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A Solution to Hammer's X-ray Reconstruction Problem

by Richard J. Gardner and Markus Kiderlen
Research Reports Number 482 (August 2006)
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 $k\to\infty$. This solves, for the first time in the strongest sense, Hammer's X-ray problem published in 1963.
Format available: PDF (287 KB)
Published in Adv. Math. 214, 323-343 (2007).
The definitive full-text version is for instance available at ScienceDirect
This primarily serves as Thiele Research Reports number 13-2006, but was also published in Research Reports