Polyominoes Simulating Arbitrary-Neighborhood Zippers and Tilings

This paper provides a bridge between the classical tiling theory and the complex neighborhood self-assembling situations that exist in practice. The neighborhood of a position in the plane is the set of coordinates which are considered adjacent to it. This includes classical neighborhoods of size four, as well as arbitrarily complex neighborhoods. A generalized tile system consists of a set of tiles, a neighborhood, and a relation which dictates which are the "admissible" neighboring tiles of a given tile. Thus, in correctly formed assemblies, tiles are assigned positions of the plane in accordance to this relation. We prove that any validly tiled path defined in a given but arbitrary neighborhood (a zipper) can be simulated by a simple "ribbon" of microtiles. A ribbon is a special kind of polyomino, consisting of a non-self-crossing sequence of tiles on the plane, in which successive tiles stick along their adjacent edge. Finally, we extend this construction to the case of traditional tilings, proving that we can simulate arbitrary-neighborhood tilings by simple-neighborhood tilings, while preserving some of their essential properties.Comment: Submitted to Theoretical Computer Scienc

Reconstruction of Convex Sets from One or Two X-rays

We consider a class of problems of Discrete Tomography which has been deeply investigated in the past: the reconstruction of convex lattice sets from their horizontal and/or vertical X-rays, i.e. from the number of points in a sequence of consecutive horizontal and vertical lines. The reconstruction of the HV-convex polyominoes works usually in two steps, first the filling step consisting in filling operations, second the convex aggregation of the switching components. We prove three results about the convex aggregation step: (1) The convex aggregation step used for the reconstruction of HV-convex polyominoes does not always provide a solution. The example yielding to this result is called \textit{the bad guy} and disproves a conjecture of the domain. (2) The reconstruction of a digital convex lattice set from only one X-ray can be performed in polynomial time. We prove it by encoding the convex aggregation problem in a Directed Acyclic Graph. (3) With the same strategy, we prove that the reconstruction of fat digital convex sets from their horizontal and vertical X-rays can be solved in polynomial time. Fatness is a property of the digital convex sets regarding the relative position of the left, right, top and bottom points of the set. The complexity of the reconstruction of the lattice sets which are not fat remains an open question.Comment: 31 pages, 24 figure

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

A benchmark set for the reconstruction of hv-convex discrete sets

AbstractIn this paper we summarize the most important generation methods developed for the subclasses of hv-convex discrete sets. We also present some new generation techniques to complement the former ones thus making it possible to design a complete benchmark set for testing the performance of reconstruction algorithms on the class of hv-convex discrete sets and its subclasses. By using this benchmark set the paper also collects several statistics on hv-convex discrete sets, which are of great importance in the analysis of algorithms for reconstructing such kinds of discrete sets

Approximate X-rays reconstruction of special lattice sets

Sometimes the inaccuracy of the measurements of the X-rays can give rise to an inconsistent reconstruction problem. In this paper we address the problem of reconstructing special lattice sets in Z2 from their approximate X-rays in a finite number of prescribed lattice directions. The class of "strongly Q-convex sets" is taken into consideration and a polynomial time algorithm for reconstructing members of that class with line sums having possibly some bounded differences with the given X-ray values is provided. In particular, when these differences are zero, the algorithm exactly reconstructs any set. As a result, this algorithm can also be used to reconstruct convex subsets of Z2 from their exact X-rays in a finite set of suitable prescribed lattice directions

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

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

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