Development of block сycling inversion method in computer tomography

A. V. Khovanskiy

Matematičeskoe modelirovanie, Tome 24 (2012) no. 5, pp. 65-80

Résumé

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.

Détail

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