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Coiflet
Coiflets are discrete wavelets designed by Ingrid Daubechies, at the request of Ronald Coifman, to have scaling functions with vanishing moments. The wavelet is near symmetric, their wavelet functions have N/3 vanishing moments and scaling functions N/3-1, and has been used in many applications using Calderón–Zygmund operators.G. Beylkin, R. Coifman, and V. Rokhlin (1991),''Fast wavelet transforms and numerical algorithms'', Comm. Pure Appl. Math., 44, pp. 141–183Ingrid Daubechies, ''Ten Lectures on Wavelets'', Society for Industrial and Applied Mathematics, 1992, Theory Some theorems about Coiflets: Theorem 1 For a wavelet system \, the following three equations are equivalent: : \begin \mathcal(0,l] = 0 & \textl =0,1,\ldots,L-1 \\ \sum_n (-1)^n n^l h 0 & \textl =0,1,\ldots,L-1 \\ H^(\pi)=0 & \textl=0,1,\ldots,L-1 \end and similar equivalence holds between \psi and \tilde Theorem 2 For a wavelet system \, the following six equations are equivalent: : \begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wavelet Coif1
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 t ... [...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] |
Ingrid Daubechies
Baroness Ingrid Daubechies ( ; ; born 17 August 1954) is a Belgian physicist and mathematician. She is best known for her work with wavelets in image compression. Daubechies is recognized for her study of the mathematical methods that enhance image-compression technology. She is a member of the National Academy of Engineering, the National Academy of Sciences and the American Academy of Arts and Sciences. She is a 1992 MacArthur Fellow. She also served on the Mathematical Sciences jury for the Infosys Prize from 2011 to 2013. The name Daubechies is widely associated with the orthogonal Daubechies wavelet and the biorthogonal CDF wavelet. A wavelet from this family of wavelets is now used in the JPEG 2000 standard. Her research involves the use of automatic methods from both mathematics, technology, and biology to extract information from samples such as bones and teeth. She also developed sophisticated image processing techniques used to help establish the authenticity and ... [...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] |
Moment (mathematics)
In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the center of mass, and the second moment is the moment of inertia. If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics. For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all orders, from to ) uniquely determines the distribution (Hausdorff moment problem). The same is not true on unbounded intervals ( Hamburger moment problem). In the mid-nineteenth century, Pafnuty Chebyshev became the first person to think sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Integral
In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator : T(f)(x) = \int K(x,y)f(y) \, dy, whose kernel function ''K'' : R''n''×R''n'' → R is singular along the diagonal ''x'' = ''y''. Specifically, the singularity is such that , ''K''(''x'', ''y''), is of size , ''x'' − ''y'', −''n'' asymptotically as , ''x'' − ''y'', → 0. Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over , ''y'' − ''x'', > ε as ε → 0, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on ''L''''p''(R''n''). The Hilbert transform The archetypal singular integral operator is ... [...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 poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Wavelets
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics * In optics, polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right- and left-handed circular polarization. * In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
