A Note on Tiling under Tomographic Constraints

Given a tiling of a 2D grid with several types of tiles, we can count for every row and column how many tiles of each type it intersects. These numbers are called the_projections_. We are interested in the problem of reconstructing a tiling which has given projections. Some simple variants of this problem, involving tiles that are 1x1 or 1x2 rectangles, have been studied in the past, and were proved to be either solvable in polynomial time or NP-complete. In this note we make progress toward a comprehensive classification of various tiling reconstruction problems, by proving NP-completeness results for several sets of tiles.Comment: added one author and a few theorem

Tile Packing Tomography is NP-hard

Discrete tomography deals with reconstructing finite spatial objects from lower dimensional projections and has applications for example in timetable design. In this paper we consider the problem of reconstructing a tile packing from its row and column projections. It consists of disjoint copies of a fixed tile, all contained in some rectangular grid. The projections tell how many cells are covered by a tile in each row and column. How difficult is it to construct a tile packing satisfying given projections? It was known to be solvable by a greedy algorithm for bars (tiles of width or height 1), and NP-hardness results were known for some specific tiles. This paper shows that the problem is NP-hard whenever the tile is not a bar

Reconstructing Binary Matrices underWindow Constraints from their Row and Column Sums

The present paper deals with the discrete inverse problem of reconstructing binary matrices from their row and column sums under additional constraints on the number and pattern of entries in specified minors. While the classical consistency and reconstruction problems for two directions in discrete tomography can be solved in polynomial time, it turns out that these window constraints cause various unexpected complexity jumps back and forth from polynomial-time solvability to $\mathbb{N}\mathbb{P}$-hardness

ON DOUBLE-RESOLUTION IMAGING AND DISCRETE TOMOGRAPHY

Super-resolution imaging aims at improving the resolution of an image by enhancing it with other images or data that might have been acquired using different imaging techniques or modalities. In this paper we consider the task of doubling, in each dimension, the resolution of grayscale images of binary objects by fusion with double-resolution tomographic data that have been acquired from two viewing angles. We show that this task is polynomial-time solvable if the gray levels have been reliably determined. The problem becomes $\mathbb{N}\mathbb{P}$-hard if the gray levels of some pixels come with an error of $\pm1$ or larger. The $\mathbb{N}\mathbb{P}$-hardness persists for any larger resolution enhancement factor. This means that noise does not only affect the quality of a reconstructed image but, less expectedly, also the algorithmic tractability of the inverse problem itself.Comment: 26 pages, to appear in SIAM Journal on Discrete Mathematic

Reciprocity in Social Networks with Capacity Constraints

Directed links -- representing asymmetric social ties or interactions (e.g., "follower-followee") -- arise naturally in many social networks and other complex networks, giving rise to directed graphs (or digraphs) as basic topological models for these networks. Reciprocity, defined for a digraph as the percentage of edges with a reciprocal edge, is a key metric that has been used in the literature to compare different directed networks and provide "hints" about their structural properties: for example, are reciprocal edges generated randomly by chance or are there other processes driving their generation? In this paper we study the problem of maximizing achievable reciprocity for an ensemble of digraphs with the same prescribed in- and out-degree sequences. We show that the maximum reciprocity hinges crucially on the in- and out-degree sequences, which may be intuitively interpreted as constraints on some "social capacities" of nodes and impose fundamental limits on achievable reciprocity. We show that it is NP-complete to decide the achievability of a simple upper bound on maximum reciprocity, and provide conditions for achieving it. We demonstrate that many real networks exhibit reciprocities surprisingly close to the upper bound, which implies that users in these social networks are in a sense more "social" than suggested by the empirical reciprocity alone in that they are more willing to reciprocate, subject to their "social capacity" constraints. We find some surprising linear relationships between empirical reciprocity and the bound. We also show that a particular type of small network motifs that we call 3-paths are the major source of loss in reciprocity for real networks

Programmation mathématique en tomographie discrète

La tomographie est un ensemble de techniques visant à reconstruirel intérieur d un objet sans toucher l objet lui même comme dans le casd un scanner. Les principes théoriques de la tomographie ont été énoncéspar Radon en 1917. On peut assimiler l objet à reconstruire à une image,matrice, etc.Le problème de reconstruction tomographique consiste à estimer l objet àpartir d un ensemble de projections obtenues par mesures expérimentalesautour de l objet à reconstruire. La tomographie discrète étudie le cas où lenombre de projections est limité et l objet est défini de façon discrète. Leschamps d applications de la tomographie discrète sont nombreux et variés.Citons par exemple les applications de type non destructif comme l imageriemédicale. Il existe d autres applications de la tomographie discrète, commeles problèmes d emplois du temps.La tomographie discrète peut être considérée comme un problème d optimisationcombinatoire car le domaine de reconstruction est discret et le nombrede projections est fini. La programmation mathématique en nombres entiersconstitue un outil pour traiter les problèmes d optimisation combinatoire.L objectif de cette thèse est d étudier et d utiliser les techniques d optimisationcombinatoire pour résoudre les problèmes de tomographie.The tomographic imaging problem deals with reconstructing an objectfrom a data called a projections and collected by illuminating the objectfrom many different directions. A projection means the information derivedfrom the transmitted energies, when an object is illuminated from a particularangle. The solution to the problem of how to reconstruct an object fromits projections dates to 1917 by Radon. The tomographic reconstructingis applicable in many interesting contexts such as nondestructive testing,image processing, electron microscopy, data security, industrial tomographyand material sciences.Discete tomography (DT) deals with the reconstruction of discret objectfrom limited number of projections. The projections are the sums along fewangles of the object to be reconstruct. One of the main problems in DTis the reconstruction of binary matrices from two projections. In general,the reconstruction of binary matrices from a small number of projections isundetermined and the number of solutions can be very large. Moreover, theprojections data and the prior knowledge about the object to reconstructare not sufficient to determine a unique solution. So DT is usually reducedto an optimization problem to select the best solution in a certain sense.In this thesis, we deal with the tomographic reconstruction of binaryand colored images. In particular, research objectives are to derive thecombinatorial optimization techniques in discrete tomography problems.PARIS-CNAM (751032301) / SudocSudocFranceF

On the Reconstruction of Static and Dynamic Discrete Structures

We study inverse problems of reconstructing static and dynamic discrete structures from tomographic data (with a special focus on the `classical' task of reconstructing finite point sets in $\mathbb{R}^d$). The main emphasis is on recent mathematical developments and new applications, which emerge in scientific areas such as physics and materials science, but also in inner mathematical fields such as number theory, optimization, and imaging. Along with a concise introduction to the field of discrete tomography, we give pointers to related aspects of computerized tomography in order to contrast the worlds of continuous and discrete inverse problems