Using Tomography in Digital Plane to solve problems of Geometric Tomography

We study the problem of determining in a constructive way a convex body in the plane from its tomographic projections. For this, we consider the similar problem in digital plane: reconstructing a lattice convex set from its discrete tomographic projection. We show that we can use a reconstruction algorithm for the discrete problem to solve the continuous reconstruction to any precision. The proof of this result uses stability properties of geometric tomography. An extension to point-source tomographic projections is also investigated. Parallel sources case If F is a convex body of R and p = (a, b) is a direction then the (parallel source) tomographic projection of F with respect to p (denoted XpF ) is defined by: XpF (u) = length({(x, y) ∈ F : bx− ay = u}). Hammer’s X-ray problem consists to reconstruct a convex body of R from a minimum number of tomographic projections. This is the most classical problem of Geometric Tomography ([5]). In [8] the authors characterize completely the sets of directions which permit to reconstruct all the convex sets.(such a set is called a Gardner-McMullen set of directions. In particular it is proved that all the sets of directions which provide uniqueness have a cardinal not less than four. But this result is not constructive, it does not give a method to reconstruct the convex body from its projections. In the literature there are several descriptions of constructive methods which attempt to make this reconstruction: In [12] the authors are able to reconstruct and infinite set of points of the border of the convex body but until now we are not able to prove that this infinite set is dense in the border. In [11], the authors construct a subset and a superset of approximative solution by making some choices and use “filling operations”, but unfortunately there is no proof that the sequence of the obtained solution converges to the good set even when the directions provide uniqueness. In [7], the method reconstructs a polygon which tomographic projections has the least square distance with the projection. It is proven that the polygon tends to the set when the number of vertices tends to infinity but it lacks an efficient method to find the polygon. ∗LSIIT CNRS UMR 7005, Université de Strasbourg, Pôle API, Boulevard Sébastien Brant, 67400 Illkirch-Graffenstaden, France, [email protected]