LTB curves with Lipschitz turn are par-regular
This work provides a theoretical equivalence for digital geometry researchers, but it is incremental as it builds on prior definitions and proofs.
The paper tackles the problem of characterizing par-regular curves, which are crucial for preserving topology in digital geometry, by proving that locally turn-bounded curves with Lipschitz turn are equivalent to par-regular curves, as stated in Theorem 3.1 and Lemma 3.2.
Preserving the topology during a digitization process is a requirement of first importance. To this end, it is classical in Digital Geometry to assume the shape borders to be par-regular. Par-regularity was proved to be equivalent to having positive reach or to belong to the class C 1,1 of curves with Lipschitz derivative. Recently, we proposed to use a larger class that encompasses polygons with obtuse angles, the locally turn-bounded curves. The aim of this technical report is to define the class of par-regular curves inside the class of locally turn-bounded curves using only the notion of turn, that is of integral curvature. To be more precise, in a previous article, we have already proved that par-regular curves are locally turn-bounded. Incidentally this proof lead us to show that the turn of par-regular curves is a Lipschitz function of their length. We call the class of curves verifying this latter property the curves with Lipschitz turn. In this technical report, we prove the converse assertion : locally turn-bounded curves with Lipschitz turn are par-regular. The equivalence is stated in Theorem 3.1 and the converse assertion is proved in Lemma 3.2. In section 1, we recall the definition of par-regularity and equivalently of sets with positive reach. In section 2, we present the notions of curves locally turn-bounded and of curves with Lipschitz turn. Throughout this latter section, some of intermediate steps (Lemmas 2.3 and 2.11) are proved just after the introduction of their related notions. The last section (section 3) is dedicated to the proof of the equivalence of the notions.