Higher Order Transport of Intensity Equation Methods: Comparisons and Their Hybrid Application for Noise Adaptive Phase Imaging

As a non-interferometric method, the transport of intensity equation (TIE) method suits for quantitative phase imaging with the commercial microscope platform, especially those higher order TIE approaches which can realize precise phase retrieval by eliminating undesired higher order derivatives are widely used. However, these approaches are mostly adopted separately without considering their phase retrieval precision in various noise levels. In this paper, we first compared these classical higher order TIE approaches through theoretical analysis, numerical simulations, and experiments. Then, based on the quantitative comparisons mainly focusing on their phase retrieval accuracy and noise suppression capability, we determine the application scope corresponding to different noise levels of each higher order TIE approach. Finally, in order to deal with different noisy cases, we design the hybrid higher order TIE application, in which the specific higher order TIE approach is selected for phase retrieval according to the precise noise estimation, and the performance of the hybrid higher order TIE application is tested by both the numerical simulations and the experiments, proving it can perform high-quality phase imaging by balancing the tradeoff between the phase retrieval accuracy and the noise influence. The paper not only provides a systematic reference for analysis and comparisons on different higher order TIE approaches, but also proposes the hybrid application for noise adaptive phase imaging, which can be a potential tool in biological observations and medical diagnostics.