Alberto Del Lungo

The number of convex polyominoes reconstructible from their orthogonal projections

Abstract Many 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 ∈ N n and H ∈ N m . 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.

Random generation of trees and other combinatorial objects

Abstract In this paper, we present a general method for the random generation of some classes of combinatorial objects. Our basic idea is to translate ECO method (Enumerating Combinatorial Objects) from a method for the enumeration of combinatorial objects into a random generation method. The algorithms we illustrate are based on the concepts of succession rule and generating tree: the former is a law that predicts the combinatorial object class growth according to a given parameter. The generating tree related to a given succession rule is a particular labelled plane tree that represents the rule in an extensive way. Each node of a generating tree can also be seen as a particular combinatorial object and so a random path in the generating tree coincides with the random generation of that combinatorial object. The generation is uniform if we take the probability of each branch to be selected into account when the path is generated. We also give the formulae evaluating complexity. Finally, we take the class of m -ary trees into consideration in order to illustrate our general method. In this case, the average time complexity of the generating algorithm can be estimated as O( mn ).

Comparison of algorithms for reconstructing hv-convex discrete sets

Three reconstruction algorithms to be used for reconstructing hv-convex discrete sets from their row and column sums are compared. All these algorithms have two versions: one for reconstructing hv-convex polyominoes and another one for reconstructing hv-convex 8-connected discrete sets. In both classes of discrete sets the algorithms are compared from the viewpoints of average execution time and memory complexities. Discrete sets with given sizes are generated with uniform distribution, and then reconstructed from their row and column sums. First we have implemented two previously published algorithms. According to our comparisons, the algorithm which was better from the viewpoint of worst time complexity was the worse from the viewpoint of average time complexity and memory requirements. Then, as a new method, a combination of the two algorithms was also implemented and it is shown that it inherits the best properties of the other two methods.

X-rays characterizing some classes of discrete sets

In this paper, we study the problem of determining discrete sets by means of their X-rays. An X-ray of a discrete set F in a direction u counts the number of points in F on each line parallel to u. A class F of discrete sets is characterized by the set U of directions if each element in F is determined by its X-rays in the directions of U. By using the concept of switching component introduced by Chang and Ryser [Comm. ACM 14 (1971) 21; Combinatorial Mathematics, The Carus Mathematical Monographs, No. 14, The Mathematical Association of America, Rahway, 1963] and extended in [Discrete Comput. Geom. 5 (1990) 223], we prove that there are some classes of discrete sets that satisfy some connectivity and convexity conditions and that cannot be characterized by any set of directions. Gardner and Gritzmann [Trans. Amer. Math. Soc. 349 (1997) 2271] show that any set U of four directions having cross ratio that does not belong to {4/3,3/2,2,3,4}, characterizes the class of convex sets. We prove the converse, that is, if Us cross ratio is in {4/3,3/2,2,3,4}, then the hv-convex sets cannot be characterized by U. We show that if the horizontal and vertical directions do not belong to U, Gardner and Gritzmanns result cannot be extended to hv-convex polyominoes. If the horizontal and vertical directions belong to U and Us cross ratio is not in {4/3,3/2,2,3,4}, we believe that U characterizes the class of hv-convex polyominoes. We give experimental evidence to support our conjecture. Moreover, we prove that there is no number δ such that, if |U|⩾δ, then U characterizes the hv-convex polyominoes. This number exists for convex sets and is equal to 7 (see [Trans. Amer. Math. Soc. 349 (1997) 2271]).

Discrete Tomography: Reconstruction under Periodicity Constraints

This paper studies the problem of reconstructing binary matrices that are only accessible through few evaluations of their discrete X-rays. Such question is prominently motivated by the demand in material science for developing a tool for the reconstruction of crystalline structures from their images obtained by high-resolution transmission electron microscopy. Various approaches have been suggested for solving the general problem of reconstructing binary matrices that are given by their discrete X-rays in a number of directions, but more work have to be done to handle the ill-posedness of the problem. We can tackle this ill-posedness by limiting the set of possible solutions, by using appropriate a priori information, to only those which are reasonably typical of the class of matrices which contains the unknown matrix that we wish to reconstruct. Mathematically, this information is modelled in terms of a class of binary matrices to which the solution must belong. Several papers study the problem on classes of binary matrices on which some connectivity and convexity constraints are imposed.We study the reconstruction problem on some new classes consisting of binary matrices with periodicity properties, and we propose a polynomial-time algorithm for reconstructing these binary matrices from their orthogonal discrete X-rays.

Reconstruction of lattice sets from their horizontal, vertical and diagonal X-rays

Abstract In this paper, we study the problem of reconstructing a lattice set from its X-rays in a finite number of prescribed directions. The problem is NP-complete when the number of prescribed directions is greater than two. We provide a polynomial-time algorithm for reconstructing an interesting subclass of lattice sets (having some connectivity properties) from its X-rays in directions (1,0), (0,1) and (1,1). This algorithm can be easily extended to contexts having more than three X-rays.

ECO method and the exhaustive generation of convex polyominoes

ECO is a method for the enumeration of classes of combinatorial objects based on recursive constructions of such classes. In this paper we use the ECO method and the concept of succession rule to develop an algorithm for the exhaustive generation of convex polyominoes. Then we prove that this algorithm runs in constant amortized time.

Reconstructing permutation matrices from diagonal sums

In this paper, we present a result concerning the reconstruction of permutation matrices from their diagonal sums. The problem of reconstructing a sum of k permutation matrices from its diagonal sums is NP-complete. We prove that a simple variant of this problem in which the permutation matrices lie on a cylinder instead of on a plane can be solved in polynomial time. We give an exact, algebraic characterization of the diagonal sums that correspond to a sum of permutation matrices. Then, we derive an O(kn2)-time algorithm for reconstructing the sum of k permutation matrices of order n from their diagonal sums. We obtain these results by means of a generalization of a classical theorem of Hall on the finite abelian groups.

Medians of Discrete Sets according to a Linear Distance

Abstract. In this paper we present some results concerning the median points of a discrete set according to a distance defined by means of two directions p and q . We describe a local characterization of the median points and show how these points can be determined from the projections of the discrete set along directions p and q . We prove that the discrete sets having some connectivity properties have at most four median points according to a linear distance, and if there are four median points they form a parallelogram. Finally, we show that the 4-connected sets which are convex along the diagonal directions contain their median points along these directions.

Reconstruction of Binary Matrices from Absorbed Projections

A generalization of the classical discrete tomography problem is considered: Reconstruct binary matrices from their absorbed row and column sums. We show that this reconstruction problem can be linked to a 3SAT problem if the absorption is characterized with the constant � = ln((1 + �5)/2).