The Saturated Subpaths Decomposition in Z 2 : a short note on generalized Tangential Cover

TL;DR

This paper generalizes the Tangential Cover in Digital Geometry to use broad geometric predicates, proving linear size bounds and discussing computational complexity without relying on point connectivity.

Contribution

It introduces a generalized framework for the Tangential Cover using conservative predicates, extending its applicability to various digital primitives.

Findings

The size of the Tangential Cover is linear in the number of input points.

The approach does not depend on point connectivity.

Computational complexity depends on predicate recognition complexity.

Abstract

In this short note, we generalized the Tangential Cover used in Digital Geometry in order to use very general geometric predicates. We present the required notions of saturated $\alpha$-paths of a digital curve as well as conservative predicates which indeed cover nearly all geometric digital primitives published so far. The goal of this note is to prove that under a very general situation, the size of the Tangential Cover is linear with the number of points of the input curve. The computation complexity of the Tangential Cover depends on the complexity of incremental recognition of geometric predicates. Moreover, in the discussion, we show that our approach does not rely on connectivity of points as it might be though first.