This thesis focuses on the mathematical analysis of the optical phase reconstruction problem. Phase information of light waves has played an important role in many optical applications. However, the phase function of a light wave cannot be measured directly. In 1983, Teague proposed the idea of an intensity senor for measuring phase functions of light waves. It uses an elliptic partial differential equation called the Transport of Intensity Equation (TIE), which relates intensity to the phase function of a light wave. Teague’s study was followed by Roddier and others. When intensity decreases to zero at the boundary, the equation has singularity since the diffusion coefficient vanishes. In 1996, Gureyev and Nugent claimed that no boundary conditions are needed for getting a unique solution of the TIE in this singular case. We present in this thesis the theoretical analysis of the necessity of boundary conditions for solving the TIE. A hybrid theoretical-numerical boundary condition is also derived for solving the TIE numerically in the case of vanishing intensity at the boundary. Numerical tests and optical simulations over discs verified the potency of this theoretical-numerical hybrid boundary condition and the algorithm. Another approach studied is the Weighted Least Action Principle (WLAP), which is proposed by Rubinstein and Wolansky in the year 2004. The WLAP states a variational principle for finding the light rays mapping between two planes using the intensity profiles on the planes, and it writes the problem of phase reconstruction in the functional form. Minimizing the associated functional, we obtain the ray mapping of the light wave in question. The phase function can be derived from the optimized ray mapping. A numerical algorithm was designed to carry out the process. Simulations and tests are reported to show the feasibility of the methods proposed.