konvolúcia

Konvolúcia is a mathematical operation on two functions that produces a third function. This third function expresses how the shape of one of the original functions is modified by the other. It is commonly used in signal processing, image processing, and probability theory. In essence, it describes the process of combining two signals to produce a new signal. Think of it as a weighted average of one function as it slides over another.

The formal definition of the convolution of two functions, f and g, is given by the integral:

(f * g)(t) = ∫ f(τ)g(t - τ) dτ, where the integral is taken over all possible values of τ.

In image processing, convolution is a fundamental operation used for tasks like blurring, sharpening, and edge detection. A small matrix called a kernel is convolved with an image. Each pixel in the output image is calculated based on a weighted sum of the surrounding pixels in the input image, with the weights determined by the kernel.

In signal processing, convolution is used to describe the output of a linear time-invariant (LTI) system. The impulse response of the system acts as the kernel, and convolving the input signal with the impulse response gives the output signal. This operation is crucial for understanding how systems affect signals.

In probability, the convolution of two probability density functions gives the probability density function of the sum of two independent random variables. This is a key concept in understanding the behavior of combined random events. The concept extends to discrete sequences as well, where summation replaces integration.

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