Geodesic
On phase correction in tomographic research
Sibirskij žurnal industrialʹnoj matematiki, Tome 23 (2020) no. 4, pp. 18-29.

Voir la notice de l'article provenant de la source Math-Net.Ru

Under consideration is the problem of improving the contrast of the image obtained by processing tomographic projections with phase distortion. The study is based on the well-known intensity transfer equation. Unlike other works, this equation is solved in a bounded region of variation of the tomographic parameters. In a domain, a boundary value problem is posed for the intensity transfer equation which is then specialized for a three-dimensional parallel tomographic scheme. The case of two-dimensional tomography is also considered, together with the corresponding boundary value problem for the intensity transfer equation. We propose numerical methods for solving the boundary value problems of phase correction. The results are given of the numerical experiments on correction of tomographic projections and reconstruction of the structure of the objects under study (in particular, a slice of a geological sample) by using piecewise uniform regularization.
Keywords: tomography, phase correction, intensity transfer, regularization, ill-posed problem. . 
@article{SJIM_2020_23_4_a1,
     author = {Ya. Wang and A. S. Leonov and D. V. Lukyanenko and V. D. Shinkarev and A. G. Yagola},
     title = {On phase correction in tomographic research},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {18--29},
     publisher = {mathdoc},
     volume = {23},
     number = {4},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2020_23_4_a1/}
}
TY  - JOUR
AU  - Ya. Wang
AU  - A. S. Leonov
AU  - D. V. Lukyanenko
AU  - V. D. Shinkarev
AU  - A. G. Yagola
TI  - On phase correction in tomographic research
JO  - Sibirskij žurnal industrialʹnoj matematiki
PY  - 2020
SP  - 18
EP  - 29
VL  - 23
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJIM_2020_23_4_a1/
LA  - ru
ID  - SJIM_2020_23_4_a1
ER  - 
%0 Journal Article
%A Ya. Wang
%A A. S. Leonov
%A D. V. Lukyanenko
%A V. D. Shinkarev
%A A. G. Yagola
%T On phase correction in tomographic research
%J Sibirskij žurnal industrialʹnoj matematiki
%D 2020
%P 18-29
%V 23
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJIM_2020_23_4_a1/
%G ru
%F SJIM_2020_23_4_a1
Ya. Wang; A. S. Leonov; D. V. Lukyanenko; V. D. Shinkarev; A. G. Yagola. On phase correction in tomographic research. Sibirskij žurnal industrialʹnoj matematiki, Tome 23 (2020) no. 4, pp. 18-29. http://geodesic.mathdoc.fr/item/SJIM_2020_23_4_a1/

[1] D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object”, J. Microscopy, 206 (2001), 33–40 | DOI | MR

[2] S. C. Mayo, T. J. Davis, T. E. Gureyev, P. R. Miller, D. Paganin, A. Pogany, A. W. Stevenson, S. W. Wilkins, “X-ray phase-contrast microscopy and microtomography”, Optics Express, 11:19 (2003), 2289–2302 | DOI

[3] M. A. Beltran, D. M. Paganin, K. Uesugi, M. J. Kitchen, “2D and 3D X-ray phase retrieval of multimaterial objects using a single defocus distance”, Optics Express, 18:7 (2010), 6423–6436 | DOI

[4] A. A. Samarskii, Introduction to the theory of difference schemes, Nauka, M., 1971 (in Russian) | MR

[5] N. N. Kalitkin, Numerical methods, Nauka, M., 1978 (in Russian)

[6] A. N. Tikhonov, V. Ya. Arsenin, Method of solving ill-posed problems, Nauka, M., 1979 (in Russian)

[7] V. K. Ivanov, V. V. Vasin, V. P. Tanana, Theory of linear ill-posed problems and its application, Nauka, M., 1978 (in Russian)

[8] V. A. Morozov, Regular methods of solving ill-posed problems, Nauka, M., 1987 (in Russian)

[9] A. N. Tikhonov, A. V. Goncharskii, V. V. Stepanov, A. G. Yagola, Numerical methods of solving ill-posed problems, Nauka, M., 1990 (in Russian) ; Курс, М., 2017 | MR

[10] A. N. Tikhonov, A. S. Leonov, A. G. Yagola, Nonlinear ill-posed problems, Nauka, M., 1995; Kurs, M., 2017

[11] H. W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996 | MR | Zbl

[12] A. S. Leonov, Solution of ill-posed inverse problems. An outline of the theory, practical algorithms, and demonstrations by MATLAB, LIBROKOM, M., 2009 (in Russian)

[13] A. S. Leonov, “Functions of several variables of bounded variation in the ill-posed problems”, Zh. Vychisl. Mat. Mat. Fiz., 36:9 (1996), 35–49 | MR | Zbl

[14] A. S. Leonov, “Some remarks on complete variation of functions of several variables and a multidimensional analog of Helly's choice principle”, Mat. Zametki, 63:1 (1998), 69–80 (in Russian) | MR | Zbl

[15] A. S. Leonov, “Numerical piecewise-uniform regularization for two-dimensional ill-posed problems”, Inverse Probl., 15 (1999), 1165–1176 | DOI | MR | Zbl

[16] A. S. Leonov, Y. Wang, A. G. Yagola, “Piecewise uniform regularization for the inverse problem of microtomography with a-posteriori error estimate”, Inverse Probl. Sci. Engrg, 27 (2019), 1–11 | DOI | MR

[17] Y. F. Wang, S. F. Fan, A. S. Leonov, D. V. Lukyanenko, A. G. Yagola, L. H. Wang, Wang Yu, J. Q. Wang, “Non-smooth regularization and fast optimization algorithm for micropore reconstruction of shale”, Chinese J. Geophysics, 63:5 (2019), 2036–2042