konvuláció

Konvolúció is a mathematical operation on two functions that produces a third function. This third function expresses how the shape of one of the input functions is modified by the other. It is commonly used in signal processing, image processing, and probability theory.

In signal processing, convolution is used to describe the output of a linear time-invariant (LTI) system. If one function represents the input signal and the other represents the system's impulse response, their convolution yields the output signal. This is fundamental for understanding how systems filter or alter signals.

In image processing, convolution is the basis of many filtering operations. A small matrix, called a kernel or a filter, is convolved with an image. This kernel defines the operation to be performed, such as blurring, sharpening, or edge detection. Each pixel in the output image is calculated by a weighted sum of pixels in the input image, with the weights determined by the kernel.

Mathematically, the convolution of two functions f and g is denoted by (f * g) and is defined by the integral:

(f * g)(t) = ∫ f(τ)g(t - τ) dτ

where the integral is taken over all possible values of τ. For discrete functions, the integral becomes a summation. The operation involves flipping one function (g in this case) and sliding it across the other function (f), calculating the integral (or sum) of their product at each position. This operation is commutative, meaning f * g = g * f.