Computing moments by prefix sums

The goal of many image processing tasks is to recognise objects, to match scenes and to classify images. The shape, gray or colour features of objects are mainly used to describe objects. The most efficient method used to describe such features of an object is to use various transformed moments. We express the moments as a linear combination of higher order prefix sums, obtained by iterating the prefix sum computation on previous prefix sums, starting with the original function values. Thus the p´th moment m_p=Σ_x=1^N x^pf(x) can be computed by O(N·p) additions followed by p multiply-adds. The prefix summations can be realized in time O(N) using p+1 simple adders, and in time O(p·log N) using parallel prefix computation and O(N) adders. In 1986 Hatamian published a computationally equivalent algorithm, based on a cascade of filters performing the summations. Our recursive derivation allows for explicit expressions and recursive equations for the coefficients used in the final moment calculation. Thus a number of alternative forms for the moment computation can be derived, based on different sets of prefix sums, allowing some simplifications in the implementations