Robust two-step phase estimation using the Simplified Lissajous Ellipse Fitting method with Gabor Filters Bank preprocessing

Abstract

We present the Simplified Lissajous Ellipse Fitting (SLEF) method for the calculation of the random phase step and the phase distribution from two phase-shifted interferograms. We consider interferograms with spatial and temporal dependency of background intensities, amplitude modulations and noise. Given these problems, the use of the Gabor Filters Bank (GFB) allows us to filter-out the noise, normalize the amplitude and eliminate the background. The normalized patterns permit to implement the SLEF algorithm, which is based on reducing the number of estimated coefficients of the ellipse equation, from five terms to only two. Our method consists of three stages. First, we preprocess the interferograms with GFB methodology in order to normalize the fringe patterns. Second, we calculate the phase step by using the proposed SLEF technique and third, we estimate the phase distribution using a two-steps formula. For the calculation of the phase step, we present two alternatives: the use of the Least Squares (LS) method to approximate the values of the coefficients and, in order to improve the LS estimation, a robust estimation based on the Leclerc's potential. The SLEF method's performance is evaluated through synthetic and experimental data to demonstrate its feasibility.

Two-step phase shifting algorithms: Where are we?

“…If the case that the Interferograms present spatial and temporal dependency of the background illumination and amplitude term, the normalization process is required. In such case, Flores and Rivera [21] proposed the SLEF algorithm which consists on using normalized fringe patterns to calculate the coefficients. Given the normalization process, the conical equation is simplified to…”

“…The phase step between the interferograms is still calculated by (24). For more details, please refer to [20,21].…”

“…The RP and SLEF-RE methods were originally proposed with the use of GFB as normalization process, as shown in Figures 21m and 20f [13,21]. Here, we present their variations by using DNNs and HHT.…”

“…The use of this components in the experimental setup can lead to an erroneous step tuning since if the implementation of the arrangement does not include characterized components. This detuning of the phase displacement causes the presence of harmonics in the recovered phase [6], for this reason, the Two-Steps Phase Shifting Algorithms (TS-PSAs) consider that the phase step is unknown and include, implicitly or explicitly, its calculation [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24].…”

“…In this study, we will compare eleven up-to-date algorithms against our proposals. The considered algorithms are the following methods: the Fourier based method proposed by Kreis [7], the Phase-Step Calibration (PSC) of Van Brug [8], the Gram-Schmidt orthonormalization (GS) proposed by Vargas et al [9] by using the phase step estimation proposed in Ref [16], the Extreme Value of Interference (EVI) proposed by Deng et al [12], the estimation based on Random Points (RP) proposed by Dalmau et al [13], the Iterative Robust Estimator (IRE) algorithm proposed by Rivera et al [14], the Quadratic Phase Parameter estimation algorithm (QPP) proposed by Kulkarni and Rastogi [15], the Tilt-Shift Error estimation proposed by Wielgus et al [17], the Diamond Diagonal Vectors demodulation algortihm (DDV) proposed by Lou et al [18], the General Phase-Shifting Interferometry method of Meng et al [19] and the Simplified Lissajous Ellipse Fitting by Robust Estimator (SLEF-RE) of Flores and Rivera [21].…”