# A Variational Reconstruction Method for Undersampled Dynamic X-ray Tomography based on Physical Motion Models

## TL;DR

This paper introduces a variational reconstruction method for undersampled dynamic X-ray tomography that incorporates physical motion models and optical flow to improve image quality from limited projections.

## Contribution

It develops a joint variational model for simultaneous image reconstruction and motion estimation using realistic measurement protocols and analyzes its mathematical properties.

## Key findings

- Random sampling with the proposed model achieves comparable quality to static reconstructions.
- The method effectively reconstructs moving objects from limited angular projections.
- Numerical results demonstrate the approach's robustness on simulated and real data.

## Abstract

In this paper we study the reconstruction of moving object densities from undersampled dynamic X-ray tomography in two dimensions. A particular motivation of this study is to use realistic measurement protocols for practical applications, i.e. we do not assume to have a full Radon transform in each time step, but only projections in few angular directions. This restriction enforces a space-time reconstruction, which we perform by incorporating physical motion models and regularization of motion vectors in a variational framework. The methodology of optical flow, which is one of the most common methods to estimate motion between two images, is utilized to formulate a joint variational model for reconstruction and motion estimation. We provide a basic mathematical analysis of the forward model and the variational model for the image reconstruction. Moreover, we discuss the efficient numerical minimization based on alternating minimizations between images and motion vectors. A variety of results are presented for simulated and real measurement data with different sampling strategy. A key observation is that random sampling combined with our model allows reconstructions of similar amount of measurements and quality as a single static reconstruction.