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Second-generation Wavelet Transform
{{Short description, Type of wavelet transform In signal processing, the second-generation wavelet transform (SGWT) is a wavelet transform where the filter (signal processing), filters (or even the represented wavelets) are not designed explicitly, but the transform consists of the application of the Lifting scheme. Actually, the sequence of lifting steps could be converted to a regular discrete wavelet transform, but this is unnecessary because both design and application is made via the lifting scheme. This means that they are not designed in the frequency domain, as they are usually in the ''classical'' (so to speak ''first generation'') transforms such as the discrete wavelet transform, DWT and continuous wavelet transform, CWT). The idea of moving away from the Fourier analysis, Fourier domain was introduced independently by David Donoho and Harten in the early 1990s. Calculating transform The input signal f is split into odd \gamma _1 and even \lambda _1 samples using shifti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signal Processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomography, seismic signals, Altimeter, altimetry processing, and scientific measurements. Signal processing techniques are used to optimize transmissions, Data storage, digital storage efficiency, correcting distorted signals, improve subjective video quality, and to detect or pinpoint components of interest in a measured signal. History According to Alan V. Oppenheim and Ronald W. Schafer, the principles of signal processing can be found in the classical numerical analysis techniques of the 17th century. They further state that the digital refinement of these techniques can be found in the digital control systems of the 1940s and 1950s. In 1948, Claude Shannon wrote the influential paper "A Mathematical Theory of Communication" which was publis ... [...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] |
Filter (signal Processing)
In signal processing, a filter is a device or process that removes some unwanted components or features from a Signal (electronics), signal. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal. Most often, this means removing some frequency, frequencies or frequency bands. However, filters do not exclusively act in the frequency domain; especially in the field of image processing many other targets for filtering exist. Correlations can be removed for certain frequency components and not for others without having to act in the frequency domain. Filters are widely used in electronics and telecommunication, in radio, television, audio recording, radar, control systems, music synthesis, image processing, computer graphics, and structural dynamics. There are many different bases of classifying filters and these overlap in many different ways; there is no simple hierarchical classification. Fil ... [...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 s ... [...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] |
Frequency Domain
In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time series. While a time-domain graph shows how a signal changes over time, a frequency-domain graph shows how the signal is distributed within different frequency bands over a range of frequencies. A complex valued frequency-domain representation consists of both the magnitude and the phase of a set of sinusoids (or other basis waveforms) at the frequency components of the signal. Although it is common to refer to the magnitude portion (the real valued frequency-domain) as the frequency response of a signal, the phase portion is required to uniquely define the 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 transfo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Wavelet Transform
In mathematics, the continuous wavelet transform (CWT) is a formal (i.e., non-numerical) tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the wavelets vary continuously. Definition The continuous wavelet transform of a function x(t) at a scale a\in\mathbb and translational value b\in\mathbb is expressed by the following integral :X_w(a,b)=\frac \int_^\infty x(t)\overline\psi\left(\frac\right)\,\mathrmt where \psi(t) is a continuous function in both the time domain and the frequency domain called the mother wavelet and the overline represents operation of complex conjugate. The main purpose of the mother wavelet is to provide a source function to generate the daughter wavelets which are simply the translated and scaled versions of the mother wavelet. To recover the original signal x(t), the first inverse continuous wavelet transform can be exploited. :x(t)=C_\psi^\int_^\int_^ X_w(a,b)\frac\tilde\psi\left(\frac\righ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. The subject of Fourier analysis encompasses a vast spectrum of mathematics. In the sciences and engineering, the process of decomposing a function into oscillatory components is often called Fourier analysis, while the operation of rebuilding the function from these pieces is known as Fourier synthesis. For example, determining what component frequencies are present in a musical note would involve computing the Fourier transform of a sampled musical note. One could then re-synthesize the same sound by including the frequency components as revealed in the Fourier analysis. In mathematics, the term ''Fourier an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Donoho
David Leigh Donoho (born March 5, 1957) is an American statistician. He is a professor of statistics at Stanford University, where he is also the Anne T. and Robert M. Bass Professor in the Humanities and Sciences. His work includes the development of effective methods for the construction of low-dimensional representations for high-dimensional data problems ( multiscale geometric analysis), development of wavelets for denoising and compressed sensing. He was elected a Member of the American Philosophical Society in 2019. Academic biography Donoho did his undergraduate studies at Princeton University, graduating in 1978. His undergraduate thesis advisor was John W. Tukey. Donoho obtained his Ph.D. from Harvard University in 1983, under the supervision of Peter J. Huber.. He was on the faculty of the University of California, Berkeley, from 1984 to 1990 before moving to Stanford. He has been the Ph.D. advisor of at least 20 doctoral students, including Jianqing Fan and Emmanu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harten
Harten is a surname of German or Dutch origin. Notable people with the surname include: *Ami Harten (1946–1994), American-Israeli applied mathematician * James Harten (1924–2001), Australian cricketer * Jo Harten (born 1989), English netball player *Patrick Harten, American air traffic controller * Uwe Harten (born 1944), German musicologist See also * Gus Van Harten * '' Wilde Harten'' * Young Hearts (1936 film) ''Young Hearts'' or ''Jonge Harten'' is a 1936 in film, 1936 Netherlands, Dutch film directed and written by and the German exile Heinz Josephson.Günter Peter Straschek ''Handbuch wider das Kino'' 1975 p.308 " mit Jonge harten (1936; Junge Herze ... References Surnames of German origin Surnames of Dutch origin {{surname ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Downsampling
In digital signal processing, downsampling, compression, and decimation are terms associated with the process of ''resampling'' in a multi-rate digital signal processing system. Both ''downsampling'' and ''decimation'' can be synonymous with ''compression'', or they can describe an entire process of bandwidth reduction ( filtering) and sample-rate reduction. When the process is performed on a sequence of samples of a ''signal'' or a continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a lower rate (or density, as in the case of a photograph). ''Decimation'' is a term that historically means the '' removal of every tenth one''. But in signal processing, ''decimation by a factor of 10'' actually means ''keeping'' only every tenth sample. This factor multiplies the sampling interval or, equivalently, divides the sampling rate. For example, if compact disc audio at 44,100 samples/second is ''decimated'' by a factor o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second Generation Wavelet Transform
{{Short description, Type of wavelet transform In signal processing, the second-generation wavelet transform (SGWT) is a wavelet transform where the filters (or even the represented wavelets) are not designed explicitly, but the transform consists of the application of the Lifting scheme. Actually, the sequence of lifting steps could be converted to a regular discrete wavelet transform, but this is unnecessary because both design and application is made via the lifting scheme. This means that they are not designed in the frequency domain, as they are usually in the ''classical'' (so to speak ''first generation'') transforms such as the DWT and CWT). The idea of moving away from the Fourier domain was introduced independently by David Donoho and Harten in the early 1990s. Calculating transform The input signal f is split into odd \gamma _1 and even \lambda _1 samples using shifting and downsampling. The detail coefficients \gamma _2 are then interpolated using the values of \ga ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
