In the RIXS experiment these two pieces of information come together in a convolved manner, strongly perturbed by the core-hole potential in the intermediate state. RIXS studies can be performed using both soft and hard X-rays.
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343–347, 1988. for application to decomposition of one-dimensional electromyography convolved signals via de-convolution. This design was modified in 1989 to other de-convolution-based designs.Daniel Graupe, Boris Vern, G. Gruener, Aaron Field, and Qiu Huang.
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There are two common methods used to implement discrete convolution: the definition of convolution and fast Fourier transformation (FFT and IFFT) according to the convolution theorem. To calculate the optical broad-beam response, the impulse response of a pencil beam is convolved with the beam function. As shown by Equation 4, this is a 2-D convolution. To calculate the response of a light beam on a plane perpendicular to the z axis, the beam function (represented by a b × b matrix) is convolved with the impulse response on that plane (represented by an a × a matrix).
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To determine the position of an audio source in 3D space, the ear input signals are convolved with the inverses of all possible HRTF pairs, where the correct inverse maximizes cross-correlation between the convolved right and left signals. In the case of multiple simultaneous sound sources, the transmission of sound from source to ears can be considered a multiple-input and multiple-output. Here, the HRTFs the source signals were filtered with en route to the microphones can be found using methods such as convolutive blind source separation, which has the advantage of efficient implementation in real-time systems. Overall, these approaches using HRTFs can be well optimized to localize multiple moving sound sources.
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This is due to blind deconvolution common mode of usage in digital communications systems, as a means to extract the continuously transmitted signal from the received signal, with the channel impulse response being of secondary intrinsic importance. The estimated equalizer is then convolved with the received signal to yield an estimation of the transmitted signal.
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For auralizations to be perceived as realistic, it is critical to emulate the human hearing in terms of position and orientation of the listener's head with respect to the sources of sound. For IR data to be convolved convincingly, the acoustic events are captured using a dummy head where two microphones are positioned on each side of the head to record an emulation of sound arriving at the locations of human ears, or using an ambisonics microphone array and mixed down for binaurality. Head-related transfer functions (HRTF) datasets can be used to simplify the process insofar as a monaural IR can be measured or simulated, then audio content is convolved with its target acoustic space. In rendering the experience, the transfer function corresponding to the orientation of the head is applied to simulate the corresponding spatial emanation of sound.
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Accurate wavelet estimation requires the accurate tie of the impedance log to the seismic. Errors in well tie can result in phase or frequency artifacts in the wavelet estimation. Once the wavelet is identified, seismic inversion computes a synthetic log for every seismic trace. To ensure quality, the inversion result is convolved with the wavelet to produce synthetic seismic traces which are compared to the original seismic.
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However, in general, expansion is not applied for Gabor wavelets, since this requires computation of bi-orthogonal wavelets, which may be very time-consuming. Therefore, usually, a filter bank consisting of Gabor filters with various scales and rotations is created. The filters are convolved with the signal, resulting in a so-called Gabor space. This process is closely related to processes in the primary visual cortex.
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As an example, when a picture is taken indoors, the brightness of outdoor objects seen through a window may be 70 or 80 times brighter than objects inside the room. If exposure levels are set for objects inside the room, the bright image of the windows will bleed past the window frames when convolved with the Airy disc of the camera being used to produce the image.
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In this case, the intensity is integrated over the slit width. The resulting measurement is equivalent to the original cross section convolved with the profile of the slit. These techniques can measure very small spot sizes down to 1 μm, and can be used to directly measure high power beams. They do not offer continuous readout, although repetition rates as high as twenty hertz can be achieved.
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When data is convolved with a function with wide support, such as for downsampling by a large sampling ratio, because of the Convolution theorem and the FFT algorithm, it may be faster to transform it, multiply pointwise by the transform of the filter and then reverse transform it. Alternatively, a good filter is obtained by simply truncating the transformed data and re-transforming the shortened data set.
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Results, 208: College Station, TX (Ocean Drilling Program), 1–27 This acoustic impedance log is combined with the velocity data to generate a reflection coefficient series in time. This series is convolved with a seismic wavelet to produce the synthetic seismogram. The input seismic wavelet is chosen to match as closely as possible to that produced during the original seismic acquisition, paying particular attention to phase and frequency content.
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The entries in the triangle satisfy the identity :H(n, i) = F(i + 1) × F(n − i + 1). Thus, the two outermost diagonals are the Fibonacci numbers, while the numbers on the middle vertical line are the squares of the Fibonacci numbers. All the other numbers in the triangle are the product of two distinct Fibonacci numbers greater than 1. The row sums are the first convolved Fibonacci numbers.
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Current generation gaming systems are able to render 3D graphics using floating point frame buffers, in order to produce HDR images. To produce the bloom effect, the HDRR images in the frame buffer are convolved with a convolution kernel in a post- processing step, before converting to RGB space. The convolution step usually requires the use of a large gaussian kernel that is not practical for realtime graphics, causing programmers to use approximation methods.
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The most exact way to do this is to loop through the frequencies of interest, and for each frequency, calculate the radiance at this frequency. For this, one needs to calculate the contribution of each spectral line for all molecules in the atmospheric layer; this is called a line-by-line calculation. For an instrument response, this is then convolved with the spectral response of the instrument. A faster but more approximate method is a band transmission.
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Fluorescence lifetimes can be determined in the time domain by using a pulsed source. When a population of fluorophores is excited by an ultrashort or delta pulse of light, the time-resolved fluorescence will decay exponentially as described above. However, if the excitation pulse or detection response is wide, the measured fluorescence, d(t), will not be purely exponential. The instrumental response function, IRF(t) will be convolved or blended with the decay function, F(t).
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In stochastic analysis, the usual way to model a random process, or field, is done by specifying the dynamics of the process through a stochastic (partial) differential equation (SPDE). It is known, that solutions of (partial) differential equations can in some cases be given as an integral of a Green's function convolved with another function – if the differential equation is stochastic, i.e. contaminated by random noise (e.g. white noise) the corresponding solution would be a stochastic integral of the Green's function.
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In addition to compactly supported functions and integrable functions, functions that have sufficiently rapid decay at infinity can also be convolved. An important feature of the convolution is that if f and g both decay rapidly, then f∗g also decays rapidly. In particular, if f and g are rapidly decreasing functions, then so is the convolution f∗g. Combined with the fact that convolution commutes with differentiation (see #Properties), it follows that the class of Schwartz functions is closed under convolution .
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To perform anti-aliasing in computer graphics, the anti-aliasing system requires a key piece of information: which objects cover specific pixels at any given time in the animation. One approach used is to derive a high resolution (i.e. larger than the output image) temporal intensity function from object attributes which can then be convolved with an averaging filter to compute the final anti-aliased image. In this approach, there are two methods available for computing the temporal intensity function.
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A two dimensional convolution matrix is precomputed from the formula and convolved with two dimensional data. Each element in the resultant matrix new value is set to a weighted average of that elements neighborhood. The focal element receives the heaviest weight (having the highest Gaussian value) and neighboring elements receive smaller weights as their distance to the focal element increases. In Image processing, each element in the matrix represents a pixel attribute such as brightness or a color intensity, and the overall effect is called Gaussian blur.
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The combined electron mean free path results in an image blur, which is usually modeled as a Gaussian function (where σ = blur) that is convolved with the expected image. As the desired resolution approaches the blur, the dose image becomes broader than the aerial image of the incident X-rays. The blur that matters is the latent image that describes the making or breaking of bonds during the exposure of resist. The developed image is the final relief image produced by the selected high contrast development process on the latent image.
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In audio signal processing, convolution reverb is a process used for digitally simulating the reverberation of a physical or virtual space through the use of software profiles; a piece of software (or algorithm) that creates a simulation of an audio environment. It is based on the mathematical convolution operation, and uses a pre-recorded audio sample of the impulse response of the space being modeled. To apply the reverberation effect, the impulse-response recording is first stored in a digital signal-processing system. This is then convolved with the incoming audio signal to be processed.
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There are three main processes in seismic data processing: deconvolution, common-midpoint (CMP) stacking and migration. Deconvolution is a process that tries to extract the reflectivity series of the Earth, under the assumption that a seismic trace is just the reflectivity series of the Earth convolved with distorting filters. This process improves temporal resolution by collapsing the seismic wavelet, but it is nonunique unless further information is available such as well logs, or further assumptions are made. Deconvolution operations can be cascaded, with each individual deconvolution designed to remove a particular type of distortion.
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Flows can be given as derivations over the algebra of smooth functionals over the configuration space. If we have a flow distribution (i.e. flow-valued distribution) such that the flow convolved over a local region only affects the field configuration in that region, we call the flow distribution a gauge flow. Given that we are only interested in what happens on shell, we would often take the quotient by the ideal generated by the Euler–Lagrange equations, or in other words, consider the equivalence class of functionals/flows which agree on shell.
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Seismic wavelet For example, a wavelet could be created to have a frequency of Middle C and a short duration of roughly a 32nd note. If this wavelet were to be convolved with a signal created from the recording of a song, then the resulting signal would be useful for determining when the Middle C note was being played in the song. Mathematically, the wavelet will correlate with the signal if the unknown signal contains information of similar frequency. This concept of correlation is at the core of many practical applications of wavelet theory.
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The nonlinear interaction mixes ultrasonic tones in air to produce sum and difference frequencies. A DSB-AM modulation scheme with an appropriately large baseband DC offset, to produce the demodulating tone superimposed on the modulated audio spectrum, is one way to generate the signal that encodes the desired baseband audio spectrum. This technique suffers from extremely heavy distortion as not only the demodulating tone interferes, but also all other frequencies present interfere with one another. The modulated spectrum is convolved with itself, doubling its bandwidth by the length property of the convolution.
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In nonparametric statistics, a kernel is a weighting function used in non-parametric estimation techniques. Kernels are used in kernel density estimation to estimate random variables' density functions, or in kernel regression to estimate the conditional expectation of a random variable. Kernels are also used in time-series, in the use of the periodogram to estimate the spectral density where they are known as window functions. An additional use is in the estimation of a time-varying intensity for a point process where window functions (kernels) are convolved with time- series data.
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In the window design method, one first designs an ideal IIR filter and then truncates the infinite impulse response by multiplying it with a finite length window function. The result is a finite impulse response filter whose frequency response is modified from that of the IIR filter. Multiplying the infinite impulse by the window function in the time domain results in the frequency response of the IIR being convolved with the Fourier transform (or DTFT) of the window function. If the window's main lobe is narrow, the composite frequency response remains close to that of the ideal IIR filter.
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The convolutional layer is the core building block of a CNN. The layer's parameters consist of a set of learnable filters (or kernels), which have a small receptive field, but extend through the full depth of the input volume. During the forward pass, each filter is convolved across the width and height of the input volume, computing the dot product between the entries of the filter and the input and producing a 2-dimensional activation map of that filter. As a result, the network learns filters that activate when it detects some specific type of feature at some spatial position in the input.
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There are a number of advantages to computing symmetric convolutions in DSTs and DCTs in comparison with the more common circular convolution with the Fourier transform. Most notably the implicit symmetry of the transforms involved is such that only data unable to be inferred through symmetry is required. For instance using a DCT-II, a symmetric signal need only have the positive half DCT-II transformed, since the frequency domain will implicitly construct the mirrored data comprising the other half. This enables larger convolution kernels to be used with the same cost as smaller kernels circularly convolved on the DFT.
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The audio stream is sampled and convolved with a triangle function, and interpolated later during playback. The techniques employed, including the sampling of signals with a finite rate of innovation, were developed by a number of researchers over the preceding decade, including Pier Luigi Dragotti and others. MQA- encoded content can be carried via any lossless file format such as FLAC or ALAC; hence, it can be played back on systems either with or without an MQA decoder. In the latter case, the resulting audio has easily identifiable high- frequency noise occupying 3 LSB bits, thus limiting playback on non-MQA devices effectively to 13 bit.
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The distribution of the product of two random variables which have lognormal distributions is again lognormal. This is itself a special case of a more general set of results where the logarithm of the product can be written as the sum of the logarithms. Thus, in cases where a simple result can be found in the list of convolutions of probability distributions, where the distributions to be convolved are those of the logarithms of the components of the product, the result might be transformed to provide the distribution of the product. However this approach is only useful where the logarithms of the components of the product are in some standard families of distributions.
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In actuality, though, the range of secondary electron scattering is quite far, sometimes exceeding 100 nm, but becoming very significant below 30 nm. The proximity effect is also manifest by secondary electrons leaving the top surface of the resist and then returning some tens of nanometers distance away. Proximity effects (due to electron scattering) can be addressed by solving the inverse problem and calculating the exposure function E(x,y) that leads to a dose distribution as close as possible to the desired dose D(x,y) when convolved by the scattering distribution point spread function PSF(x,y). However, it must be remembered that an error in the applied dose (e.g.
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The deterministic nature of linear inversion requires a functional relationship which models, in terms of the earth model parameters, the seismic variable to be inverted. This functional relationship is some mathematical model derived from the fundamental laws of physics and is more often called a forward model. The aim of the technique is to minimize a function which is dependent on the difference between the convolution of the forward model with a source wavelet and the field collected seismic trace. As in the field of optimization, this function to be minimized is called the objective function and in convectional inverse modeling, is simply the difference between the convolved forward model and the seismic trace.
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The family of CLEAN algorithms, a chapter from the MAPPING software manual The algorithm assumes that the image consists of a number of point sources. It will iteratively find the highest value in the image and subtract a small gain of this point source convolved with the point spread function ("dirty beam") of the observation, until the highest value is smaller than some threshold. Astronomer T. J. Cornwell writes, "The impact of CLEAN on radio astronomy has been immense", both directly in enabling greater speed and efficiency in observations, and indirectly by encouraging "a wave of innovation in synthesis processing that continues to this day." It has also been applied in other areas of astronomy and many other fields of science.
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A core member of this class is the Wigner–Ville distribution (WVD), as all other TFDs can be written as a smoothed or convolved versions of the WVD. Another popular member of this class is the spectrogram which is the square of the magnitude of the short- time Fourier transform (STFT). The spectrogram has the advantage of being positive and is easy to interpret, but also has disadvantages, like being irreversible, which means that once the spectrogram of a signal is computed, the original signal can't be extracted from the spectrogram. The theory and methodology for defining a TFD that verifies certain desirable properties is given in the "Theory of Quadratic TFDs".B. Boashash, “Theory of Quadratic TFDs”, Chapter 3, pp.
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Principle of the SOFI auto-cumulant calculation (A) Schematic depiction of a CCD-pixel grid containing several emitter-signals (B) Cut-out of two fluorophores with their signals convolved with the system's PSF, recorded in an image stack (C) The signals on every pixel are evaluated by cumulant calculation (a process that can be understood in terms of a correlation and integration) Likewise to other super-resolution methods SOFI is based on recording an image time series on a CCD- or CMOS camera. In contrary to other methods the recorded time series can be substantially shorter, since a precise localization of emitters is not required and therefore a larger quantity of activated fluorophores per diffraction-limited area is allowed. The pixel values of a SOFI-image of the n-th order are calculated from the values of the pixel time series in the form of a n-th order cumulant, whereas the final value assigned to a pixel can be imagined as the integral over a correlation function. The finally assigned pixel value intensities are a measure of the brightness and correlation of the fluorescence signal.
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In practice in its simplest two- speaker implementation, ambiophonic reproduction unlocks auditory cues for images of up to 150° horizontally (azimuth), depending on the binaural cues captured in existing stereo recordings. Multi-channel recordings made with ambiophone-like microphone arrays to make 5.1-compatible DVD/SACD recordings can be reproduced using just four speakers (a center speaker is obviated in ambiophonic layouts). Allowing for the human hearing “cone of confusion” at each side, a full 360° degree circle of perceived sound localization has been measured within ±5° of actual source azimuth, reproducing lifelike spatial envelopment and timbre (contributed by accurate directional provenance of early reflections) of multi-channel music, movies, and game content.Glasgal, Ralph, “Ambiophonics, 2nd Edition”Glasgal, Ralph, “The Ambiophone, Derivation of a Recording Methodology Optimized for Ambiophonic Reproduction,” AES 19th Conference, Schloss Elmau, June 2001 Especially in the case of stereo content where ambience has been purposely reduced (because a natural level coming from front 60°-only is perceived as too much), additional signals for surround speakers can be produced using a measured hall impulse response, convolved in a PC with the two front channel signals.
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