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Markus Kiderlen

A Solution to Hammer's X-ray Reconstruction Problem

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  • Richard J. Gardner, Western Washington University, United States
  • Markus Kiderlen
  • Department of Mathematical Sciences
We propose algorithms for reconstructing a planar convex body K from
possibly noisy measurements of either its parallel X-rays taken in a fixed finite set of
directions or its point X-rays taken at a fixed finite set of points, in known situations
that guarantee a unique solution when the data is exact. The algorithms construct a
convex polygon P(k) whose X-rays approximate (in the least squares sense) k equally
spaced noisy X-ray measurements in each of the directions or at each of the points.
It is shown that these procedures are strongly consistent, meaning that, almost
surely, P(k) tends to K in the Hausdorff metric as k tends to infinity. This solves, for the first time in the strongest sense, Hammer’s X-ray problem published in 1963.
Original languageEnglish
JournalAdvances in Mathematics
Volume214
Pages (from-to)323-343
Number of pages21
ISSN0001-8708
DOIs
Publication statusPublished - 2007

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