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1

Error Correction for Discrete Tomography

Ceko Matthew, Hajdu Lajos, Tijdeman Rob

Fundamenta Informaticae

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2022

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Vol. 189, nr 2

91--112

EN

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. [

2

Regions of Uniqueness Quickly Reconstructed by Three Directions in Discrete Tomography

Dulio P., Pagani S. M. C., Frosini A.

Fundamenta Informaticae

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2017

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Vol. 155, nr 4

407--423

EN

In discrete tomographic image reconstruction, projections are taken along a finite set S of valid directions for a working grid A. In general, uniqueness cannot be achieved in the whole grid A. Usually, some information on the object to be reconstructed is introduced, that, sometimes, allows possible ambiguities to be removed. From a different perspective, one aims in finding subregions of A where uniqueness can be guaranteed, and obtained in linear time, only from the knowledge of S. When S consists of two lattice directions, the shape of any such region of uniqueness, say ROU, have been completely characterized in previous works by means of a double Euclidean division algorithm called DEDA. Results have been later extended to special triples of directions, under a suitable assumption on their entries. In this paper we remove the previous assumption, so providing a complete characterization of the shape of the ROU for such kind of triples. We also show that the employed strategy can be even applied to more general sets of three directions, where the corresponding ROU can be characterized as well. Independently of the combinatorial interest of the problem, the result can be exploited to define in advance, namely before using any kind of radiation, suitable sets of directions that allow regions of interest to be included in the corresponding ROU. Results have been proved in all details, and several experiments are considered, in order to support the theoretical steps and to clarify possible applications.

3

Reconstructing Binary Matrices under Window Constraints from their Row and Column Sums

Alpers A., Gritzmann P.

Fundamenta Informaticae

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2017

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Vol. 155, nr 4

321--340

EN

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 polynomialtime solvability to NP-hardness.

4

Consistency Conditions for Discrete Tomography

Hajdu L., Tijdeman R.

Fundamenta Informaticae

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2017

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Vol. 155, nr 4

425--447

EN

For continuous tomography Helgason and Ludwig developed consistency conditions. They were used by others to overcome defects in the measurements. In this paper we introduce a consistency criterion for discrete tomography. We indicate how the consistency criterion can be used to overcome defects in measurements.

5

Experimental Study on Multivalued Phantoms Using Different Filters in the DART Algorithm

Nagy A.

Fundamenta Informaticae

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2015

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Vol. 141, nr 2/3

207--231

EN

In our previous paper [10] we proposed new variants of the Discrete Algebraic Reconstruction Technique with combined filtering technique and performed experiments on binary software phantoms within a new test framework to investigate the effect of the filters. Continuing our work, in this paper we extend our study to multivalued phantoms for a deeper investigation in this field. We create a new test phantom set with different intensity levels and perform a comprehensive experimental study. The results are evaluated with Relative Mean Error which is extended to multivalued discrete phantoms. We use a ranking system to create different views to our quantitative data. Finally, the achievements are discussed.

6

How Random is Your Tomographic Noise? A Number Theoretic Transform (NTT) Approach

Fiorini R. A.

Fundamenta Informaticae

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2014

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Vol. 135, nr 1/2

135--170

EN

Discrete Tomography (DT), differently from GT and CT, focuses on the case where only few specimen projections are known and the images contain a small number of different colours (e.g. black-and-white). A concise review on main contemporary physical and mathematical CT system problems is offered. Stochastic vs. Combinatorially Optimized Noise generation is compared and presented by two visual examples to emphasise a major double-bind problem at the core of contemporary most advanced instrumentation systems. Automatic tailoring denoising procedures to real dynamic system characteristics and performance can get closer to ideal self-registering and selflinearizing system to generate virtual uniform and robust probing field during its whole designed service life-cycle. The first attempt to develop basic principles for system background low-level noise source automatic characterization, profiling and identification by CICT, from discrete system parameter, is presented. As a matter of fact, CICT can supply us with cyclic numeric sequences perfectly tuned to their low-level multiplicative source generators, related to experimental high-level overall perturbation (according to high-level classic perturbation computational model under either additive or multiplicative perturbation hypothesis). Numeric examples are presented. Furthermore, a practical NTT example is given. Specifically, advanced CT system, HRO and Mission Critical Project (MCP) for very low Technological Risk (TR) and Crisis Management (CM) system will be highly benefitted mostly by CICT infocentric worldview. The presented framework, concepts and techniques can be used to boost the development of next generation algorithms and advanced applications quite conveniently.

7

A New Approach for the Reconstruction of Object-Based Images in Discrete Tomography

Brocchi S.

Fundamenta Informaticae

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2014

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Vol. 135, nr 1/2

43--57

EN

In this paper we tackle the problem of the reconstruction of object based images, specifically formed by a set of circles inside a ring. By analyzing the projections of the image, we are able to determine some coordinates corresponding to interest points that give significant information about features of the image of aid in the reconstruction. Our approach yields promising results in comparison to other methods in literature. Finally, we discuss how a similar approach could be extended to more complex problems deriving from tomographic applications, in order to develop an efficient method exploiting the prior knowledge assumed on an image.

8

Solving Multicolor Discrete Tomography Problems by Using Prior Knowledge

Barcucci E., Brocchi S.

Fundamenta Informaticae

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2013

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Vol. 125, nr 3/4

313--328

EN

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.

9

Discrete Tomography in MRI : a Simulation Study

Segers H., Palenstijn W. J., Batenburg K. J., Sijbers J.

Fundamenta Informaticae

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2013

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Vol. 125, nr 3/4

223--237

EN

Discrete tomography (DT) is concerned with the tomographic reconstruction of images that consist of only a small number of grey levels. Recently, DART, a practical algorithm was introduced for discrete tomography, which was validated in the domain of X-ray computed and electron tomography. In this paper, DART is introduced for magnetic resonance imaging. Using simulation experiments, it is shown that the proposed MRI-DART algorithm is capable of computing high quality reconstructions from substantially fewer data compared to state-of-the-art MRI reconstruction methods.

10

Discrete Tomography Data Footprint Reduction via Natural Compression

Fiorini R. A., Condorelli A., Laguteta G.

Fundamenta Informaticae

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2013

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Vol. 125, nr 3/4

273--284

EN

In Discrete Tomography (DT) by electron microscopy, 2-D projection images are acquired from various angles, by tilting the sample, generating new challenges associated with the problem of formation, acquisition, compression, transmission, and analysis of enormous quantity of data. Data Footprint Reduction (DFR) is the process of employing one or more techniques to store a given set of data in less storage space. Modern lossless compressors use classical probabilistic models only, and are unable to match high end application requirements like “Arbitrary Bit Depth” (ABD) resolution and information “Dynamic Upscale Regeneration” (DUR). Traditional \mathbbQ Arithmetic can be regarded as a highly sophisticated open logic, powerful and flexible bidirectional (LTR and RTL) formal language of languages, according to brand new “Information Conservation Theory” (ICT). This new awareness can offer competitive approach to guide more convenient algorithm development and application for combinatorial lossless compression, we named “Natural Compression” (NC). To check practical implementation performance, a first raw example is presented, benchmarked to standard, more sophisticate lossless JPEG2000 algorithm, and critically discussed. NC raw overall lossless compression performance compare quite well to standard one, but offering true ABD and DUR at no extra computational cost.

11

Discrete Tomography Data Footprint Reduction by Information Conservation

Fiorini R. A., Laguteta G.

Fundamenta Informaticae

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2013

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Vol. 125, nr 3/4

261--272

EN

The first impact of Discrete Tomography (DT) applied to nanoscale technology has been to generate enormous quantity of data. Data Footprint Reduction (DFR) is the process of employing one or more techniques to store a given set of data in less storage space. The very best modern lossless compressors use classical probabilistic models only, and are unable to match high end application requirements, like “Arbitrary Bit Depth” (ABD) resolution and “Dynamic Upscale Regeneration” (DUR), with full information conservation. This paper explores, at core level, the basic properties and relationships of Q Arithmetic to achieve full numeric information conservation and regeneration, algorithmically. That knowledge shows strong connections to modular group theory and combinatorial optimization. Traditional Q Arithmetic can be even regarded as a highly sophisticated open logic, powerful and flexible LTR and RTL formal numeric language of languages, with self-defining consistent word and rule, starting from elementary generator and relation. This new awareness can guide the development of successful more convenient algorithm and application.

12

Approximate Discrete Reconstruction Algorithm

Batenburg K. J., Fortes W., Tijdeman R.

Fundamenta Informaticae

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2013

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Vol. 125, nr 3/4

239--259

EN

Discrete tomography deals with tomographic reconstruction of greyscale images for which the set of possible grey levels is discrete and small. Here, we develop a discrete approximate reconstruction algorithm. Our algorithm computes an image that has only grey values belonging to a given finite set. It also guarantees that the difference between the given projections and the projections of the reconstructed discrete image is bounded. The bound, which is computable, is independent of the image size. We present reconstruction experiments for a range of phantom images and a varying number of grey values.

13

Global extremes analysis in applications of discrete tomography

Nowak L., Zieliński B.

Bio-Algorithms and Med-Systems

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2010

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Vol. 6, no. 12

21--24

EN

One of the most important problems in discrete tomography is to reconstruct function f:a→{0,1}, where a is a finite subset of Z˄n (n ≥2), from the finite set of projections. There are a lot of methods dedicated for this problem, which employ basic methods of discrete mathematic, distribution theory, and even evolutionary algorithms. In this paper, new approach to this problem, based on global extremes analysis, is presented. It is competitive with the other algorithms, due to the fact that, it returns projections identical with the original ones and is most effective in case of images with consistent objects.