# A least-squares functional for joint exit wave reconstruction and image registration

## TL;DR

This paper introduces a novel joint approach for reconstructing the exit wave and registering TEM image series using a least-squares functional, addressing non-convexity and proving the existence of minimizers with promising numerical results.

## Contribution

It generalizes existing methods by simultaneously reconstructing the exit wave and registering images, and establishes the theoretical existence of minimizers for the proposed functional.

## Key findings

- Existence of minimizers for the non-convex functional.
- The method successfully reconstructs exit waves from simulated data.
- The approach is validated through numerical experiments.

## Abstract

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.