Issue No.09 - September (1977 vol.26)
I.S. Reed , Department of Electrical Engineering, University of Southern California
A transform analogous to the discrete Fourier transform is defined on the Galois field GF(p), where p is a prime of the form k X 2n + 1, where k and n are integers. Such transforms offer a substantial variety of possible transform lengths and dynamic ranges. The fast Fourier transform (FFT) algorithm of this transform is faster than the conventional radix-2 FFT. A transform of this type is used to filter a two-dimensional picture (e.g., 256 X 256 samples), and the results are presented with a comparison to the standard FFT. An absence of roundoff errors is an important feature of this technique.
Convolution, cyclic groups, dynamic range, FFT, Fermat prime, Fourier transform, Galois field, Mersenne prime.
T.K. Truong, Y.S. Kwoh, I.S. Reed, "Image Processing by Transforms Over a Finite Field", IEEE Transactions on Computers, vol.26, no. 9, pp. 874-881, September 1977, doi:10.1109/TC.1977.1674935