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Continuous Wavelet Transform
In mathematics, the continuous wavelet transform (CWT) is a formal (i.e., non-numerical) tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the wavelets vary continuously. Definition The continuous wavelet transform of a function x(t) at a scale a\in\mathbb and translational value b\in\mathbb is expressed by the following integral :X_w(a,b)=\frac \int_^\infty x(t)\overline\psi\left(\frac\right)\,\mathrmt where \psi(t) is a continuous function in both the time domain and the frequency domain called the mother wavelet and the overline represents operation of complex conjugate. The main purpose of the mother wavelet is to provide a source function to generate the daughter wavelets which are simply the translated and scaled versions of the mother wavelet. To recover the original signal x(t), the first inverse continuous wavelet transform can be exploited. :x(t)=C_\psi^\int_^\int_^ X_w(a,b)\frac\tilde\psi\left(\frac\righ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific properties that make them useful for signal processing. For example, a wavelet could be created to have a frequency of middle C and a short duration of roughly one tenth of a second. If this wavelet were to be convolved with a signal created from the recording of a melody, then the resulting signal would be useful for determining when the middle C note appeared in the song. Mathematically, a wavelet correlates with a signal if a portion of the signal is similar. Correlation is at the core of many practical wavelet applications. As a mathematical tool, wavelets can be used to extract information from many kinds of data, including audio signals and images. Sets of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Wavelet Transform
In mathematics, the continuous wavelet transform (CWT) is a formal (i.e., non-numerical) tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the wavelets vary continuously. Definition The continuous wavelet transform of a function x(t) at a scale a\in\mathbb and translational value b\in\mathbb is expressed by the following integral :X_w(a,b)=\frac \int_^\infty x(t)\overline\psi\left(\frac\right)\,\mathrmt where \psi(t) is a continuous function in both the time domain and the frequency domain called the mother wavelet and the overline represents operation of complex conjugate. The main purpose of the mother wavelet is to provide a source function to generate the daughter wavelets which are simply the translated and scaled versions of the mother wavelet. To recover the original signal x(t), the first inverse continuous wavelet transform can be exploited. :x(t)=C_\psi^\int_^\int_^ X_w(a,b)\frac\tilde\psi\left(\frac\righ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Wavelet
In numerical analysis, continuous wavelets are functions used by the continuous wavelet transform. These functions are defined as analytical expressions, as functions either of time or of frequency. Most of the continuous wavelets are used for both wavelet decomposition and composition transforms. That is they are the continuous counterpart of orthogonal wavelets. The following continuous wavelets have been invented for various applications: * Poisson wavelet * Morlet wavelet * Modified Morlet wavelet * Mexican hat wavelet * Complex Mexican hat wavelet * Shannon wavelet * Meyer wavelet * Difference of Gaussians * Hermitian wavelet * Beta wavelet * Causal wavelet * μ wavelets * Cauchy wavelet * Addison wavelet See also *Wavelet References {{reflist Continuous wavelets, Numerical analysis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theory Of Continuous Functions
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, and research. Theories can be scientific, falling within the realm of empirical and testable knowledge, or they may belong to non-scientific disciplines, such as philosophy, art, or sociology. In some cases, theories may exist independently of any formal discipline. In modern science, the term "theory" refers to scientific theories, a well-confirmed type of explanation of nature, made in a way consistent with the scientific method, and fulfilling the criteria required by modern science. Such theories are described in such a way that scientific tests should be able to provide empirical support for it, or empirical contradiction (" falsify") of it. Scientific theories are the most reliable, rigorous, and comprehensive form of scientif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stéphane Mallat
Stéphane Georges Mallat (born 24 October 1962) is a French applied mathematician, concurrently appointed as Professor at Collège de France and École normale supérieure. He made fundamental contributions to the development of wavelet theory in the late 1980s and early 1990s. He has additionally done work in applied mathematics, signal processing, music synthesis and image segmentation. With Yves Meyer, he developed the multiresolution analysis (MRA) construction for compactly supported wavelets. His MRA wavelet construction made the implementation of wavelets practical for engineering applications by demonstrating the equivalence of wavelet bases and conjugate mirror filters used in discrete, multirate filter banks in signal processing. He also developed (with Sifen Zhong) the wavelet transform modulus maxima method for image characterization, a method that uses the local maxima of the wavelet coefficients at various scales to reconstruct images. He introduced the scatterin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Wavelet
In mathematics, Cauchy wavelets are a family of continuous wavelets, used in the continuous wavelet transform. Definition The Cauchy wavelet of order p is defined as: \psi_p(t) = \frac\left ( \frac \right ) ^ where p > 0 and j = \sqrt therefore, its Fourier transform is defined as \hat(\xi) = \xi^e^I_. Sometimes it is defined as a function with its Fourier transform \hat(\xi) = \rho(\xi)\xi^e^I_ where \rho(\xi) \in L^(\mathbb) and \rho(\xi) = \rho(a\xi) for \xi \in \mathbb almost everywhere and \rho(\xi) \neq 0 for all \xi \in \mathbb. Also, it had used to be defined as \psi_p(t) = (\frac)^ in previous research of Cauchy wavelet. If we defined Cauchy wavelet in this way, we can observe that the Fourier transform of the Cauchy wavelet \int_^ \hat(\xi) \,d\xi = \int_^ \frac \xi^e^ \,d\xi = 2\pi Moreover, we can see that the maximum of the Fourier transform of the Cauchy wavelet of order p is happened at \xi = p and the Fourier transform of the Cauchy wavelet is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
S Transform
''S'' transform as a time–frequency distribution was developed in 1994 for analyzing geophysics data.Stockwell, RG (1999). ''S''-transform analysis of gravity wave activity from a small scale network of airglow imagers. PhD thesis, University of Western Ontario, London, Ontario, Canada. In this way, the ''S'' transform is a generalization of the short-time Fourier transform (STFT), extending the continuous wavelet transform and overcoming some of its disadvantages. For one, modulation sinusoids are fixed with respect to the time axis; this localizes the scalable Gaussian window dilations and translations in ''S'' transform. Moreover, the ''S'' transform doesn't have a cross-term problem and yields a better signal clarity than Gabor transform. However, the ''S'' transform has its own disadvantages: the clarity is worse than Wigner distribution function and Cohen's class distribution function. A fast ''S'' transform algorithm was invented in 2010. It reduces the computational compl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electroencephalography
Electroencephalography (EEG) is a method to record an electrogram of the spontaneous electrical activity of the brain. The biosignal, bio signals detected by EEG have been shown to represent the postsynaptic potentials of pyramidal neurons in the neocortex and allocortex. It is typically non-invasive, with the EEG electrodes placed along the scalp (commonly called "scalp EEG") using the 10–20 system (EEG), International 10–20 system, or variations of it. Electrocorticography, involving surgical placement of electrodes, is sometimes called Electrocorticography, "intracranial EEG". Clinical interpretation of EEG recordings is most often performed by visual inspection of the tracing or quantitative EEG, quantitative EEG analysis. Voltage fluctuations measured by the EEG bioamplifier, bio amplifier and electrodes allow the evaluation of normal Brain activity and meditation, brain activity. As the electrical activity monitored by EEG originates in neurons in the underlying Huma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Epilepsy
Epilepsy is a group of Non-communicable disease, non-communicable Neurological disorder, neurological disorders characterized by a tendency for recurrent, unprovoked Seizure, seizures. A seizure is a sudden burst of abnormal electrical activity in the brain that can cause a variety of symptoms, ranging from brief lapses of awareness or muscle jerks to prolonged convulsions. These episodes can result in physical injuries, either directly, such as broken bones, or through causing accidents. The diagnosis of epilepsy typically requires at least two unprovoked seizures occurring more than 24 hours apart. In some cases, however, it may be diagnosed after a single unprovoked seizure if clinical evidence suggests a high risk of recurrence. Isolated seizures that occur without recurrence risk or are provoked by identifiable causes are not considered indicative of epilepsy. The underlying cause is often unknown, but epilepsy can result from brain injury, stroke, infections, Brain tumor, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
