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Discrete Wavelet Transform
In numerical analysis and functional analysis, 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). Examples Haar wavelets The first DWT was invented by Hungarian mathematician Alfréd Haar. For an input represented by a list of 2^n numbers, the Haar wavelet transform may be considered to pair up input values, storing the difference and passing the sum. This process is repeated recursively, pairing up the sums to prove the next scale, which leads to 2^n-1 differences and a final sum. Daubechies wavelets The most commonly used set of discrete wavelet transforms was formulated by the Belgian mathematician Ingrid Daubechies in 1988. This formulation is based on the use of recurrence relations to generate progressively finer discrete samplings o ...
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Jpeg2000 2-level Wavelet Transform-lichtenstein
JPEG ( ) is a commonly used method of lossy compression for digital images, particularly for those images produced by digital photography. The degree of compression can be adjusted, allowing a selectable tradeoff between storage size and image quality. JPEG typically achieves 10:1 compression with little perceptible loss in image quality. Since its introduction in 1992, JPEG has been the most widely used image compression standard in the world, and the most widely used digital image format, with several billion JPEG images produced every day as of 2015. The term "JPEG" is an acronym for the Joint Photographic Experts Group, which created the standard in 1992. JPEG was largely responsible for the proliferation of digital images and digital photos across the Internet, and later social media. JPEG compression is used in a number of image file formats. JPEG/Exif is the most common image format used by digital cameras and other photographic image capture devices; along with JPEG/J ...
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Ali Naci Akansu
Ali Naci Akansu (born May 6, 1958) is a Turkish-American Professor of electrical & computer engineering and scientist in applied mathematics. He is best known for his seminal contributions to the theory and applications of linear subspace methods including sub-band and wavelet transforms, particularly the binomial QMF (also known as Daubechies wavelet) and the multivariate framework to design statistically optimized filter bank (eigen filter bank). Biography Akansu received his B.S. degree from the Istanbul Technical University, Turkey, in 1980, his M.S. and PhD degrees from the Polytechnic University (now New York University), Brooklyn, New York, in 1983 and 1987, respectively, all in Electrical Engineering. Since 1987, he has been with the New Jersey Institute of Technology where he is a Professor of Electrical and Computer Engineering. He was a Visiting Professor at Courant Institute of Mathematical Sciences of the New York University, 2009–2010. In 1990, he showed th ...
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Noise Wavelet
Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference arises when the brain receives and perceives a sound. Acoustic noise is any sound in the acoustic domain, either deliberate (e.g., music or speech) or unintended. In contrast, noise in electronics may not be audible to the human ear and may require instruments for detection. In audio engineering, noise can refer to the unwanted residual electronic noise signal that gives rise to acoustic noise heard as a hiss. This signal noise is commonly measured using A-weighting or ITU-R 468 weighting. In experimental sciences, noise can refer to any random fluctuations of data that hinders perception of a signal. Measurement Sound is measured based on the amplitude and frequency of a sound wave. Amplitude measures how forceful the wave is. The e ...
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Data Compression
In information theory, data compression, source coding, or bit-rate reduction is the process of encoding information using fewer bits than the original representation. Any particular compression is either lossy or lossless. Lossless compression reduces bits by identifying and eliminating statistical redundancy. No information is lost in lossless compression. Lossy compression reduces bits by removing unnecessary or less important information. Typically, a device that performs data compression is referred to as an encoder, and one that performs the reversal of the process (decompression) as a decoder. The process of reducing the size of a data file is often referred to as data compression. In the context of data transmission, it is called source coding; encoding done at the source of the data before it is stored or transmitted. Source coding should not be confused with channel coding, for error detection and correction or line coding, the means for mapping data onto a signal. ...
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Signal Coding
In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' includes audio, video, speech, image, sonar, and radar as examples of signal. A signal may also be defined as observable change in a quantity over space or time (a time series), even if it does not carry information. In nature, signals can be actions done by an organism to alert other organisms, ranging from the release of plant chemicals to warn nearby plants of a predator, to sounds or motions made by animals to alert other animals of food. Signaling occurs in all organisms even at cellular levels, with cell signaling. Signaling theory, in evolutionary biology, proposes that a substantial driver for evolution is the ability of animals to communicate with each other by developing ways of signaling. In human engineering, signals are ty ...
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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 algo ...
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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^ ...
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Complex Wavelet Transform
The complex wavelet transform (CWT) is a complex-valued extension to the standard discrete wavelet transform (DWT). It is a two-dimensional wavelet transform which provides multiresolution, sparse representation, and useful characterization of the structure of an image. Further, it purveys a high degree of shift-invariance in its magnitude, which was investigated in. However, a drawback to this transform is that it exhibits 2^ (where d is the dimension of the signal being transformed) redundancy compared to a separable (DWT). The use of complex wavelets in image processing was originally set up in 1995 by J.M. Lina and L. Gagnoin the framework of the Daubechies orthogonal filters bank It was then generalized in 1997 by Nick Kingsbury, Prof. Nick Kingsbury of Cambridge University. In the area of computer vision, by exploiting the concept of visual contexts, one can quickly focus on candidate regions, where objects of interest may be found, and then compute additional features thr ...
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Wavelet Packet Decomposition
Originally known as optimal subband tree structuring (SB-TS), also called wavelet packet decomposition (WPD) (sometimes known as just wavelet packets or subband tree), is a wavelet transform where the discrete-time (sampled) signal is passed through more filters than the discrete wavelet transform (DWT). Introduction In the DWT, each level is calculated by passing only the previous wavelet approximation coefficients (''cAj'') through discrete-time low- and high-pass quadrature mirror filters. However, in the WPD, both the detail (''cDj'' (in the 1-D case), ''cHj'', ''cVj'', ''cDj'' (in the 2-D case)) and approximation coefficients are decomposed to create the full binary tree.Daubechies, I. (1992), Ten lectures on wavelets, SIAM. For ''n'' levels of decomposition the WPD produces 2''n'' different sets of coefficients (or nodes) as opposed to sets for the DWT. However, due to the downsampling process the overall number of coefficients is still the same and there is no redundanc ...
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Frequency Space
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a signal changes over time, whereas a frequency-domain graph shows how much of the signal lies within each given frequency band over a range of frequencies. A frequency-domain representation can also include information on the phase shift that must be applied to each sinusoid in order to be able to recombine the frequency components to recover the original time signal. A given function or signal can be converted between the time and frequency domains with a pair of mathematical operators called transforms. An example is the Fourier transform, which converts a time function into a complex valued sum or integral of sine waves of different frequencies, with amplitudes and phases, each of which represents a frequency component. The "spectrum ...
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Top-hat Filter
The name Top-hat filter refers to several real-space or Fourier space filtering techniques (not to be confused with the top-hat transform). The name top-hat originates from the shape of the filter, which is a rectangle function, when viewed in the domain in which the filter is constructed. Real space In real-space the filter performs nearest-neighbour filtering, incorporating components from neighbouring y-function values. However, despite their ease of implementation their practical use is limited as the real-space representation of a top-hat filter is the sinc function, which has the often undesirable effect of incorporating non-local frequencies. Analogue implementations Exact non-digital implementations are only theoretically possible. Top-hat filters can be constructed by chaining theoretical low-band and high-band filters. In practice, an approximate top-hat filter can be constructed in analogue hardware using approximate low-band and high-band filters. Fourier space In ...
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Orthonormal
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis. Intuitive overview The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the Cartesian plane, two vectors are said to be ''perpendicular'' if the angle between them is 90° (i.e. if they form a right angle). This definition can be formalized in Cartesian space by defining the dot product and specifying that two vectors in the plane are orthogonal if their dot product is zero. Similarly, the construction of the norm of a vector is motivated by a desire to extend the intuitive notion of the length of a vector to higher-dimensional spaces. In Cartes ...
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