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Fast Wavelet Transform
The fast wavelet transform is a mathematical algorithm designed to turn a waveform or signal in the time domain into a sequence of coefficients based on an orthogonal basis of small finite waves, or wavelets. The transform can be easily extended to multidimensional signals, such as images, where the time domain is replaced with the space domain. This algorithm was introduced in 1989 by Stéphane Mallat. It has as theoretical foundation the device of a finitely generated, orthogonal multiresolution analysis (MRA). In the terms given there, one selects a sampling scale ''J'' with sampling rate of 2''J'' per unit interval, and projects the given signal ''f'' onto the space V_J; in theory by computing the scalar products :s^_n:=2^J \langle f(t),\varphi(2^J t-n) \rangle, where \varphi is the scaling function of the chosen wavelet transform; in practice by any suitable sampling procedure under the condition that the signal is highly oversampled, so : P_J x):=\sum_ s^_n\,\varphi(2^Jx ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavelets - DWT
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 aud ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ronald Coifman
Ronald Raphael Coifman is the Sterling professor of Mathematics at Yale University. Coifman earned a doctorate from the University of Geneva in 1965, supervised by Jovan Karamata. Coifman is a member of the American Academy of Arts and Sciences, the Connecticut Academy of Science and Engineering, and the National Academy of Sciences. He is a recipient of the 1996 DARPA Sustained Excellence Award, the 1996 Connecticut Science Medal, the 1999 Pioneer Award of the International Society for Industrial and Applied Science, and the 1999 National Medal of Science. In 2013, he co-founded ThetaRay, a cyber security and big data analytics company. In 2018, he received the Rolf Schock Prize for Mathematics. References External links Scientific Data Has Become So Complex, We Have to Invent New Math to Deal With It Wired ''Wired'' (stylized as ''WIRED'') is a monthly American magazine, published in print and online editions, that focuses on how emerging technologies affect culture, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gregory Beylkin
Gregory Beylkin (born 16 March 1953) is an applied mathematician. Education and career He studied from 1970 to 1975 at the University of Leningrad, with Diploma in Mathematics in November 1975. From 1976 to 1979 he was a research scientist at the Research Institute of Ore Geophysics, Leningrad. From 1980 to 1982 he was a graduate student at New York University, where he received his PhD under the supervision of Peter Lax. From 1982 to 1983 Beylkin was an associate research scientist at the Courant Institute of Mathematical Sciences. From 1983 to 1991 he was a member of the professional staff of Schlumberger-Doll Research in Ridgefield, Connecticut. Since 1991 he has been a professor in the Department of Applied Mathematics at the University of Colorado Boulder. He was a visiting professor at Yale University, the University of Minnesota, and the Mittag-Leffler Institute and participated in 2012 and 2015 in the summer seminar on "Applied Harmonic Analysis and Sparse Approximation" at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barbara Burke Hubbard
Barbara Burke Hubbard (born 1948) is an American science journalist, mathematics popularizer, textbook author, and book publisher, known for her books on wavelet transforms and multivariable calculus. Life Burke Hubbard is the daughter of ''Los Angeles Times'' reporter Vincent J. Burke, and spent a year in high school living in Moscow when Burke was stationed there in 1964. She was an undergraduate at Harvard University, initially majoring in biology but switching to English, and graduating in 1969. She became a science writer for the Massachusetts Institute of Technology and a journalist for ''The Ithaca Journal'', and was the 1981 winner of the AAAS Westinghouse Science Journalism Award in the small newspaper category, for her articles on acid rain in ''The Ithaca Journal''. She married mathematician John H. Hubbard, with whom she has four children, and with her family has split her time between Ithaca, New York and Marseille, France, with shorter-term stays elsewhere. Books ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cristina Pereyra
María Cristina Pereyra (born 1964) is a Venezuelan mathematician. She is a professor of mathematics and statistics at the University of New Mexico, and the author of several books on wavelets and harmonic analysis. Education and employment Pereyra was a member of the Venezuelan team for the 1981 and 1982 International Mathematical Olympiads. She earned a licenciado (the equivalent of a bachelor's degree) in mathematics in 1986 from the Central University of Venezuela. She went to Yale University for graduate studies, completing her Ph.D. there in 1993. Her dissertation, ''Sobolev Spaces On Lipschitz Curves: Paraproducts, Inverses And Some Related Operators'', was supervised by Peter Jones. After working for three years as an instructor at Princeton University, she joined the University of New Mexico faculty in 1996. Books Pereyra is the author or editor of: *''Lecture Notes on Dyadic Harmonic Analysis'' (Second Summer School in Analysis and Mathematical Physics, Cuernavaca, 2000 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fast Fourier Transform
A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in the frequency domain and vice versa. The DFT is obtained by decomposing a sequence of values into components of different frequencies. This operation is useful in many fields, but computing it directly from the definition is often too slow to be practical. An FFT rapidly computes such transformations by factorizing the DFT matrix into a product of sparse (mostly zero) factors. As a result, it manages to reduce the complexity of computing the DFT from O\left(N^2\right), which arises if one simply applies the definition of DFT, to O(N \log N), where N is the data size. The difference in speed can be enormous, especially for long data sets where ''N'' may be in the thousands or millions. In the presence of round-off error, many FFT algorithm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lifting Scheme
The lifting scheme is a technique for both designing wavelets and performing the discrete wavelet transform (DWT). In an implementation, it is often worthwhile to merge these steps and design the wavelet filters ''while'' performing the wavelet transform. This is then called the second-generation wavelet transform. The technique was introduced by Wim Sweldens. The lifting scheme factorizes any discrete wavelet transform with finite filters into a series of elementary convolution operators, so-called lifting steps, which reduces the number of arithmetic operations by nearly a factor two. Treatment of signal boundaries is also simplified. The discrete wavelet transform applies several filters separately to the same signal. In contrast to that, for the lifting scheme, the signal is divided like a zipper. Then a series of convolution–accumulate operations across the divided signals is applied. Basics The simplest version of a forward wavelet transform expressed in the lifting sc ... [...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 under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upsampling
In digital signal processing, upsampling, expansion, and interpolation are terms associated with the process of resampling in a multi-rate digital signal processing system. ''Upsampling'' can be synonymous with ''expansion'', or it can describe an entire process of ''expansion'' and filtering (''interpolation''). When upsampling is performed on a sequence of samples of a ''signal'' or other continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a higher rate (or density, as in the case of a photograph). For example, if compact disc audio at 44,100 samples/second is upsampled by a factor of 5/4, the resulting sample-rate is 55,125. Upsampling by an integer factor Rate increase by an integer factor ''L'' can be explained as a 2-step process, with an equivalent implementation that is more efficient: #Expansion: Create a sequence, x_L comprising the original samples, x separated by ''L'' − 1 zeros.&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, linear subspaces, flats, and affine subspaces are also called ''linear manifolds'' for emphasizing that there are also manifolds. is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a ''subspace'' when the context serves to distinguish it from other types of subspaces. Definition If ''V'' is a vector space over a field ''K'' and if ''W'' is a subset of ''V'', then ''W'' is a linear subspace of ''V'' if under the operations of ''V'', ''W'' is a vector space over ''K''. Equivalently, a nonempty subset ''W'' is a subspace of ''V'' if, whenever are elements of ''W'' and are elements of ''K'', it follows that is in ''W''. As a corollary, all vector spaces are equipped with at least two ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Filter
In signal processing, the adjoint filter mask h^* of a filter mask h is reversed in time and the elements are complex conjugated. :(h^*)_k = \overline Its name is derived from the fact that the convolution with the adjoint filter is the adjoint operator of the original filter, with respect to the Hilbert space \ell_2 of the sequences in which the inner product is the Euclidean norm. :\langle h*x, y \rangle = \langle x, h^* * y \rangle The autocorrelation Autocorrelation, sometimes known as serial correlation in the discrete time case, is the correlation of a signal with a delayed copy of itself as a function of delay. Informally, it is the similarity between observations of a random variable ... of a signal x can be written as x^* * x. Properties * ^* = h * (h*g)^* = h^* * g^* * (h\leftarrow k)^* = h^* \rightarrow k References Digital signal processing {{signal-processing-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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