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Boundary Ghosts for Discrete Tomography

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Abstract

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 non-unique 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 (\(\pm 1\)) points over N directions. Maximal ghosts obfuscate image reconstruction and find application in secure storage of digital data.

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Acknowledgements

The School of Physics and Astronomy at Monash University, Australia, has provided funds for this work. M.C. has the support of the Australian government’s Research Training Program (RTP) and the J. L. William scholarship from the School of Physics and Astronomy at Monash University. The authors thank the referees for their critical remarks which led to improvements in the paper.

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Authors and Affiliations

  1. School of Physics and Astronomy, Monash University, Melbourne, Australia

    Matthew Ceko, Timothy Petersen & Imants Svalbe

  2. Monash Centre for Electron Microscopy, Monash University, Melbourne, Australia

    Timothy Petersen

  3. Mathematical Institute, Leiden University, Leiden, Netherlands

    Rob Tijdeman

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  1. Matthew Ceko
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  2. Timothy Petersen
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  4. Rob Tijdeman
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Correspondence to Matthew Ceko.

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Ceko, M., Petersen, T., Svalbe, I. et al. Boundary Ghosts for Discrete Tomography. J Math Imaging Vis 63, 428–440 (2021). https://doi.org/10.1007/s10851-020-01010-2

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  • DOI: https://doi.org/10.1007/s10851-020-01010-2

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