Image characterization by fast calculation of low-order Legendre moments

Moments, which are projections of the input signal onto the polynomial function space, have found wide applications in image processing and computer vision. By use of orthogonal Legendre polynomial bases, the calculation can be reduced, the error is easier to estimate, and the reconstruction can be more simple also. In the present paper, we propose the fast calculation to characterise images by Legendre moments. We first present the recursive property of Legendre moments and analyse their recursive calculation. The implementation of the recursive calculation of Legendre moments in discrete case is presented. The fast algorithm is then generalized to 2D cases. We show that with our algorithm, the computational complexity to calculate the Legendre moments is much reduced, and it is independent of the window size. Moreover, we show that the higher order Legendre moments are just linear combinations of Legendre moments of order 0 and 1 of the integrated signals, so we can only use these low order Legendre moments to characterise the input signal, and the computational complexity is thus further reduced.