Binary Matrices under the Microscope: A Tomographical Problem

A binary matrix can be scanned by moving a fixed rectangular window (submatrix) across it, rather like examining it closely under a microscope. With each viewing, a convenient measurement is the number of 1s visible in the window, which might be thought of as the luminosity of the window. The rectangular scan of the binary matrix is then the collection of these luminosities presented in matrix form. We show that, at least in the technical case of a smooth m x n binary matrix, it canbe reconstructed from its rectangular scan in polynomial time in the parameters m and n, where the degree of the polynomial depends on the size of the window of inspection. For an arbitrary binary matrix, we then extend this result by determining the entries in its rectangular scan that preclude the smoothness of the matrix.Comment: 25 pages, 15 figures, submitte

The NP-completeness of a tomographical problem on bicolored domino tilings

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On Some Geometric Aspects of the Class of hv-Convex Switching Components

In the usual aim of discrete tomography, the reconstruction of an unknown discrete set is considered, by means of projection data collected along a set U of discrete directions. Possible ambiguous reconstructions can arise if and only if switching components occur, namely, if and only if non-empty images exist having null projections along all the directions in U. In order to lower the number of allowed reconstructions, one tries to incorporate possible extra geometric constraints in the tomographic problem. In particular, the class P of horizontally and vertically convex connected sets (briefly, hv-convex polyominoes) has been largely considered. In this paper we introduce the class of hv-convex switching components, and prove some preliminary results on their geometric structure. The class includes all switching components arising when the tomographic problem is considered in P, which highly motivates the investigation of such configurations. It turns out that the considered class can be partitioned in two disjointed subclasses of closed patterns, called windows and curls, respectively. It follows that all windows have a unique representation, while curls consist of interlaced sequences of sub-patterns, called Z-paths, which leads to the problem of understanding the combinatorial structure of such sequences. We provide explicit constructions of families of curls associated to some special sequences, and also give additional details on further allowed or forbidden configurations by means of a number of illustrative examples

The reconstruction of a subclass of domino tilings from two projections

AbstractWe present a new way of studying the classical and still unsolved problem of the reconstruction of a domino tiling from its row and column projections. After giving a simple greedy strategy for solving the problem from one projection, we introduce the concept of degree of a domino tiling. We generalize an algorithm for the reconstruction of domino tilings of degree two from two projections, to domino tilings of degree three and four

A Brief Introduction to Multidimensional Persistent Betti Numbers

In this paper, we propose a brief overview about multidimensional persistent Betti numbers (PBNs) and the metric that is usually used to compare them, i.e., the multidimensional matching distance. We recall the main definitions and results, mainly focusing on the 2-dimensional case. An algorithm to approximate n-dimensional PBNs with arbitrary precision is described

Characterization of hv-Convex Sequences

Reconstructing a discrete object by means of X-rays along a finite set U of (discrete) directions represents one of the main task in discrete tomography. Indeed, it is an ill-posed inverse problem, since different structures exist having the same projections along all lines whose directions range in U. Characteristic of ambiguous reconstructions are special configurations, called switching components, whose understanding represents a main issue in discrete tomography, and an independent interesting geometric problem as well. The investigation of switching component usually bases on some kind of prior knowledge that is incorporated in the tomographic problem. In this paper, we focus on switching components under the constraint of convexity along the horizontal and the vertical directions imposed to the unknown object. Moving from their geometric characterization in windows and curls, we provide a numerical description, by encoding them as special sequences of integers. A detailed study of these sequences leads to the complete understanding of their combinatorial structure, and to a polynomial-time algorithm that explicitly reconstructs any of them from a pair of integers arbitrarily given

On the Reconstruction of 3-Uniform Hypergraphs from Degree Sequences of Span-Two

A nonnegative integer sequence is k-graphic if it is the degree sequence of a k-uniform simple hypergraph. The problem of deciding whether a given sequence π is 3-graphic has recently been proved to be NP-complete, after years of studies. Thus, it acquires primary relevance to detect classes of degree sequences whose graphicality can be tested in polynomial time in order to restrict the NP-hard core of the problem and design algorithms that can also be useful in different research areas. Several necessary and few sufficient conditions for π to be k-graphic, with k≥ 3 , appear in the literature. Frosini et al. defined a polynomial time algorithm to reconstruct k-uniform hypergraphs having regular or almost regular degree sequences. Our study fits in this research line providing a combinatorial characterization of span-two sequences, i.e., sequences of the form π= (d, … , d, d- 1 , … , d- 1 , d- 2 , … , d- 2 ) , d≥ 2 , which are degree sequences of some 3-uniform hypergraphs. Then, we define a polynomial time algorithm to reconstruct one of the related 3-uniform hypergraphs. Our results are likely to be easily generalized to k≥ 4 and to other families of degree sequences having simple characterization, such as gap-free sequences