A solution to Hammer's X-ray reconstruction problem

Abstract

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 → ∞ . This solves, for the first time in the strongest sense, Hammer's X-ray problem published in 1963.