Using the Transport of Intensity and the Transport of Phase Equation for Phase Retrieval

TL;DR

This paper explores the use of the transport of intensity and phase equations for phase retrieval, comparing their advantages, and proposing hybrid and numerical solution methods including finite element approaches.

Contribution

It introduces a comparative discussion of TIE and TPE, and proposes hybrid use and numerical solutions with finite element methods for phase retrieval.

Findings

TIE is generally preferred due to nonlinearity of TPE.

Hybrid use of TIE and TPE can be beneficial.

Finite element methods effectively solve the equations numerically.

Abstract

We investigate the transport of intensity equation (TIE) and the transport of phase equation (TPE) for solving the phase retrieval problem. Both the TIE and the TPE are derived from the paraxial Helmholtz equation and relate phase information to the intensity. The TIE is usually favored since the TPE is nonlinear. The main contribution of this paper is a discussion of preferential use of either one of the two equations or potential benefits of a hybrid use. Moreover, we discuss the solution of the TPE with the method of characteristics and with viscosity methods. Both the TIE and the viscosity method are numerically implemented with finite element methods.