Exact Image Reconstruction from a Single Projection through Real Computation

Abstract

In Discrete Tomography one aims to reconstruct a function (image) with a known discrete range from its projection along certain directions. By modern electron-microscopy techniques, one can count the number of atoms laying on a line representing, e.g., an X-ray. The so obtained data is used in the integer programming formulation. However, in real applications the size of the problem, that is well-known to be NP-hard, is so large that no method seems to be applicable to it. Other natural restrictions can make the problem even harder. In an attempt to avoid such kind of difficulties, we present an alternative approach to the problem. With this, we also aim to shed more light on the theoretical limitations for efficient computation in Discrete Tomography. Our approach is based on image reconstruction from a single projection, under the hypothesis that all computations take place in an algebraic computation model. In terms of computational efficiency, the proposed algorithm is significantly superior to the known algorithms for the problem. We also discuss on the possibilities for practical implementation of our method.