Transport of intensity phase imaging from multiple noisy intensities measured in unequally-spaced planes

The noise problem is generally inevitable for phase retrieval by solving the transport of intensity equation (TIE). The noise effect can be alleviated by using multiple intensities to estimate the axial intensity derivative in the TIE. In this study, a method is proposed for estimating the intensity derivative by using multiple unevenly-spaced noisy measurements. The noise-minimized intensity derivative is approximated by a linear combination of the intensity data, in which the coefficients are obtained by solving a constrained optimization problem. The performance of the method is investigated by both the error analysis and the numerical simulations, and the results show that the method can reduce the noise effect on the retrieved phase. In addition, guidelines for the choice of the number of the intensity planes are given.