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An analytical formula for the covariance matrix of basis material projection estimates in spectral x-ray computed tomography

Spectral x-ray computed tomography with photon counting detectors involves the estimation of basis material projections from energy-resolved measurements, which is also known as sinogram material decomposition. The statistically optimal image reconstruction requires knowledge of the full covariance matrix of the basis material projection estimates. The covariance matrix is block diagonal (assuming no crosstalk), with each block having dimensions equal to the number of basis materials. The maximum likelihood estimate is efficient, meaning it will attain the Cramer-Rao lower bound so that the covariance matrix is equal to the inverse of the Fisher information matrix. Using the Fisher information matrix, we derive an analytical formula for the covariance matrix of the estimates of the basis material projections from photon counting measurements. The derivation assumes Poisson statistics, which is valid if pulse pileup is negligible or the detector electronics possesses a pileup rejection mechanism. We compare the derived analytical formula to the covariance calculated using multiple realizations of maximum likelihood estimation. The detector response model in this study assumes perfect pileup rejection and a FWHM energy resolution of 9 keV. Several flux levels with two different sets of basis material lengths were simulated. We found good agreement (less than 4% relative error) between the formula and simulations.