Fractional Fourier ridges for demodulation of interferograms with quadratic phase

A method based on the fractional Fourier ridges for accurate phase demodulation of a single interferogram with quadratic phase is presented. The interferograms being analyzed may contain circular, elliptic or astigmatic fringes. In signal processing field, such interferograms can be called 2-D chirp-type signals. Since the fractional Fourier transform (FRFT) of a chirp signal is a function under the matched angle that is determined by chirp rates of the signal, so the method can be used to match the multiple chirp rates in chirp-type signals with multiple chirp components. In this work, the FRFT of all row (column) signals are firstly calculated, and the ridge of the FRFT amplitude of each row (column) signal in FRFT domain is recorded. Repeat the above process for each angle of a searching range. Then a ridge tracking approach is employed to determine the matched angle, which can be used to calculate the coefficient of the square term of row (column) coordinates. Moreover, under the matched angle, the ridge of the FRFT amplitude of each row (column) signal all lie on a straight line. The slope and constant term of the line can be used to calculate the coefficient of the linear term of row (column) coordinates and the coefficient of cross term, respectively. The same procedures are implemented to all column (row) signals to determine the coefficients of the square and liner term of column (row) coordinates. According to the obtained coefficients, the phase of the fringe pattern can be constructed without phase unwrapping operation. Furthermore, the present procedure is also capable of analysis of interferograms with or without circularly symmetry fringe distribution instead of using complex and time consuming algorithms for recovering phase from fringe patterns with closed fringes. Finally, the method is tested in simulated and real data.