Papers by Robert Tijdeman
Fundamenta Informaticae
Discrete tomography focuses on the reconstruction of functions from their line sums in a finite n... more 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 correct...
The reconstruction of an unknown function f from its line sums is the aim of discrete tomography.... more The reconstruction of an unknown function f from its line sums is the aim of discrete tomography. However, two main aspects prevent reconstruction from being an easy task. In general, many solutions are allowed due to the presence of the switching functions. Even when uniqueness conditions are available, results about the NP-hardness of reconstruction algorithms make their implementation inefficient when the values of f are in certain sets. We show that this is not the case when f takes values in a field or a unique factorization domain, such as or . We present a linear time reconstruction algorithm (in the number of directions and in the size of the grid), which outputs the original function values for all points outside of the switching domains. Freely chosen values are assigned to the other points, namely, those with ambiguities. Examples are provided.
The goal of discrete tomography is to reconstruct an unknown function f via a given set of line s... more 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 switching functions, or ghosts, which allow many solutions to exist in general. Previous work has shown that it is possible to determine all solutions in linear time (with respect to the number of directions and grid size) regardless of whether the solution is unique. In this work, we show that a similar linear algorithm exists in three dimensions. This is achieved by viewing the three-dimensional grid along each 2D coordinate plane, effectively solving the problem with a series of 2D linear algorithms. By that, it is possible to solve the problem of 3D discrete tomography in linear time.
In earlier papers we have developed an algebraic theory of discrete tomography. In those papers t... more In earlier papers we have developed an algebraic theory of discrete tomography. In those papers the structure of the functions f: A →{0,1} and f: A →Z having given line sums in certain directions have been analyzed. Here A was a block in Z^n with sides parallel to the axes. In the present paper we assume that there is noise in the measurements and (only) that A is an arbitrary or convex finite set in Z^n. We derive generalizations of earlier results. Furthermore we apply a method of Beck and Fiala to obtain results of he following type: if the line sums in k directions of a function h: A → [0,1] are known, then there exists a function f: A →{0,1} such that its line sums differ by at most k from the corresponding line sums of h.
At a conference in Debrecen in October 2010 Nathanson announced some results concerning the arith... more At a conference in Debrecen in October 2010 Nathanson announced some results concerning the arithmetic diameters of certain sets. He proposed some related results on the representation of integers by sums or differences of powers of 2 and 3. In this note we prove some results on this problem and the more general problem about the representation by linear combinations of powers of some fixed integers.
Let P be a finite set of at least two prime numbers, and A the set of positive integers that are ... more Let P be a finite set of at least two prime numbers, and A the set of positive integers that are products of powers of primes from P. Let F(k) denote the smallest positive integer which cannot be presented as sum of less than k terms of A. In a recent paper Nathanson asked to determine the properties of the function F(k), in particular to estimate its growth rate. In this paper we derive several results on F(k) and on the related function which denotes the smallest positive integer which cannot be presented as sum of less than k terms from the union of A and -A.
Journal of Mathematical Imaging and Vision
Discrete tomography reconstructs an image of an object on a grid from its discrete projections al... more Discrete tomography reconstructs an image of an object on a grid from its discrete projections along relatively few directions. When the resulting system of linear equations is under-determined, the reconstructed image is not unique. Ghosts are arrays of signed pixels that have zero sum projections along these directions; they define the image pixel locations that have nonunique solutions. In general, the discrete projection directions are chosen to define a ghost that has minimal impact on the reconstructed image. Here we construct binary boundary ghosts, which only affect a thin string of pixels distant from the object centre. This means that a large portion of the object around its centre can be uniquely reconstructed. We construct these boundary ghosts from maximal primitive ghosts, configurations of 2 N connected binary (±1) points over N directions. Maximal ghosts obfuscate image reconstruction and find application in secure storage of digital data.
The goal of discrete tomography is to reconstruct an unknown function f via a given set of line s... more 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 switching functions, or ghosts, which allow many solutions to exist in general. In this paper we consider the case of a function f : A→ R where A is a finite grid in Z3. 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. This is achieved by viewing the threedimensional grid along each 2D coordinate plane, effectively solving the problem with a series of 2D linear algorithms. We show that the condition of nonproportionali...
In this paper we answer Spijker’s question in a slightly generalizedand sharpened form. Spijker a... more In this paper we answer Spijker’s question in a slightly generalizedand sharpened form. Spijker asked the question because of applica-tions to numerical analysis. Linear multistep methods (LMMs) form awell-known class of numerical step-by-step methods for solving initial-value problems for certain systems of ordinary differential equations.In many applications of such methods it is essential that the LMMhas specific stability properties. An important property of this kindis named boundedness and has recently been studied by Hundsdorfer,Mozartova and Spijker [2]. In that paper the stepsize-coefficient γ is acrucial parameter in the study of boundedness. In [3] Spijker attemptsto single out all LMMs with a positive stepsize-coefficient γ for bound-edness. By using Corollary 1 below he is able to nicely narrow the classof such LMMs.Theorem1. Let n be a positive integer. Put m = ⌊n/2⌋. Let be givennonzero complex numbers b
Indagationes Mathematicae, 2013
Indagationes Mathematicae, 2013
In this note we prove results of the following types. Let be given distinct complex numbers z j s... more In this note we prove results of the following types. Let be given distinct complex numbers z j satisfying the conditions |z j | = 1, z j ̸ = 1 for j = 1,. .. , n and for every z j there exists an i such that z i = z j. Then inf k n j=1 z k j ≤ −1. If, moreover, none of the ratios z i /z j with i ̸ = j is a root of unity, then inf k n j=1 z k j ≤ − 1 π 4 log n. The constant −1 in the former result is the best possible. The above results are special cases of upper bounds for inf k n j=1 b j z k j obtained in this paper. c
arXiv: Number Theory, 2016
This paper provides a survey of results on the greatest prime factor, the number of distinct prim... more This paper provides a survey of results on the greatest prime factor, the number of distinct prime factors, the greatest squarefree factor and the greatest m-th powerfree part of a block of consecutive integers, both without any assumption and under assumption of the abc-conjecture. Finally we prove that the explicit abc-conjecture implies the Erdős–Woods conjecture for each k ≥ 3.
Journal of Mathematical Imaging and Vision
Hardy-Ramanujan Journal
International audience This paper aims to show two things. Firstly the importance of Alan Baker&#... more International audience This paper aims to show two things. Firstly the importance of Alan Baker's work on linear forms in logarithms for the development of the theory of exponential Diophantine equations. Secondly how this theory is the culmination of a series of greater and smaller discoveries.
Hardy-Ramanujan Journal
International audience Tomography is the theory behind scans, e.g. MRI-scans. Most common is cont... more International audience Tomography is the theory behind scans, e.g. MRI-scans. Most common is continuous tomography where an object is reconstructed from numerous projections. In some cases this is not applicable, because the object changes too quickly or is damaged by making hundreds of projections (by X-rays). In such cases discrete tomography may apply where only few projections are made. The present paper shows how number theory helps to provide insight in the application and structure of discrete tomography.
Discrete Applied Mathematics
Fundamenta Informaticae
For continuous tomography Helgason and Ludwig developed consistency conditions. They were used by... more 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.
In many applications of digital picture processing such as segmentation, skeletonizing and cluste... more In many applications of digital picture processing such as segmentation, skeletonizing and clustering, it is important to compute approximations of the Euclidean distance ecien tly. Borgefors, Verwer, Coquin and Bolon, and others have developed a method to do so based on choosing local distances for small neighborhood sizes. These local distances are integer multiples of some xed scaling factor. In
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Papers by Robert Tijdeman