Geodesic
Development of block сycling inversion method in computer tomography
Matematičeskoe modelirovanie, Tome 24 (2012) no. 5, pp. 65-80.

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

The development of block-cycling Radon inversion method (BCI) [1] in computer tomography for spiral-fan scheme of scanning (SFSS) and cylinder inspection domain is presented. 3-dim inverse Radon problem is reduced to series of $P$ 2-dim inverse Radon problems with the same Radon matrix. Taking into account a priori information about circle invariability for fan scheme of scanning (FSS) allows to apply direct block-cycling inversion of 2-dim Radon matrix by block-Greville-1 method instead of a classical block-teoplitz inversion (BTI) [2,3] based on the notion of teoplitz rang. The time complexity of the BCI algorithm $N$ times better by performance at the stage of the preliminary inversion, so as on the flow due to the vectotization. Memory volume required is also 6 times better. But it’s main advantage — the simplicity of implementation due to the absence of main minor degeneration problem. The BCI algorithm was numerically simulated with the space resolution up to $201\times201$ (with – 2 sec. on the flow with 20 sec. for preliminary inversion of Radon matrix with spatial resolution $101\times101$ at the PC PENTIUM–4, Visual Fortran 90). Stability coefficient $\sim 10$, 75(in metric $\mathrm{L}_2$, $\mathrm{C}$) — 3–10 times better comparing with result in [23] due to the filtration of noise in Radon projection, smoothing of the solution and some other improvements. Singularity problem mentioned in [1] is also solved. The results obtained in this work may be applied for fourth generation tomography soft ware.
Keywords: (accuracy, complexity, parallel processing, stability) of algorithm, computer tomography (FFT(Fast Fourier Transform), BCI(Block Cycling Inversion), BTI(Block Teoplitz Inversion), GB(Glassman-de Boor), BP(Back Projection), LS(Least Square) and others) methods, operator, problem, projection), (Wedderburn, convolution) theorem, (fan-spyral, parallel) Scheme of Scanning.
Mots-clés : G(Greville), NN(Neuron Nets), Radon(equation, image, matrix, Gauss–Markoff 
@article{MM_2012_24_5_a5,
     author = {A. V. Khovanskiy},
     title = {Development of block {\cyrs}ycling inversion method in computer tomography},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {65--80},
     publisher = {mathdoc},
     volume = {24},
     number = {5},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2012_24_5_a5/}
}
TY  - JOUR
AU  - A. V. Khovanskiy
TI  - Development of block сycling inversion method in computer tomography
JO  - Matematičeskoe modelirovanie
PY  - 2012
SP  - 65
EP  - 80
VL  - 24
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2012_24_5_a5/
LA  - ru
ID  - MM_2012_24_5_a5
ER  - 
%0 Journal Article
%A A. V. Khovanskiy
%T Development of block сycling inversion method in computer tomography
%J Matematičeskoe modelirovanie
%D 2012
%P 65-80
%V 24
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2012_24_5_a5/
%G ru
%F MM_2012_24_5_a5
A. V. Khovanskiy. Development of block сycling inversion method in computer tomography. Matematičeskoe modelirovanie, Tome 24 (2012) no. 5, pp. 65-80. http://geodesic.mathdoc.fr/item/MM_2012_24_5_a5/

[1] A. V. Khovanskii, A. M. Demkin, “Metody blochno-tsiklicheskogo obrascheniya v kompyuternoi tomografii”, Matematicheskoe modelirovanie, 23:1 (2001), 51–64 | MR

[2] V. V. Voevodin, E. E. Tyrtyshnikov, Vychislitelnye protsessy s tëplitsevymi matritsami, Nauka, M., 1987, 320 pp. | MR | Zbl

[3] F. Natterer, Matematicheskie aspekty kompyuternoi tomografii, Mir, M., 1990, 280 pp. | MR | Zbl

[4] A. V. Likhachev, V. V. Pikalov, “Sintezirovannyi algoritm trekhmernoi tomografii”, Matematicheskoe modelirovanie, 10:1 (1998), 1–73 | MR

[5] V. V. Pikalov, I. G. Kazantsev, V. P. Golubyatnikov, Voprosy programmirovaniya, 7:2 (2006), 180–184

[6] A. N. Tikhonov, V. Ya. Arsenin, Metody resheniya nekorrektnykh zadach, Nauka, M., 1979, 288 pp. | MR

[7] A. N. Tikhonov, V. Ya. Arsenin, A. A. Timonov, Matematicheskie zadachi kompyuternoi tomografii, Nauka, M., 1987, 159 pp. | MR

[8] I. N. Troitskii, Statisticheskaya teoriya tomografii, Radio i svyaz, M., 1989, 239 pp. | MR

[9] D. K. Faddeev, V. N. Faddeeva, Vychislitelnye metody lineinoi algebry, Fizmatlit, M., 1963, 734 pp. | MR

[10] G. A. Fedorov, S. A. Tereschenko, Vychislitelnaya emissionnaya tomografiya, Energoatomizdat, M., 1990, 183 pp.

[11] G. Khermen, Vosstanovlenie izobrazhenii po proektsiyam: osnovy rekonstruktivnoi tomografii, Mir, M., 1983 | MR

[12] A. V. Khovanskii, “Metody postroeniya pochti adamarovykh tsirkulyantov i vozmozhnosti ikh prilozheniya”, Matematicheskoe modelirovanie, 8:1 (1996), 69–76 | MR

[13] A. V. Khovanskii, “Regulyarizovannyi algoritm Grevillya i ego primenenie v transmissionnoi kompyuternoi tomografii”, Matematicheskoe modelirovanie, 8:11 (1996), 109–118 | MR

[14] V. V. Pikalov, T. S. Melnikova, Nizkotemperaturnaya plazma, v. 13, Tomografiya plazmy, Nauka, Novosibirsk, 1995, 224 pp.

[15] I. Abaffi, E. Spedikato, Matematicheskie metody dlya lineinykh i nelineinykh uravnenii. Proektsionnye ABS-algoritmy, Mir, M., 1996, 268 pp. | MR

[16] G. Nolet (red.), Seismicheskaya tomografiya, Mir, M., 1990, 416 pp.

[17] R. Bleikhut, Bystrye algoritmy tsifrovoi obrabotki signalov, Mir, M., 1989 | MR

[18] V. Yu. Terebizh, Vvedenie v statisticheskuyu teoriyu obratnykh zadach, Fizmatlit, M., 2005, 375 pp.

[19] N. S. Bakhvalov, Chislennye metody, Nauka, M., 1975, 631 pp. | MR

[20] G. Kramer, Matematicheskie metody statistiki, Mir, M., 1975 | MR

[21] F. R. Gantmakher, Teoriya matrits, Nauka, M., 1967, 575 pp. | MR

[22] T. Kokhonen, Assotsiativnaya pamyat, Mir, M., 1980, 238 pp. | MR

[23] K. Brammer, G. Ziffling, Filtr Kalmana–Byusi, Nauka, M., 1982, 199 pp.