Tomographic task solution using a dichotomous discretization scheme in polar coordinates and partial system matrices invariant to rotations

Objectives. The purpose of this work was to create an effective iterative algorithm for the tomographic reconstruction of objects with large volumes of initial data. Unlike the convolutional projection algorithm, widely used in commercial industrial and medical tomographic devices, algebraic iterative reconstruction methods use significant amounts of memory and typically involve long reconstruction times. At the same time, iterative methods enable a wider range of diagnostic tasks to be resolved where greater accuracy of reconstruction is required, as well as in cases where a limited amount of data is used for sparse-view angle shooting or shooting with a limited angular range.
Methods. A feature of the algorithm thus created is the use of a polar coordinate system in which the projection system matrices are invariant with respect to the rotation of the object. This enables a signification reduction of the amount of memory required for system matrices storage and the use of graphics processors for reconstruction. Unlike the simple polar coordinate system used earlier, we used a coordinate system with a dichotomous division of the reconstruction field enabling us to ensure invariance to rotations and at the same time a fairly uniform distribution of spatial resolution over the reconstruction field.
Results. A reconstruction algorithm was developed on the basis of the use of partial system matrices corresponding to the dichotomous division of the image field into partial annular reconstruction regions. A 2D and 3D digital phantom was used to show the features of the proposed reconstruction algorithm and its applicability to solving tomographic problems.
Conclusions. The proposed algorithm allows algebraic image reconstruction to be implemented using standard libraries for working with sparse matrices based on desktop computers with graphics processors.

1. Feldkamp L.A., Davis L.C., Kress J.W. Practical cone-beam algorithm J. Opt. Soc. Am. A. 1984;1(6):612–619. https://doi.org/10.1364/JOSAA.1.000612

2. Zou Y., Pan X. Image reconstruction on PI-lines by use of filtered backprojection in helical cone-beam CT. Phys. Med. Biol. 2004;49:2717–2731. https://doi.org/10.1088/0031-9155/49/12/017

3. Parker D.L. Optimal short scan convolution reconstruction for fan-beam CT. Med. Phys. 1982;9(2):245–257. https://doi.org/10.1118/1.595078

4. Chen Z., Jin X., Li L., Wang G. A limited-angle CT reconstruction method based on anisotropic TV minimization. Phys. Med. Biol. 2013;58:2119–2141. https://doi.org/10.1088/0031-9155/58/7/2119

5. Wang C., Tao M., Nagy J.G., Lou Y. Limited-angle CT reconstruction via the L1/L2 minimization. SIAM Journal on Imaging Sciences. 2021;14(2):749–777. https://doi.org/10.1137/20M1341490

6. Li M., Zhang C., Peng C., Guan Y., Xu P., Sun M., Zheng J. Smoothed l0 norm regularization for sparse-view X-ray CT reconstruction. BioMed Res. Int. 2016;2016:Article ID 2180457. https://doi.org/10.1155/2016/2180457

7. Sun Y., Chen H., Tao J., Lei L. Computed tomography image reconstruction from few views via Log-norm total variation minimization. Digital Signal Processing. 2019;88:172–181. https://doi.org/10.1016/j.dsp.2019.02.009

8. Sun Y., Tao J. Few views image reconstruction using alternating direction method via ℓ0-norm minimization. Int. J. Syst. Technol. 2014;24(3):215–223. https://doi.org/10.1002/ima.22097

9. Xu Z., Chang X., Xu F., Zhang H. L1/2 regularization: A thresholding representation theory and a fast solver. IEEE Trans. Neural Networks Learn. Syst. 2012;23(7):1013–1027. https://doi.org/10.1109/TNNLS.2012.2197412

10. Wang C., Yan M., Rahimi Y., Lou Y. Accelerated schemes for L1/L2 minimization. IEEE Trans. Signal Processing. 2020;68:2660–2669. https://doi.org/10.1109/TSP.2020.2985298

11. Jumanazarov D., Koo J., Kehres, J., Poulsen H.F., Olsen U.L., Iovea M. Material classification from sparse spectral X-ray CT using vectorial total variation based on L infinity norm. Mater. Charact. 2022;187:111864. https://doi.org/10.1016/j.matchar.2022.111864

12. Hegazy M.A.A., Cho M.H., Cho M.H., Lee S.Y. Metal artifact reduction in dental CBCT Images using direct sinogram correction combined with metal path-length weighting. Sensors. 2023;23(3):1288. https://doi.org/10.3390/s23031288

13. Bigury A., Dosanjh M., Hancock S., Soleimani M. Tigre: A MATLAB-GPU toolbox for CBCT image reconstruction. Biomed. Phys. Eng. Express. 2016;2(5):055010. http://doi.org/10.1088/2057-1976/2/5/055010

14. Siddon R.L. Fast calculation of the exact radiological path for a three-dimensional CT array. Med. Phys. 1985;12(2):252–255. https://doi.org/10.1118/1.595715

15. Landweber L. An iteration formula for Fredholm integral equations of the first kind. Am. J. Math. 1951;73(3):615–624. https://doi.org/10.2307/2372313

16. Tuy H.K. An inversion formula for cone-beam reconstruction. SIAM. J. Appl. Math. 1983;43(3):546–552. https://doi.org/10.1137/0143035

17. Osipov S.P., Chakhlov S.V., Zhvyrblia V.Y., Sednev D.A., Osipov O.S., Usachev E.Y. The Nature of Metal Artifacts in X-ray Computed Tomography and Their Reduction by Optimization of Tomography Systems Parameters. Appl. Sci. 2023;13(4):2666. https://doi.org/10.3390/app13042666

18. Hashem N., Pryor M., Haas D., Hunter J. Design of a Computed Tomography Automation Architecture. Appl. Sci. 2021;11(6):2858. https://doi.org/10.3390/app11062858

19. Jian L., Litao L., Peng C., Qi S., Zhifang W. Rotating polar-coordinate ART applied in industrial CT image reconstruction. NDT&E International. 2007;40(4):333–336. https://doi.org/10.1016/j.ndteint.2006.11.005