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Topic: Convolution

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  PlanetMath: convolution
Convolution is an important tool in data processing, in particular in digital signal and image processing.
The (Dirichlet) convolution of multiplicative functions considered in number theory does not quite fit the above definition, since there the functions are defined on a commutative monoid (the natural numbers under multiplication) rather than on an abelian group.
The convolution of an exponential and a normal distribution is approximated by another exponential distribution.
planetmath.org /encyclopedia/Convolution.html   (412 words)

 Convolution - Wikipedia, the free encyclopedia
In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g.
An out-of-focus photograph is the convolution of the sharp image with the shape of the iris diaphragm.
In acoustics, an echo is the convolution of the original sound with a function representing the various objects that are reflecting it.
en.wikipedia.org /wiki/Convolution   (895 words)

Convolution has been a standard topic in engineering and computing science for some time, but only since the early 1990s has it been widely available to computer music composers, thanks largely to the theoretical descriptions by Curtis Roads (1996), and the SoundHack software of Tom Erbe that made this technique accessible.
In fact, convolution in this example is simply a mathematical description of what happens when any sound is "coloured" by the acoustic space within which it occurs, which is in fact true of all sounds in all spaces except an anechoic chamber.
In this case we are convolving their spectra which is why ring modulation results in the sum and difference frequencies of each component being present in the output, though an understanding of this result depends on the mathematics of the complex domain.
www.sfu.ca /~truax/conv.html   (789 words)

 The convolution theorem and its applications
Mathematically, a convolution is defined as the integral over all space of one function at x times another function at u-x.
The only difference with the convolution theorem is in the presence of a complex conjugate, which reverses the phase and corresponds to the inversion of the argument u-x.
Convolution with a Gaussian will shift the origin of the function to the position of the peak of the Gaussian, and the function will be smeared out, as illustrated above.
www-structmed.cimr.cam.ac.uk /Course/Convolution/convolution.html   (2266 words)

 Glossary - Convolution
Convolution is a simple mathematical operation which is fundamental to many common image processing operators.
Convolution provides a way of `multiplying together' two arrays of numbers, generally of different sizes, but of the same dimensionality, to produce a third array of numbers of the same dimensionality.
The convolution is performed by sliding the kernel over the image, generally starting at the top left corner, so as to move the kernel through all the positions where the kernel fits entirely within the boundaries of the image.
homepages.inf.ed.ac.uk /rbf/HIPR2/convolve.htm   (500 words)

 Convolution Background
Convolution can be used to calculate the response of a system to arbitrary inputs by using the impulse response of a system.
Using the convolution integral it is possible to calculate the output, y(t), of any linear system given only the input, f(t), and the impulse response, h(t).
Another excellent web-based demonstration of convolution is The Joy of Convolution at Johns Hopkins.
www.swarthmore.edu /NatSci/echeeve1/Ref/Convolution/Convolution.html   (911 words)

 SMM: Convolution; spreadsheet models
Convolution is widely used in the physical sciences, in signal theory, in probability theory, in statistics and in communications.
For those who understand what convolution is, it's much easier to understand a model implemented with convolution than with any other alternative technique.
Convolution gives you a way to find the time response — often called the "temporal response" — of a system as it responds to a specified input, provided that you know the "base response" of the system, which is its response to a single-event input.
www.chacocanyon.com /smm/readings/convolve.shtml   (1805 words)

Convolutions are a really cool way to smooth and even differentiate data.
Sure, I could sling the exponential integrals the academics loved, but really had no idea the convolution is the simple process of using one function, in a narrow sliding window against another function, to generate a third that is the combination of the two.
The convolution is a different way of looking at noise reduction, potentially improving response time by performing a smart average over the time dimension.
www.ganssle.com /articles/aconvolv.htm   (1966 words)

 4.2 Continuous Convolution   (Site not responding. Last check: 2007-10-11)
This process is graphically illustrated in Figure 4.3 by the classical example of the convolution of two rectangle functions (or "boxcar functions") to give a triangle function.
Convolution of a signal, s(t) with a Dirac comb (equation 2.3.5) is
Convolution with a Dirac comb yields an infinite periodic series of replicas of the original function.
www-rohan.sdsu.edu /~jiracek/digital/filtering/continuousconvolution.html   (763 words)

 Oilfield Glossary: Term 'convolution'
Convolution can be applied to any two functions of time or space (or other variables) to yield a third function, the output of the convolution.
For example, a convolution can be used to model the filtering of seismic energy by the various rock layers in the Earth; deconvolution is used extensively in seismic processing to counteract that filtering.
Convolution assumes a particular model for the pressure-transient response, usually infinite-acting radial flow.
www.glossary.oilfield.slb.com /Display.cfm?Term=convolution   (211 words)

 Convolution in DSP
Convolution is used in DSP to find the system response of Linear Time-Invariant [LTI] systems, to arbitrary inputs.
The polynomial method of convolution is based on the idea of assuming the two sequences as polynomials and then multiplying the two polynomials we have.
The Convolution theorem is the basis of making a convolution in time domain as a simple multiplication in the frequency domain.
cybernetics.freewebspace.com /dsp   (1294 words)

 Discrete-Time Convolution   (Site not responding. Last check: 2007-10-11)
Convolution is a concept that extends to all systems that are both linear and time-invariant (LTI).
Recall that convolution is a very powerful tool in determining a system's output from knowledge of an arbitrary input and the system's impulse response.
It will also be helpful to see convolution graphically with your own eyes and to play around with it some, so experiment with the applets available on the internet.
cnx.org /content/m10087/latest   (833 words)

 Processing Image Pixels, Applying Image Convolution in Java, Part 2
Convolution in two dimensions involves essentially the same steps except that in this case we are dealing with three different 3D sampled surfaces and a 3D convolution filter surface instead of a simple sampled time series.
The results of the DFT computation on the convolution filter are shown in the fifth graph in Figure 2.
Although the mean was removed prior to convolution, it is possible that arithmetic inaccuracies during the convolution process could result in a slightly non-zero mean in the filtered image.
www.developer.com /java/other/article.php/3596351   (6974 words)

 Sussex Computer Vision: TEACH VISION2   (Site not responding. Last check: 2007-10-11)
The table of numbers that specifies a convolution is known as a mask, a kernel, an operator or a template, depending on the author, and the numbers in the mask are often called weights.
The reason is that with this convention, the operation of convolution is associative - that is, if two masks are convolved one with the other, then the result can be used as a mask that has the same effect as convolving each of the two masks with the image, one after the other.
In other words, the convolution operation has found the part of the image with the structure most closely corresponding to the template - not surprisingly, this is the part from which the template was originally copied.
www.cogs.susx.ac.uk /users/davidy/teachvision/vision2.html   (3425 words)

 AMS Glossary
This effect is more readily understood by taking the Fourier transform of the convolution equation, in effect transforming from the time domain to the frequency domain.
An example of a convolution process in radar is the smoothing of the spatial pattern of reflectivity as a consequence of the finite size of the pulse volume.
Spatial irregularities with scales smaller than the pulse volume are attenuated in the measurement process by a convolution of the reflectivity field with the pulse volume.
amsglossary.allenpress.com /glossary/search?p=1&query=convolution   (263 words)

 Convolution - Analog
In order to fully understand convolution, you may find it useful to look at the discrete-time convolution as well.
Taking a closer look at the convolution integral, we find that we are multiplying the input signal by the time-reversed impulse response and integrating.
Convolution is a truly important concept, which must be well understood.
cnx.org /content/m11540/latest   (762 words)

Convolution is the term given to the mathematical technique for determining a system output given an input signal and the system impulse response.
The product and convolution data will change as the time-reversed input signal is moved relative to the system impulse response.
As you perform the convolution, as before, you will find that the output is a copy of the input signal, except that it is delayed by 2ms.
www.see.ed.ac.uk /~mjj/dspDemos/EE4/tutConv.html   (530 words)

 Introduction to DSP - Time domain processing: Convolution
Convolution is a weighted moving average with one signal flipped back to front:
The reason convolution is preferred to correlation for filtering has to do with how the frequency spectra of the two signals interact.
Correlation is equivalent to multiplying the complex conjugate of the frequency spectrum of one signal by the frequency spectrum of the other.
www.bores.com /courses/intro/time/2_conv.htm   (274 words)

 Convolution Theorem
It is the basis of a large number of applications of the FFT.
Since the FFT provides a fast Fourier transform, it also provides fast convolution, thanks to the convolution theorem.
convolution, the FFT is approximately 300 times faster in Octave, and 30 times faster in Matlab.
ccrma.stanford.edu /~jos/mdft/Convolution_Theorem.html   (453 words)

 Convolution reverb | How convolution works, how to make recordings suitable for use with convolution reverbs
Convolution software offers huge potential for sonic manipulation, and as you will see, even those on a small budget can blaze new trails into sonic territory using convolution and their own recordings.
Convolution, in practice, is the process of multiplying two audio signals in the frequency domain.
Convolution plug-ins are commonly billed as being able to place your sounds in a “real acoustic environment.” It's a hefty claim, but the sound they can provide often measures up.
emusician.com /tutorials/emusic_acting_impulse   (4456 words)

 Filters and Convolution
The impulse is typically denoted by the Greek delta symbol δ and is often called the \'Dirac delta function\', after Paul Dirac who pioneered its use (in addition to being a lead contributor in the development of Quantum Mechanics).
, which we introduced as the convolution representation of a filter, has been shown to be more specifically the impulse response of a system is its output signal in response to the impulse signal.
In other words, every LTI system has a convolution representation in terms of its impulse response.
www-ccrma.stanford.edu /~jos/mdft/LTI_Filters_Convolution.html   (376 words)

 Convolution Number Nine
Convolution is therefore impractical for variable or dynamic filtering, because it doesn’t offer the parametric control available on the average synthesizer filter or equalizer.
This may be the most interesting feature of this convolution, because there is potential for a “morph.” The original blocks and the cymblock tracks have common rhythmic, accentual, and pitch features; therefore, a careful crossfade can produce an interesting transformation in which the temple block appears to “turn into” the cymblock.
Convolution, unlike FM synthesis, is not a widely explored, well-documented electronic-music technique.
emusician.com /convolution_number_nine/index.html   (2678 words)

 Convolution & VideoScript   (Site not responding. Last check: 2007-10-11)
Perform a convolution of the source parameter with the kernel, this convolution uses cyclic co-ordinates.
A convolution kernel is just an image, for historical reasons, convolutios kernels are usually specified using integer elements.
A Gaussian blur, is a convolution of an image with a Gaussian Filter.Kernel The radius parameter specifies the standard deviation of the Gaussian.
www.videoscript.com /VideoScript/VideoSource/Convolution.html   (849 words)

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