Image Moment - Raw Moments

Raw Moments

For a 2D continuous function f(x,y) the moment (sometimes called "raw moment") of order (p + q) is defined as

for p,q = 0,1,2,... Adapting this to scalar (greyscale) image with pixel intensities I(x,y), raw image moments M_ij are calculated by

In some cases, this may be calculated by considering the image as a probability density function, i.e., by dividing the above by

A uniqueness theorem (Hu ) states that if f(x,y) is piecewise continuous and has nonzero values only in a finite part of the xy plane, moments of all orders exist, and the moment sequence (M_pq) is uniquely determined by f(x,y). Conversely, (M_pq) uniquely determines f(x,y). In practice, the image is summarized with functions of a few lower order moments.