Fast and exact computation of Cartesian geometric moments using discrete Green's theorem

Green's theorem evaluates a double integral over a planar object by a simple integration along the boundary of the object. It can be used in fast computation of Cartesian geometric moments of two-dimensional binary objects, since the object shape is totally determined by the boundary. Li and Shen [Pattern Recognition 24, 807-813 (1991)] used an approximation of Green's theorem to deal with discrete objects. This method is fast, but not accurate. In this paper, we present a discrete version of Green's theorem, which evaluates a double sum over a two-dimensional discrete object by a simple summation along the discrete boundary of the object. By using the discrete Green's theorem, we propose a new algorithm for fast computation of the geometric moments. The new algorithm is faster than the previous methods, and gives exactly the same results as if a double sum were used. The importance of the exact computation is discussed by examining the accuracy of Hu's moment invariants and some other moment based shape features, which are frequently used in various shape description and pattern recognition tasks. A fast method for computing moments of regions in gray level images, using the discrete Green's theorem, is also presented. We show that the moments of this type are useful in structural texture analysis. Copyright (C) 1996 Pattern Recognition Society.