Mathematical Problems of Tomography

Description

As early as 1917, Radon derived an explicit formula for the reconstruction of a function on the plane, given its integrals over all lines. In the late 1960s, the first applications of the Radon formula appeared, in radio astronomy and then in electron micrography. The use of the Radon formula for constructing tomograms, made possible by the advent of the computer, saw its first use in clinical medicine in 1970 and earned its developers the Nobel Prize in medicine. Today, practical application of the Radon transform, especially in medical tomography, has continued to capture the attention of mathematicians, partly because of the range of new applications that have been found. But the most fascinating aspect for mathematicians may be the opportunity to apply deep mathematics to tackle new problems arising from real-world applications. The papers in this volume cover various problems arising from and related to computerized tomography. The main idea unifying the papers is that the methods used satisfy strong requirements imposed by practical applications of computerized tomography, such as reconstruction of non-smooth functions, pointwise convergence, and discretization in computational algorithms. The papers draw upon a broad range of mathematical areas, including integral geometry, the theory of several complex variables, the theory of distributions, and integral transformations. In addition, applications to reconstruction of biological objects and mathematical economics are given Content: Introduction : integral geometry and tomography / I.M. Gelfand and S.G. Gindikin -- On the convergence of a class of algorithms for the inversion of the numerical Radon transform / D.A. Popov -- Three-dimensional reconstruction of arbitrarily arranged identical particles given their projections / A.B. Goncharov -- Spatial rotational alignment of identical particles given their projections : theory and practice / M.S. Gelfand and A.B. Goncharov -- Some singular problems in tomography / V.P. Palamodov -- Discrete Radon transform and image reconstruction / N.D. Vvedenskaya and S.G. Gindikin -- Bernstein theorems and the Radon transform : application to the theory of production functions / G.M. Henkin and A.A. Shananin -- Mathematical models and algorithms of tomographic synthesis of wave fields and inhomogeneous media / V.M. Buchstaber and V.K. Maslov.