This paper proposes a new approach for solving ill-posed nonlinear inverse problems. For ease of explanation of the proposed approach, we use the example of lung electrical impedance tomography (EIT), which is known to be a nonlinear and ill-posed inverse problem. Conventionally, penaltybased regularization methods have been used to deal with the ill-posed problem. However, experiences over the last three decades have shown methodological limitations in utilizing prior knowledge about tracking expected imaging features for medical diagnosis. The proposed method’s paradigm is completely different from conventional approaches; the proposed reconstruction uses a variety of training data sets to generate a low dimensional manifold of approximate solutions, which allows conversion of the ill-posed problem to a well-posed one. Variational autoencoder was used to produce a compact and dense representation for lung EIT images with a low dimensional latent space. Then, we learn a robust connection between the EIT data and the low dimensional latent data. Numerical simulations validate the effectiveness and feasibility of the proposed approach.
Bibliographical noteFunding Information:
\ast Received by the editors October 24, 2018; accepted for publication (in revised form) April 23, 2019; published electronically July 2, 2019. https://doi.org/10.1137/18M1222600 Funding: The work of the first and second authors was supported by the National Research Foundation of Korea (NRF) grants 2015R1A5A1009350 and 2017R1A2B20005661. The work of the third and fourth authors was supported by the National Research Foundation of Korea (NRF) grant 2017R1E1A1A03070653. \dagger Department of Computational Science and Engineering, Yonsei University, Seoul, 120-749, Korea (seoj@ yonsei.ac.kr, firstname.lastname@example.org, email@example.com, firstname.lastname@example.org). \ddagger Department of Mathematics, Goethe University Frankfurt, 60325 Frankfurt am Main, Germany (harrach@ math.uni-frankfurt.de).
© 2019 Society for Industrial and Applied Mathematics.
All Science Journal Classification (ASJC) codes
- Applied Mathematics