On local definitions of length of digital curves

In this paper we investigate the 'local' definitions of length of digital curves in the digital space rZ 2 where r is the resolution of the discrete space. We prove that if μ r is any local definition of the length of digital curves in rZ 2 , then for almost all segments S of R 2 , the measure μ r (S r ) does not converge to the length of S when the resolution r converges to 0, where S r is the Bresenham discretization of the segment S in rZ 2 . Moreover, the average errors of classical local definitions are estimated, and we define a new one which minimizes this error.