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 images from discrete X-rays

We present a new algorithm for reconstructing binary images from their projections along a small number of directions. Our algorithm performs a sequence of related reconstructions, each using only two projections. The algorithm makes extensive use of network flow algorithms for solving the two-projection subproblems. Our experimental results demonstrate that the algorithm can compute reconstructions which resemble the original images very closely from a small number of projections, even in the presence of noise. Although the effectiveness of the algorithm is based on certain smoothness assumptions about the image, even tiny, non-smooth details are reconstructed exactly. The class of images for which the algorithm is most effective includes images of convex objects, but images of objects that contain holes or consist of multiple components can also be reconstructed with great accurac

Non-convex image reconstruction via Expectation Propagation

Tomographic image reconstruction can be mapped to a problem of finding solutions to a large system of linear equations which maximize a function that includes \textit{a priori} knowledge regarding features of typical images such as smoothness or sharpness. This maximization can be performed with standard local optimization tools when the function is concave, but it is generally intractable for realistic priors, which are non-concave. We introduce a new method to reconstruct images obtained from Radon projections by using Expectation Propagation, which allows us to reframe the problem from an Bayesian inference perspective. We show, by means of extensive simulations, that, compared to state-of-the-art algorithms for this task, Expectation Propagation paired with very simple but non log-concave priors, is often able to reconstruct images up to a smaller error while using a lower amount of information per pixel. We provide estimates for the critical rate of information per pixel above which recovery is error-free by means of simulations on ensembles of phantom and real images.Comment: 12 pages, 6 figure

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

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