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 l2 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, leastsquares 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 loglikelihood 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.