Faster image template matching in the sum of the absolute value of differences measure

Given an m x m image I and a smaller n x n image P, the computation of an (m-n+1) x (m-n+1) matrix C where C(i, j) is of the form C(i,j)=Sigma(k=0)(n-1)Sigma(k'=0)(n-1)f(I(i+k,j+k'), P(k,k')), 0= or <i, j= or <m-n for some function f, is often used in template matching. Frequent choices for the function f are f(x,y)=(x-y)(2) and f(x,y)=/m-y/. For the case when f(x,y)=(x-y)(2), it is well known that C is computable in O(m(2) log n) time. For the case f(x,y)=/-y/, on the other hand, the brute force O((m-n+1)(2)n(2)) time algorithm for computing C seems to be the best known. This paper gives an asymptotically faster algorithm for computing C when f(x,y)=/x-y/, one that runs in time O(min{s,n/square root log n}m(2) log n) time, where s is the size of the alphabet, i.e., the number of distinct symbols that appear in I and P. This is achieved by combining two algorithms, one of which runs in O(sm(2) log n) time, the other in O(m(2)n square root log n) time. We also give a simple Monte Carlo algorithm that runs in O(m(2) log n) time and gives unbiased estimates of C.