Fractional Fourier ridges for demodulation of a single-closed-fringe interferogram with quadratic phase

Closed-fringe patterns with quadratic phases play a crucial role in optical interferometry, but their analysis presents significant challenges. We propose a method based on fractional Fourier ridges. First, we compute the fractional Fourier transform (FRFT) of all row (or column) signals and search for the matched rotational angle within a predefined angle range. The matched angle, slope, and constant term are then used to estimate the coefficients of each phase term. These coefficients are applied to recover the phase of the fringe pattern without the need for traditional phase unwrapping. The proposed method accurately recovers the phase of fringe patterns with quadratic phases in various forms, including circular, elliptical, and astigmatic fringes. In addition, the phase of the interferogram can be either symmetric or asymmetric. Due to the energy concentration property of fringe pattern signals in the FRFT domain, the method achieves high precision and demonstrates strong robustness to noise. Therefore, the proposed approach offers an efficient and accurate solution for fringe pattern analysis.