konvoluution

Konvoluution (often called convolution in English) is a mathematical operation that combines two functions to produce a third function that expresses how the shape of one is modified by the other. In continuous form, the convolution of functions f and g is defined as (f * g)(t) = ∫_{-∞}^{∞} f(τ) g(t − τ) dτ. In discrete form, used for sequences, it is (f * g)[n] = Σ_k f[k] g[n − k]. Convolutions are linear and can be interpreted as applying a sliding kernel, or impulse response, to a signal.

Key properties include commutativity (f * g = g * f), associativity ((f * g) * h = f * (g * h)), and distributivity over addition (f * (g + h) = f * g + f * h). The operation is shift-invariant, meaning a shift in the input produces an equivalent shift in the output. Boundary handling, such as zero-padding or reflective padding, affects the result at edges in finite-length signals.

A fundamental result is the convolution theorem: the Fourier transform of a convolution equals the pointwise product of the Fourier transforms, and conversely the inverse transform of a product is a convolution. This links time-domain filtering to frequency-domain multiplication and underpins efficient computation using the fast Fourier transform (FFT). In two dimensions, convolution is used for image filtering, with (f * g)(x, y) = ∬ f(u, v) g(x − u, y − v) du dv.

Applications span signal processing, image processing, probability (the sum of independent random variables), and the solution of linear differential equations via Green’s functions. Common kernels include the Gaussian, which produces smoothing, and various edge-detection kernels in computer vision.