This paper addresses the image reconstruction problem in discrete tomography, particularly under challenging imaging conditions such as sparse-view and limited-angle geometries commonly encountered in computed tomography (CT). These conditions often result in low-quality reconstructions due to insufficient projection data and incomplete angular coverage. To overcome these limitations, we propose a novel reconstruction framework that integrates compressed sensing (CS) with a parametric level set (PLS) method tailored for discrete images. The proposed approach leverages prior knowledge of discrete gray-level values and employs a parametric level set function to represent boundaries in both binary and multi-gray-level images. Unlike previous methods, our PLS is constructed using a dictionary of basis functions composed of single-scale or multiscale Gaussian functions. Reconstruction is formulated as 𝚤1-norm minimization of Gaussian coefficients, promoting sparsity. We assess the method's robustness by introducing varying levels of Gaussian noise into the projection data under both sparse-view and limited-angle conditions. Quantitative evaluations using PSNR, SSIM, and Dice coefficients demonstrate that the proposed method preserves boundary sharpness and accurately reconstructs discrete intensity levels, even in highly undersampled and noisy scenarios. Simulations and experiments on both synthetic and real CT data confirm that the proposed approach consistently outperforms conventional methods in terms of reconstruction quality, boundary accuracy, and noise robustness.

Many different descriptors have been proposed to measure the convexity of digital shapes. Most of these are based on the definition of continuous convexity and exhibit both advantages and drawbacks when applied in the digital domain. In contrast, within the field of Discrete Tomography, a special type of convexity—called Quadrant-convexity—has been introduced. This form of convexity naturally arises from the pixel-based representation of digital shapes and demonstrates favorable properties for reconstruction from projections. In this paper, we present an overview of using Quadrant-convexity as the basis for designing shape descriptors. We explore two different approaches: the first is based on the geometric features of Quadrant-convex objects, while the second relies on the identification of Quadrant-concave pixels. For both approaches, we conduct extensive experiments to evaluate the strengths and limitations of the proposed descriptors. In particular, we show that all our descriptors achieve an average accuracy of approximately 95% to 97.5% on noisy retina images for a binary classification task. Furthermore, in a multiclass classification setting using a dataset of desmids, all our descriptors outperform traditional low-level shape descriptors, achieving an accuracy of 76.74%.

The question whether there exists a hypergraph whose degrees are equal to a given sequence of integers is a well-known reconstruction problem in graph theory, which is motivated by discrete tomography. In this paper we approach the problem by randomized algorithms which generate the required hypergraph with positive probability if the sequence satisfies certain constraints.

Computed tomography (CT) reconstruction problems are always framed as inverse problems, where the attenuation map of an imaged object is reconstructed from the sinogram measurement. In practice, these inverse problems are often ill-posed, especially under few-view and limited-angle conditions, which makes accurate reconstruction challenging. Existing solutions use regularizations such as total variation to steer reconstruction algorithms to the most plausible result. However, most prevalent regularizations rely on the same priors, such as piecewise constant prior, hindering their ability to collaborate effectively and further boost reconstructionprecision. This study aims to overcome the aforementioned challenge a prior previously limited to discrete tomography. This enables more accurate reconstructions when the proposed method is used in conjunction with most existing regularizations as they utilize different priors. The improvements will be demonstrated through experiments conducted under variousconditions. Inspired by the discrete algebraic reconstruction technique (DART) algorithm for discrete tomography, we find out that pixel grayscale values in CT images are not uniformly distributed and are actually highly clustered. Such discovery can be utilized as a powerful prior for CT reconstruction. In this paper, we leverage the collaborative filtering technique to enable the collaboration of the proposed prior and most existing regularizations, significantly enhancing the reconstructionaccuracy. Our experiments show that the proposed method can work with most existing regularizations and significantly improve the reconstruction quality. Such improvement is most pronounced under limited-angle and few-view conditions. Furthermore, the proposed regularization also has the potential for further improvement and can be utilized in other image reconstructionareas. We propose improving the performance of iterative CT reconstruction algorithms by applying the collaborative filtering technique along with a prior based on the densely clustered distribution of pixel grayscale values in CT images. Our experimental results indicate that the proposed methodology consistently enhances reconstruction accuracy when used in conjunction with most existing regularizations, particularly under few-view and limited-angleconditions.

For patients with metal implants, computed tomography (CT) imaging results suffer from metal artifacts, which seriously affect image evaluation and even lead to misdiagnosis. Because spectral CT technology has the advantage of quantitative imaging, basis material decomposition, and so on, the current metal artifact reduction methods are utilizing spectral information to reduce metal artifacts with good results. However, they usually require projection data from multiple spectra or energy-windows, which is difficult to realize in conventional CT. To satisfy the status quo, the aim of this work is to propose a metal artifact reduction (MAR) method based on single spectral CT (MARSS). By incorporating prior information, the average density of some base materials, and a constrained image reconstruction model is established. It forces the solution spaces of the materials to be discrete and finite, making the model easier tosolve. The MARSS method uses the idea of discrete tomography to establish a constrained reconstruction model. By incorporating priori knowledge, the constraint forces the solution spaces of some materials to be discrete, which can effectively downsize the solution space and reduce the ill-posedness of this problem. Then, an iteration algorithm is developed to solve this model. This algorithm iterates alternately between reconstruction and discretization. It ensures that the solution spaces are discrete while the polychromatic projection of the reconstructed image converges to that of the scannedobject. The MRASS method significantly reduces artifacts and restores structures near metal to a large extent. Unlike the comparison MAR methods, it effectively prevents the introduction of new artifacts and distortion of thestructure. The MARSS method can achieve MAR based on single spectral CT. Subjective and quantitative evaluation of the results show that the method significantly improves image quality compared to competingmethods.

Background: Huntington's disease (HD) is still often defined by the onset of motor symptoms, inversely associated with the size of the CAG repeat expansion in the huntingtin gene. Although the cause of HD is known, much remains unknown about mechanisms underlying clinical symptom development, disease progression, and specific clinical subtypes/endophenotypes. Objective: In the iMarkHD study, we aim to investigate four discrete molecular positron emission tomography (PET) tracers and magnetic resonance imaging (MRI) markers as biomarkers for disease and symptom progression. Methods: Following MRI optimization in five healthy volunteers (cohort 1), we aim to recruit 108 participants of whom 72 are people with HD (PwHD) and 36 healthy volunteers (cohort 2). Pending interim analysis, these numbers could increase to 96 PwHD and 48 healthy controls. Participants will complete a total of 10 study visits, consisting of a screening visit followed by a clinical and MRI visit and PET visits at baseline, year 1, and year 2. PET targets include the cannabinoid 1, histamine 3, and serotonin 2A receptors, and phosphodiesterase 10A, whereas MRI will be multimodal, including, but not limited to, the assessment of cerebral blood flow, functional connectivity, and brain iron. Results: Recruitment is currently active and started in September 2022. Conclusions: By combining PET and multi-modal MRI assessments we expect to provide a comprehensive examination of the molecular, functional, and structural framework of HD progression. As such, the iMarkHD study will provide a solid base for the identification of treatment targets and novel outcome measures for future clinical trials.

Alternative Approaches of Solving the Discrete Tomography

The goal of discrete tomography is to reconstruct an unknown function $f$ via a given set of line sums. In addition to requiring accurate reconstructions, it is favourable to be able to perform the task in a timely manner. This is complicated by the presence of ghosts, which allow many solutions to exist in general. In this paper we consider the case of a function $f : A \to \mathbb{R}$ where $A$ is a finite grid in $\mathbb{Z}^3$. Previous work has shown that in the two-dimensional case it is possible to determine all solutions in parameterized form in linear time (with respect to the number of directions and the grid size) regardless of whether the solution is unique. In this work, we show that a similar linear method exists in three dimensions under the condition of nonproportionality. We show that the condition of nonproportionality is fulfilled in the case of three-dimensional boundary ghosts.

Spectral-domain optical coherence tomography is a pervasive, non-invasive, in vivo biomedical imaging platform that currently utilizes incoherent broadband superluminescent diodes to generate interferograms from which depth and structural information are extracted. Advancements in laser frequency microcombs have enabled the chip-scale broadband generation of discrete frequency sources, with prior soliton and chaotic comb states examined in discrete spectral-domain optical coherence tomography at 1.3 μm. In this work, we demonstrate coherence tomography through Si3N4 microresonator laser frequency microcombs at 1 μm, achieving imaging qualities on-par with or exceeding the equivalent commercial optical coherence tomography system. We characterize the noise performance of our frequency comb states and additionally show that inherent comb line amplitude fluctuations in a chaotic state and the resultant tomograms can be compensated via multi-scan averaging.

Discrete tomography is a specific case of computerized tomography that deals with the reconstruction of objects made of a few density values on a discrete lattice of points (integer valued coordinates). In the general case of computerized tomography, several hundreds of projections are required to obtain a single high-resolution slice of the object; in the case of discrete tomography, projections of an object made by just one homogeneous material are sums along very few angles of the pixel values, which can be thought to be 0’s or 1’s without loss of generality. Genetic algorithms are global optimization techniques with an underlying random approach and, therefore, their convergence to a solution is provided in a probabilistic sense. We present here a genetic algorithm able to straightforwardly reconstruct binary objects in the three-dimensional space. To the best of our knowledge, our methodology is the first to require no model of the shape (e.g., periodicity, convexity or symmetry) to reconstruct. Experiments were carried out to test our new approach in terms of computational time and correctness of the solutions. Over the years, discrete tomography has been studied for many interesting applications to computer vision, non-destructive reverse engineering and industrial quality control, electron microscopy, X-rays crystallography, biplane angiography, data coding and compression.

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.

In this paper, we are interested in digital convexity. This notion is applied in several domains like image processing and discrete tomography. We choose to study the inflation and deflation of digital convex sets while maintaining the convexity property. Knowing that any digital convex set can be read and identified by its boundary word, we use combinatorics on words perspective instead of a purely geometric approach. In this context, we characterize the points that can be added or removed over the digital convex sets without losing their convexity. Some algorithms are given at the end of each section with examples of each process.

Discrete tomography focuses on the reconstruction of functions from their line sums in a finite number d of directions. In this paper we consider functions f : A → R where A is a finite subset of ℤ2 and R an integral domain. Several reconstruction methods have been introduced in the literature. Recently Ceko, Pagani and Tijdeman developed a fast method to reconstruct a function with the same line sums as f. Up to here we assumed that the line sums are exact. Some authors have developed methods to recover the function f under suitable conditions by using the redundancy of data. In this paper we investigate the case where a small number of line sums are incorrect as may happen when discrete tomography is applied for data storage or transmission. We show how less than d/2 errors can be corrected and that this bound is the best possible. Moreover, we prove that if it is known that the line sums in k given directions are correct, then the line sums in every other direction can be corrected provided that the number of wrong line sums in that direction is less than k/2.

We consider a class of problems of Discrete Tomography which has been deeply investigated in the past: the reconstruction of convex lattice sets from their horizontal and/or vertical X-rays, i.e. from the number of points in a sequence of consecutive horizontal and vertical lines. The reconstruction of the HV-convex polyominoes works usually in two steps, first the filling step consisting in filling operations, second the convex aggregation of the switching components. We prove three results about the convex aggregation step: (1) The convex aggregation step used for the reconstruction of HV-convex polyominoes does not always provide a solution. The example yielding to this result is called the bad guy and disproves a conjecture of the domain. (2) The reconstruction of a digital convex lattice set from only one X-ray can be performed in polynomial time. We prove it by encoding the convex aggregation problem in a Directed Acyclic Graph. (3) With the same strategy, we prove that the reconstruction of fat digital convex sets from their horizontal and vertical X-rays can be solved in polynomial time. Fatness is a property of the digital convex sets regarding the relative position of the left, right, top and bottom points of the set. The complexity of the reconstruction of the digital convex sets which are not fat remains an open question.

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.

Abstract The 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.

Discrete Tomography (DT) is a technology that uses image projection to reconstruct images. Its reconstruction problem, especially the binary image (0–1 matrix) has attracted strong attention. In this study, a fixed point iterative method of integer programming based on intelligent optimization is proposed to optimize the reconstructed model. The solution process can be divided into two procedures. First, the DT problem is reformulated into a polyhedron judgment problem based on lattice basis reduction. Second, the fixed-point iterative method of Dang and Ye is used to judge whether an integer point exists in the polyhedron of the previous program. All the programs involved in this study are written in MATLAB. The final experimental data show that this method is obviously better than the branch and bound method in terms of computational efficiency, especially in the case of high dimension. The branch and bound method requires more branch operations and takes a long time. It also needs to store a large number of leaf node boundaries and the corresponding consumption matrix, which occupies a large memory space.

A Binary Valued Reconstruction Algorithm for Discrete TDLAS Tomography of Dynamic Flames

Beam arrangement with limited projections based on tunable diode laser absorption spectroscopy is the key to achieving a more accurate two-dimensional reconstruction of the combustion. Using fractional calculus theory, a beam optimization method based on fractional Tikhonov regularization is proposed. The beam arrangement function based on fractional Tikhonov regularization is established by extending the standard Tikhonov regularization to fractional modes. The genetic algorithm is used to analyze the calculation results of different orders in a range of (0, 1), and the beam arrangement is obtained. Using 20 laser beams to scan the characteristic absorption spectrum of H<sub>2</sub>O in the near-infrared band 7185.6 cm<sup>–1</sup>, modeling the calculations in a 10×10 element discrete tomography domain, and comparing the reconstruction results of the five beam arrangements for different Gaussian distribution models, the beam arrangement based on fractional Tikhonov regularization shows more obvious advantages. This design method proposed in this work is valuable for the theoretical study of the optimal design of two-dimensional measurement beams based on the tunable diode laser absorption spectroscopy technique, which can promote the application of this technique in the two-dimensional reconstruction of complex engine combustion and combustion efficiency improvement.

In X-ray computed tomography, discrete tomography (DT) algorithms have been successful at reconstructing objects composed of only a few distinct materials. Many DT-based methods rely on a divide-and-conquer procedure to reconstruct the volume in parts, which improves their run-time and reconstruction quality. However, this procedure is based on static rules, which introduces redundant computation and diminishes the efficiency. In this work, we introduce an update strategy framework that allows for dynamic rules and increases control for divide-and-conquer methods for DT. We illustrate this framework by introducing Tabu-DART, which combines our proposed framework with the Discrete Algebraic Reconstruction Technique (DART). Through simulated and real data reconstruction experiments, we show that our approach yields similar or improved reconstruction quality compared to DART, with substantially lower computational complexity.