The paper presented here describes a new practical approach to the reconstruction problem applied to 3D spiral x-ray tomography. The concept we propose is based on a continuous-to-continuous data model, and the reconstruction problem is formulated as a shift invariant system. This original reconstruction method is formulated taking into consideration the statistical properties of signals obtained by the 3D geometry of a CT scanner. It belongs to the class of nutating reconstruction methods and is based on the advanced single slice rebinning (ASSR) methodology. The concept shown here significantly improves the quality of the images obtained after reconstruction and decreases the complexity of the reconstruction problem in comparison with other approaches. Computer simulations have been performed, which prove that the reconstruction algorithm described here does indeed significantly outperforms conventional analytical methods in the quality of the images obtained.
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 G. N. Ramachandran, A. V. Lakshminarayanan, Three-dimensional reconstruction from radio-graphs and electron micrographs: II. Application of convolutions instead of Fourier transforms, Proc. Nat. Acad. Sci. of USA, vol. 68, 1971, pp. 2236–2240.
 R. M. Lewitt, Reconstruction algorithms: transform methods, Proc. of the IEEE, vol. 71, 1983, pp. 390–408.
 J. D. Mathews et al., Cancer risk in 680 peope expose to computed tomography scans in childhood or adolescent: data inkage study of 11 million Australians, British Medical Journal, f2360, 2013, pp. 346-360.
 K. Sauer, C. Bouman, A local update strategy for iterative reconstruction from projections, IEEE Tran. Signal Proc., vol. 41, 1993, pp. 534–548.
 C. A. Bouman, K. Sauer, A unified approach to statistical tomography using coordinate descent optimization, IEEE Tran. Image Proc., vol. 5, 1996, pp. 480–492.
 Ding Q., Long Y., Zhang X., Fessler J.A.: Modeling mixed Poisson-Gaussian noise in statistical image reconstruction for x-ray CT. In: Proc. of the 4th International Conference on Image Formation in X-Ray Computed Tomography, Bamberg, Germany, 399–402 (2016)
 Geyer, L.L., et al.: State of the art: iterative CT reconstruction techniques. Radiology 276, 339–357 (2017)
 J. -B Thibault, K. D. Sauer, C. A. Bouman, J. Hsieh, A three-dimensional statistical approach to improved image quality for multislice helical CT, Med. Phys., vol. 34, 2007, pp. 4526–4544.
 B. DeMan, S. Basu, Distance-driven projection and backprojection in three dimensions, Phys. Med. Biol., vol. 49, 2004, pp. 2463–2475.
 Y. Zhou, J.-B Thibault, C.A. Bouman, J. Hsieh, K.D. Sauer, Fast model-based x-ray CT reconstruction using spatially non-homogeneous ICD optimization, IEEE Tran. Image Proc., vol. 20, 2011, pp. 161–175.
 R. Cierniak, A. Lorent, Comparison of algebraic and analytical approaches to the formulation of the statistical model-based reconstruction problem for x-ray computed tomography, Computerized Medical Imaging and Graphics, vol. 52, 2016, pp. 19-27.
 R. Cierniak, A new approach to tomographic image reconstruction using a Hopfield-type neural network, International Journal Artificial Intelligence in Medicine, vol. 43, 2008, pp. 113–125.
 R. Cierniak, A new approach to image reconstruction from projections problem using a recurrent neural network, International Journal of Applied Mathematics and Computer Science, vol. 183, 2008, pp. 147–157.
 R. Cierniak, New neural network algorithm for image reconstruction from fan-beam projections, Neurocomputing, vol. 72, 2009, pp. 3238–3244.
 R. Cierniak, A three-dimensional neural network based approach to the image reconstruction from projections problem, Lecture Notes in Artificial Intelligence, 6113, 2010, pp.505–514.
 F. Noo, M. Defrise, R. Clackdoyle, Single-slice rebinning method for helical cone-beam CT, Phys. Med. Biol., vol. 44, 1999, pp. 561–570.
 H. Bruder, M. Kachelrieß, S. Schaller, K. Stierstorfer, T. Flohr, Single-slice rebinning reconstruction in spiral cone-beam computed tomography, IEEE Trans. Med. Imag., vol. 9, 2000, pp. 873–887.
 M. Kachelrieß, S. Schaller, W. A. Kalender, Advanced single-slice rebinning in cone-beam spiral CT, Med. Phys., vol. 27, 2000, pp.754–773.
 M. Kachelrieß;, T. Fuchs, S. Schaller, et al, Advanced single-slice rebinning for tilted spiral cone-beam CT, Med. Phys., vol. 28, 2001, pp.1033–1041.
 R. Cierniak, A novel approach to image reconstruction from projections using Hopfield-type neural network, Lecture Notes in Artificial Intelligence, 4029, 2006, pp. 890–898.
 J. -B. Thibault, C. A. Bouman, K. D. Sauer, J. Hsieh, A recursive filter noise reduction in statistical iterative tomographic imaging, Proc. of SPIE-IS&T Symposium on Electronic Imaging Science and Technology–Computational Imaging, vol. 6065, 2006, pp. 15–19.
 J. -F. Aujol, Some first-order algorithms for total variation based image restoration, J. Math. Im. Vision, vol. 34, 2009, pp. 307–327.
 L.I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D: Nonlinear Phenomena, vol. 60, 1992, pp. 259–268.
 A. C. Kak, M. Slanley, Principles of computerized tomographic imaging, IEEE Press, New York, 1988.