Annals of Mathematical Sciences and Applications

Volume 7 (2022)

Number 2

Solving optical tomography with deep learning

Pages: 195 – 220



Yuwei Fan (Stanford University, Stanford, California, U.S.A.)

Lexing Ying (Stanford University, Stanford, California, U.S.A.)


This paper presents a neural network approach for solving two-dimensional optical tomography (OT) problems based on the radiative transfer equation. The mathematical problem of OT is to recover the optical properties of an object based on the albedo operator that is accessible from boundary measurements. Both the forward map from the optical properties to the albedo operator and the inverse map are high-dimensional and nonlinear. For the circular tomography geometry, a perturbative analysis shows that the forward map can be approximated by a vectorized convolution operator in the angular direction. Motivated by this, we propose effective neural network architectures for the forward and inverse maps based on convolution layers, with weights learned from training datasets. Numerical results demonstrate the efficiency of the proposed neural networks.


inverse problems, deep learning, optical tomography, radiative transfer equation

2010 Mathematics Subject Classification

65N21, 74J25, 85A25

The work of Y.F. and L.Y. is partially supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Scientific Discovery through Advanced Computing (SciDAC) program. The work of L.Y. is also partially supported by the National Science Foundation under award DMS-1818449. This work is also supported AWS Cloud Credits for Research program from Amazon.

Received 12 January 2022

Accepted 17 March 2022

Published 12 September 2022