Wavelet System Research Articles

Evolution of single-lead ECG for STEMI detection using a deep learning approach

BackgroundWhile ST-Elevation Myocardial Infarction (STEMI) door-to-balloon times are often below 90 min, symptom to door times remain long at 2.5-h, due at least in part to a delay in diagnosis. ObjectivesTo develop and validate a machine learning-guided algorithm which uses a single‑lead electrocardiogram (ECG) for STEMI detection to speed diagnosis. MethodsData was extracted from the Latin America Telemedicine Infarct Network (LATIN), a population-based Acute Myocardial Infarction (AMI) program that provides care to patients in Brazil, Colombia, Mexico, and Argentina through telemedicine. Sample: the first dataset was comprised of 8511 ECGs that were used for various machine learning experiments to test our Deep Learning approach for STEMI diagnosis. The second dataset of 2542 confirmed STEMI diagnosis EKG records, including specific ischemic heart wall information (anterior, inferior, and lateral), was derived from the previous dataset to test the STEMI localization model. Preprocessing: Detection of QRS complexes by wavelet system, segmentation of each EKG record into individual heartbeats with fixed window of 0.4 s to the left and 0.9 s to the right of main. Training & Testing: 90% and 10% of the total dataset, respectively, were used for both models. Classification: two 1-D convolutional neural networks were implemented, two classes were considered for first models (STEMI/Not-STEMI) and three classes for the second model (Anterior/Inferior/Lateral) each corresponding to the heart wall affected. These individual probabilities were aggregated to generate the final label for each model. ResultsThe single‑lead ECG strategy was able to provide an accuracy of 90.5% for STEMI detection with Lead V2, which also yielded the best results overall among individual leads. STEMI Localization model provided promising results for anterior and inferior wall STEMIs but remained suboptimal for Lateral STEMI. ConclusionsAn Artificial Intelligence-enhanced single‑lead ECG is a promising screening tool. This technology provides an autonomous and accurate STEMI diagnostic alternative that can be incorporated into wearable devices, potentially providing patients reliable means to seek treatment early and offers the potential to thereby improve STEMI outcomes in the long run.

Jackson-type theorem on approximation by non-stationary periodic wavelets

In this paper, sufficient conditions on non-stationary periodic wavelet systems to provide good approximation properties of wavelet expansions are established. The approximation error is estimated in terms of modulus of continuity in [Formula: see text]-spaces.

Wavelet decomposition and embeddings of generalised Besov–Morrey spaces

We study embeddings between generalised Besov–Morrey spaces Nφ,p,qs(Rd). Both sufficient and necessary conditions for the embeddings are proved. Embeddings of the Besov–Morrey spaces into the Lebesgue spaces Lr(Rd) are also considered. Our approach requires a wavelet characterisation of the spaces which we establish for the system of Daubechies wavelets.

Sparsity enhancing wavelets design for ECG and fetal ECG compression

This paper details the design of a set of wavelet filters that are bi orthogonal in nature. The designed wavelets are experimented in publicly available MIT data base. In Compressed Sensing (CS) paradigm reconstruction quality can be improved by improving the sparsity. From experiments it was found that ECG records especially fetal ECGs have a better sparse representation in the newly designed domain as compared to commonly used wavelets. This paper also details the fundamentals of CS in a much easier way and briefly reviews some state of art methods in CS based on ECG reconstruction scenarios. The best state of art method in terms of reconstruction quality is selected for testing the designed wavelets. The designed wavelet is named as ‘dew2’.The design details of this bi orthogonal wavelet system are detailed in different sections with the help of standard wavelet system design rules.

Alpert wavelet system for solving fractional nonlinear Fredholm integro-differential equations

In this paper, we first construct Alpert wavelet system and propose a computational method for solving a fractional nonlinear Fredholm integro-differential equation. Then we create an operational matrix of fractional integration and use it to simplify the equation to a system of algebraic equations. By using Newtons iterative method, this system is solved and the solution of the fractional nonlinear Fredholm integro-differential equations is achieved. Thresholding parameter is used to increase the sparsity of matrix coefficients and the speed of computations. Finally, the method is demonstrated by examples, and then compared results with CAS wavelet method show that our proposed method is more effective and accurate.

Hyperbolic Wavelet Analysis of Classical Isotropic and Anisotropic Besov\u2013Sobolev Spaces

In this paper we introduce new function spaces which we call anisotropic hyperbolic Besov and Triebel-Lizorkin spaces. Their definition is based on a hyperbolic Littlewood-Paley analysis involving an anisotropy vector only occurring in the smoothness weights. Such spaces provide a general and natural setting in order to understand what kind of anisotropic smoothness can be described using hyperbolic wavelets (in the literature also sometimes called tensor-product wavelets), a wavelet class which hitherto has been mainly used to characterize spaces of dominating mixed smoothness. A centerpiece of our present work are characterizations of these new spaces based on the hyperbolic wavelet transform. Hereby we treat both, the standard approach using wavelet systems equipped with sufficient smoothness, decay, and vanishing moments, but also the very simple and basic hyperbolic Haar system. The second major question we pursue is the relationship between the novel hyperbolic spaces and the classical anisotropic Besov–Lizorkin-Triebel scales. As our results show, in general, both approaches to resolve an anisotropy do not coincide. However, in the Sobolev range this is the case, providing a link to apply the newly obtained hyperbolic wavelet characterizations to the classical setting. In particular, this allows for detecting classical anisotropies via the coefficients of a universal hyperbolic wavelet basis, without the need of adaption of the basis or a-priori knowledge on the anisotropy.

Multivariate Haar systems in Besov function spaces

We determine all cases for which the -dimensional Haar wavelet system on the unit cube is a conditional or unconditional Schauder basis in the classical isotropic Besov function spaces , , , defined in terms of first-order -moduli of smoothness. We obtain similar results for the tensor-product Haar system , and characterize the parameter range for which the dual of is trivial for . Bibliography: 31 titles.

An Algorithm with the Even-odd Splitting of the Wavelet Transform of Non-Hermitian Splines of the Seventh Degree, II

In this study, the zeroing property of the first two moments is used to construct an algorithm for splitting spline wavelets of the seventh degree. The presentation is based on the system of basic spline wavelets of the seventh degree, constructed in the previous article, which implements the conditions of orthogonality to all polynomials of any degree. Then, using homogeneous Dirichlet boundary conditions, the system is adapted to orthogonality to all polynomials up to the first degree on a finite interval. Implicit finite relationships are obtained between the spline coefficients in the original scale, on the one hand, and the spline coefficients and wavelet coefficients in the nested scale, on the other hand. After eliminating the even rows of the system, the transformation matrix has seven diagonals instead of five, as in the previous case studied. The resulting system has been modified to ensure strict diagonal dominance and, hence, computational stability, in contrast to the fivediagonal case.

Spline Wavelet Decomposition in Weighted Function Spaces

Battle-Lemarie wavelet systems of natural orders are established in the paper. The main result of the work is decomposition theorem in Besov and Triebel-Lizorkin spaces with local Muckenhoupt weights, which is performed in terms of bases generated by the systems of such a type. Battle-Lemarie wavelets are splines and suit very well for the study of integration operators.

A note on discrete wave packet frames in ℂN

We show that for every nonzero vector [Formula: see text] in [Formula: see text], the discrete wave packet system [Formula: see text] constitutes a frame for the unitary space [Formula: see text]. An application of this result is given, where frame conditions cannot be derived from discrete wavelet systems in [Formula: see text]. The canonical dual of discrete wave packet frame is also discussed.

The dynamics of COVID-19 in the UAE based on fractional derivative modeling using Riesz wavelets simulation

The well-known novel virus (COVID-19) is a new strain of coronavirus family, declared by the World Health Organization (WHO) as a dangerous epidemic. More than 3.5 million positive cases and 250 thousand deaths (up to May 5, 2020) caused by COVID-19 and has affected more than 280 countries over the world. Therefore studying the prediction of this virus spreading in further attracts a major public attention. In the Arab Emirates (UAE), up to the same date, there are 14,730 positive cases and 137 deaths according to national authorities. In this work, we study a dynamical model based on the fractional derivatives of nonlinear equations that describe the outbreak of COVID-19 according to the available infection data announced and approved by the national committee in the press. We simulate the available total cases reported based on Riesz wavelets generated by some refinable functions, namely the smoothed pseudosplines of types I and II with high vanishing moments. Based on these data, we also consider the formulation of the pandemic model using the Caputo fractional derivative. Then we numerically solve the nonlinear system that describes the dynamics of COVID-19 with given resources based on the collocation Riesz wavelet system constructed. We present graphical illustrations of the numerical solutions with parameters of the model handled under different situations. We anticipate that these results will contribute to the ongoing research to reduce the spreading of the virus and infection cases.

Application of wavelets and conformal reflections to finding optimal scheme of fiber placement at 3d printing constructions from composition materials

The article is devoted to finding the optimal schemes of fiber placement at the production of constructions, reinforced with continuous fibers by 3D printing method. As the optimization of the objective function one of the criteria for the destruction of the composite was chosen. For the process acceleration of multiple solution of the system of partial differential equations describing the stress-strain state of the structure, a computational algorithm based on wavelets built through subdivision schemes is proposed. To set the local coordinate system, it is proposed to use analytical functions, which will be constructed using the well-known Dini and Cisotti formulas, just by specifying the direction of laying the fiber at the product boundary. The article also presents a lifting scheme (lifting scheme) allowing to construct biorthogonal wavelet systems with specified properties using some initial biorthogonal wavelet systems with filters.

Sufficient Conditions for a Multidimensional System of Periodic Wavelets to be a Frame

We study multidimensional periodic wavelet systems with matrix dilations. We obtain conditions sufficient for such a system to be Bessel. The conditions are given in terms of Fourier coefficients. We propose a method for constructing a wavelet Riesz basis that starts with a suitable sequence of trigonometric polynomials.

A wavelet-based tool to modulate variance in predictors: An application to predicting drought anomalies

This work presents an open-source tool to predict natural system responses by transforming the frequency spectrum of predictor variables to create a response that better resembles observations. The R package, namely WAvelet System Prediction (WASP), is based on a discrete wavelet transform (DWT)-based variance transformation method. We further introduce the maximal overlap DWT (MODWT)-based variance transformation which allows the method to be used in forecasting applications. We also develop the method to include an unbiased estimator that mitigates the well-known issue of edge effects in wavelet transforms. The predictive model in the method is a k-nearest neighbor (knn) approach. The main functionalities of the software include: (1) transforming the system predictors, (2) identifying significant predictors corresponding to the response, (3) predicting target response using the knn. Results of predicting sustained drought anomalies across Australia show clear improvements in predictive skill compared to the use of untransformed predictors.

New Hybrid Wavelet and CNN-Based Indirect Tire-Pressure Monitoring System for Autonomous Vehicles

Since the tire pressure has a significant influence on driving safety, even self-driving vehicles need to be aware of their current tire pressures. Two major types of methods for estimating tire pressures exist: direct and indirect methods. In spite of recent advancements in direct Tire Pressure Monitoring Systems (TPMSs), indirect pressure monitoring systems still play a significant role due to their low costs. Indirect systems rely on the processing of signals from wheel speed sensors. In most cases, a transformation is applied to generate a frequency spectrum from which the tire pressure-dependent eigenfrequency can be extracted. The most accurate methods apply the Fourier transform, but these require the highest computational power. After the spectrum of signals from the wheel speed sensor is created, the eigenfrequency must be extracted. Several methods are available to extract significant frequency components. One of the easiest methods is peak searching, however, it is susceptible to noise. On the other hand, more accurate methods that are less sensitive to noise require more computational power. If a transform that consumes less computational power can be applied, then the freed resources can be used by a better eigenfrequency identification method. In this paper, a Hybrid Wavelet-Fourier Transform and Convolutional Neural Network-based method is presented, which exhibits a promising level of noise tolerance.

On Changes of Variable Preserving the Convergence and Absolute Convergence of Fourier Series in the Haar Wavelet System

It is established that, among all continuously differentiable homeomorphic changes of variable, the absolute convergence of Fourier series in the Haar wavelet system is preserved by only those for which $$\varphi^{-1}(0)$$ is binary-rational and $$\varphi'(x)=\pm 2^m$$ , where $$m$$ is an integer and $$x\in\mathbb R$$ . It is also established that this condition is necessary for a continuously differentiable homeomorphic change of variable to preserve the convergence of Fourier series in the Haar wavelet system.

Continuous Empirical Wavelets Systems

The recently proposed empirical wavelet transform was based on a particular type of filter. In this paper, we aim to propose a general framework for the construction of empirical wavelet systems in the continuous case. We define a well-suited formalism and then investigate some general properties of empirical wavelet systems. In particular, we provide some sufficient conditions to the existence of a reconstruction formula. In the second part of the paper, we propose the construction of empirical wavelet systems based on some classic mother wavelets.

Multi-window dilation-and-modulation frames on the half real line

Wavelet and Gabor systems are based on translation-and-dilation and translation-and-modulation operators, respectively. They have been extensively studied. However, dilation-and-modulation systems have not, and they cannot be derived from wavelet or Gabor systems. In this paper, we investigate a class of dilation-and-modulation systems in the causal signal space $L^{2}(\Bbb R_{+})$. $L^{2}(\Bbb R_{+})$ can be identified a subspace of $L^{2}(\Bbb R)$ consisting of all $L^{2}(\Bbb R)$-functions supported on $\Bbb R_{+}$, and is unclosed under the Fourier transform. So the Fourier transform method does not work in $L^{2}(\Bbb R_{+})$. In this paper, we introduce the notion of $\Theta_{a}$-transform in $L^{2}(\Bbb R_{+})$, using $\Theta_{a}$-transform we characterize dilation-and-modulation frames and dual frames in $L^{2}(\Bbb R_{+})$; and present an explicit expression of all duals with the same structure for a general dilation-and-modulation frame for $L^{2}(\Bbb R_{+})$. Interestingly, we prove that an arbitrary frame of this form is always nonredundant whenever the number of the generators is $1$, and is always redundant whenever it is greater than $1$. Some examples are also provided to illustrate the generality of our results.

Dilations of dual g-frame generators for an abstract wavelet system

In this paper, we study the dilation of g-frame generators and the dual g-frame generators for an abstract wavelet system. By the semi-orthogonality of a special unitary system, we prove the dual g-frame generators can be dilated to a pair of dual g-Riesz basis generators for some larger Hilbert space. We first show the existence of the dual g-frame generators with the same structure of a g-frame generator for a unitary system. We then get a sufficient and necessary condition for the existing of the dual g-frame generators for unitary groups on a subspace.

Unconditional bases of wavelets in local fields

We provide sufficient conditions on the wavelets on a local field K for the wavelet system to form an unconditional basis for the Hardy space H1(K) and the Lebesgue spaces Lp(K), 1 < p < ∞.