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 = (alpha)x + beta, (0 less than or equal to alpha, beta less than 1). An asymptotic estimate for L(N) that might prove useful for many applications is proved; namely, L(N) = (N cubed)/(pi squared) + O(N squared log N). An application to an image registration problem is given.