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Journal of Mathematical Imaging and Vision

, Volume 49, Issue 3, pp 569–582 | Cite as

Reconstruction of hv-Convex Sets by Their Coordinate X-Ray Functions

  • Ábris NagyEmail author
  • Csaba Vincze
Article

Abstract

Gardner and Kiderlen (Adv. Math. 214:323–343, 2007) presented an algorithm for reconstructing convex bodies from noisy X-ray measurements with a full proof of convergence in 2007. We would like to present some new steps into the direction of reconstructing not necessarily convex bodies by the help of the continuity properties of so-called generalized conic functions. Such a function measures the average taxicab distance of the points from a given compact set \(K\subset \mathbb {R}^{N}\) by integration. The basic result (Vincze and Nagy in J. Approx. Theory 164:371–390, 2012) is that the generalized conic function associated to a compact planar set determines the coordinate X-rays and vice versa. Vincze and Nagy (Submitted to Aequationes Math., 2014) proved continuity properties of the mapping which sends connected compact hv-convex sets having the same axis parallel bounding box to the associated generalized conic functions. We use these results to present an algorithm for the reconstruction of compact connected hv-convex planar bodies given by their coordinate X-rays. The basic method is varied with the quota system scheme. Greedy and anti-greedy versions are also presented with examples.

Keywords

Parallel X-ray Generalized conic Geometric tomography Linear integer programming 

Notes

Acknowledgements

Ábris Nagy has been supported, in part, by the Hungarian Academy of Sciences, the European Union and the State of Hungary, co-financed by the European Social Fund in the framework of TÁMOP 4.2.4. A/2-11-1-2012-0001 ‘National Excellence Program’.

Csaba Vincze was partially supported by the European Union and the European Social Fund through the project Supercomputer, the national virtual lab (grant No.: TÁMOP-4.2.2.C-11/1/KONV-2012-0010).

This work is supported by the University of Debrecen’s internal research project RH/885/2013.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Institute of Mathematics, MTA-DE Research Group “Equations Functions and Curves”Hungarian Academy of Sciences and University of DebrecenDebrecenHungary
  2. 2.Institute of MathematicsUniversity of DebrecenDebrecenHungary

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