On the number of digital straight line segments

A closed-form expression has been reported in the literature for L_N, the number of digital line segments of length N that correspond to lines of the form y=ax+β, O⩽α, β<1. The authors prove an asymptotic estimate for L_N that might prove useful for many applications, namely, L_N=N ³/π²+O(N² log N). An application to an image registration problem is given