Maximum-likelihood method for focus-variation image reconstruction in high resolution transmission electron microscopy

This paper deals with image reconstruction in high-resolution transmission electron microscopy (HRTEM) via focus variation. This technique aims at the reconstruction of the complex-valued electron wave at the exit plane of the specimen including correction for the aberrations of the microscope, with a focal series of HRTEM images as input. The general case with linear and non-linear contributions to the images is considered, requiring recursive optimization of the wave function, which is obtained by a non-linear least-squares fit to the measured image intensities. Based upon the pioneering work of Kirkland (1984) in this field, we discuss the following improvements. In the first place, a solution of the maximum-likelihood (MAL) equations for image reconstruction is derived in which the - for HRTEM inherent - coupling between the wave function and its complex conjugate is accounted for. Secondly, it is outlined how optimum convergence for the MAL solution is obtained. Thirdly, a computationally efficient implementation of the recursive algorithm to solve the MAL equations is presented. The algorithm is based on a factorization of the spatial coherence envelope, and on focal integration approach for the temporal coherence envelope in the transmission-cross-coefficient. The convergence behavior of this algorithm is studied using simulated focal series as input.