From the Radon Transform to Deep Learning in Medical Image Reconstruction

Abstract

Medical image reconstruction began with an exact mathematical result. In 1917, Radon showed how to recover a function from its line integrals, providing the mathematical basis for computed tomography, positron emission tomography, and single-photon emission computed tomography. This review follows the development of reconstruction from analytical inversion to modern deep learning. We first present the Radon transform, its attenuated and three-dimensional generalizations, and the corresponding inversion formulae, including filtered backprojection. We then review iterative and optimization-based methods, including maximum-likelihood expectation-maximization, variational regularization, and compressed sensing. The discussion proceeds to deep learning, covering convolutional and transformer architectures, algorithm unrolling, physics-informed networks, neural operators, and generative diffusion priors. Our central argument is that analytical, iterative, and learned reconstruction are different formulations of the same inverse problem rather than competing paradigms. Many learned components derive from classical formulations, and many classical operators admit learnable counterparts. We conclude by discussing open challenges related to theoretical guarantees, trustworthiness, data efficiency, federated learning, and regulatory validation. The most reliable learned systems are likely to retain the mathematical and physical structure of established reconstruction methods.