.1985.325532. K. Thygesen, J. S. Alpert, and H. D. White. Universal definition of myocardial infarction. Journal of the American College of Cardiology, 50(22):2173–2195, 2007. doi:10.1016/j.jacc.2007.09.011. U.S. Department of Health and Human Services, Food and Drug Administration, Center for Drug Evaluation and Research (CDER), Center for Biologics Evaluation and Research (CBER). Guidance for Industry E14 Clinical Evaluation of QT/QTc Interval Prolongation and Proarrhythmic Potential for Non-Antiarrhythmic Drugs, October 2005. http://www.fda.gov/downloads/RegulatoryInformation/Guidances/ucm129357.pdf. W. Zong, G. B. Moody, and D. Jiang. A robust open-source algorithm to detect onset and duration of QRS complexes. Computers in Cardiology, 30:737–740, 2003. doi:10.1109/CIC.2003.1291261. W. Zong, M. Saeed, and T. Heldt. A QT interval detection algorithm based on ECG curve length transform. Computers in Cardiology, 33:377–380, 2006. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4511867.

An Arc Fault Detection Method Based on Wavelet Feature Extraction and The Design & Realization by LabWindows/CVI

Arc fault is one of the primary reasons that cause electrical fire. When arc fault occurs in power supply line, the current can not make protective equipments act and the arc fault can not be found and cut off easily, so electrical fire comes into being. In the comparisons, the current signal is used for the detection physical parameter and the wavelet transform is used to study the arc fault detection. First according to the select principle of wavelet singularity detection, select and construct orthogonal quadratic spline wavelet as wavelet function, and use the porous algorithm dyadic wavelet transform to realize wavelet transform fast algorithm. Then carried out the arc fault's wavelet singularity detection through modulus maxima detection method, finally analyzed the current signal by the method of wavelet approximation. LabWindows/CVI is used as development platform to design and implement the above analysis. Based on LabWindows/CVI, the upper computer program uses multithreading technology to detect and analyze the current signal. By arc fault detection algorithm, the system judges whether there has arc fault or not. The algorithm judges whether there has arc fault or not through detect if there has periodicity singularity points or not. The experiments show that this detection method of arc fault can detect arc fault in power supply circuits exactly and efficaciously.

A 0.83-<formula formulatype="inline"><tex Notation="TeX">$\mu {\rm W}$</tex></formula> QRS Detection Processor Using Quadratic Spline Wavelet Transform for Wireless ECG Acquisition in 0.35-<formula formulatype="inline"> <tex Notation="TeX">$\mu{\rm m}$</tex></formula> CMOS

Healthcare electronics count on the effectiveness of the on-patient signal preprocessing unit to moderate the wireless data transfer for better power efficiency. In order to reduce the system power in long-time ECG acquisition, this work describes an on-patient QRS detection processor for arrhythmia monitoring. It extracts the concerned ECG part, i.e., the RR-interval between the QRS complex for evaluating the heart rate variability. The processor is structured by a scale-3 quadratic spline wavelet transform followed by a maxima modulus recognition stage. The former is implemented via a symmetric FIR filter, whereas the latter includes a number of feature extraction steps: zero-crossing detection, peak (zero-derivative) detection, threshold adjustment and two finite state machines for executing the decision rules. Fabricated in 0.35-μm CMOS the 300-Hz processor draws only 0.83 μW, which is favorably comparable with the prior arts. In the system tests, the input data is placed via an on-chip 10-bit SAR analog-to-digital converter, while the output data is emitted via an off-the-shelf wireless transmitter (TI CC2500) that is configurable by the processor for different data transmission modes: 1) QRS detection result, 2) raw ECG data or 3) both. Validated with all recordings from the MIT-BIH arrhythmia database, 99.31% sensitivity and 99.70% predictivity are achieved. Mode 1 with solely the result of QRS detection exhibits 6× reduction of system power over modes 2 and 3.

PCA and ICA processing methods for removal of artifacts and noise in electrocardiograms: A survey and comparison

Electrocardiogram (ECG) signals are affected by various kinds of noise and artifacts that may hide important information of interest. Wavelet transform (WT) technique is used to identify the characteristic points of the electrocardiogram (ECG) signal with fairly good accuracy, even in the presence of severe high frequency and low frequency noise. Independent component analysis (ICA) is a new technique suitable for separating independent components from ECG complex signals, whereas principal component analysis (PCA) is used to reduce dimensionality and for feature extraction of the ECG data prior to or at times after performing ICA in special circumstances. In this analysis, PCA is analyzed from three points of view, variance maximization, the singular value decomposition and ECG data compression. The sensitivity of the different ECG components with respect to the ECG data dimensions has been studied using PCA screen plots. The validity and performance of the approaches used are confirmed through computer simulations on common standards for electrocardiography (CSE) base ECG data. Standard or instantaneous ICA, which is the most commonly, accepted ICA technique is first compared with PCA technique and then with constrained ICA, which enables the estimation of only one component close to a particular reference ECG signal. The ICA method can also be extended for QRS detection and reference signal generation, using constrained ICA and as well for multichannel ECG separation after removing noise and artifacts, further favoring segment classification. The results were obtained using Matlab environment. Using composite WT based PCA–ICA methods helps for redundant data reduction as well for better feature extraction. The efficacy of the combined PCA–ICA algorithm lies on the fact that the location of the R-peaks is accurately determined, and none of the peaks are ignored or missed, as Quadratic Spline wavelet is also used.

Improved Multi-***scale Product Method for Glottal Opening Instants From Speech Signal

In this paper, we improve the glottal opening instant (GOI) detection from speech signal by a new approach based on Multi-***scale Product Method (MPM). We compute wavelet transform coefficients products at three scales using the quadratic spline wavelet. Glottal closure and opening instants appear respectively as strong maximum and weak minimum on the product signal. The GOI localisation is enhanced by refining the time interval containing this instant and based on the open quotient value. The proposed method is applied and evaluated on the Keele University database. Evaluation results show that the MPM detects GOIs with a performance rate equals to 96% for the entire database.

DESIGN AND EVALUATION OF MATCHED WAVELETS WITH MAXIMUM CODING GAIN AND MINIMUM APPROXIMATION ERROR CRITERIA FOR R PEAK DETECTION IN ECG

Recently, several wavelet-based algorithms have been proposed for feature extraction in non-stationary signals such as ECG. These methods, however, have mainly used general purpose (unmatched) wavelet bases such as Daubechies and Quadratic Spline. In this paper, five new matched wavelet bases, with minimum approximation error and maximum coding gain criteria, are designed and applied to ECG signal analysis. To study the effect of using different wavelet bases for this application, two different wavelet-based R peak detection algorithms are implemented: (1) a conventional wavelet-based method; and (2) a modified wavelet-based R peak detection algorithm. Both algorithms are evaluated using the MIT-BIH Arrhythmia database. Experimental results show lower computational complexity (up to 76%) of the proposed R peak detection method compared to the conventional method. They also show considerable decrease in the number of failed detections (up to 55%) for both the conventional and the proposed algorithms when using matched wavelets instead of Quadratic Spline wavelet which, according to the literature, has generated the best detection results among all conventional wavelet bases studied previously for ECG signal analysis.

PARAMETRIZATION AND CORRECTION OF ELECTROCARDIOGRAM SIGNALS USING INDEPENDENT COMPONENT ANALYSIS

Electrocardiogram (ECG) signals are largely employed as a diagnostic tool in clinical practice in order to assess the cardiac status of a specimen. Independent component analysis (ICA) of measured ECG signals yields the independent sources, provided that certain requirements are fulfilled. Properly parametrized ECG signals provide a better view of the extracted ECG signals, while reducing the amount of ECG data. Independent components (ICs) of parametrized ECG signals may also be more readily interpretable than original ECG measurements or even their ICs. The purpose of this analysis is to evaluate the effectiveness of ICA in removing artifacts and noise from ECG signals for a clear interpretation of ECG data in diagnostic applications. In this work, ICA is tested on the Common Standards for Electrocardiography (CSE) database files corrupted by abrupt changes, high frequency noise, power line interference, etc. The joint approximation for diagonalization of eigen matrices (JADE) algorithm for ICA is applied to three-channel ECG, and the sources are separated as ICs. In this analysis, an extension is applied to the algorithm for further correction of the extracted components. The values of R-peak before and after application of ICA are found using quadratic spline wavelet, which facilitates the estimation of the reconstruction errors. The results indicate that, in most of the cases, the percentage reconstruction error is small at around 3%. The paper also highlights the advantages, limitations, and diagnostic feature extraction capability of ICA for clinicians and medical practitioners. Kurtosis is varied in the range of 3.0–7.0, and variance of variance (Varvar) is varied in the range of 0.2–0.5.

Parameterization and R‐Peak Error Estimations of ECG Signals Using Independent Component Analysis

Principal component analysis (PCA) is used to reduce dimensionality of electrocardiogram (ECG) data prior to performing independent component analysis (ICA). A newly developed PCA variance estimator by the author has been applied for detecting true, actual and false peaks of ECG data files. In this paper, it is felt that the ability of ICA is also checked for parameterization of ECG signals, which is necessary at times. Independent components (ICs) of properly parameterized ECG signals are more readily interpretable than the measurements themselves, or their ICs. The original ECG recordings and the samples are corrected by statistical measures to estimate the noise statistics of ECG signals and find the reconstruction errors. The capability of ICA is justified by finding the true, false and actual peaks of around 25–50, CSE (common standards for electrocardiography) database ECG files. In the present work, joint approximation for diagonalization of the eigen matrices (Jade) algorithm is applied to 3‐channel ECG. ICA processing of different cases is dealt with and the R‐peak magnitudes of the ECG waveforms before and after applying ICA are found and marked. ICA results obtained indicate that in most of the cases, the percentage error in reconstruction is very small. The developed PCA variance estimator along with the quadratic spline wavelet gave a sensitivity of 97.47% before applying ICA and 98.07% after ICA processing.

Quadratic spline wavelets with arbitrary simple knots on the sphere

In this paper, we extend the method for fitting functions on the sphere, described in Lyche and Schumaker (SIAM J. Sci. Comput. 22 (2) (2000) 724) to the case of nonuniform knots. We present a multiresolution method leading to C 1 -functions on the sphere, which is based on tensor products of quadratic polynomial splines and trigonometric splines of order three with arbitrary simple knot sequences. We determine the decomposition and reconstruction matrices corresponding to the polynomial and trigonometric spline spaces. We describe the tensor product decomposition and reconstruction algorithms in matrix forms which are convenient for the compression of surfaces. We give the different steps of computer implementation and finally we present a test example by using two knot sequences: a uniform one and a sequence of Chebyshev points.

ECG analysis using wavelet transform: application to myocardial ischemia detection

In this paper, we propose a method for the detection of myocardial ischemic events from electrocardiogram (ECG) signal using the wavelet transform technique. The wavelet transform is obtained using the quadratic spline wavelet. Then, based on the wavelet transform values, the characteristic points of the ECG signal are found out. These characteristic points are used to identify any ischemic episodes present in the ECG signal. This technique can be extended for other types of cardiac abnormality detections, which induce changes in the ECG.

The application of spline wavelet transform to the determination of electroosmotic mobility in capillary electrophoresis

The rule of current change was studied during capillary electrophoresis (CE) separation process while the conductivity of the sample solution was different from that of the buffer. Using a quadratic spline wavelet of compact support, the wavelet transforms (WTs) of capillary electrophoretic currents were performed. The time corresponding to the maximum of WT coefficients was chosen as the time of current inflection to calculate electroosmotic mobility. The proposed method was suitable for different CE modes, including capillary zone electrophoresis, nonaqueous CE and micellar electrokinetic chromatography. Compared with the neutral marker method, the relative errors of the developed method for the determination of electroosmotic mobility were all below 2.5%.