A recursive updating rule for efficient computation of linear moments in sliding-window applications

The computation of linear moment matrices, whose elements are defined as zeroth order integration values of an image, was recently introduced as a tool to reduce the computational cost required to obtain the geometric moments of an image. The main relevance of these matrices is twofold: on one hand, they can be efficiently obtained by means of accumulation filters, which only require additions; on the other one, their relation to geometric moments, as well as to discrete Fourier spectrum coefficients, allows the exchange and interpretation of many results from different areas of image processing and pattern recognition. Taking into account the relevance of these matrices, a new recursive property that allows their efficient computation in sliding-window processes is presented here. First, a scalar recursive updating rule is formulated. It relates the value of each element of a linear moment matrix to those calculated in the previous location of the sliding-window. Then, this result is reformulated to obtain an explicit matrix formula. The obtained recursive updating rule has a straightforward application in many different fields involving sliding window processes in order to efficiently obtain local features related to geometric moments and discrete Fourier spectrum coefficients