Fast numerically stable computation of orthogonal Fourier--Mellin moments

An efficient algorithm for the computation of the orthogonal Fourier-Mellin moments (OFMMs) is presented. The proposed method computes the fractional parts of the orthogonal polynomials, which consist of fractional terms, recursively, by eliminating the number of factorial calculations. The recursive computation of the fractional terms makes the overall computation of the OFMMs a very fast procedure in comparison with the conventional direct method. Actually, the computational complexity of the proposed method is linear O(p) in multiplications, with p being the moment order, while the corresponding complexity of the direct method is O(p²). Moreover, this recursive algorithm has better numerical behaviour, as it arrives at an overflow situation much later than the original one and does not introduce any finite precision errors. These are the two major advantages of the algorithm introduced in the current work, establishing the computation of the OFMMs to a very high order as a quite easy and achievable task. Appropriate simulations on images of different sizes justify the superiority of the proposed algorithm over the conventional algorithm currently used