Lyndon Christoffel digitally convex

Abstract

Discrete geometry redefines notions borrowed from Euclidean geometry creating a need for new algorithmical tools. The notion of convexity does not translate trivially, and detecting if a discrete region of the plane is convex requires a deeper analysis. To the many different approaches of digital convexity, we propose the combinatorics on words point of view, unnoticed until recently in the pattern recognition community. In this paper we provide first a fast optimal algorithm checking digital convexity of polyominoes coded by their contour word. The result is based on linear time algorithms for both computing the Lyndon factorization of the contour word, and the recognition of Christoffel factors that are approximations of digital lines. By avoiding arithmetical computations the algorithm is much simpler to implement and much faster in practice. We also consider the convex hull computation and relate previous work in combinatorics on words with the classical Melkman algorithm.

Arithmetic Discrete Planes Are Quasicrystals

“…The use of word combinatorics and the study of boundary words has recently proven to be particularly useful in discrete geometry. Let us also quote in particular [BLPR08] which gives a nice characterization of digitally convex polyominoes in terms of the Lyndon decomposition of the word coding their boundary. See also [BKP09] which gives an efficient recognition algorithm of nonintersecting paths based on the use of suffix trees.…”

A Linear Time and Space Algorithm for Detecting Path Intersection

“…It suffices to check that the last visited node is the starting one. This does not penalize the linear algorithms for determining, for instance, if a discrete figure is digitally convex [7], or if it tiles the plane by translation [8]. In the case of a self intersecting path, it also allows the decomposition of a discrete figure in elementary components, not necessarily disjoint.…”

“…A convenient way to represent them is to use the well-known Freeman chain code [1,2] which encodes the contour by a word w on the four letter alphabet Σ = {a, b, a, b}, corresponding to the unit displacements in the four directions (right, up, left, down) on a square grid. Among the many problems that have been considered in the literature, we mention : computations of statistics such as area, moment of inertia [3,4], digital convexity [5,6,7], and tiling of the plane by translation [8,9]. All of the above mentioned problems are solved by using algorithms that are linear in the length of the contour word, but often it is assumed that the path encoded by this word does not intersect itself.…”

Geometric Preservation of 2D Digital Objects Under Rigid Motions

“…The tangential cover of the curve [31] can be used to obtain this decomposition. Alternatively, an approach presented in [32] uses tools of combinatorics on words to study contour words: the linear Lyndon factorization algorithm [33] and the Christoffel words. A linear time algorithm decides convexity of polyominoes and can also compute the convex hull of a digital object (it is presented as a discrete version of the classical Melkman algorithm [34]).…”