Transport of intensity phase imaging with error correction using transport of phase equation

Abstract

The transport of intensity equation (TIE) has been widely applied to phase imaging. A variety of methods for solving the TIE have been proposed. One of the most popular methods is an FFT-based method, which is simple and fast. However, this method has a strict restriction that the intensity of the object is assumed to be uniform. Otherwise, the accuracy of phase retrieval results may drop significantly. The transport of phase equation (TPE) is an equation coupled with the TIE, and both are derived from the Helmholtz equation. Few works have studied the role of the TPE in 3D imaging. In this work, a non-iterative FFT-based TIE with TPE correction is proposed. The phases at the object plane and four defocused planes are first calculated by the TIE. Ideally, the object intensity and computed phases are supposed to satisfy both the TIE and TPE. But the non-uniformity of object intensity, as well as the use of finite differences of intensities in the calculation of longitudinal derivatives for the FFT-based TIE method, introduce errors in phase retrieval result. We show that by using the TPE, the local refractive index during propagation can be updated and used as a correction in the TIE. The TIE is solved once again using the updated refractive index, and shows reduction of errors. This proposed technique can be extended to amplitude and phase imaging. It also offers the advantage of yielding the unwrapped phase with good accuracy performance and can be potentially applied to medical high-definition 3D imaging, for cells, micro bubbles, etc.