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

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