Stability and Instability in Discrete Tomography

Alpers, Andreas; Gritzmann, Peter; Thorens, Lionel

doi:10.1007/3-540-45576-0_11

Cited by 34 publications

(71 citation statements)

References 11 publications

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“…Our results are somewhat related to the stability problem in discrete tomography, which has been studied by several authors [1][2][3]8,[15][16][17] for the grid model. The stability problem deals with the question whether a small perturbation of the observed data results in a small perturbation of the reconstruction.…”

Section: Introductionmentioning

confidence: 73%

“…The bounds on the difference between binary solutions that will be introduced in the upcoming sections are functions of the Euclidean norm of the solutions of the binary solution problem (1). In this section we present lower and upper bounds on the Euclidean norm (also referred to as length) of all binary solutions of the equation system (1).…”

Section: The Euclidean Norm Of Binary Solutionsmentioning

confidence: 99%

“…In this section we present lower and upper bounds on the Euclidean norm (also referred to as length) of all binary solutions of the equation system (1).…”

Section: The Euclidean Norm Of Binary Solutionsmentioning

confidence: 99%

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Quality bounds for binary tomography with arbitrary projection matrices

Fortes

Batenburg

2015

Discrete Applied Mathematics

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a b s t r a c tBinary tomography deals with the problem of reconstructing a binary image from a set of its projections. The problem of finding binary solutions of underdetermined linear systems is, in general, very difficult and many such solutions may exist. In a previous paper we developed error bounds on differences between solutions of binary tomography problems restricted to projection models where the corresponding matrix has constant column sums. In this paper, we present a series of computable bounds that can be used with any projection model. In fact, the study presented here is not restricted to tomography and works for more general linear systems.We report the results of computational experiments for some phantom images, focused on parallel and fan beam projection models. Our results show that in some cases the computed bounds can be used to prove that the difference between binary solutions must be small, even if the corresponding linear system is severely underdetermined.

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Section: Introductionmentioning

confidence: 73%

Section: The Euclidean Norm Of Binary Solutionsmentioning

confidence: 99%

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Quality bounds for binary tomography with arbitrary projection matrices

Fortes

Batenburg

2015

Discrete Applied Mathematics

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“…In practice, these X-rays are affected by noise. In [3] the authors investigate the stability of the problem of reconstruction from X-rays taken from m ≥ 3 lattice directions. In this case they show that a small change in the data (of 2(m − 1) measured in the 1 -norm) can lead to a dramatic change in the reconstruction.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we analyze the case m = 2 with the same kind of data errors as in [3], i.e., where 2(m−1) = 2. We show in Theorem 17 that lattice sets which are uniquely determined by their X-rays along two directions can only have stable reconstructions even if the X-rays are changed by 2 (in the 1 -norm).…”

Section: Introductionmentioning

confidence: 99%

On the Stability of Reconstructing Lattice Sets from X-rays Along Two Directions

Alpers

Brunetti

2005

Discrete Geometry for Computer Imagery

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Abstract. We consider the stability problem of reconstructing lattice sets from their noisy X-rays (i.e. line sums) taken along two directions. Stability is of major importance in discrete tomography because, in practice, these X-rays are affected by errors due to the nature of measurements. It has been shown that the reconstruction from noisy X-rays taken along more than two directions can lead to dramatically different reconstructions. In this paper we prove a stability result showing that the same instability result does not hold for the reconstruction from two directions. We also show that the derived stability result can be carried over by similar techniques to lattice sets with invariant points.

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Discrete Tomography: Reconstruction under Periodicity Constraints

Lungo

Frosini

Nivat

et al. 2002

Automata, Languages and Programming

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This paper studies the problem of reconstructing binary matrices that are only accessible through few evaluations of their discrete X-rays. Such question is prominently motivated by the demand in material science for developing a tool for the reconstruction of crystalline structures from their images obtained by high-resolution transmission electron microscopy. Various approaches have been suggested for solving the general problem of reconstructing binary matrices that are given by their discrete X-rays in a number of directions, but more work have to be done to handle the ill-posedness of the problem. We can tackle this ill-posedness by limiting the set of possible solutions, by using appropriate a priori information, to only those which are reasonably typical of the class of matrices which contains the unknown matrix that we wish to reconstruct. Mathematically, this information is modelled in terms of a class of binary matrices to which the solution must belong. Several papers study the problem on classes of binary matrices on which some connectivity and convexity constraints are imposed. We study the reconstruction problem on some new classes consisting of binary matrices with periodicity properties, and we propose a polynomialtime algorithm for reconstructing these binary matrices from their orthogonal discrete X-rays.keywords: combinatorial problem, discrete tomography, binary matrix, polyomino, periodic constraint, discrete X-rays.

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Stability and Instability in Discrete Tomography

Cited by 34 publications

References 11 publications

Quality bounds for binary tomography with arbitrary projection matrices

Quality bounds for binary tomography with arbitrary projection matrices

On the Stability of Reconstructing Lattice Sets from X-rays Along Two Directions

Discrete Tomography: Reconstruction under Periodicity Constraints

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