Xavier Provençal

Lyndon + Christoffel = digitally convex

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.

On the tiling by translation problem

On square or hexagonal lattices, tiles or polyominoes are coded by words. The polyominoes that tile the plane by translation are characterized by the Beauquier-Nivat condition. By using the constant time algorithms for computing the longest common extensions in two words, we provide a linear time algorithm in the case of pseudo-square polyominoes, improving the previous quadratic algorithm of Gambini and Vuillon. We also have a linear algorithm for pseudo-hexagon polyominoes not containing arbitrarily large square factors. The results are extended to more general tiles.

A linear time and space algorithm for detecting path intersection in Zd

The Freeman chain code is a common and useful way for representing discrete paths by means of words such that each letter encodes a step in a given direction. In the discrete plane Z^2 such a coding is widely used for representing connected discrete sets by their contour which forms a closed and intersection free path. In this paper, we use a multidimensional radix tree like data structure for storing paths in the discreted-dimensional space Z^d. It allows to design a simple and efficient algorithm for detecting path intersection. Even though an extra initialization is required, the time and space complexities remain linear for any fixed dimension d. Several problems that are solved by adapting our algorithm are also discussed.

Two linear-time algorithms for computing the minimum length polygon of a digital contour

The Minimum Length Polygon (MLP) is an interesting first order approximation of a digital contour. For instance, the convexity of the MLP is characteristic of the digital convexity of the shape, its perimeter is a good estimate of the perimeter of the digitized shape. We present here two novel equivalent definitions of MLP, one arithmetic, one combinatorial, and both definitions lead to two different linear time algorithms to compute them.

An optimal algorithm for detecting pseudo-squares

We consider the problem of determining if a given word, which encodes the boundary of a discrete figure, tiles the plane by translation These words have been characterized by the Beauquier-Nivat condition, for which we provide a linear time algorithm in the case of pseudo-square polyominoes, improving the previous quadratic algorithm of Gambini and Vuillon.

Critical connectedness of thin arithmetical discrete planes

The critical thickness of an arithmetical discrete plane refers to the infimum thickness that preserves its 2-connectedness. This infimum thickness can be computed thanks to a multidimensional continued fraction algorithm, namely the fully subtractive algorithm. We provide a characterization of the discrete planes with critical thickness that contain the origin and that are 2-connected.

A linear time and space algorithm for detecting path intersection

For discrete sets coded by the Freeman chain describing their contour, several linear algorithms have been designed for determining their shape properties. Most of them are based on the assumption that the boundary word forms a closed and non-intersecting discrete curve. In this article, we provide a linear time and space algorithm for deciding whether a path on a square lattice intersects itself. This work removes a drawback by determining efficiently whether a given path forms the contour of a discrete figure. This is achieved by using a radix tree structure over a quadtree, where nodes are the visited grid points, enriched with neighborhood links that are essential for obtaining linearity.

Combinatorial view of digital convexity

The notion of convexity translates non-trivially from Euclidean geometry to discrete geometry, and detecting if a discrete region of the plane is convex requires analysis. In this paper we study digital convexity from the combinatorics on words point of view, and provide a fast optimal algorithm checking digital convexity of polyominoes coded by the contour word. The result is based on the Lyndon factorization of the contour word, and the recognition of Christoffel factors that are approximations of digital lines.

Palindromic Complexity of Trees

We consider finite trees with edges labeled by letters on a finite alphabet \(\varSigma \). Each pair of nodes defines a unique labeled path whose trace is a word of the free monoid \(\varSigma ^*\). The set of all such words defines the language of the tree. In this paper, we investigate the palindromic complexity of trees and provide hints for an upper bound on the number of distinct palindromes in the language of a tree.

Facet Connectedness of Discrete Hyperplanes with Zero Intercept: The General Case

A digital discrete hyperplane in ℤ d is defined by a normal vector v, a shift μ, and a thickness θ. The set of thicknesses θ for which the hyperplane is connected is a right unbounded interval of ℝ + . Its lower bound, called the connecting thickness of v with shift μ, may be computed by means of the fully subtractive algorithm. A careful study of the behaviour of this algorithm allows us to give exhaustive results about the connectedness of the hyperplane at the connecting thickness in the case μ = 0. We show that it is connected if and only if the sequence of vectors computed by the algorithm reaches in finite time a specific set of vectors which has been shown to be Lebesgue negligible by Kraaikamp & Meester.