Fast computation of convolution operations via low-rank approximation

Methods for the approximation of 2D discrete convolution operations are derived for the case when a low-rank approximation of one of the input matrices is available. Algorithms based on explicit computation of discrete convolution and on the Fast Fourier Transform are both described. Applications of the described methods to the computation of cross-correlation and autocorrelation are discussed and illustrated by examples. Both theory and numerical experiments show that the use of low-rank approximations makes it possible to determine accurate approximations of convolution, cross-correlation, and autocorrelation operations at competitive speeds.