Characterization of Maximum Likelihood Solutions to Image Reconstruction in Photon Emission Tomography

For photon emission tomography, the maximum likelihood (ML) estimator for image reconstruction is generally solution to a nonlinear equation involving the vector of measured data. No explicit closed-form solution is known in general for such a nonlinear ML equation, and numerical resolution is usually implemented, with a very popular iterative method formed by the expectation-maximization algorithm. The numerical character of such resolutions usually makes it difficult to obtain a general characterization of the performance of the ML solution. We show that the nonlinear ML equation can be replaced by an equivalent system of two dual linear equations nonlinearly coupled. This formulation allows us to exhibit explicit (to some extent) forms for the solutions to the ML equation, in general conditions corresponding to the various possible configurations of the imaging system, and to characterize their performance with expressions for the mean-squared error, bias and Cramér-Rao bound. The approach especially applies to characterize the ML solutions obtained numerically, and offers a theoretical framework to contribute to better appreciation of the capabilities of ML reconstruction in photon emission tomography.