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

TL;DR

This work analyzes phase retrieval via two PDE-based routes derived from the paraxial Helmholtz equation: the linear Transport of Intensity Equation (TIE) and the nonlinear Transport of Phase Equation (TPE). It derives both equations from $A=\sqrt{I}\,e^{i\phi}$ and compares their modeling errors, highlighting that the TIE is linear in $\phi$ while the TPE retains more nonlinear structure, yet both depend on boundary data for well-posed reconstructions. The paper investigates numerical solutions of the TIE (with finite element methods) and of the TPE through the method of characteristics and a vanishing-viscosity (Cole–Hopf) approach, applying them to constant-intensity and Gaussian-beam test cases. A key finding is that boundary conditions strongly govern reconstruction quality, and a hybrid TIE/TPE strategy can localize phase retrieval while mitigating boundary-data requirements. The study underscores the practical implications for 3D phase retrieval in microscopy and related fields, where boundary information and computational approaches must be carefully balanced.

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.

Figures (16)

• Figure 1: Two boundary cases for the TIE and the TPE.
• Figure 2: Ground-truth
• Figure 3: $\phi\vert_{\Gamma_0}=0$
• Figure 4: $\phi\vert_{\Gamma_0}= \lfloor 10 x \rfloor$
• Figure 6: Reconstruction

Theorems & Definitions (2)

• Example 1
• Remark 1