# Automatic alignment for three-dimensional tomographic reconstruction

## TL;DR

This paper introduces a general algorithmic framework to automatically correct for misalignments in 3D tomographic reconstruction, improving image quality when calibration errors are present.

## Contribution

It presents a novel combined approach for parameter retrieval and algebraic reconstruction, applicable to noisy and incomplete measurement data.

## Key findings

- Effective correction of alignment errors demonstrated on simulated data.
- Improved reconstruction quality shown on electron tomography dataset.
- Framework adaptable to various sampling and error conditions.

## Abstract

In tomographic reconstruction, the goal is to reconstruct an unknown object from a collection of line integrals. Given a complete sampling of such line integrals for various angles and directions, explicit inverse formulas exist to reconstruct the object. Given noisy and incomplete measurements, the inverse problem is typically solved through a regularized least-squares approach. A challenge for both approaches is that in practice the exact directions and offsets of the x-rays are only known approximately due to, e.g. calibration errors. Such errors lead to artifacts in the reconstructed image. In the case of sufficient sampling and geometrically simple misalignment, the measurements can be corrected by exploiting so-called consistency conditions. In other cases, such conditions may not apply and we have to solve an additional inverse problem to retrieve the angles and shifts. In this paper we propose a general algorithmic framework for retrieving these parameters in conjunction with an algebraic reconstruction technique. The proposed approach is illustrated by numerical examples for both simulated data and an electron tomography dataset.