Revisiting Digital Straight Segment Recognition

TL;DR

This paper investigates digital straight segments, providing new analytical formulas and relations that improve understanding of their recognition process and geometric properties, especially in relation to the Stern-Brocot tree.

Contribution

It introduces new parameters and relations for digital straight segments, enhancing the analysis of their evolution during recognition and their geometric characteristics.

Findings

Derived formulas for parameter evolution during recognition

Connected digital segment properties to Stern-Brocot tree structure

Bounded slope differences between consecutive segments

Abstract

This paper presents new results about digital straight segments, their recognition and related properties. They come from the study of the arithmetically based recognition algorithm proposed by I. Debled-Rennesson and J.-P. Reveill\`es in 1995 [Debled95]. We indeed exhibit the relations describing the possible changes in the parameters of the digital straight segment under investigation. This description is achieved by considering new parameters on digital segments: instead of their arithmetic description, we examine the parameters related to their combinatoric description. As a result we have a better understanding of their evolution during recognition and analytical formulas to compute them. We also show how this evolution can be projected onto the Stern-Brocot tree. These new relations have interesting consequences on the geometry of digital curves. We show how they can for instance be used to bound the slope difference between consecutive maximal segments.