Ábris Nagy

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

Gardner and Kiderlen (Adv. Math. 214:323–343, 2007) 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

A Robbins–Monro-type algorithm for computing global minimizer of generalized conic functions

K\subset \mathbb {R}^{N}

On a special class of generalized conics with infinitely many focal points

by integration. The basic result (Vincze and Nagy in J. Approx. Theory 164:371–390, 2012) is that the generalized conic function associated to a compact planar set determines the coordinate X-rays and vice versa. Vincze and Nagy (Submitted to Aequationes Math., 2014) 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.

Full length article: On the theory of generalized conics with applications in geometric tomography

We generalize the notion and some properties of the conic function introduced by Vincze and Nagy in 2012. We provide a stochastic algorithm for computing the global minimizer of generalized conic functions, we prove almost sure and -convergence of this algorithm.

An introduction to the theory of generalized conics and their applications

Let a continuous, piecewise smooth curve in the Euclidean space be given. We are going to investigate the surfaces formed by the vertices of generalized cones with such a curve as the common directrix and the same area. The basic geometric idea in the background is when the curve runs through the sides of a non-void triangle ABC. Then the sum of the areas of some triangles is constant for any point of such a surface. By the help of a growth condition we prove that these are convex compact surfaces in the space provided that the points A, B and C are not collinear. The next step is to introduce the general concept of awnings spanned by a curve. As an important example awnings spanned by a circle will be considered. Estimations for the volume of the convex hull will be also given.