Mathematical modelling of reconstruction of volumetric images in x-ray computed tomography using holographic methods

We consider a computational algorithm for solving the inverse problem of x-ray tomography to reconstruct the internal structure of micro objects in the near Fresnel zone using holographic methods of visualization of three-dimensional images. Holographic methods do not give a direct solution to the problem on reconstruction of the internal structure of the object. These methods can only solve the problem on volumetric mapping of some object surface. However, using the data on the absorption of both x-ray radiation of the object and phase contrast holographic signals in the near Fresnel zone, we show the possibility to obtain a volumetric holographic image of the inner layers of the object. In order to solve this complex problem, it is necessary to use a three-dimensional (3D) Radon transformation of the internal function of the object and a two-dimensional (2D) Radon transformation of the phase contrast holographic projection. We obtain an algorithm to reconstruct phase-contrast tomographic images of the internal structure of an object. Based on the algorithm, we construct a computational algorithm for the practical reconstruction of volumetric tomographic images of the internal structure of microobjects. The results of the research were confirmed by the mathematical modelling of the algorithm to reconstruct three-dimensional images. To this end, we develop a mathematical model of the test phantom, and simulate the phase contrast projections for the test phantom. Then, we develop a software in order to reconstruct the phase contrast tomographic images by tomographic methods on the basis of the obtained phase contrast projections.