The filter response F is given by The standard computation of F takes O(n) operations; however, for the special case of binary filters like the ones shown in figures 1a) and 1b), we can compute this response in constant time with some preprocessing as follows.
let I be the integral signal, where each element of I is given by this summation can be computed with two accesses to memory, after pre-computing and storing the values of I in an array.