Least-squares and maximum-likelihood in Computed Tomography

Statistical reconstruction methods in X-ray Computed Tomography (XCT) are well-regarded for their ability to produce more accurate and artefact-free reconstructed volumes, in the presence of measurement noise. Maximum-likelihood methods are particularly salient and have been shown to result in superior reconstruction quality, compared with methods that minimise the l(2) residual between measured and projected line attenuations. Least-squares more generally may refer to the minimisation of quadratic forms of the projected attenuation residuals. Early maximum-likelihood methods showed promising reconstruction capabilities but were not practical to implement due to very slow convergence, especially compared with least-squares methods. More recently, least-squares methods have been adapted to minimise quadratic approximations to (negative) log-likelihood, thereby attaining the speed of least-squares minimisation in service of likelihood maximisation for superior reconstruction fidelity. Quadratic approximation to the log-likelihood under Poisson measurement statistics has been demonstrated several times in the literature. In this publication we describe an approach to quadratically expanding log-likelihood under an arbitrary noise model, and demonstrate via simulation that this can be implemented practically to maximise likelihood under mixed Poisson-Gaussian models that describe a broad range of transmission XCT imaging systems.