First Steps in the Algorithmic Reconstruction of Digital Convex Sets

Paolo Dulio , Andrea Frosini (1) , Simone Rinaldi (1) , Lama Tarsissi (2) , Laurent Vuillon (3)

1 DSMI - Department of Mathematics and Computer Science / Dipartimento di Scienze Matematiche e Informatiche "Roberto Magari"
2 LAMA - Laboratoire de Mathématiques
3 LIAFA - Laboratoire d'informatique Algorithmique : Fondements et Applications

Digital convex (DC) sets plays a prominent role in the framework of digital geometry providing a natural generalization to the concept of Euclidean convexity when we are dealing with polyominoes, i.e., finite and connected sets of points. A result by Brlek, Lachaud, Provençal and Reutenauer (see [4]) on this topic sets a bridge between digital con-vexity and combinatorics on words: the boundary word of a DC poly-omino can be divided in four monotone paths, each of them having a Lyndon factorization that contains only Christoffel words. The intent of this paper is to provide some local properties that a boundary words has to fulfill in order to allow a single point modifications that preserves the convexity of the polyomino.

Domaines: Mathématique discrète [cs.DM]

Paolo Dulio, Andrea Frosini, Simone Rinaldi, Lama Tarsissi, Laurent Vuillon. First Steps in the Algorithmic Reconstruction of Digital Convex Sets. Word, Aug 2017, Montreal, Canada. ⟨10.1007/978-3-319-66396-8⟩. ⟨hal-02315353⟩