Variational Phase Imaging Using the Transport-of-Intensity Equation

We introduce a variational phase retrieval algorithm for the imaging of transparent objects. Our formalism is based on the transport-of-intensity equation (TIE), which relates the phase of an optical field to the variation of its intensity along the direction of propagation. TIE practically requires one to record a set of defocus images to measure the variation of intensity. We first investigate the effect of the defocus distance on the retrieved phase map. Based on our analysis, we propose a weighted phase reconstruction algorithm yielding a phase map that minimizes a convex functional. The method is nonlinear and combines different ranges of spatial frequencies—depending on the defocus value of the measurements—in a regularized fashion. The minimization task is solved iteratively via the alternating-direction method of multipliers. Our simulations outperform commonly used linear and nonlinear TIE solvers. We also illustrate and validate our method on real microscopy data of HeLa cells.