A generalization on the solvability of integral geometry problems along plane curves
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A generalization on the solvability of integral geometry problems along plane curves Zekeriya Ustaoglu* *
Correspondence: [email protected] Department of Mathematics, Bulent Ecevit University, Zonguldak, 67100, Turkey
Abstract This paper is concerned with a general condition for the solvability of integral geometry problems along the plane curves of given curvatures. As two important results, the solvabilities of integral geometry problems along the family of circles with fixed radius and along the family of circles of varying radius centered on a fixed circle are given. By using some extension of the class of unknown functions, the proofs are based on the solvabilities of equivalent inverse problems for transport-like equation. MSC: 35R30; 53C65; 65N30 Keywords: integral geometry problem; inverse problem; Galerkin method; transport-like equation
1 Introduction The problems of integral geometry are to determine a function, given (weighted) integrals of this function over a family of manifolds, and there has been significant progress in the classical Radon problem when manifolds are hyperplanes and the weight function is unity, there are interesting results in the plane case when a family of curves is regular or in the case of a family of straight lines with arbitrary regular attenuation [, Chapter ]. It is assumed that the basis of the integral geometry problems is the Radon transform []. The Radon transform R integrates a function f on Rn over hyperplanes. Let H(s, ) = {x ∈ Rn : x · = s} be the hyperplane perpendicular to ∈ Sn– (unit sphere) with signed distance s ∈ R from the origin, and the Radon transform (Rf )(s, ) is defined as the integral of f over H(s, ), i.e., (Rf )(s, ) =
f (x) dx H(s,)
(see [, Chapter ]). The problems of integral geometry have important applications in imaging and provide the mathematical background of tomography, where the main goal is to recover the interior structure of a nontransparent object using external measurements. The object under investigation is exposed to radiation at different angles, and the radiation parameters are measured at the points of observation. The basic problem in computerized tomography is the reconstruction of a function from its line or plane integrals, and there are many applications related with computerized tomography: medical imaging, geophysics, diagnostic radiology, astronomy, seismology, radar and many other fields (see, e.g., []). © 2013 Ustaoglu; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Ustaoglu Boundary Value Problems 2013, 2013:202 http://www.boundaryvalueproblems.com/content/2013/1/202
From the applied point of view, the importance of integral geometry problem over a family of straight lines in the plane is indicated in [], where the probl
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