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Continuous Wavelets
{{Unreferenced, date=December 2009 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 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Meyer Wavelet
The Meyer wavelet is an orthogonal wavelet proposed by Yves Meyer. As a type of a continuous wavelet, it has been applied in a number of cases, such as in adaptive filters, fractal random fields, and multi-fault classification. The Meyer wavelet is infinitely differentiable with infinite support and defined in frequency domain in terms of function \nu as : \Psi(\omega) := \begin \frac \sin\left(\frac \nu \left(\frac -1\right)\right) e^ & \text 2 \pi /3<, \omega, < 4 \pi /3, \\ \frac \cos\left(\frac \nu \left(\frac-1\right)\right) e^ & \text 4 \pi /3<, \omega, < 8 \pi /3, \\ 0 & \text, \end where : There are many different ways for defining this auxiliary function, which yields variants of the Meyer wavelet. For instance, another standard implementation adopts : < ... [...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 different kinds of data, including but not limited to au ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Addison Wavelet
Addison may refer to: Places Canada *Addison, Ontario United States *Addison, Alabama *Addison, Illinois *Addison Street in Chicago, Illinois which runs by Wrigley Field *Addison, Kentucky *Addison, Maine *Addison, Michigan *Addison, New York **Addison (village), New York *Addison, Ohio *Addison, Pennsylvania *Addison, Tennessee, an unincorporated community in McMinn County, Tennessee, McMinn County *Addison, Texas *Addison, Vermont *Addison, West Virginia, the official name of the town commonly called Webster Springs, WV *Addison, Wisconsin, a town **Addison (community), Wisconsin, an unincorporated community *Addison County, Vermont *Addison Township (other), several places Other uses *Addison (given name) *Addison (surname) *Addison (restaurant), a Michelin-starred restaurant in San Diego *Addison Road (band), an American band *Addison Motor Company, British car manufacturer *Addison's disease, endocrine disorder *Sports Beanie Babies#Addison, Addison, a Beanie Baby b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Wavelet
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors. He almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had a great influence over his contemporaries and successors; Hans Freudenthal stated: "More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)." Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks on a variety of topics in the fields of mathematics and mathematical physics. Biogra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
μ Wavelet
Mu (uppercase Μ, lowercase μ; Ancient Greek , ell, μι or μυ—both ) is the 12th letter of the Greek alphabet, representing the voiced bilabial nasal . In the system of Greek numerals it has a value of 40. Mu was derived from the Egyptian hieroglyphic symbol for water, which had been simplified by the Phoenicians and named after their word for water, to become 𐤌 (mem). Letters that derive from mu include the Roman M and the Cyrillic М. Names Ancient Greek In Ancient Greek, the name of the letter was written and pronounced Modern Greek In Modern Greek, the letter is spelled and pronounced . In polytonic orthography, it is written with an acute accent: . Use as symbol The lowercase letter mu (μ) is used as a special symbol in many academic fields. Uppercase mu is not used, because it appears identical to Latin M. Measurement *the SI prefix ''micro-'', which represents one millionth, or 10−6. Lowercase letter "u" is often substituted for "μ" w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Causal Wavelet
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cause is partly responsible for the effect, and the effect is partly dependent on the cause. In general, a process has many causes, which are also said to be ''causal factors'' for it, and all lie in its past. An effect can in turn be a cause of, or causal factor for, many other effects, which all lie in its future. Some writers have held that causality is metaphysically prior to notions of time and space. Causality is an abstraction that indicates how the world progresses. As such a basic concept, it is more apt as an explanation of other concepts of progression than as something to be explained by others more basic. The concept is like those of agency and efficacy. For this reason, a leap of intuition may be needed to grasp it. Accordingly, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beta Wavelet
Continuous wavelets of compact support alpha can be built, which are related to the beta distribution. The process is derived from probability distributions using blur derivative. These new wavelets have just one cycle, so they are termed unicycle wavelets. They can be viewed as a ''soft variety'' of Haar wavelets whose shape is fine-tuned by two parameters \alpha and \beta. Closed-form expressions for beta wavelets and scale functions as well as their spectra are derived. Their importance is due to the Central Limit Theorem by Gnedenko and Kolmogorov applied for compactly supported signals. Beta distribution The beta distribution is a continuous probability distribution defined over the interval 0\leq t\leq 1. It is characterised by a couple of parameters, namely \alpha and \beta according to: P(t)=\fract^\cdot (1-t)^,\quad 1\leq \alpha ,\beta \leq +\infty . The normalising factor is B(\alpha ,\beta )=\frac, where \Gamma (\cdot ) is the generalised factorial function o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Wavelet
Hermitian wavelets are a family of continuous wavelets, used in the continuous wavelet transform. The n^\textrm Hermitian wavelet is defined as the n^\textrm derivative of a Gaussian distribution: \Psi_(t)=(2n)^c_He_\left(t\right)e^ where He_\left(\right) denotes the n^\textrm Hermite polynomial. The normalisation coefficient c_ is given by: c_ = \left(n^\Gamma(n+\frac)\right)^ = \left(n^\sqrt2^(2n-1)!!\right)^\quad n\in\mathbb. The prefactor C_ in the resolution of the identity of the continuous wavelet transform for this wavelet is given by: C_=\frac i.e. Hermitian wavelets are admissible for all positive n. In computer vision and image processing, Gaussian derivative operators of different orders are frequently used as a basis for expressing various types of visual operations; see scale space and N-jet. Examples of Hermitian wavelets: Starting from Gaussian function with \mu=0, \sigma=1: f(t) = \pi^e^ the first 3 derivatives read as, :\begin f'(t) & = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Difference Of Gaussians
In imaging science, difference of Gaussians (DoG) is a feature enhancement algorithm that involves the subtraction of one Gaussian blurred version of an original image from another, less blurred version of the original. In the simple case of grayscale images, the blurred images are obtained by convolving the original grayscale images with Gaussian kernels having differing width (standard deviations). Blurring an image using a Gaussian kernel suppresses only high-frequency spatial information. Subtracting one image from the other preserves spatial information that lies between the range of frequencies that are preserved in the two blurred images. Thus, the DoG is a spatial band-pass filter that attenuates frequencies in the original grayscale image that are far from the band center. \sigma_1. In one dimension, \Gamma is defined as: :\Gamma_(x) = I*\frac \, e^-I*\frac \, e^. and for the centered two-dimensional case: :\Gamma_(x,y) = I*\frac e^ - I*\frac e^ which is formally eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shannon Wavelet
In functional analysis, the Shannon wavelet (or sinc wavelets) is a decomposition that is defined by signal analysis by ideal bandpass filters. Shannon wavelet may be either of real number, real or complex number, complex type. Shannon wavelet is not well-localized(noncompact) in the time domain,but its Fourier transform is band-limited(compact support). Hence Shnnon wavelet has poor time localization but has good frequency localization. These characteristics are in stark contrast to those of the Haar wavelet. The Haar and sinc systems are Fourier duals of each other. Definition Sinc funcition is the starting point for the definition of the shannon wavelet. Scaling function First, we define the scaling function to be the sinc function. \phi^(t) := \frac = \operatorname(t). And define the dilated and translated intances to be \phi^n_k(t) := 2^\phi^(2^n t-k) where the parameter n,k means the dilation and the translation for the wavelet respectively. Then we can derive th ... [...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 different kinds of data, including but not limited to au ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
