# Learning a stable approximation of an existing but unknown inverse mapping: application to the half-time circular Radon transform

## TL;DR

This paper explores using deep learning to reconstruct images from a specific type of Radon transform data where a stable inverse mapping is known to exist but not yet known analytically.

## Contribution

The novel approach uses a convolutional neural network to learn a stable inverse mapping for the half-time circular Radon transform.

## Key findings

- A learned filtered backprojection method using a CNN approximates the unknown filtering operation effectively.
- The method demonstrates stability and generalization to data different from the training set.
- This approach could be applied to wave-based imaging like photoacoustic computed tomography.

## Abstract

Supervised deep learning-based methods have inspired a new wave of image reconstruction methods that implicitly learn effective regularization strategies from a set of training data. While they hold potential for improving image quality, they have also raised concerns regarding their robustness. Instabilities can manifest when learned methods are applied to find approximate solutions to ill-posed image reconstruction problems for which a unique and stable inverse mapping does not exist, which is a typical use case. In this study, we investigate the performance of supervised deep learning-based image reconstruction in an alternate use case in which a stable inverse mapping is known to exist but is not yet analytically available in closed form. For such problems, a deep learning-based method can learn a stable approximation of the unknown inverse mapping that generalizes well to data that differ significantly from the training set. The learned approximation of the inverse mapping eliminates the need to employ an implicit (optimization-based) reconstruction method and can potentially yield insights into the unknown analytic inverse formula. The specific problem addressed is image reconstruction from a particular case of radially truncated circular Radon transform (CRT) data, referred to as ‘half-time’ measurement data. For the half-time image reconstruction problem, we develop and investigate a learned filtered backprojection method that employs a convolutional neural network to approximate the unknown filtering operation. We demonstrate that this method behaves stably and readily generalizes to data that differ significantly from training data. The developed method may find application to wave-based imaging modalities that include photoacoustic computed tomography.