An Algorithm for the Reconstruction of hv-convex Planar Bodies by Finitely Many and Noisy Measurements of their Coordinate X-rays

Issue title: Tomography and Applications

Article type: Research Article

Authors: Vincze, Csaba^{a; *} | Nagy, Ábris^b

Affiliations: [a] Institute of Mathematics, University of Debrecen, P. O. Box 12, 4010 Debrecen, Hungary. [email protected] | [b] Institute of Mathematics, MTA-DE Research Group ”Equations Functions and Curves”, Hungarian Academy of Sciences and University of Debrecen, P. O. Box 12, 4010 Debrecen, Hungary. [email protected]

Note: [*] Address for correspondence: Institute of Mathematics, University of Debrecen, P. O. Box 12, 4010 Debrecen, Hungary

Abstract: Parallel X-rays are functions that measure the intersection of a given set with lines parallel to a fixed direction in ℝ2. The reconstruction problem concerning parallel X-rays is to reconstruct the set if the parallel X-rays into some directions are given. There are several algorithms to give an approximate solution of this problem. In general we need some additional knowledge on the object to obtain a unique solution. By assuming convexity a suitable finite number of directions is enough for all convex planar bodies to be uniquely determined by their X-rays in these directions [13]. Gardner and Kiderlen [12] presented an algorithm for reconstructing convex planar bodies from noisy X-ray measurements belonging to four directions. For a reconstruction algorithm assuming convexity we can also refer to [17]. An algorithm for the reconstruction of hv-convex planar sets by their coordinate X-rays (two directions) can be found in [18]: given the coordinate X-rays of a compact connected hv-convex planar set K the algorithm gives a sequence of polyominoes Ln all of whose accumulation points (with respect to the Hausdorff metric) have the given coordinate X-rays almost everywhere. If the set is uniquely determined by the coordinate X-rays then Ln tends to the solution of the problem. This algorithm is based on generalized conic functions measuring the average taxicab distance by integration [21]. Now we would like to give an extension of this algorithm that works in the case when only some measurements of the coordinate X-rays are given. Following the idea in [12] we extend the algorithm for noisy X-ray measurements too.

DOI: 10.3233/FI-2015-1270

Journal: Fundamenta Informaticae, vol. 141, no. 2-3, pp. 169-189, 2015