Deep learning-based solvability of underdetermined inverse problems in medical imaging

Recently, with the significant developments in deep learning techniques, solving underdetermined inverse problems has become one of the major concerns in the medical imaging domain, where underdetermined problems are motivated by the willingness to provide high resolution medical images with as little data as possible, by optimizing data collection in terms of minimal acquisition time, cost-effectiveness, and low invasiveness. Typical examples include undersampled magnetic resonance imaging(MRI), interior tomography, and sparse-view computed tomography(CT), where deep learning techniques have achieved excellent performances. However, there is a lack of mathematical analysis of why the deep learning method is performing well. This study aims to explain about learning the causal relationship regarding the structure of the training data suitable for deep learning, to solve highly underdetermined problems. We present a particular low-dimensional solution model to highlight the advantage of deep learning methods over conventional methods, where two approaches use the prior information of the solution in a completely different way. We also analyze whether deep learning methods can learn the desired reconstruction map from training data in the three models (undersampled MRI, sparse-view CT, interior tomography). This paper also discusses the nonlinearity structure of underdetermined linear systems and conditions of learning (called M-RIP condition). (c) 2021 Published by Elsevier B.V.