Development of the block-cycling inversion in computer tomography

Abstract

This paper describes an improved block-cyclic inversion (BCI) method [1] for the three-dimensional (3D) inverse Radon problem: a major problem in computer tomography. A spiral-fan scheme of scanning (SFSS) and the cylindrical domain of inspection are taken. The 3D problem is reduced by decomposition to the series P of 2D problems (with the same Radon matrix). The account taken for a priori information about the circular FSS invariance allows the direct block-cyclic inversion of the Radon matrix by use of the Greville-1 block algorithm instead of the classic block Toeplitz inversion (BTI) [2,3], based on the concept of the Teoplitz range. The performance of the BCI versus BTI algorithm is better by a factor N both at the stage of preliminary inversion and in the flow due to vectorization. The required volume of memory is a sixth, but the main advantage of BCI is its simplicity of implementation, because of the absence of the degeneration problem, which is inherent in the classic method, as well as a higher stability. This has allowed a space resolution of up to 201 × 201. In 101 × 101 resolution, the counting time of a 2 s order in the variant of the PC PENTIUM-4 model (language Visual Fortran 90), and the time of the Radon matrix inversion itself is 20 s with the stability coefficient of ∼10 (in L ₂ metrics), 75 (in C metrics), i.e., by a factor of 3–10 better than in [23]. This is achieved due to the filtration of noise in the right-hand part, smoothing the solution itself and some other improvements of the algorithm. The problem mentioned in [1] was also solved. The obtained results can be applied to the software support of 4th generation tomographs.