Abstract
In the present paper, we first describe the principal basis of non-integral geometry. Non-integral geometry is a new field of generalized function (distribution) theory, where the effects breaking the symmetry of integration measure have been investigated. In turn, the non-symmetric integration measure (the non-invariant measure) leads to the complex form of the universal, dimension-independent inverse operator with the additional contributions compared to the methods of integral geometry. The additional term with the complex integration measure serves to the extension that improves the image reconstruction procedure. Then, we prove that this additional term fA in the universal inverse Radon transforms plays a role of the regularizing contribution. In particular, we show that owing to the presence of fA, the corresponding complex singularities can be eliminated in the image reconstruction process.
Acknowledgements
References
[1] I. M. Gelfand, G. E. Shilov, Generalized Functions, Vol. 1: Properties and Operations, Academic Press, 1964.
[2] I. M. Gelfand, M. I. Graev, N. Ya. Vilenkin, Generalized Functions, Vol. 5: Integral Geometry and Representation Theory, AMS Chelsea Publishing: An Imprint of the American Mathematical Society, 1966, 449 pp.
[3] S. R. Deans, The Radon Transform and Some of Its Applications, Wiley, 1983, 299 pp.
[4] I. V. Anikin and L. Szymanowski, Phys. Rev. D 100 (9) (2019) 094034. doi:10.1103/PhysRevD.100.094034; [arXiv:1909.00017 [hep-ph]].
[5] I. V. Anikin and X. Chen, Mod. Phys. Lett. A 39 (38) (2024) 2450181. doi:10.1142/S0217732324501815; [arXiv:2405.14897 [physics.comp-ph]].
[6] I. V. Anikin, Natural Sci. Rev. 2 (5) (2025) 100501. doi:10.54546/NaturalSciRev.100501; [arXiv:2506.18911 [math.FA]].
[7] I. V. Anikin, Mod. Phys. Lett. A 40 (32) (2025) 2550141. doi:10.1142/S021773232550141X; [arXiv:2504.01744 [math.CA]].
[8] R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II, Interscience, New York, 1962.
[9] I. V. Anikin, A. I. Anikina, and O. I. Streltsova, Reconstruction Problem: Computational Scheme (in preparation).