Discrete Tomography Research Articles

Discrete tomography of demanding samples based on a modified SIRT algorithm

The 3D structure of three particularly challenging samples was reconstructed by electron tomography. Due to sample limitations resulting in a large missing wedge and large tilt increments respectively the 3D structure could not be reconstructed by standard iterative algorithms; even a recently developed discrete algorithm failed until the input parameters for discrete reconstruction were improved. These challenges were addressed by adding a mask in each step of the preceding standard iterative reconstruction, setting all voxels known to be vacuum as zero, thus improving the segmentation and the 3D starting model. The position of these vacuum voxels is obtained from TEM images or other measurement data.

Fuel concentration in isothermal Diesel sprays through structured planar laser imaging measurements

The mixing of isothermal liquid sprays in engine-like conditions has been investigated by applying the structured planar laser imaging technique to remove multiple light scattering. The intensity of the illumination plane has been recovered by removing multiply scattered light and mapping the spray three-dimensionally via discrete tomography. Based on the extinction of light within the illumination plane, the number density has been extracted. Coupled with 2-D maps of droplet diameters obtained through LIF/Mie ratio, the number density allowed to calculate the fuel concentration in the sprays. The mixture fraction of DI Diesel sprays injected into an inert environment held at room temperature has been evaluated and compared to a 2-D model based on gas-jet theory. The experimental results showed good agreement with the predictions when a Gaussian radial distribution is assumed and the Schmidt number is correctly tuned. Differences in the radial distribution has been observed and related to incomplete momentum transfer between the liquid spray and the surrounding gases. For different testing conditions, while the influence of ambient density on mixing was expected, the effect of injection pressure has been found to provide additional information concerning the global mixing of liquid sprays.

Accurate segmentation of dense nanoparticles by partially discrete electron tomography

Accurate segmentation of nanoparticles within various matrix materials is a difficult problem in electron tomography. Due to artifacts related to image series acquisition and reconstruction, global thresholding of reconstructions computed by established algorithms, such as weighted backprojection or SIRT, may result in unreliable and subjective segmentations. In this paper, we introduce the Partially Discrete Algebraic Reconstruction Technique (PDART) for computing accurate segmentations of dense nanoparticles of constant composition. The particles are segmented directly by the reconstruction algorithm, while the surrounding regions are reconstructed using continuously varying gray levels. As no properties are assumed for the other compositions of the sample, the technique can be applied to any sample where dense nanoparticles must be segmented, regardless of the surrounding compositions. For both experimental and simulated data, it is shown that PDART yields significantly more accurate segmentations than those obtained by optimal global thresholding of the SIRT reconstruction.

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.

Three-Dimensional Atomic Imaging of Colloidal Core–Shell Nanocrystals

Colloidal core-shell semiconductor nanocrystals form an important class of optoelectronic materials, in which the exciton wave functions can be tailored by the atomic configuration of the core, the interfacial layers, and the shell. Here, we provide a trustful 3D characterization at the atomic scale of a free-standing PbSe(core)-CdSe(shell) nanocrystal by combining electron microscopy and discrete tomography. Our results yield unique insights for understanding the process of cation exchange, which is widely employed in the synthesis of core-shell nanocrystals. The study that we present is generally applicable to the broad range of colloidal heteronanocrystals that currently emerge as a new class of materials with technological importance.

Direction-dependency of binary tomographic reconstruction algorithms

In this work we study the relation between the quality of a binary tomographic reconstruction and the choice of angles of the projections. We conduct experiments on a set of software phantoms by reconstructing them from different projection sets using three different discrete tomography reconstruction algorithms, and compare the accuracy of the corresponding reconstructions with suitable approaches. To validate our results for possible real-world applications, we conduct the experiments by adding random noise of different characteristics to the simulated projection data, and by applying small topological changes on the phantom images as well. In addition, we also discuss some consequences of the angle-selection dependency and possible practical applications arising from the field of non-destructive testing, too.

A full row-rank system matrix generated along two directions in discrete tomography

Abstract A full row-rank system matrix generated by the strip-based projection model along one scanning direction was studied recently in [9] . In this paper, we generalize the result to multiple directions. Let Cu = h be a reduced binary linear system generated along two distinct scanning directions by the strip-based projection model in discrete tomography, where C is row-rank deficient. We identify all the linearly dependent rows explicitly through a partition of the rows of C into minimal linearly dependent sets. The removal of these linearly dependent rows results in a full-rank matrix. Consequently, the computational cost for image reconstruction is reduced.

DART: A Practical Reconstruction Algorithm for Discrete Tomography

In this paper, we present an iterative reconstruction algorithm for discrete tomography, called discrete algebraic reconstruction technique (DART). DART can be applied if the scanned object is known to consist of only a few different compositions, each corresponding to a constant gray value in the reconstruction. Prior knowledge of the gray values for each of the compositions is exploited to steer the current reconstruction towards a reconstruction that contains only these gray values. Based on experiments with both simulated CT data and experimental μCT data, it is shown that DART is capable of computing more accurate reconstructions from a small number of projection images, or from a small angular range, than alternative methods. It is also shown that DART can deal effectively with noisy projection data and that the algorithm is robust with respect to errors in the estimation of the gray values.

Tile-Packing Tomography Is $\mathbb{NP}$ -hard

Discrete tomography deals with reconstructing finite spatial objects from their projections. The objects we study in this paper are called tilings or tile-packings, and they consist of a number of disjoint copies of a fixed tile, where a tile is defined as a connected set of grid points. A row projection specifies how many grid points are covered by tiles in a given row; column projections are defined analogously. For a fixed tile, is it possible to reconstruct its tilings from their projections in polynomial time? It is known that the answer to this question is affirmative if the tile is a bar (its width or height is 1), while for some other types of tiles \(\mathbb {NP}\)-hardness results have been shown in the literature. In this paper we present a complete solution to this question by showing that the problem remains \(\mathbb {NP}\)-hard for all tiles other than bars.

ADART: An Adaptive Algebraic Reconstruction Algorithm for Discrete Tomography

In this paper we suggest an algorithm based on the Discrete Algebraic Reconstruction Technique (DART) which is capable of computing high quality reconstructions from substantially fewer projections than required for conventional continuous tomography. Adaptive DART (ADART) goes a step further than DART on the reduction of the number of unknowns of the associated linear system achieving a significant reduction in the pixel error rate of reconstructed objects. The proposed methodology automatically adapts the border definition criterion at each iteration, resulting in a reduction of the number of pixels belonging to the border, and consequently of the number of unknowns in the general algebraic reconstruction linear system to be solved, being this reduction specially important at the final stage of the iterative process. Experimental results show that reconstruction errors are considerably reduced using ADART when compared to original DART, both in clean and noisy environments.

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.

Modeling of phase interfaces during pre-critical crack growth in concrete

Abstract Recent developments in the Rigid-Body-Spring Network (RBSN) modeling of concrete are presented, with candid assessments of research needs for the modeling of pre-critical crack growth under multi-axial stress conditions. Fracture of micro-concrete specimens is investigated through the combined use of such discrete (lattice) models, X-ray tomography, and imaging techniques. The specimens contain small amounts (10–50 wt.% fraction) of spherical glass aggregates. Acid-etching of the glass aggregate surfaces is done to vary the bond properties of the matrix–aggregate interface. Tomographic images of the unloaded specimens provide the initial configurations of the three-dimensional lattice models, which are based on a three-phase representation of the material meso-structure: fine-grained mortar matrix, glass aggregate inclusions, and matrix–aggregate interfaces. The numerical and physical test results agree well with respect to peak loads and the associated crack patterns. Material structure and interface properties affect pre-critical cracking, in accordance with expectations. Dependence of composite strength on aggregate content and arrangement is studied through simulations of large sets of nominally identical models, which differ only in random positioning of the aggregates.

Uniqueness in Discrete Tomography: Three Remarks and a Corollary

Discrete tomography is concerned with the retrieval of finite point sets in some $\mathbbm{R}^d$ from their X-rays in a given number m of directions $u_1,\ldots, u_m$. In the present paper we focus on uniqueness issues. The first remark gives a uniform treatment and extension of known uniqueness results. In particular, we introduce the concept of J-additivity and give conditions when a subset J of possible positions is already determined by the given data. As a by-product, we settle a conjecture of Brunetti and Daurat on planar lattice convex sets. Remark 2 resolves a problem of Kuba posed in 1997 on the uniqueness in the case $d=m=3$ with $u_1,u_2,u_3$ being the standard unit vectors. Remark 3 determines the computational complexity of finding a smallest set J of positions whose disclosure yields uniqueness. As a corollary, we obtain a hardness result for 0-1-polytopes.

Projection Selection Dependency in Binary Tomography

It has already been shown that the choice of projection angles can significantly influence the quality of reconstructions in discrete tomography. In this contribution we summarize and extend the previous results by explaining and demonstrating tile effects of projection selection dependency, in a set of experimental software tests. We perform reconstructions of software phantoms, by using different binary tomography reconstruction algorithms, from different equiangular and non-equiangular projections sets, under various conditions (i.e., when the objects to be reconstructed undergo slight topological changes, or the projection data is affected by noise) and compare the results with suitable approaches. Based on our observations, we reveal regularities in the resulting data and discuss possible consequences of such projection selection dependency in binary tomography.

Boundary Length of Reconstructions in Discrete Tomography

We consider possible reconstructions of a binary image of which the row and column sums are given. For any reconstruction we can define the length of the boundary of the image. In this paper we prove a new lower bound on the length of this boundary. In contrast to simple bounds that have been derived previously, in this new lower bound the information of both row and column sums is combined.

Three-dimensional reconstruction of the atomic arrangement of icosahedral quasicrystals by binary discrete tomography

A three-dimensional reconstruction of the structure of icosahedral quasicrystals from high-resolution transmission electron microscope images observed along high-symmetry zone-axis projections is studied. As in discrete tomography, the quasicrystalline structure is treated as a discrete set of atoms arranged on the icosahedral lattice, including atom flip sites, which is a structural disorder originating from the degrees of freedom in the internal space. The occupancy of the atoms on each site of the lattice is determined using only the zone-axis projection images without any knowledge of the image intensity. Simulation studies using icosahedral model lattices composed of several thousands of atoms were carried out to test the present reconstruction method. In the case of the ideal icosahedral lattice, a reconstruction rate of 90% is achieved from 10 projection images and a reconstruction rate of 99% is achieved from 18 projections. The reconstruction rate degrades when the projection images are misaligned. Reconstruction of the three-dimensional distribution of the atom flips is discussed.

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.

A semi-automatic algorithm for grey level estimation in tomography

Discrete tomography focuses on the reconstruction of images that contain only a few grey levels from their projections. By incorporating prior knowledge about the set of grey levels, the required number of projections can be reduced substantially. In practical applications, however, the number of grey levels is often known in advance, yet the actual grey level values are unknown. Moreover, it can be difficult to estimate these grey levels, particularly if only a small number of projections are available. In this paper, we propose a semi-automatic approach for grey level estimation that can be used as a preprocessing step before applying discrete tomography algorithms. After an initial, non-discrete reconstruction has been computed, the user first selects some regions that are likely to correspond with the respective grey levels. The fact that these regions should be constant in the original image is then used as prior knowledge in the grey level estimation algorithm. We present the results of a series of simulation experiments, demonstrating the accuracy and robustness of our approach.

Dynamical symmetry reduction and discrete tomography of a Ξ atom

When implemented using a reasonable Hamiltonian, the tomography of a three-level Ξ atom is complicated by the equidistant energy levels of the atom. This restricts the possible transformations to those in the SO(3) subgroup of SU(3). Although complete reconstruction is possible for a single Ξ atom using a continuous set of tomograms, the discrete optimal set of tomograms, related to mutually unbiased bases in dimension 3, are not accessible by time evolution. We discuss here the search for an optimal set of discrete basis states compatible with the reduced SO(3) symmetry of the system.

A reconstruction algorithm for a subclass of instances of the 2-color problem

In the field of Discrete Tomography, the 2-color problem consists in reconstructing a matrix whose elements are of two different types, starting from its horizontal and vertical projections. It is known that the 1-color problem admits a polynomial time reconstruction algorithm, while the c-color problem, with c ≥ 2 , is NP-hard. Thus, the 2-color problem constitutes an interesting example of a problem just in the frontier between hard and easy problems. In this paper we define a linear time algorithm (in the size of the output matrix) to solve a subclass of its instances, where some values of the horizontal and vertical projections are constant, while the others are upper bounded by a positive number proportional to the dimension of the problem. Our algorithm relies on classical studies for the solution of the 1-color problem.