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 that we discuss situations in which it is possible to use the two equations in a hybrid manner: We show that 2-dimensional phase information retrieved by the TIE can be used as initial data for the TPE, enabling the acquisition of 3-dimensional phase information. The latter is solved using the method of characteristic and viscosity methods. Both the TIE and the viscosity method are numerically implemented with finite element methods.

翻译：本文研究用于解决相位恢复问题的强度传输方程(TIE)和相位传输方程(TPE)。TIE与TPE均从近轴亥姆霍兹方程推导而来，将相位信息与强度分布相关联。由于TPE具有非线性特性，通常更倾向于使用TIE。本论文的主要贡献在于探讨了将两个方程混合使用的情形：我们证明通过TIE获取的二维相位信息可作为TPE的初始数据，从而能够获得三维相位信息。后者采用特征线法和粘性法进行求解。TIE与粘性法均通过有限元方法进行数值实现。