Research on the Wirtinger Flow algorithm based on quadratic distribution initial value

Phase retrieval algorithms, such as the Wirtinger Flow (WF) algorithm, are widely used in various fields. As a nonconvex optimization algorithm for phase retrieval, WF is commonly employed in the reconstruction of holograms in holographic image projection. These types of algorithms typically involve two stages: an initialization stage and an iterative optimization stage. In the initialization stage, an initial value is provided, and a spectral method is used to calculate an approximate solution as the initial guess. The iterative optimization stage then utilizes the Wirtinger gradient to iteratively compute and converge the initial guess to a nearby real solution, thereby obtaining the global optimal solution. However, due to the random nature of the initial values, the computed results often exhibit significant instability. To address this issue, this paper proposes an approach based on a quadratic distribution for improving the stability of the results. In the initialization stage, the initial value is set as the quadratic distribution initial value. Then, the spectral method is applied again to calculate the initial guess. Since the quadratic distribution initial value is artificially assigned, it enhances the stability of the computed results. To validate this method, the paper applies the quadratic distribution initial value to both the initialization stage of the WF algorithm and the Truncated Amplitude Flow (TAF) algorithm. A comparison is made between the results obtained using random initial value and those obtained using the quadratic distribution initial values. The results demonstrate that compared to random initial values, the quadratic distribution initial values can achieve faster and equally accurate computation results with higher stability. Finally, this method is applied to simulation experiments of in-line digital holography, and the reconstruction results from the experiments further confirm the effectiveness of our approach.