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. 2020 Nov 19;10:20146. doi: 10.1038/s41598-020-77156-1

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© The Author(s) 2020

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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Reconstruction process with the present method (QURT) using all improved reconstruction procedures. (a) General model susceptible to the missing wedge used in the simulation. (b) Reconstructed result with the conventional SIRT from ± 70°, 5° step projection data. Missing wedge causes considerable blur. (c) Fourier transform pattern of (b) showing a ‘missing wedge’ and radial spectral lines based on the projection slice theorem. (d) Fourier transform pattern of the present reconstruction (QURT) result of (h), which recovers the missing wedge spectral region and all the gaps among those radial lines. Typical ring-like diffraction pattern generated from the doughnut model is seen. (e–h) Convergence process from low to high image definition (16 × 16, 32 × 32, 64 × 64, and 512 × 512 pixels (final)). In this process, three base tilt angles Θ are utilised − 70°, 0°, and + 70°, effectively improving the recovery of the structure rather than the single Θ case shown in Fig. 4h. Final image (h) recovers the model without artificial blur. (i–l) Error maps corresponding to (e)–(h), respectively. As the model is gradually recovered, the error map converges to the flat zero distribution.