Some recent developments in the Radon transform theory

In this survey several papers related to the Radon transform appeared after 2013 are discussed. For the short survey of the …rst 100 years of the history of the theory, I refer to historical notes in the books of Helgason [7] Natterer [5] and in [50]. Pioneering papers of A. Cormack 1963-64 on "Representation of a function by its line integrals" were devoted to medical applications [3]. Few years later he wrote that "his …rst mathematical problem had been already solved by Radon’s theory". These papers triggered numerous applications of the Radon transform to inverse problems. A fragment of the prehistory of these applications was described by A. Cormack [4]: "In 1906 H. B. A. Bockwinkel used a reconstruction formula of H. A. Lorentz in a paper on propagation of light in biaxial crystals. Lorentz’s result was generalized by G. Uhlenbeck in 1925. In 1936 H. Cramér and H. Wold proved their theorem on marginal distribution (which is widely used in probability theory). Also in 1936 V. A. Ambartsumyan found reconstruction procedures and used these to calculate the distribution of velocities of stars from their radial velocities in various directions. This is the …rst numerical inversion of the Radon transform and it gives the lie to the often made statement that computed tomography would be impossible without computers.” In 1947 J. Szarski and T. Wa· zewski presented a reconstruction procedure by starting from an elementary method which the Polish physician M. S. Majerek 1932 used for the reconstruction of the human head from X-ray pictures of it taken from various directions. R. N. Bracewell ran into Radon’s problem investigating the sun about 1956."