Recognition of Arithmetic Line Segments and Hyperplanes Using the Stern-Brocot Tree

The classic problem of discrete structure recognition is revisited in this article. We focus on naive digital straight segments (DSS) and, more generally, naive arithmetic hyperplanes, and we present a new approach to recognise these discrete structures based on the Stern-Brocot tree. The algorithm for DSS recognition proposes an alternative method to the state of the art, keeping the linear complexity and incremental character. While most of the concepts can be generalised to planes in dimension 3 and hyperplanes in higher dimensions, certain points in the process of descending in the Stern-Brocot tree need to be explored further. The proposed algorithm calculates separating chords characterising the membership of planes to cones generated by the branch of the Stern-Brocot tree. This generalisation shows the close link between arithmetic hyperplanes and the generalised Stern-Brocot tree and opens up interesting perspectives for the recognition of pieces of arithmetic hyperplanes.