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Wavelet Analysis
Published in Nong Ye, Data Mining, 2013
Hence, the Haar wavelet transform of times series data allows us to transform time series data to the data in the time–frequency domain and observe the characteristics of the wavelet data pattern (e.g., a step change for the Haar wavelet) in the time–frequency domain. For example, the wavelet transform of the time series data 0, 2, 0, 2, 6, 8, 6, 8 in Equation 20.12 reveals that the data have the average of 4, a step increase of 6 at four data points (at the lowest frequency of step change), no step change at every two data points (at the medium frequency of step change), and a step increase of 2 at every data point (at the highest frequency of step change). In addition to the Haar wavelet that captures the data pattern of a step change, there are many other wavelet forms, for example, the Paul wavelet, the DoG wavelet, the Daubechies wavelet, and Morlet wavelet as shown in Figure 20.3, which capture other types of data patterns. Many wavelet forms are developed so that an appropriate wavelet form can be selected to give a close match to the data pattern of time series data. For example, the Daubechies wavelet (Daubechies, 1990) may be used to perform the wavelet transform of time series data that shows a data pattern of linear increase or linear decrease. The Paul and DoG wavelets may be used for time series data that show wave-like data patterns.
Image Interpolation for Pattern Recognition
Published in Fathi E. Abd El-Samie, Mohiy M. Hadhoud, Said E. El-Khamy, Image Super-Resolution and Applications, 2012
Fathi E. Abd El-Samie, Mohiy M. Hadhoud, Said E. El-Khamy
The Haar wavelet is the simplest type of wavelet. In the discrete form, Haar wavelets are related to a mathematical operation called the Haar transform that serves as a prototype for all other wavelet transforms [73]. Like all wavelet transforms, the Haar decomposes a discrete signal into two sub-signals of half its length. One is a running average or trend; the other is a running difference or fluctuation. This uses the simplest possible Pt(Z) with a single zero at Z= −1. It is represented as follows [73]: Pt(Z)=1+Z and Z=12(z+z−1)
Artificial neural networks
Published in A. W. Jayawardena, Environmental and Hydrological Systems Modelling, 2013
Wavelets, meaning small waves, which can be thought of as a spinoff from STFT, overcome many such problems. Unlike in Fourier analysis, the wavelet analysis does not require assumptions about stationarity and periodicity of data. The basic approach in Fourier as well as wavelet decomposition is to convolute the signal function by a basis function. In the case of Fourier approach, the basis function is a combination of sines and cosines. In the case of wavelet approach, there exist a number of different basis functions, such as the Haar wavelet (Haar, 1910), the Daubechies wavelet (Daubechies, 1988, 1992), the Mexican Hat wavelet (normalized second derivative of a Gaussian function), and the Morlet wavelet (Goupillaud et al., 1984). Of these, the Haar wavelet has the major advantages of being conceptually simple, computationally fast, and exactly reversible. It has a rectangular shape and has been used in many fields of study. One of the shortcomings of the Haar wavelet is its discontinuity, which makes it impossible to directly apply to solve differential equations. The most popular wavelet in signal processing is the Daubechies wavelet (Daubechies, 1988), which is continuous and symmetric. Other wavelets that have been widely used include the Morlet wavelet (Goupillaud et al., 1984), which is symmetric and has the advantage of minimizing the Heisenberg uncertainty principle, and the Mexican hat wavelet, which is the second derivative of the Gaussian function and is also symmetric. Among the many popular uses of the Haar wavelet is its application in the JPEG format of digital image compression.
Cat Swarm Fractional Calculus optimization-based deep learning for artifact removal from EEG signal
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2020
Jayalaxmi Anem, G. Sateesh Kumar, R. Madhu
After the pre-processing, the EEG signals are subjected to the feature extraction. The EEG signal has small SNR and amplitude. Thus, it is necessary to obtain the information very accurately without losing the EEG source signal. The feature extraction process finds suitable information from the EEG signal by applying the necessary transformation and provides the extracted features for the training. The wavelet features depict the EEG signal for the artefact removal, and hence, this work extracts the wavelet features from the EEG signal. For extracting the wavelet features from the EEG signal, the Haar wavelet transform is applied to the signal. The Haar wavelet is the simplest possible wavelet. The disadvantage of the Haar wavelet is that it is not continuous, and therefore not differentiable. This property is an advantage for the analysis of signals with sudden transitions, such as monitoring of tool failure in machines. The wavelets extracted from the EEG signal are represented as follows: .
Monitoring air pollution by deep features and extreme learning machine
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2019
Seyyed Amirhosein Rahimi, Hedieh Sajedi
The 2D wavelet decomposition is an edge extractor method, which searches the corresponding density levels, extracted by two low- and high-pass filters. The Haar function is implemented in this research because of its orthogonality property, which is able to show the features efficiently with only a few wavelet coefficients (Khatami, 2017). In addition, the Haar wavelet transform has been implemented widely for digital image compression, denoising, edge detection, feature extraction, texture analysis and image segmentation (Yimsiri, 2010). Wavelet transform uses many diverse filters. A sky image is decomposed by these filters into several frequencies. These frequencies are LL, HL, LH and HH bands that represent the approximation image and the horizontal, vertical and diagonal edge components, respectively. The mechanism of discrete wavelet transform (DWT) is executed by passing the input image through filters with different scales and frequencies (Einshoka, 2018). In two-band wavelet transform, signal f(x) can be represented by the wavelet and scaling basis functions at distinctive scales in a hierarchical manner as shown in Equation(3).
Fractional gravitational search-radial basis neural network for bone marrow white blood cell classification
Published in The Imaging Science Journal, 2018
Namdev Devidas Pergad, Satish T. Hamde
In this section, the feature extraction of the proposed methodology FGS-RBNN is deliberated. The feature sets in this work are the integrated features. Figure 4 illustrates the feature extraction procedure in the proposed methodology. The statistical features from the segmented image, region of interest nucleus of the blood cells, are extracted initially. The statistical features include the area of the nucleus, mean, variance, moment, and entropy. The wavelet features are also extracted from the pre-processed image on the application of the Haar wavelet transform. The statistical features and wavelet features are combined to form the integrated feature sets. The integrated feature sets are also called the combined features.