Application of a Deep Neural Network to Phase Retrieval in Inverse Medium Scattering Problems

Soojong Lim,Jaemin Shin

doi:10.3390/computation9050056

Abstract

We address the inverse medium scattering problem with phaseless data motivated by nondestructive testing for optical fibers. As the phase information of the data is unknown, this problem may be regarded as a standard phase retrieval problem that consists of identifying the phase from the amplitude of data and the structure of the related operator. This problem has been studied intensively due to its wide applications in physics and engineering. However, the uniqueness of the inverse problem with phaseless data is still open and the problem itself is severely ill-posed. In this work, we construct a model to approximate the solution operator in finite-dimensional spaces by a deep neural network assuming that the refractive index is radially symmetric. We are then able to recover the refractive index from the phaseless data. Numerical experiments are presented to illustrate the effectiveness of the proposed model.

Highlights

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We address the phase retrieval arising in the 2-dimensional inverse medium scattering problem, which is governed by the following Helmholtz equation:
The recovered coefficient cm (m > M ) is very small in X M ( M = 2, 4 < 5), and in the case of X M ( M = 6, 8 > 5), our model acts as the projection of X M onto X5 by ignoring the actual cm (m > 5)—i.e., our model is an approximation of the projection of T −1 onto X5

Summary

Introduction

The goal of the inverse medium scattering problem is to reconstruct the refractive index q from the measurement of scattered fields. This problem has been studied analytically as well as numerically in many fields, such as medical imaging, nondestructive testing, optics, radar, and seismology. Several results have been provided for the uniqueness of the inverse problem without phase information under certain restrictions (e.g., [14,15,16,17,18]), to the authors’ best knowledge, identification of the refractive index from the measured data set {|u( x, d)| : | x | = R, d ∈ S1 }.

Mathematical Model

Deep Neural Network for the Inverse Problem

Discussion and Conclusions