page_title: Convolution
section_title: Applications
hierarchical_section_title: Convolution / Applications
caption_attribution_description: English: Halftone images can be smoothed digitaly by applying a Gaussian blur filter. Original image taken from an interbellic newspaper.
context_page_description: In mathematics convolution is a mathematical operation on two functions that produces a third function expressing how the shape of one is modified by the other. The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reversed and shifted. And the integral is evaluated for all values of shift, producing the convolution function.
  Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, it differs from cross-correlation only in that either f(x) or g(x) is reflected about the y-axis; thus it is a cross-correlation of f(x) and g, or f and g(x). For continuous functions, the cross-correlation operator is the adjoint of the convolution operator.
  Convolution has applications that include probability, statistics, computer vision, natural language processing, image and signal processing, engineering, and differential equations.
  The convolution can be defined for functions on Euclidean space, and other groups.
context_section_description: Convolution and related operations are found in many applications in science, engineering and mathematics.
  In image processing
  In digital image processing convolutional filtering plays an important role in many important algorithms in edge detection and related processes.
  In optics, an out-of-focus photograph is a convolution of the sharp image with a lens function. The photographic term for this is bokeh.
  In image processing applications such as adding blurring.
  In digital data processing
  In analytical chemistry, Savitzky–Golay smoothing filters are used for the analysis of spectroscopic data. They can improve signal-to-noise ratio with minimal distortion of the spectra
  In statistics, a weighted moving average is a convolution.
  In acoustics, reverberation is the convolution of the original sound with echoes from objects surrounding the sound source.
  In digital signal processing, convolution is used to map the impulse response of a real room on a digital audio signal.
  In electronic music convolution is the imposition of a spectral or rhythmic structure on a sound. Often this envelope or structure is taken from another sound. The convolution of two signals is the filtering of one through the other.
  In electrical engineering, the convolution of one function (the input signal) with a second function (the impulse response) gives the output of a linear time-invariant system (LTI). At any given moment, the output is an accumulated effect of all the prior values of the input function, with the most recent values typically having the most influence (expressed as a multiplicative factor). The impulse response function provides that factor as a function of the elapsed time since each input value occurred.
  In physics, wherever there is a linear system with a "superposition principle", a convolution operation makes an appearance. For instance, in spectroscopy line broadening due to the Doppler effect on its own gives a Gaussian spectral line shape and collision broadening alone gives a Lorentzian line shape. When both effects are operative, the line shape is a convolution of Gaussian and Lorentzian, a Voigt function.
  In time-resolved fluorescence spectroscopy, the excitation signal can be treated as a chain of delta pulses, and the measured fluorescence is a sum of exponential decays from each delta pulse.
  In computational fluid dynamics, the large eddy simulation (LES) turbulence model uses the convolution operation to lower the range of length scales necessary in computation thereby reducing computational cost.
  In probability theory, the probability distribution of the sum of two independent random variables is the convolution of their individual distributions.
  In kernel density estimation, a distribution is estimated from sample points by convolution with a kernel, such as an isotropic Gaussian. (Diggle 1995).
  In radiotherapy treatment planning systems, most part of all modern codes of calculation applies a convolution-superposition algorithm.
  Convolutional neural networks apply multiple cascaded convolution kernels with applications in machine vision and artificial intelligence. Though these are actually cross correlations.
  In structural reliability, the reliability index can be defined based on the convolution theorem.
  The definition of reliability index for limit state functions with nonnormal distributions can be established corresponding to the joint distribution function. In fact, the joint distribution function can be obtained using the convolution theory. (Ghasemi-Nowak 2017).
caption: Gaussian blur can be used to obtain a smooth grayscale digital image of a halftone print.
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