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In two dimensions, we consider the problem of inversion of the attenuated \begin{document}$ X $$\end{document}-ray transform of a compactly supported function from data restricted to lines leaning on a given arc. We provide a method to reconstruct the function on the convex hull of this arc. The attenuation is assumed known. The method of proof uses the Hilbert transform associated with \begin{document}$$ A $$\end{document}$-analytic functions in the sense of Bukhgeim. 

This paper presents a new algorithm for X-ray Computerized Tomography (CT) based on Bukhgeim’s theory of analytic maps. The reconstruction relies on a Cauchy-type integral formula, where the integration over the boundary replaces the integration in the back- projection operator used in existing algorithms. From the numerical computation stand point, the proposed method recovers the attenuation coefficient at arbitrarily points by utilizing the boundary integration without internal global meshes. This means that it achieves high-parallel efficiency, and it reduces computational resources. Some numerical examples are presented to show feasibility of the proposed algorithm. 

This work concerns the numerical realization of a Cauchy-type integral formula for sequence valued analytic functions in the sense of Bukhgeim, and its applications to the source reconstruction problem in inverse radiative transport through a non-absorbing and non-scattering medium. The inverse source problem is mathematically equivalent to the classical X-ray Computed Tomography (CT), where a function is to be determined from its line integrals. The proposed algorithms have the added advantage to extend to the source determination problems in media with absorbing and scattering properties. Such extensions cannot be achieved in the existing X-ray CT algorithms. The numerical experiments demonstrate the feasibility of our new tomographic algorithms. 

Abstract We revisit the inverse source problem in a two dimensional absorbing and scattering medium and present a direct reconstruction method, which does not require iterative solvability of the forward problem, using measurements of the radiating flux at the boundary. The attenuation and scattering coefficients are known and the unknown source is isotropic. The approach is based on the Cauchy problem for a Beltrami-like equation for the sequence valued maps, and extends the original ideas of Bukhgeim from the non-scattering to the scattering media. We demonstrate the feasibility of the method in a numerical experiment in which the scattering is modeled by the two dimensional Henyey–Greenstein kernel with parameters meaningful in optical tomography. 

Ultrafine-grained and heterostructured materials are currently of high interest due to their superior mechanical and functional properties. Severe plastic deformation (SPD) is one of the most effective methods to produce such materials with unique microstructure-property relationships. In this review paper, after summarizing the recent progress in developing various SPD methods for processing bulk, surface and powder of materials, the main structural and microstructural features of SPD-processed materials are explained including lattice defects, grain boundaries and phase transformations. The properties and potential applications of SPD-processed materials are then reviewed in detail including tensile properties, creep, superplasticity, hydrogen embrittlement resistance, electrical conductivity, magnetic properties, optical properties, solar energy harvesting, photocatalysis, electrocatalysis, hydrolysis, hydrogen storage, hydrogen production, CO2 conversion, corrosion resistance and biocompatibility. It is shown that achieving such properties is not currently limited to pure metals and conventional metallic alloys, and a wide range of materials are processed by SPD, including high-entropy alloys, glasses, semiconductors, ceramics and polymers. It is particularly emphasized that SPD has moved from a simple metal processing tool to a powerful means for the discovery and synthesis of new superfunctional metallic and nonmetallic materials. The article ends by declaring that the borders of SPD have been extended from materials science and it has become an interdisciplinary tool to address scientific questions such as the mechanism of geological and astronomical phenomena and the origin of life. Keywords: Severe plastic deformation (SPD); Nanostructured materials; Ultrafine grained (UFG) materials; Gradient-structured materials, High-pressure torsion (HPT)