Digital segments and Hausdorff discretization

In this paper we investigate some properties of digital segments in floor and Hausdorff discretizations. We characterize the Hausdorff discretization of straight lines and we prove that the frequency of digital segment in a digital straight line is continuous and piecewise affine function relatively to the slope. It allows to prove some combinatorial properties of digital segments. In particular we give a new proof of the results in [3,2,8] corresponding to the frequencies and the numbers of digital segments of size m.