On the computation of integrals over fixed-size rectangles of arbitrary dimension

We present a formula for computing integrals over rectangles of arbitrary dimension in the special case that the size of the rectangles of interest is constant. Compared to the popular method of integral images, our approach reduces the amount of auxiliary memory and alleviates the effect of precision loss when computing large integral images. As an application of this technique we present a fast implementation of the Normalized Cross Correlation (NCC) metric for deformable image registration and test it with the popular Symmetric Normalization (SyN) registration algorithm dramatically reducing the computational cost in terms of both time and temporary memory.