  Images generated by a transmission electron microscope (TEM) are blurred by
aberrations from the objective lens and can be difficult to interpret
correctly. One possible solution to this problem is to reconstruct the
so-called exit wave, i.e. the electron wave in the microscope right before it
passes the objective lens, from a series of TEM images acquired with varying
focus. While the forward model of simulating a TEM image from a given exit wave
is known and easy to evaluate, it is in general not possible to reconstruct the
exit wave from a series of images analytically. The corresponding inverse
problem can be formulated as a minimization problem, which is done in the well
known MAL and MIMAP methods. We propose a generalization of these methods by
performing the exit wave reconstruction and the registration of the image
series simultaneously. We show that our objective functional is not convex with
respect to the exit wave, which also carries over to the MAL and MIMAP
functionals. The main result is the existence of minimizers of our objective
functional. These results are based on the properties of a generalization of
the cross-correlation. Finally, the applicability of our method is verified
with a numerical experiment on simulated input data.
