Accurate two-step random phase retrieval approach without pre-filtering based on hyper ellipse fitting

In this work, we propose a hyper ellipse fitting-based high-precision random two-frame phase shifting algorithm to improve the accuracy of phase retrieval. This method includes a process of Gram-Schmidt orthonormalization, followed by a hyper ellipse fitting procedure. The Gram-Schmidt orthonormalization algorithm constructs a quadrature fringe pattern relative to the original fringe pattern. These two quadrature fringe patterns are then fed into the hyper ellipse fitting procedure, which reconstructs the phase map and refines the background light to produce the final accurate phase of interest. Due to the hyper ellipse fitting procedure, the demodulation results are significantly improved in many cases. This method allows us to design a two-shot phase reconstruction algorithm without the need for least squares iteration or pre-filtering, effectively mitigating residual background to the greatest extent. It combines the advantages of both the Gram-Schmidt orthonormalization method and the Lissajous ellipse fitting method, making our hyper ellipse fitting approach a simple, flexible, and accurate phase retrieval algorithm. Experiments show that by using the weighted least squares method and adjusting the weights, this method can prioritize data points with more significant information or higher reliability, ensuring more accurate estimation of the ellipse parameters.