Digital straight line segments and curves

Abstract: This thesis consists of two papers:Paper A. Chord properties of digital straight line segments.This paper treats digital straight line segments in two different cases, in the 8-connected plane and in the Khalimsky plane. We investigate them using a new classification, dividing them into a union of horizontal and diagonal segments. Then we study necessary and sufficient conditions for straightness in both cases, using vertical distances for certain points. We also establish necessary and sufficient conditions in the 8-connected plane as well as in the Khalimsky plane by transforming their chain codes. Using this technique we can transform Khalimsky lines to the 8-connected case.Paper B. The number of Khalimsky-continuous functions on intervals.This paper deals with Khalimsky-continuous functions. We consider these functions when they have two, three or four points in their codomain. In the case of two points in the codomain, we see a new example of the classical Fibonacci sequence. In the study of functions with three and four points in their codomain, we find some new sequences, the asymptotic behavior of which we investigate. Finally, we consider Khalimsky-continuous functions with one fixed endpoint. In this case, we get a sequence which has the same recursion relation as the Pell numbers but different initial values. We also obtain a new example of the Delannoy numbers.