Computation of Convolutions and Discrete Fourier Transforms by Polynomial Transforms

Discrete transforms are introduced and are defined in a ring of polynomials. These polynomial transforms are shown to have the convolution property and can be computed in ordinary arithmetic, without multiplications. Polynomial transforms are particularly well suited for computing discrete two-dimensional convolutions with a minimum number of operations. Efficient algorithms for computing one-dimensional convolutions and Discrete Fourier Transforms are then derived from polynomial transforms. Introduction The calculation of two-dimensional digital convolutions technique. applicable to one-dimensional nd multihas many applications, particularly in image processing. The main problem associated with these convolutions relates to the huge processing load required for their computation. Direct calculation of a two-dimensional convolution of dimension N x N corresponds to N4 multiplications and N 2 ( N 2 1) additions. A substantial reduction in the number of operations can be achieved if direct computation is replaced by a Fast Fourier Transform (FFT) [ I ] or a Number Theoretic Transform (NTT) [2-61 approach. These two techniques, however, do not provide an optimum solution for evaluating two-dimensional convolutions. Computation by means of FFTs introduces a significant amount of roundoff errors, requires ome means of processing trigonometric functions, and corresponds to still a large amount of multiplying. NTTs can be calculated without multiplications, and allow the computation of convolutions without quantization errors. However, such transforms suffer severe word length and transform length limitations. Moreover, implementation of modular arithmetic in general purpose computers is sometimes inefficient. Recently, Agarwal and Cooley [7, 81 introduced new algorithms for digital convolutions. Their method is based dimensional convolutions, is attractive because i t appears to be computationally more efficient than the FFT for lengths up to 400 and because it does not place any restrictions on the arithmetic nor require any manipulation of trigonometric functions. In this paper, we extend earlier work by Nussbaumer [9] on polynomial transforms to cover the various polynomial transforms that can be calculated without multiplications. We show that these transforms hdve the convolution property and yield efficient algorithms for computing two-dimensional convolutions in ordinary arithmetic. These algorithms correspond to a number of multiplications equal to M,N2N3 . . . rather than M1M2M3 . . . with the Agarwal-Cooley method and are therefore particularly efficient for large convolutions. These algorithms are then extended to one-dimensional convolutions. ‘ I ; In this case, since two-dimensional convolutions computed by polynomial transforms are usually such that both dimensions have a common factor, the efficient twodimensional to one-dimensional mapping introduced by Agarwal and Cooley [7, 81 is not applicable, and the computation of one-dimensional convolutions is done at the expense of some reduction in computing efficiency by usupon a nesting of several short convolutions having ing the approach proposed by Agarwal and Burrus in [lo]. lengths N , , N,, N,, . . ., which are relatively prime, and We show also that polynomial transforms can be used yields a total number of multiplications equal to M1M2M3, for computing Discrete Fourier Transforms (DFTs) and . . ., where M , , M,, M,, . . . are the number of multiplicaallow, in some cases, a sigdificant reduction in number of tions required to calculate the short convolutions of operations when compared to the Winograd Fourier lengths N , , N,, N 3 , . . ., and are such that Mi = 2Ni. This Transform Algorithm (WFTA) [ll-131. Copyright 1978 by International Business Machines Corporation. Copying is permitted without payment of royalty provided that (1) each reproduction is done without alteration and (2) theJournu1 reference and IBM copyright notice are included on the first page. The title and abstract may be used without further permission in computer-based and other information-service systems. Permission to republish other 134 excerpts should be obtained from the Editor. H. J . NUSSBAUMER ND P. QUANDALLE 1BM J. RES. DEVELOP. VOL. 22 NO. 2 MAR 1978 Polynomial transforms Let yU,/ be a two-dimensional circular convolution of dimension q X q with In polynomial notations, yU,/ can be obtained from a set of q polynomials Y,(Z), by taking the coefficient of Z" in the polynomial YJZ) with Y1 Y,(Z) = 1 yu,/zu 1 = 0, I , . . . , q -1. ( 2 )