Discrete Tomography Problem Research Articles

Some Discrete Tomography Problems in Hypergraph Model Interpretation

Abstract In this paper we consider discrete tomography problems with an additional requirement of non-repeatability of rows of the binary matrix to be reconstructed; as well as discrete tomography problems with given pairwise projections. Representing the problems in the hypergraph model and pointing out their equivalence to the basic definitions, we state the following results: (i) nonconvexity of the set of hypergraphic sequences of simple hypergraphs with $$n$$ vertices and $$m$$ hyperedges in the $$n$$-dimensional $$m + 1$$-valued lattice, (ii) characterization of monotone Boolean functions associated with degree sequences of 3-/(n–3)-uniform hypergraphs, (iii) formulation of discrete tomography problems with paired projections, their connection to hypergraph degree sequence problem with generalized degrees, a solution for a particular case.

An efficient implementable inexact entropic proximal point algorithm for a class of linear programming problems

We introduce a class of specially structured linear programming (LP) problems, which has favorable modeling capability for important application problems in different areas such as optimal transport, discrete tomography, and economics. To solve these generally large-scale LP problems efficiently, we design an implementable inexact entropic proximal point algorithm (iEPPA) combined with an easy-to-implement dual block coordinate descent method as a subsolver. Unlike existing entropy-type proximal point algorithms, our iEPPA employs a more practically checkable stopping condition for solving the associated subproblems while achieving provable convergence. Moreover, when solving the capacity constrained multi-marginal optimal transport (CMOT) problem (a special case of our LP problem), our iEPPA is able to bypass the underlying numerical instability issues that often appear in the popular entropic regularization approach, since our algorithm does not require the proximal parameter to be very small in order to obtain an accurate approximate solution. Numerous numerical experiments show that our iEPPA is efficient and robust for solving some large-scale CMOT problems on synthetic data. The preliminary experiments on the discrete tomography problem also highlight the potential modeling capability of our model.

Battleship, tomography and quantum annealing

AbstractThe classic game of Battleship involves two players taking turns attempting to guess the positions of a fleet of vertically or horizontally positioned enemy ships hidden on a $10\times 10$ grid. One variant of this game, also referred to as Battleship Solitaire, Bimaru or Yubotu, considers the game with the inclusion of X-ray data, represented by knowledge of how many spots are occupied in each row and column in the enemy board. This paper considers the Battleship puzzle problem: the problem of reconstructing an enemy fleet from its X-ray data. We generate non-unique solutions to Battleship puzzles via certain reflection transformations akin to Ryser interchanges. Furthermore, we demonstrate that solutions of Battleship puzzles may be reliably obtained by searching for solutions of the associated classical binary discrete tomography problem which minimise the discrete Laplacian. We reformulate this optimisation problem as a quadratic unconstrained binary optimisation problem and approximate solutions via a simulated annealer, emphasising the future practical applicability of quantum annealers to solving discrete tomography problems with predefined structure.

Computational complexity of three-dimensional discrete tomography with missing data

Discrete tomography deals with problems of determining shape of a discrete object from a set of projections. In this paper, we deal with a fundamental problem in discreet tomography: reconstructing a discrete object in $$\mathbb {R}^3$$ from its orthogonal projections, which we call three-dimensional discrete tomography. This problem has been mostly studied under the assumption that complete data of the projections are available. However, in practice, there might be missing data in the projections, which come from, e.g., the lack of precision in the measurements. In this paper, we consider the three-dimensional discrete tomography with missing data. Specifically, we consider the following three fundamental problems in discrete tomography: the consistency, counting, and uniqueness problems, and classify the computational complexities of these problems in terms of the length of one dimension. We also generalize these results to higher-dimensional discrete tomography, which has applications in operations research and statistics.

Ambiguous reconstructions of hv-convex polyominoes

The core problem of discrete tomography, i.e., the faithful reconstruction of an unknown discrete object from projections along a given set of directions, is ill-posed in general. When further constraints are imposed on the object or on the employed directions, uniqueness of the reconstruction can be obtained. It is the case, for example, of convex lattice sets in Z2, for which a theorem by Gardner and Gritzmann assures the faithful reconstruction when suitable sets of directions are considered. It was conjectured that a similar result holds for the class of hv-convex polyominoes. In this paper we are concerned with this conjecture, providing new 4-tuples of discrete directions that do not lead to a unique reconstruction of hv-convex polyominoes, underlining the relevant structural difference with the class of convex sets. Our result is based on the recursive definition of new hv-convex switching components on discrete sets along four directions.

On Polyatomic Tomography over Abelian Groups: Some Remarks on Consistency, Tree Packings and Complexity

The paper deals with an inverse problem of reconstructing matrices from their marginal sums. More precisely, we are interested in the existence of rtimes s matrices for which only the following information is available: The entries belong to known subsets of c distinguishable abelian groups, and the row and column sums of all entries from each group are given. This generalizes Ryser’s classical problem of characterizing the set of all 0–1-matrices with given row and column sums and is a basic problem in (polyatomic) discrete tomography. We show that the problem is closely related to packings of trees in bipartite graphs, prove consistency results, give algorithms and determine its complexity. In particular, we find a somewhat unusual complexity behavior: the problem is hard for “small” but easy for “large” matrices.

Regular switching components

We consider a problem of Discrete Tomography which consists of reconstructing a finite lattice set S⊂Z2 from given horizontal and vertical X-rays (i.e. with prescribed numbers of points in each row and column). Without additional requirements, the problem can be solved in polynomial time. Many variants require the solution to be in a chosen class A. For instance, the problem is NP-complete for the class A=H∩V of HV-convex lattice sets and it is polynomial for the class A=H∩V∩P of HV-convex polyominoes. Twenty years after these results, the problem's complexity remains unknown for A=C, the class of C-convex lattice sets (i.e. two-dimensional lattice polytopes).The main difficulty to solve this problem comes from combinatorial structures called switching components. Switching components are closed path with horizontal and vertical edges such that the solutions S are composed of either the elements of even or of odd indices. This binary choice can be encoded by a Boolean variable associated with the switching component and the convexity constraints are simply encoded by SAT clauses (2-clauses for HV-convexity and 3-clauses for C-convexity).The purpose of the paper is to investigate the properties of the switching components and the consequences of the convexity requirements. We divide the switching components in two classes: regular if their turning angle is constant, irregular otherwise. We prove that adjacent regular switching components have the same Boolean values. This property allows us to merge them into extended switching components. We prove that if all switching components are regular, then the extended switching components are all independent (then the number of solutions with the considered feet is 2n, where n is the number of extended switching components). Finally, we prove that they are geometrically ordered.

Dynamic discrete tomography

We consider the problem of reconstructing the paths of a set of points over time, where, at each of a finite set of moments in time the current positions of points in space are only accessible through some small number of their x-rays. This particular particle tracking problem, with applications, e.g. in plasma physics, is the basic problem in dynamic discrete tomography. We introduce and analyze various different algorithmic models. In particular, we determine the computational complexity of the problem (and various of its relatives) and derive algorithms that can be used in practice. As a byproduct we provide new results on constrained variants of min-cost flow and matching problems.

Multivalued Discrete Tomography Using Dynamical System That Describes Competition

Multivalued discrete tomography involves reconstructing images composed of three or more gray levels from projections. We propose a method based on the continuous-time optimization approach with a nonlinear dynamical system that effectively utilizes competition dynamics to solve the problem of multivalued discrete tomography. We perform theoretical analysis to understand how the system obtains the desired multivalued reconstructed image. Numerical experiments illustrate that the proposed method also works well when the number of pixels is comparatively high even if the exact labels are unknown.

A geometrical characterization of regions of uniqueness and applications to discrete tomography

In the reconstruction problem of discrete tomography, projections are considered from a finite set of lattice directions. Employing a limited number of projections implies that the injectivity of the Radon transform is lost, and, in general, images consistent with a given set of projections form a huge class. In order to lower the number of allowed solutions, one usually tries to include in the problem some a priori information. This suggests that modelling the tomographic reconstruction problem as a linear system of equations is preferable. In this paper we propose to restrict the usual notion of uniqueness, related to the solutions of the linear system, and to provide, for each set a geometrical characterization of the shape of a lattice subset, say region of uniqueness (ROU), forming a partial, fast reconstructible, solution. Any selected set intrinsically determines its ROU inside an arbitrary lattice grid. For instance, trivially, if the ROU is represented by two rectangles having sizes equal to the absolute values of the entries of the unique direction in and placed at two opposite corners of the chosen grid. Surprisingly, if the problem becomes much more complicated. Our purpose is to provide a geometrical characterization of the ROU. This is based on a double Euclidean division algorithm (DEDA), which runs in polynomial time. It turns out that the ROU is delimited by a zigzag profile obtained by means of numerical relations among the entries of the employed directions. According to different inputs in DEDA, the shape of the ROU can change consistently, as it can be easily observed from the provided examples. Moreover, after selecting a region of interest (ROI) from a given phantom, we exploit DEDA to reconstruct the part of the ROI which falls in the ROU and, with a few further a priori knowledge, even parts of the ROI which are outside the ROU.

Extremal Theorems for Degree Sequence Packing and the Two-Color Discrete Tomography Problem

A sequence $\pi=(d_1,\ldots,d_n)$ is graphic if there is a simple graph $G$ with vertex set $\{v_1,\ldots,v_n\}$ such that the degree of $v_i$ is $d_i$. We say that graphic sequences $\pi_1=(d_1^{(1)},\ldots,d_n^{(1)})$ and $\pi_2=(d_1^{(2)},\ldots,d_n^{(2)})$ pack if there exist edge-disjoint $n$-vertex graphs $G_1$ and $G_2$ such that for $j\in\{1,2\}$, $d_{G_j}(v_i)=d_i^{(j)}$ for all $i\in\{1,\ldots,n\}$. Here, we prove several extremal degree sequence packing theorems that parallel central results and open problems from the graph packing literature. Specifically, the main result of this paper implies degree sequence packing analogues to the Bollobas--Eldridge--Catlin graph packing conjecture and the classical graph packing theorem of Sauer and Spencer. In discrete tomography, a branch of discrete imaging science, the goal is to reconstruct discrete objects using data acquired from low-dimensional projections. Specifically, in the $k$-color discrete tomography problem the goal is to color the entries ...

A formulation of the wide partition conjecture using the atom problem in discrete tomography

The Wide Partition Conjecture (WPC) was introduced by Chow and Taylor as an attempt to prove inductively Rota’s Basis Conjecture, and in the simplest case tries to characterize partitions whose Young diagram admits a “Latin” filling. Chow et al. (2003) showed how the WPC is related to problems such as edge-list coloring and multi-commodity flow. As far as we know, the conjecture remains widely open.We show that the WPC can be formulated using the k-atom problem in Discrete Tomography, introduced in Gardner et al. (2000). In this approach, the WPC states that the sequences arising from partitions admit disjoint realizations if and only if any combination of them can be realized independently. This realizability condition can be checked in polynomial time, although is not sufficient in general Chen and Shastri (1989), Guiñez et al. (2011). In fact, the problem is NP-hard for any fixed k⩾2 Dürr et al. (2012). A stronger condition, called the saturation condition, was introduced in Guiñez et al. (2011) to solve instances where the realizability condition fails. We prove that in our case, the saturation condition is implied by the realizability condition. Moreover, we show that the saturation condition can be obtained as the Lagrangian dual of the linear programming relaxation of a natural integer programming formulation of the k-atom problem.

The Wide Partition Conjecture and the Atom Problem in Discrete Tomography

The Wide Partition Conjecture (WPC) was introduced by Chow and B. Taylor as an attempt to prove inductively Rotaʼs Basis Conjecture, and in the simplest case tries to characterize partitions whose Young diagram admits a “Latin” filling. Chow et al. [T. Chow, C. K. Fan, M. Goemans, and J. Vondrak. Wide partitions, latin tableaux, and rotaʼs basis conjecture. Adv. in Appl. Math., 31(2):334–358, 2003] showed how the WPC is related to problems such as edge-list coloring and multi commodity flow. As far as we know, the conjecture remains widely open.We show that the WPC can be formulated using the k-atom problem in Discrete Tomography [C. Dürr, F. Guíñez, and M. Matamala. Reconstructing 3-Colored Grids from Horizontal and Vertical Projections is NP-Hard: A Solution to the 2-Atom Problem in Discrete Tomography. SIAM J Discrete Math, 26(1):330, 2012.]. In this approach, the WPC states that the sequences arising from partitions admit disjoint realizations if and only if any combination of them can be realizable independently. This realizability condition is not sufficient in general. A stronger condition, the saturation condition, was used in [F. Guíñez, M. Matamala, and S. Thomassé. Realizing disjoint degree sequences of span at most two: A tractable discrete tomography problem. Discrete Appl.Math., 159(1):23–30, 2011] to solve instances were the realizability condition fails. We prove that in our case, the saturation condition is satisfied providing the realizability condition does. Moreover, we show that the saturation condition can be obtained from the Langrangean dual of a natural LP formulation of the k-atom problem.

Discrete tomography for inscribable lattice sets

In this paper we deal with uniqueness and reconstruction problems in Discrete Tomography. For a finite set D of directions in Z2, we introduce the class of D-inscribable lattice sets, and give a detailed description of their geometric structure. This shows that such sets can be considered as the natural discrete counterpart of the same notion known in the continuous case, as well as a kind of generalization of the class of the so-called L-convex polyominoes (or moon polyominoes). In view of reconstruction from projections along the directions in D, two related questions of tomographic interest are investigated, namely uniqueness and additivity. We show that both properties are fulfilled by D-inscribable lattice sets. Moreover, concerning the case D={e1,e2}, we provide an explicit reconstruction algorithm from the knowledge of directed horizontal and vertical X-rays, jointly with a few preliminary results towards a possible sharp stability inequality.

Solving Multicolor Discrete Tomography Problems by Using Prior Knowledge

Discrete tomography deals with the reconstruction of discrete sets with given projections relative to a limited number of directions, modeling the situation where a material is studied through x-rays and we desire to reconstruct an image representing the scanned object. In many cases it would be interesting to consider the projections to be related to more than one distinguishable type of cell, called atoms or colors, as in the case of a scan involving materials of different densities, as a bone and a muscle. Unfortunately the general n-color problem with n > 1 is NP-complete, but in this paper we show how several polynomial reconstruction algorithms can be defined by assuming some prior knowledge on the set to be rebuilt. In detail, we study the cases where the union of the colors form a set without switches, a convex polyomino or a convex 8-connected set. We describe some efficient reconstruction algorithms and in a case we give a sufficient condition for uniqueness.

Reconstructing 3-Colored Grids from Horizontal and Vertical Projections is NP-Hard: A Solution to the 2-Atom Problem in Discrete Tomography

We consider the problem of coloring a grid using $k$ colors with the restriction that each row and each column has a specific number of cells of each color. This problem has been known as the $(k-1)$-atom problem in the discrete tomography community. In an already classical result, Ryser obtained a necessary and sufficient condition for the existence of such a coloring when two colors are considered. This characterization yields a linear time algorithm for constructing such a coloring when it exists. Gardner et al. showed that for $k\geqslant 7$ the problem is NP-hard. Afterward Chrobak and Durr improved this result by proving that it remains NP-hard for $k\geqslant 4$. We close the gap by showing that for $k=3$ colors the problem is already NP-hard. In addition, we give some results on tiling tomography problems.

DISCRETE TOMOGRAPHY AND FUZZY INTEGER PROGRAMMING

We study the problem of reconstructing binary images from four projections data in a fuzzy environment. Given the uncertainly projections,we want to find a binary image that respects as best as possible these projections. We provide an iterative algorithm based on fuzzy integer programming and linear membership functions. Discrete Tomography (DT) deals with the reconstruction of digital images from their projections. The projections of an image are defined by the number of pixels of the colors on each line. Digital images are most commonly represented by inte- ger matrices. The problem of reconstructing monocolored images (blackand white) from two projections is well known (14). However, image processing in general and image reconstruction in particular is often characterized by uncertain and possi- bly inconsistent information. Since fuzzy programming is considered appropriate for solving real-world problems it seems reasonable to apply fuzzy programming methods to the reconstruction problem. In this paper, we propose a fuzzy approach for reconstructing binary images, which, instead of reconstructing projections exactly as in the deterministic recon- struction problem, recovers them as much as possible. The remainder of this paper is organized as follows. First, in Section 2, we briefly review fuzzy set theory and fuzzy programming and then, in Section 3, we present the problem of discrete tomography and image reconstruction. In Section 4, we discuss the application of fuzzy programming in image reconstruction and finally conclude the last section with a summary of our results.

Realizing disjoint degree sequences of span at most two: A tractable discrete tomography problem

We consider the problem of coloring a grid using p colors with the requirement that each row and each column has a specific total number of entries of each color. Ryser (1957) [20], and independently Gale (1957) [10], obtained a necessary and sufficient condition for the existence of such a coloring when two colors are considered. This characterization yields a linear-time algorithm for constructing the coloring when it exists. Later, Gardner et al. (2000) [11], and Chrobak and Dürr (2001) [5], showed that the problem is NP-hard when p ⩾ 7 and p ⩾ 4 , respectively. The case p = 3 was an open problem for several years and has been recently settled by Dürr et al. (2009) [9]: it is NP-hard too. This grid coloring problem is equivalent to finding disjoint realizations of two degree sequences d 1 , d 2 in a complete bipartite graph K X , Y . These kinds of questions are well studied when one of the degree sequences has span zero or one, where the span of a function is the difference between its maximum and its minimum values. In [4], Chen and Shastri (1989) showed a necessary and sufficient condition for the existence of a coloring when d 1 + d 2 restricted to X or Y has span at most one. In terms of discrete tomography this latter condition means that for two colors, the sum of the number of occurrences of these colors in each row is k or k + 1 , for some integer k . In the present paper we prove an analog to Chen and Shastri’s characterization when d 1 + d 2 restricted to X and to Y has span at most two. That is, there exist integers k 1 and k 2 such that the sum of the number of occurrences of two of the colors in each row is k 1 − 1 , k 1 or k 1 + 1 , and in each column is k 2 − 1 , k 2 or k 2 + 1 . Our characterization relies on a new natural condition called the total saturation condition which, when not satisfied, gives a non-existence certificate of such a coloring that can be checked in polynomial time.

Complexity Results for Wireless Sensor Network Scheduling

We study the problem of scheduling multi sensors to visit and observe a group of sites at discrete time points over a planning horizon of given length. We show that scheduling under a given number of visits for each site and in each period is an NP-complete problem by providing equivalence with a problem in discrete tomography. We also give a polynomial time algorithm to schedule the sensors under a given number of visits in each period.

An Algebraic Framework for Discrete Tomography: Revealing the Structure of Dependencies

Discrete tomography is concerned with the reconstruction of images that are defined on a discrete set of lattice points from their projections in several directions. The range of values that can be assigned to each lattice point is typically a small discrete set. In this paper we present a framework for studying these problems from an algebraic perspective, based on ring theory and commutative algebra. A principal advantage of this abstract setting is that a vast body of existing theory becomes accessible for solving discrete tomography problems. We provide proofs of several new results on the structure of dependencies between projections, including a discrete analogon of the well-known Helgason-Ludwig consistency conditions from continuous tomography.