Total curvature and intersection tomography
A basic inverse problem of integral geometry is the following one: When is a ``one-dimensional'' set in the plane determined by the number of intersections with a variable line, i.e., by its Crofton transform [\textit{H. Steinhaus}, Colloq. Math. 3, 1-13 (1954; Zbl 0057.14602)]? In this article the author takes the class of \(K\)-sets, i.e., generalized sets, represented by finite collections of closed rectifiable curves of finite total absolute curvature. He shows how to reconstruct \(K\)-sets in the Euclidean plane from their projections onto variable lines (Theorem 1.1) [cf. also \textit{H. Fast}, Real. Anal. Exch. 19, 59-80 (1994; Zbl 0797.28002); \textit{T. J. Richardson}, Discrete Comput. Geom. 16, 21-31 (1996; Zbl 0859.53052)]. Furthermore, he proves a projection formula, relating the total absolute curvature of \(K\)-sets in the Euclidean plane to the average total variation of their projections (Theorem 1.2).
scientific article; zbMATH DE number 1066076
scientific article