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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] |
Wavelet Compression
In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform. Definition A function \psi \,\in\, L^2(\mathbb) is called an orthonormal wavelet if it can be used to define a Hilbert basis, that is a complete orthonormal system, for the Hilbert space L^2\left(\mathbb\right) of square integrable functions. The Hilbert basis is constructed as the family of functions \ by means of dyadic translations and dilations of \psi\,, :\psi_(x) = 2^\frac \psi\left(2^jx - k\right)\, for integers j,\, k \,\in\, \mathbb. If under the standard inner product on L^2\left(\mathbb\right), :\langle f, g\rangle = \int_^\infty f(x)\overlinedx this family is orthonormal, it is an orthonormal system: :\begin \langle\psi_,\psi_\rangle &= \int_^\infty \psi_(x)\overlinedx ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavelet Series
In mathematics, a wavelet series is a representation of a square-integrable (real- or complex-valued) function by a certain orthonormal series generated by a wavelet. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform. Definition A function \psi \,\in\, L^2(\mathbb) is called an orthonormal wavelet if it can be used to define a Hilbert basis, that is a complete orthonormal system, for the Hilbert space L^2\left(\mathbb\right) of square integrable functions. The Hilbert basis is constructed as the family of functions \ by means of dyadic translations and dilations of \psi\,, :\psi_(x) = 2^\frac \psi\left(2^jx - k\right)\, for integers j,\, k \,\in\, \mathbb. If under the standard inner product on L^2\left(\mathbb\right), :\langle f, g\rangle = \int_^\infty f(x)\overlinedx this family is orthonormal, it is an orthonormal system: :\begin \langle\psi_,\psi_\rangle &= \int_^\infty \psi_(x)\overlinedx ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Huygens–Fresnel Principle
The Huygens–Fresnel principle (named after Dutch physicist Christiaan Huygens and French physicist Augustin-Jean Fresnel) states that every point on a wavefront is itself the source of spherical wavelets, and the secondary wavelets emanating from different points mutually interfere. The sum of these spherical wavelets forms a new wavefront. As such, the Huygens-Fresnel principle is a method of analysis applied to problems of luminous wave propagation both in the far-field limit and in near-field diffraction as well as reflection. History In 1678, Huygens proposed that every point reached by a luminous disturbance becomes a source of a spherical wave; the sum of these secondary waves determines the form of the wave at any subsequent time. He assumed that the secondary waves travelled only in the "forward" direction and it is not explained in the theory why this is the case. He was able to provide a qualitative explanation of linear and spherical wave propagation, and to d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent States In Mathematical Physics
Coherent states have been introduced in a physical context, first as quasi-classical states in quantum mechanics, then as the backbone of quantum optics and they are described in that spirit in the article Coherent states (see alsoJ-P. Gazeau,''Coherent States in Quantum Physics'', Wiley-VCH, Berlin, 2009.). However, they have generated a huge variety of generalizations, which have led to a tremendous amount of literature in mathematical physics. In this article, we sketch the main directions of research on this line. For further details, we refer to several existing surveys.S.T. Ali, J-P. Antoine, J-P. Gazeau, and U.A. Mueller, Coherent states and their generalizations: A mathematical overview, ''Reviews in Mathematical Physics'' 7 (1995) 1013-1104.S.T. Ali, J-P. Antoine, and J-P. Gazeau, ''Coherent States, Wavelets and Their Generalizations'', Springer-Verlag, New York, Berlin, Heidelberg, 2000. A general definition Let \mathfrak H\, be a complex, separable Hilbert space, X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Morlet
Jean Morlet (; 13 January 1931 – 27 April 2007) was a French geophysicist who pioneered work in the field of wavelet analysis around the year 1975. He invented the term ''wavelet'' to describe the functions he was using. In 1981, Morlet worked with Alex Grossmann to develop what is now known as the Wavelet transform. Biography Morlet graduated from the École Polytechnique in 1952 and was research engineer at Elf Aquitaine when he invented wavelets to solve signal processing problems for oil prospecting. Awards He was awarded in 1997 with the Reginald Fessenden Award. He was awarded in 2001 with the first prize Prix Chéreau Lavet, from the Académie des Technologies. Legacy The Jean-Morlet Chair at the Centre International de Rencontres Mathématiques The Centre International de Rencontres Mathématiques (CIRM) is a mathematics research institute associated with the Centre National de la Recherche Scientifique (CNRS) and the Société Mathématique de France (SMF). It ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frame Of A Vector Space
In linear algebra, a frame of an inner product space is a generalization of a basis of a vector space to sets that may be linearly dependent. In the terminology of signal processing, a frame provides a redundant, stable way of representing a signal. Frames are used in error detection and correction and the design and analysis of filter banks and more generally in applied mathematics, computer science, and engineering. Definition and motivation Motivating example: computing a basis from a linearly dependent set Suppose we have a set of vectors \ in the vector space ''V'' and we want to express an arbitrary element \mathbf \in V as a linear combination of the vectors \, that is, we want to find coefficients c_k such that : \mathbf = \sum_k c_k \mathbf_k If the set \ does not span V, then such coefficients do not exist for every such \mathbf. If \ spans V and also is linearly independent, this set forms a basis of V, and the coefficients c_ are uniquely determined by \mathb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alex Grossmann
Alexander Grossmann (5 August 1930 – 12 February 2019) was a French- American physicist of Croatian origin. He travelled to the United States in 1955, working in the physics departments of the Institute for Advanced Study (IAS), Princeton, Brandeis University, and the Courant Institute, NYU, then again at the IAS until 1963. After one year at the Institut des Hautes Études Scientifiques (IHES) in Bures-sur-Yvette, France, he joined the "Centre de Physique Théorique de Marseille" (the CPT) as it was being created in 1966, at the request of Daniel Kastler. He then becomes research supervisor at the CNRS. At the Université de la Méditerranée Aix-Marseille II The University of the Mediterranean Aix-Marseille II was a French university in the Academy of Aix and Marseille. Historically, it was part of the University of Aix-Marseille based across the communes of Aix-en-Provence and Marseille in southe ... in Luminy campus he did pioneering work on wavelet analysis with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Basis Function
In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors. In numerical analysis and approximation theory, basis functions are also called blending functions, because of their use in interpolation: In this application, a mixture of the basis functions provides an interpolating function (with the "blend" depending on the evaluation of the basis functions at the data points). Examples Monomial basis for ''Cω'' The monomial basis for the vector space of analytic functions is given by \. This basis is used in Taylor series, amongst others. Monomial basis for polynomials The monomial basis also forms a basis for the vector space of polynomials. After all, every polynomial can be written as a_0 + a_1x^1 + a_2x^2 + \cdots + a_n x^n for some n \in \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavefront
In physics, the wavefront of a time-varying '' wave field'' is the set ( locus) of all points having the same '' phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal frequency (otherwise the phase is not well defined). Wavefronts usually move with time. For waves propagating in a unidimensional medium, the wavefronts are usually single points; they are curves in a two dimensional medium, and surfaces in a three-dimensional one. For a sinusoidal plane wave, the wavefronts are planes perpendicular to the direction of propagation, that move in that direction together with the wave. For a sinusoidal spherical wave, the wavefronts are spherical surfaces that expand with it. If the speed of propagation is different at different points of a wavefront, the shape and/or orientation of the wavefronts may change by refraction. In particular, lenses can change the shape of optical wavefronts from planar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. Wavelength is commonly designated by the Greek letter '' lambda'' (λ). The term ''wavelength'' is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids. Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to frequency of the wave: waves with higher frequencies have shorter wavelengths, and lower frequencies have longer wavelengths. Wavelength depends on the medium (for example, vacuum, air, or water) that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
