On local definitions of length of digital curves

Abstract

In this paper we investigate the 'local' definitions of length of digital curves in the digital space rℤ2 where r is the resolution of the discrete space. We prove that if μr is any local definition of the length of digital curves in rℤ2, then for almost all segments S of ℝ2, the measure μr(Sr) does not converge to the length of S when the resolution r converges to 0, where Sr is the Bresenham discretization of the segment 5 in rℤ2. Moreover, the average errors of classical local definitions are estimated, and we define a new one which minimizes this error. © Springer-Verlag Berlin Heidelberg 2003.