Here the convolution theorem is demonstrated. First consider the image of a cat and its Fourier spectrum (magnitude and phase):













Next assume there is a relative motion between the camera and the cat during the exposure time. For simplicity, we further assume the motion is linear and only in the horizontal direction, and the displacement of the image caused by the motion is L. Such a motion can be modeled by the convolution of the image and a convolution kernel representing the linear motion (a rectangular window of size L in the horizontal direction multiplied by a delta function in the vertical direction). In the frequency domain, this motion becomes the multiplication of the spectra of the image and the convolution kernel (a 1D Sinc function):

x =


The resulting image blurred by the linear motion is the inverse transform of the spectrum on the right.