Efficient Legendre moment computation for grey level images

Legendre orthogonal moments have been widely used in the field of image analysis. Because their computation by a direct method is very time expensive, recent efforts have been devoted to the reduction of computational complexity. Nevertheless, the existing algorithms are mainly focused on binary images. We propose here a new fast method for computing the Legendre moments, which is not only suitable for binary images but also for grey level images. We first establish a recurrence formula of one-dimensional (1D) Legendre moments by using the recursive property of Legendre polynomials. As a result, the 1D Legendre moments of order p  , Lp=Lp(0)Lp=Lp(0), can be expressed as a linear combination of Lp-1(1)Lp-1(1) and Lp-2(0)Lp-2(0). Based on this relationship, the 1D Legendre moments Lp(0)Lp(0) can thus be obtained from the arrays of L1(a)L1(a) and L0(a)L0(a), where a is an integer number less than p  . To further decrease the computation complexity, an algorithm, in which no multiplication is required, is used to compute these quantities. The method is then extended to the calculation of the two-dimensional Legendre moments LpqLpq. We show that the proposed method is more efficient than the direct method.