This paper presents a method of significantly improving the previously proposed simple, flexible, and accurate phase retrieval algorithm for the random phase-shifting interferometry named HEFS [K. Yatabe, J. Opt. Soc. Am. A 34, 87 (2017)]. Although its e ectiveness and performance were confirmed by numerical experiments in the original paper, it is found that the algorithm may not work properly if observed fringe images contains untrusted (extremely noisy) pixels. In this paper, a method of avoiding such untrusted pixels within the estimation processes of HEFS is proposed for the practical use of the algorithm. In addition to the proposal, an experiment of measuring a sound field in air was conducted to show the performance for real data, where the proposed improvement is critical for that situation. MATLAB codes (which can be downloaded from http://goo.gl/upcsFe) are provided within the paper to aid understanding the main concept of the proposed methods.
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