Three-frame Random Phase-shift Demodulation Algorithm Based on Ellipse Fitting Correction

The objective of this study is to develop a high-precision optical interferometric phase demodulation method that does not require preset phase shifts and can accurately recover phase information from only three interferograms. The proposed method is designed to overcome the inherent limitations of conventional random phase-shifting algorithms， which typically suffer from large errors， instability， and restricted accuracy improvements， particularly when the number of interference fringes is limited. At the same time， the method ensures robustness under varying fringe types， Signal-to-Noise Ratios （SNR）， and fringe numbers. In this work， a novel three-frame random phase-shifting phase demodulation algorithm is proposed， referred to as the VU&LEF method， which integrates the VU algorithm with ellipse-fitting correction. First， the VU algorithm is applied to three randomly phase-shifted interferograms to extract two orthogonal components of the interferometric signal. In practice， due to noise interference and the limited number of frames， these components are not perfectly orthogonal， and their distribution often forms an ellipse rather than an ideal circle. To correct this deviation， ellipse-fitting is introduced to estimate the ellipse coefficients. A subsequent normalization procedure maps the ellipse into a circle， thereby restoring strict orthogonality between the two components and significantly reducing systematic errors in the subsequent phase calculation. This optimization enables high-precision phase recovery even with only three interferograms， effectively overcoming the main drawbacks of the conventional VU method. To evaluate the performance of the proposed approach， extensive numerical simulations and experimental validations were conducted under diverse conditions. The results demonstrate that the VU&LEF method consistently outperforms the conventional VU algorithm， achieving significantly improved phase accuracy. Even under low-SNR conditions， the proposed method maintains stable phase recovery， whereas the traditional approach exhibits severe degradation. To further assess computational efficiency， the processing time of the VU&LEF method was compared with that of the conventional VU method and the AIA algorithm. Although the ellipse-fitting step introduces a slight additional computational cost， the proposed method remains more efficient than the AIA algorithm while simultaneously offering superior phase accuracy. Furthermore， the robustness of the proposed method was verified using representative open， closed， and complex fringe patterns with significant background fluctuations. The results clearly indicate that the method effectively corrected elliptical distributions into ideal circular forms， thereby reducing measurement errors. In contrast， the conventional VU method exhibited large phase errors and severe distortions in regions with strong background interference， whereas the VU&LEF method substantially mitigated these errors through ellipse correction. Additional tests under varying frame numbers confirmed that the proposed method yields stable and reliable phase recovery with significantly reduced error fluctuations. These findings further indicate that phase correction is indispensable when performing demodulation with only three interferograms， highlighting the wide applicability of the method in practical interferometric measurements. From an application perspective， requiring only three interferograms greatly reduces data acquisition time compared with conventional multi-frame methods. The additional computational cost introduced by ellipse-fitting correction is negligible compared with the measurement time saved. This combination of high efficiency and accuracy makes the method particularly suitable for real-time and high-speed measurement scenarios， such as dynamic surface profiling， deformation monitoring， in-situ industrial inspection， and other precision optical metrology tasks. In conclusion， the proposed three-frame random phase-shifting phase demodulation method with ellipse-fitting correction does not require the phase shifts of interferograms to be predefined or equally spaced， and thus constitutes a true random phase-shifting technique. It inherits the advantage of the VU method in not requiring prior calibration of phase steps， while enabling high-precision phase recovery from minimal data. Moreover， it demonstrates strong robustness against noise， fringe variability， and complex background interference. Both simulation and experimental results verify that the proposed method outperforms conventional approaches in terms of accuracy， stability， and computational efficiency.