Reconstruction of hv-Convex Sets by Their Coordinate X-Ray Functions

Abstract

R. J. Gardner and M. Kiderlen [17] presented an algorithm for reconstructing convex bodies from noisy X-ray measurements with a full proof of convergence in 2007. We would like to present some new steps into the direction of reconstructing not necessarily convex bodies by the help of the continuity properties of so-called generalized conic functions. Such a function measures the average taxicab distance of the points from a given compact set K ⊂ R N by integration. The basic result [28] is that the generalized conic function associated to a compact planar set determines the coordinate X-rays and vice versa. Cs. Vincze andÁ Nagy [29] proved continuity properties of the mapping which sends connected compact hv-convex sets having the same axis parallel bounding box to the associated generalized conic functions. We use these results to present an algorithm for the reconstruction of compact connected hv-convex planar bodies given by their coordinate X-rays. The basic method is varied with the quota system scheme. Greedy and anti-greedy versions are also presented with examples.

Generalized conic functions of hv-convex planar sets: continuity properties and relations to X-rays

“…where the set H is the subcollection of non-empty compact connected hvconvex sets which are constituted by the subrectangles belonging to a partition of the reference set [9]. It is typically a rectangle B with parallel sides to the coordinate axes such that K ⊂ B.…”

“…The theorem is a way to rephrase the problem of determination in terms of the function Φ. Although it is probably hard to check the continuity of Φ −1 at f K continuity properties may give answers in terms of algorithms by finding the possible alternatives: let f K be the input data and consider the optimization problem minimize f L − f K subject to L ∈ H, where the set H is the subcollection of non-empty compact connected hvconvex sets which are constituted by the subrectangles belonging to a partition of the reference set [9]. It is typically a rectangle B with parallel sides to the coordinate axes such that K ⊂ B.…”

Geometric and Visual Reconstruction of Binary Shapes

“…However, there can be found exact solutions in the case of special classes of binary im-ages [19,26,69]. Furthermore, the projection angle selection also affects the quality of the reconstructions [107,98].…”