konvolution

Konvolution is a mathematical operation that combines two functions to produce a third function expressing how the shape of one is modified by the other. It is widely used in signal processing, physics, and statistics. In its continuous form, the konvolution of functions f and g on the real line is defined by (f * g)(t) = ∫ f(τ) g(t − τ) dτ. In the discrete case, for sequences f[n] and g[n], the konvolution is (f * g)[n] = ∑ f[m] g[n − m], with indices restricted to the finite supports in practical computation. Different conventions may swap f and g or reverse one argument; nevertheless, all are instances of the same integral or sum.

Convolution has several key interpretations: it can be viewed as a sliding weighted average, where g acts as a kernel that shapes f; or as a measure of how well f matches a reversed version of g at each shift. The operation is linear and, for suitable conditions, commutative (f * g = g * f) and associative.

The Fourier transform provides a fundamental link: the Fourier transform of a convolution is the pointwise product of the transforms, i.e., F{f * g} = F{f} · F{g}. This makes convolution efficient to compute using the fast Fourier transform (FFT). In image processing, 2D convolution with kernels is used for blurring, sharpening, and edge detection. In probability, the sum of independent random variables is modeled by the convolution of their distributions.

A related operation is cross-correlation; the inverse problem of removing the effect of convolution is deconvolution.