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

Given an m×m image I and a smaller n×n image P, the computation of an (m-n+1)×(m-n+1) matrix C where C(i, j) is of the form C(i,j)=Σ_k=0^n-1Σ_k´=0 ^n-1f(I(i+k,j+k´), P(k,k´)), 0⩽i, j⩽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(m² log n) time. For the case f(x,y)=|-y|, on the other hand, the brute force O((m-n+1)²n²) 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/√log n}m² 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(sm² log n) time, the other in O(m²n√log n) time. We also give a simple Monte Carlo algorithm that runs in O(m² log n) time and gives unbiased estimates of C