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Grayscale uncertainty and errors of tomographic reconstructions based on projection geometries and projection sets

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Abstract

In certain cases of computed tomography, the projection acquisition process is limited, and thus one cannot gain a sufficient number of projections for an acceptable reconstruction. In this case, the low number of projections yields a lack of information, and uncertainty in the reconstructions. In practice, this means that the pixel values of the reconstruction are not uniquely determined by the measured data and thus can have variable values. In this paper, we provide a theoretically proven uncertainty measure that can be used for measuring the variability of pixel values in grayscale reconstructions. The uncertainty values are based on linear algebra and measure the slopes of the hyperplane of solutions in the algebraic formulation of tomography. The method can also be applied for any linear equation system that satisfies a given set of conditions. Using the uncertainty measure, we also derive upper and lower bounds on the possible pixel values in tomographic reconstructions. Finally, we show how our results can be used for modelling reconstruction error. All of the results are supported by both theoretical proofs and experimental evaluations

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  1. https://www.developer.nvidia.com/cuda-zone.

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Acknowledgements

Gábor Lékó was supported by the ÚNKP-20-4 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund. This research was supported by the project “Integrated program for training new generation of scientists in the fields of computer science”, no. EFOP-3.6.3-VEKOP16-2017-00002. This research was supported by grant NKFIH-1279-2/2020 of the Ministry for Innovation and Technology, Hungary. This research was supported by Project no. TKP2021-NVA-09. Project no. TKP2021-NVA-09 has been implemented with the support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021-NVA funding scheme. The authors would like to thank Bence Savanya for aiding by discovering numerical concepts for the formulas.

Funding

Gábor Lékó was supported by the ÚNKP-20-4 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund. This research was supported by the project “Integrated program for training new generation of scientists in the fields of computer science”, no. EFOP-3.6.3-VEKOP16-2017-00002. This research was supported by grant NKFIH-1279-2/2020 of the Ministry for Innovation and Technology, Hungary. This research was supported by Project TKP2021-NVA-09. Project no. TKP2021-NVA-09 has been implemented with the support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021-NVA funding scheme.

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  1. Department of Image Processing and Computer Graphics, University of Szeged, H-6720 Árpád tér 2, Szeged, Hungary

    László G. Varga, Gábor Lékó & Péter Balázs

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  1. László G. Varga
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  2. Gábor Lékó
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  3. Péter Balázs
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The proposed methods and given results were the results of the joint work of the authors.

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Correspondence to László G. Varga.

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Varga, L.G., Lékó, G. & Balázs, P. Grayscale uncertainty and errors of tomographic reconstructions based on projection geometries and projection sets. Vis Comput 39, 1557–1569 (2023). https://doi.org/10.1007/s00371-022-02428-y

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