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Convolution
Introduction

Convolution is a phenomenon (or a mathematical procedure, depending on one's standpoint) which is quite common and which is of fundamental importance to signal processing. Before going into mathematical detail, let us consider an example showing the principles of convolution:

A pinhole camera with a (near) zero-diameter pinhole produces a sharp (inverted) image f(x). Each point outside the camera is reproduced perfectly on the film.

If the pinhole has a diameter larger than zero, the image becomes blurred because the image of each point outside the camera is now a spot inside it. This spot contains a distribution of intensities h(x) which are superimposed on points nearby. The distribution function h(x) depends on the shape and the width of the whole.

The entire image in the camera is the superposition of "many" (in fact, an infinite number of) images which are slightly offset against each other. In mathematical terms the resulting image F(x) is the convolution of the original image f(x) and the distribution function of light intensities caused by the nonzero diameter of the camera hole. The convolution is a process which results in a blurred signal because the originally "sharp" (or precise) signal is convolved with another function of non-zero width. The second function usually arises from some measurement restrictions, such as limited resolution, or limited time for the measurement.

The inverse process of the convolution is called deconvolution, which is used to extract the sharp signal from a blurred one.

Note, that the concept of convolution is not restricted to images but may also be used in many other situations, e.g. filtering (smoothing) of time series, or correlating two signals, or when considering Fourier transforms and their relation to the time domain signals.


Last Update: 2006-Jšn-17