Karakalpak Scientific Journal


The problems of integral geometry in a strip on a family of curves of hyperbolic and spherical type are considered which have numerous applications in problems of geophysics, thermoacoustic and photoacoustic tomography. Explicit formulas are obtained for the Fourier image of the solution of integral geometry problems in the class of smooth compactly supported functions. Further, the obtained formulas are investigated for stability using numerical methods. To solve the problems, algorithms are constructed. Numerical and graphical results of applying these algorithms to solving the problems are presented.

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[1] R.A. Kruger, P. Liu, Y.R. Fang and C.R. Appledorn, “Photoacoustic ultrasound (PAUS) reconstruction tomography”, Med. Phys, vol. 22, 1995, pp. 1605–1609.

[2] D. Finch, Rakesh and S. Patch, “Determining a function from its mean values over a family of spheres”, SIAM J. Math. Anal., vol. 35, 2004, pp. 1213–1240.

[3] F. John, Plane Waves and Spherical Means. Wiley, New York, 1955.

[4] M.M. Lavrentiev, V.G. Romanov and V.G. Vasiliev, Multidimensional Inverse Problems for Differential Equations. Lecture Notes in Math. 167, Springer-Verlag, New York, 1970.

[5] M.M. Lavrentiev, V.G. Romanov and S.P. Shishatskii, Ill-posed Problems of Mathematical Physics and Analysis. Transl. Math. Monogr. 64, AMS, Providence, RI, 1986.

[6] M.L. Agranovsky and E.T. Quinto, “Injectivity sets for the Radon transform over circles and complete systems of radial functions”, J. Funct. Anal., vol. 139. 1996, pp. 383–414.

[7] M.L. Agranovsky and E.T. Quinto, “Geometry of stationary sets for the wave equation in the case of finitely supported initial data”, Duke Math. J., vol. 107, 2001, pp. 57–84.

[8] S.Moon, “Inversion of the seismic parabolic Radon transform and the seismic hyperbolic Radon transform”, Inverse Problems in Science and Engineering, vol. 24, 2016, pp. 317–327.

[9] J. Hu and S. Fomel, “A fast butterfly algorithm for the hyperbolic Radon transform”, Society of Exploration Geophysicists, 2012, pp. 1-5.

[10] A.H. Begmatov, G.M. Djaykov, “Linear problem of integral geometry in the strip with smooth weight functions and perturbation”, Vladikavkaz. Mat. J., vol. 17, no. 3, 2015, pp. 14-22.

[11] A. Begmatov, G. Djaykov, “Numerical recovery of function in a strip from given integral data on linear manifolds”, Proceedings of the International forum on strategic technology, Part 1, Novosibirsk, pp. 478-483, June 2016.

[12] N.U. Uteuliev, G.M. Djaykov, S.A. Yadgarov, “Analytical and numerical reconstruction of internal structure of the objects in a family of straight-line segments”, International Conference on Information Science and Communications Technologies: Applications, Trends and Opportunities, Tashkent 2019.

[13] N.U. Uteuliev, G.M. Djaykov, “Modeling the problem of determining the internal structure of an object from integral data on a family of line segments”, Problems of Computational and Applied Mathematics, vol. 2, 2018, pp. 47-61.

[14] N.U. Uteuliev,G.M. Djaykov, Sh. Yadgarov, “About one of the problem of integral geometry in a strip on a family of broken lines”, Science and Society, vol.4, 2019, pp. 27-29.

[15] A.H. Begmatov, G.M. Djaykov, “Reconstruction of the function by its spherical means”, Reports of the Russian Federation Higher Education Academy of Sciences, vol. 20, no.1, 2013, pp. 6-16.



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