An experimental study of the stability problem in discrete tomography

doi:10.1016/S1571-0653(04)00475-5

Electronic Notes in Discrete Mathematics

Volume 12, March 2003, Pages 70-81

https://doi.org/10.1016/S1571-0653(04)00475-5 Get rights and content

Abstract

This paper introduces the topic of discrete tomography, briefly showing its main applications, algorithms and new prospects of research. It focuses on the still open problem of stability, facing it from an experimental point of view. In particular an extensive simulation lets verify the robustness of a well known reconstruction technique for binary convex objects, calculating the probability of finding solutions compatible with a given set of noisy projections.

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Stability results for the reconstruction of binary pictures from two projections
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In the present paper we mathematically prove several stability results concerning the problem of reconstructing binary pictures from their noisy projections taken from two directions. Stability is a major requirement in practice, because projections are often affected by noise due to the nature of measurements. Reconstruction from projections taken along more than two directions is known to be a highly unstable task. Contrasting this result we prove several theorems showing that reconstructions from two directions closely resemble the original picture when the noise level is low and the original picture is uniquely determined by its projections.
Random generation of Q-convex sets
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The problem of randomly generating Q-convex sets is considered. We present two generators. The first one uses the Q-convex hull of a set of random points in order to generate a Q-convex set included in the square $[0, n)^{2}$ . This generator is very simple, but is not uniform and its performance is quadratic in n. The second one exploits a coding of the salient points, which generalizes the coding of a border of polyominoes. It is uniform, and is based on the method by rejection. Experimentally, this algorithm works in linear time in the length of the word coding the salient points. Besides, concerning the enumeration problem, we determine an asymptotic formula for the number of Q-convex sets according to the size of the word coding the salient points in a special case, and in general only an experimental estimation.
On the stability of reconstructing lattice sets from X-rays along two directions
2005, Lecture Notes in Computer Science

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ArticlesAn experimental study of the stability problem in discrete tomography

Abstract

Theoretical Computer Science

Linear Algebra and its Applications

Inform. Process. Lett.

Linear Algebra and its Applications

Linear Algebra and its Applications

Discrete Mathematics

Linear Algebra and its Applications

Discrete Mathematics

Computational Geometry, Theory and Applications

Pattern Recognition Letters

Principles of Computerized Tomography Imaging

DIMACS Mini-Symposium on Discrete Tomography

Application of Discrete Tomography to Electron Microscopy of Crystals

Medians of polyominoes: A property for the reconstruction

Int. J. Imaging Syst. Technol

Reconstruction of connected sets from two projections

Discrete Tomography: Foundations, Algorithms, and Applications, Birkhauser

Articles
An experimental study of the stability problem in discrete tomography