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Biorthogonal Wavelet
A Biorthogonal wavelet is a wavelet where the associated wavelet transform is invertible but not necessarily orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic .... Designing biorthogonal wavelets allows more degrees of freedom than orthogonal wavelets. One additional degree of freedom is the possibility to construct symmetric wavelet functions. In the biorthogonal case, there are two scaling functions \phi,\tilde\phi, which may generate different multiresolution analyses, and accordingly two different wavelet functions \psi,\tilde\psi. So the numbers ''M'' and ''N'' of coefficients in the scaling sequences a,\tilde a may differ. The scaling sequences must satisfy the following biorthogonality condition :\sum_ a_n \tilde a_=2\cdot\delta_. Then the wavelet sequences can be ... [...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] |
Discrete Wavelet Transform
In numerical analysis and functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ..., a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency ''and'' location information (location in time). Definition One level of the transform The DWT of a signal x is calculated by passing it through a series of filters. First the samples are passed through a low-pass filter with impulse response g resulting in a convolution of the two: :y[n] = (x * g)[n] = \sum\limits_^\infty The signal is also decomposed simultaneously using a high-pass filter h. The outputs give the detail coefficients (fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is a right inverse of . (An identity element is an element such that and for all and for which the left-hand sides are defined.) When the operation is associative, if an element has both a left inverse and a right inverse, then these two inverses are equal and unique; they are called the ''inverse element'' or simply the ''inverse''. Often an adjective is added for specifying the operation, such as in additive inverse, multiplicative inverse, and functional inverse. In this case (associative operation), an invertible element is an element that has an inverse. In a ring, an ''invertible element'', also called a unit, is an element that is invertible under multiplication (this is not ambiguous, as every element is invertible under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonality
In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendicular'' is more specifically used for lines and planes that intersect to form a right angle, whereas ''orthogonal'' is used in generalizations, such as ''orthogonal vectors'' or ''orthogonal curves''. ''Orthogonality'' is also used with various meanings that are often weakly related or not related at all with the mathematical meanings. 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 Optics In optics, polarization states are said to be ort ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Wavelet
An orthogonal wavelet is a wavelet whose associated wavelet transform is orthogonal. That is, the inverse wavelet transform is the adjoint of the wavelet transform. If this condition is weakened one may end up with biorthogonal wavelets. Basics The scaling function is a refinable function. That is, it is a fractal functional equation, called the refinement equation (twin-scale relation or dilation equation): :\phi(x)=\sum_^ a_k\phi(2x-k), where the sequence (a_0,\dots, a_) of real numbers is called a scaling sequence or scaling mask. The wavelet proper is obtained by a similar linear combination, :\psi(x)=\sum_^ b_k\phi(2x-k), where the sequence (b_0,\dots, b_) of real numbers is called a wavelet sequence or wavelet mask. A necessary condition for the ''orthogonality'' of the wavelets is that the scaling sequence is orthogonal to any shifts of it by an even number of coefficients: :\sum_ a_n a_=2\delta_, where \delta_ is the Kronecker delta. In this case there is the same nu ... [...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] |
Multiresolution Analysis
A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design method of most of the practically relevant discrete wavelet transforms (DWT) and the justification for the algorithm of the fast wavelet transform (FWT). It was introduced in this context in 1988/89 by Stephane Mallat and Yves Meyer and has predecessors in the microlocal analysis in the theory of differential equations (the ''ironing method'') and the pyramid methods of image processing as introduced in 1981/83 by Peter J. Burt, Edward H. Adelson anJames L. Crowley Definition A multiresolution analysis of the Lebesgue space L^2(\mathbb) consists of a sequence of nested subspaces ::\\subset \dots\subset V_1\subset V_0\subset V_\subset\dots\subset V_\subset V_\subset\dots\subset L^2(\R) that satisfies certain self-similarity relations in time-space and scale-frequency, as well as completeness and regularity relations. * ''Self-similarity'' in ''time'' demands that each subspace ''Vk'' is in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
