Discrete Tomography Problem Research Articles

On a graph coloring problem arising from discrete tomography

AbstractAn extension of the basic image reconstruction problem in discrete tomography is considered: given a graph G = (V,E) and a family $\cal {P}$ of chains Pi together with vectors h(Pi) = (h,…,h), one wants to find a partition V1,…,Vk of V such that for each Pi and each color j, |Vj ∩ Pi| = h. An interpretation in terms of scheduling is presented. We consider special cases of graphs and identify polynomially solvable cases; general complexity results are established in this case and also in the case where V1,…,Vk is required to be a proper vertex k‐coloring of G. Finally, we examine also the case of (proper) edge k‐colorings and determine its complexity status. © 2007 Wiley Periodicals, Inc. NETWORKS, 2008

Discrete tomography of Penrose model sets

Various theoretical and algorithmic aspects of inverse problems in discrete tomography of planar Penrose model sets are discussed. These are motivated by the demand of materials science for the reconstruction of quasicrystalline structures from a small number of images produced by quantitative high-resolution transmission electron microscopy.

Discrete tomography and the Hodge conjecture for certain abelian varieties of CM-type

We study a problem in discrete tomography on Z, and show that there is an intimate connection between the problem and the study of the Hodge cycles on abelian varieties of CM-type. This connection enables us to apply our results in tomography to obtain several infinite families of abelian varieties for which the Hodge conjecture holds.

Solving problems of discrete tomography

This is a summary of the most important results presented in the author's PhD thesis. This thesis, written in French, was defended on 24 November 2004 and supervised by Marie-Christine Costa and Christophe Picouleau. This thesis consists in developing the study of discrete tomography problems. It aims at reconstructing discrete objects (matrices, images, discrete sets, etc) that are accessible only through few projections. A copy of the thesis is available on "http: //cedric.cnam.fr/AfficheArticle.php?id = 698".

A sufficient condition for non-uniqueness in binary tomography with absorption

A new kind of discrete tomography problem is introduced: the reconstruction of discrete sets from their absorbed projections. A special case of this problem is discussed, namely, the uniqueness of the binary matrices with respect to their absorbed row and column sums when the absorption coefficient is μ = log ( ( 1 + 5 ) / 2 ) . It is proved that if a binary matrix contains a special structure of 0s and 1s, called alternatively corner-connected component, then this binary matrix is non-unique with respect to its absorbed row and column sums. Since it has been proved in another paper [A. Kuba, M. Nivat, Reconstruction of discrete sets with absorption, Linear Algebra Appl. 339 (2001) 171–194] that this condition is also necessary, the existence of alternatively corner-connected component in a binary matrix gives a characterization of the non-uniqueness in this case of absorbed projections.

Using graphs for some discrete tomography problems

Given a rectangular array whose entries represent the pixels of a digitalized image, we consider the problem of reconstructing an image from the number of occurrences of each color in every column and in every row. The complexity of this problem is still open when there are just three colors in the image. We study some special cases where the number of occurrences of each color is limited to small values. Formulations in terms of edge coloring in graphs and as timetabling problems are used; complexity results are derived from the model.

An evolutionary algorithm for discrete tomography

One of the main problems in discrete tomography is the reconstruction of binary matrices from their projections in a small number of directions. In this paper we consider a new algorithmic approach for reconstructing binary matrices from only two projections. This problem is usually underdetermined and the number of solutions can be very large. We present an evolutionary algorithm for finding the reconstruction which maximises an evaluation function, representing the “quality” of the reconstruction, and show that the algorithm can be successfully applied to a wide range of evaluation functions. We discuss the necessity of a problem-specific representation and tailored search-operators for obtaining satisfactory results. Our new search-operators can also be used in other discrete tomography algorithms.

Matrices of zeros and ones with given line sums and a zero block

Abstract Motivated by certain reconstruction problems in discrete tomography we study the existence of (0, 1)-matrices with given line sums and a fixed zero block. An algorithm is given to construct such a matrix which is based on three applications of the well-known Gale-Ryser algorithm for constructing (0, 1)-matrices with given line sums. A characterization in terms of a certain “structure matrix” is proved. We also briefly discuss some generalizations where zeros may be fixed in other positions.

The stability problem and noisy projections in discrete tomography

The new field of research of discrete tomography will be described in this paper. It differs from standard computerized tomography in the reduced number of projections. It needs ad hoc algorithms which usually are based on the definition of the model of the object to reconstruct. The main problems will be introduced and an experimental simulation will prove the robustness of a slightly modified version of a well known method for the reconstruction of binary planar convex sets, even in case of projections affected by error. To the best of our knowledge this is one of the first experimental study of the stability problem with a statistical approach. Prospective applications include crystallography, quality control and reverse engineering while biomedical tests, due to their important role, still require further research.

Stability Issues for Determination and Verification in Discrete Tomography

The talk focusses on the ill-posedness of inverse problems in discrete tomography that are motivated by demands from material sciences for the reconstruction of crystalline structures from images produced by quantitative high resolution transmission electron microscopy. In particular, we present new stability and instability results for the determination and verification of finite point sets from some of their noisy X-ray images.

An experimental study of the stability problem in discrete tomography

Abstract This paper introduces the topic of discrete tomography, briefly showing its main applications, algorithms and new prospects of research. It focuses on the still open problem of stability, facing it from an experimental point of view. In particular an extensive simulation lets verify the robustness of a well known reconstruction technique for binary convex objects, calculating the probability of finding solutions compatible with a given set of noisy projections.

Algebraic aspects of emission tomography with absorption

In a previous paper the authors analysed the classical discrete tomography problem to construct a 0–1-matrix with given line sums in some given directions. One of the physical representations is that material at the lattice points corresponding to 1's emit units of radiation and that the radiation is measured along the given lines. In the present paper they extend their approach to the case that the intermediate material is absorbing the radiation. They generalise results obtained by Kuba and Nivat.

Detection of the discrete convexity of polyominoes

The convexity of a discrete region is a property used in numerous domains of computational imagery. We study its detection in the particular case of polyominoes. We present a first method, directly relying on its definition. A second method, which is based on techniques for segmentation of curves in discrete lines, leads to a very simple, linear, algorithm whose correctness is proven. Correlatively, we obtain a characterisation of lower and upper frontiers of the convex hull of a discrete line segment. Finally, we evoke some applications of these results to the problem of discrete tomography.

A convergent composite mapping Fourier domain iterative algorithm for 3-D discrete tomography

The discrete tomography problem is to reconstruct a binary function defined on a discrete lattice from its sums in directions parallel to the axes of the lattice. The problem has multiple solutions; we wish to determine a solution close, in the least-squares sense, to a specified starting function. We formulate the problem in the Fourier domain, and use the Agarwal–Cooley fast convolution or Good–Thomas fast Fourier transform to unwrap the multidimensional (2-D or 3-D) problem into a long 1-D problem, in which the given projection data specify some values of the DFT of the 1-D sequence of binary values. The z-transform of the DFT values, evaluated on the unit circle, yields these binary values. The problem is now to determine the minimum perturbation of the unconstrained DFT values of the starting function which cause a Toeplitz matrix to lose rank. This type of problem is well-known in spectral estimation. We obtain a new iterative algorithm which converges to a solution of the discrete tomography problem close to the starting function.

On the computational complexity of reconstructing three-dimensional lattice sets from their two-dimensional X-rays

A generalization of a classical discrete tomography problem is considered: reconstruct three-dimensional lattice sets from their two-dimensional X-rays parallel to three coordinate planes. First, we prove that this reconstruction problem is NP-hard. Then we propose some greedy algorithms that provide approximate solutions of the problem.

Reconstructing polyatomic structures from discrete X-rays: NP-completeness proof for three atoms

We address a discrete tomography problem that arises in the study of the atomic structure of crystal lattices. A polyatomic structure T can be defined as an integer lattice in dimension D⩾2, whose points may be occupied by c distinct types of atoms. To “analyze” T, we conduct ℓ measurements that we call discrete X-rays. A discrete X-ray in direction ξ determines the number of atoms of each type on each line parallel to ξ. Given ℓ such non-parallel X-rays, we wish to reconstruct T. The complexity of the problem for c=1 (one atom type) has been completely determined by Gardner et al. (Technical Report 970.05012, Techn. Univ. München, Fak. f. Math., 1997), who proved that the problem is NP-complete for any dimension D⩾2 and ℓ⩾3 non-parallel X-rays, and that it can be solved in polynomial time otherwise Ryser (Mathematical Association of America and Quinn & Boden, Rahway, New Jersey, 1963). The NP-completeness result above clearly extends to any c⩾2, and therefore when studying the polyatomic case we can assume that ℓ=2. As shown in another article by the same authors (Gardner et al., Theoret. Comput. Sci. (1997), to appear), this problem is also NP-complete for c⩾6 atoms, even for dimension D=2 and axis-parallel X-rays. Gardner et al. (1997) conjecture that the problem remains NP-complete for c=3,4,5, although, as they point out, the proof idea in Gardner et al. (1997) does not seem to extend to c⩽5. We resolve the conjecture from Gardner et al. (1997) by proving that the problem is indeed NP-complete for c⩾3 in 2D, even for axis-parallel X-rays. Our construction relies heavily on some structure results for the realizations of 0–1 matrices with given row and column sums.

On the computational complexity of reconstructing lattice sets from their X-rays

We study the computational complexity of various inverse problems in discrete tomography. These questions are motivated by demands from the material sciences for the reconstruction of crystalline structures from images produced by quantitative high resolution transmission electron microscopy. We completely settle the complexity status of the basic problems of existence (data consistency), uniqueness (determination), and reconstruction of finite subsets of the d-dimensional integer lattice γ d that are only accessible via their line sums (discrete X-rays) in some prescribed finite set of lattice directions. Roughly speaking, it turns out that for all d ⩾ 2 and for a prescribed but arbitrary set of m ⩾ 2 pairwise nonparallel lattice directions, the problems are solvable in polynomial time if m = 2 and are NP-complete (or NP-equivalent) otherwise.

An explicit closed-form solution to the limited-angle discrete tomography problem for finite-support objects

An explicit formula is presented for reconstructing a finite-support object defined on a lattice of points and taking on integer values from a finite number of its discrete projections over a limited range of angles. Extensive use is made of the discrete Fourier transform in doing so. The approach in this article computes the object sample values directly as a linear combination of the projections sample values. The well-known ill-posedness of the limited angle tomography problem manifests itself in some very large coefficients in these linear combinations; these coefficients (which are computed off-line) provide a direct sensitivity measure of the reconstruction samples to the projections samples. The discrete nature of the problem implies that the projections must also take on integer values; this means noise can be rejected. This makes the formula practical. © 1998 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 9, 174–180, 1998

An explicit closed‐form solution to the limited‐angle discrete tomography problem for finite‐support objects

An explicit closed‐form solution to the limited‐angle discrete tomography problem for finite‐support objects