Convolution

In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions (f and g).

It produces a third function, that is typically viewed as a modified version of one of the original functions, giving the integral of the pointwise multiplication of the two functions as a function of the amount that one of the original functions is translated.

Description

Convolution is similar to cross-correlation.

It 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 groups other than Euclidean space. For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution.

A discrete convolution can be defined for functions on the set of integers.

Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing.

Other usage

For the usage in formal language theory, see Convolution (computer science).

For other uses, see Convolute.

See also

• Analog signal processing
• Circulant matrix
• Convolution for optical broad-beam responses in scattering media
• Convolution power
• Cross-correlation
• Deconvolution
• Dirichlet convolution
• Function (mathematics)
• Functional analysis
• Jan Mikusinski
• Learning to Simplify: Fully Convolutional Networks for Rough Sketch Cleanup
• List of convolutions of probability distributions
• LTI system theory#Impulse response and convolution
• Multidimensional discrete convolution
• Scaled correlation
• Titchmarsh convolution theorem
• Toeplitz matrix (convolutions can be considered a Toeplitz matrix operation where each row is a shifted copy of the convolution kernel)

External links

• Convolution @ Wikipedia