2L convex polyominoes: discrete tomographical aspects

This paper uses the theoretical material developed in a previous article by the authors in order to reconstruct a subclass of 2L-convex polyominoes. The main idea is to control the shape of these polyominoes by combining 4 types of geometries. Some modifications are made in the reconstruction algorithm of Chrobak and Durr for HV -convex polyominoes in order to impose these geometries

2L-CONVEX POLYOMINOES: GEOMETRICAL ASPECTS

International audienceA polyomino P is called 2L-convex if for every two cells there exists a monotone path included in P with at most two changes of direction. This paper studies the geometrical aspects of a sub-class of 2L-convex polyominoes called I0,0 and states a characterization of 2L it in terms of monotone paths. In a second part, four geometries are introduced and the tomographical point of view is investigated using the switching components (that is, the elements of this sub-class that have the same projections). Finally, some unicity results are given for the reconstruction of these polyominoes according to their projections

The number of convex polyominoes reconstructible from their orthogonal projections

AbstractMany problems of computer-aided tomography, pattern recognition, image processing and data compression involve a reconstruction of bidimensional discrete sets from their projections. [3–5,10,12,16,17]. The main difficulty involved in reconstructing a set Λ starting out from its orthogonal projections (V,H) is the ‘ambiguity’ arising from the fact that, in some cases, many different sets have the same projections (V,H). In this paper, we study this problem of ambiguity with respect to convex polyominoes, a class of bidimensional discrete sets that satisfy some connection properties similar to those used by some reconstruction algorithms. We determine an upper and lower bound to the maximum number of convex polyominoes having the same orthogonal projections (V,H), with V ∈ Nn and H ∈ Nm. We prove that under these connection conditions, the ambiguity is sometimes exponential. We also define a construction in order to obtain some convex polyominoes having the same orthogonal projections

On the shape of permutomino tiles

International audienceIn this paper we explore the connections between two classes of polyominoes, namely the permu- tominoes and the pseudo-square polyominoes. A permutomino is a polyomino uniquely determined by a pair of permutations. Permutominoes, and in particular convex permutominoes, have been considered in various kinds of problems such as: enumeration, tomographical reconstruction, and algebraic characterization. On the other hand, pseudo-square polyominoes are a class of polyominoes tiling the the plane by translation. The characterization of such objects has been given by Beauquier and Nivat, who proved that a polyomino tiles the plane by translation if and only if it is a pseudo-square or a pseudo- hexagon. In particular, a polyomino is pseudo-square if its boundary word may be factorized as XY Xﰅ Yﰅ, where Xﰅ denotes the path X traveled in the opposite direction. In this paper we relate the two concepts by considering the pseudo-square polyominoes which are also convex permutominoes. By using the Beauquier-Nivat characterization we provide some geometrical and combinatorial properties of such objects, and we show for any fixed X, each word Y such that XYXﰅYﰅ is pseudo-square is prefix of an infinite word Y∞ with period 4|X|N |X|E. Also, we show that XY XﰅYﰅ are centrosymmetric, i.e. they are fixed by rotation of angle π. The proof of this fact is based on the concept of pseudoperiods, a natural generalization of periods

Boundary length of reconstructions in discrete tomography

We consider possible reconstructions of a binary image of which the row and column sums are given. For any reconstruction we can define the length of the boundary of the image. In this paper we prove a new lower bound on the length of this boundary. In contrast to simple bounds that have been derived previously, in this new lower bound the information of both row and column sums is combined