Phase Step Estimation Research Articles

Estimation of the phase step between two fringe patterns based on the Gram–Schmidt algorithm

We present the evaluation of a closed form formula for the calculation of the original step between two randomly shifted fringe patterns. Our proposal extends the Gram--Schmidt orthonormalization algorithm for fringe pattern. Experimentally, the phase shift is introduced by a electro--mechanical devices (such as piezoelectric or moving mounts).The estimation of the actual phase step allows us to improve the phase shifting device calibration. The evaluation consists of three cases that represent different pre-normalization processes: First, we evaluate the accuracy of the method in the orthonormalization process by estimating the test step using synthetic normalized fringe patterns with no background, constant amplitude and different noise levels. Second, we evaluate the formula with a variable amplitude function on the fringe patterns but with no background. Third, we evaluate non-normalized noisy fringe patterns including the comparison of pre-filtering processes such as the Gabor filter banks and the isotropic normalization process, in order to emphasize how they affect in the calculation of the phase step.

Two-step phase shifting algorithms: Where are we?

Two steps phase shifting interferometry has been a hot topic in the recent years. We present a comparison study of 12 representative self–tunning algorithms based on two-steps phase shifting interferometry. We evaluate the performance of such algorithms by estimating the phase step of synthetic and experimental fringe patterns using 3 different normalizing processes: Gabor Filters Bank (GFB), Deep Neural Networks (DNNs) and Hilbert Huang Transform (HHT); in order to retrieve the background, the amplitude modulation and noise. We present the variants of state-of-the-art phase step estimation algorithms by using the GFB and DNNs as normalization preprocesses, as well as the use of a robust estimator such as the median to estimate the phase step. We present experimental results comparing the combinations of the normalization processes and the two steps phase shifting algorithms. Our study demonstrates that the quality of the retrieved phase from of two-step interferograms is more dependent of the normalizing process than the phase step estimation method.

Two-step phase shifting interferometry based on orientation selective monogenic filtering

We propose a phase step estimation algorithm in a two-step phase shifting interferometry setup based on orientation selective adaptive monogenic filtering (AMF). Initially, the AMF algorithm is used for the fringe pattern normalization. A minor modification in the same algorithm performs an orientation selective monogenic filtering which is capable of providing a reliable estimation of phase step. Due to the use of the same algorithm for normalization and phase step estimation, the proposed method offers computational efficiency and simplicity. Simulation and experimental results are provided to demonstrate the suitability of the proposed method in phase demodulation.

Statistical phase-shifting step estimation algorithm based on the continuous wavelet transform for high-resolution interferometry metrology

We propose a statistical phase-shifting estimation algorithm for temporal phase-shifting interferometry (PSI) based on the continuous wavelet transform (CWT). The proposed algorithm explores spatial information redundancy in the intraframe interferogram dataset using the phase recovery property on the power ridge of the CWT. Despite the errors introduced by the noise of the interferogram, the statistical part of the algorithm is utilized to give a sound estimation of the phase-shifting step. It also introduces the usage of directional statistics as the statistical model, which was validated, so as to offer a better estimation compared with other statistical models. The algorithm is implemented in computer codes, and the validations of the algorithm were performed on numerical simulated signals and actual phase-shifted moiré interferograms. The major advantage of the proposed algorithm is that it imposes weaker conditions on the presumptions in the temporal PSI, which, under most circumstances, requires uniform and precalibrated phase-shifting steps. Compared with other existing deterministic estimation algorithms, the proposed algorithm estimates the phase-shifting step statistically. The proposed algorithm allows the temporal PSI to operate under dynamic loading conditions and arbitrary phase steps and also without precalibration of the phase shifter. The proposed method can serve as a benchmark method for comparing the accuracy of the different phase-step estimation methods.

Phase-step estimation in interferometry via an unscented Kalman filter

We present an unscented Kalman filter to identify the phase step imparted to a piezoelectric transducer in phase shifting interferometry in the presence of Gaussian noise. The advantage of the proposed algorithm lies in its ability to determine the phase step values between -pi and pi rad without any prior calibration of the piezoelectric device. The algorithm is tested rigorously by using the simulated data in the presence of Gaussian distributed noise. Experimental validations are also performed in a holographic interferometry optical setup to verify the proposed approach. Once the phase step is identified, the interference phase can be estimated by using the least-squares fitting approach.

Statistical study of generalized nonlinear phase step estimation methods in phase-shifting interferometry

Signal processing methods based on maximum-likelihood theory, discrete chirp Fourier transform, and spectral estimation methods have enabled accurate measurement of phase in phase-shifting interferometry in the presence of nonlinear response of the piezoelectric transducer to the applied voltage. We present the statistical study of these generalized nonlinear phase step estimation methods to identify the best method by deriving the Cramér-Rao bound. We also address important aspects of these methods for implementation in practical applications and compare the performance of the best-identified method with other bench marking algorithms in the presence of harmonics and noise.

Accurate nonlinear phase step estimation in phase shifting interferometry

We propose a new method for the accurate estimation of nonlinear response of the PZT to the applied voltage. The method uses discrete chirp Fourier transform for the coarse estimation followed by a fine search method for the accurate estimation of the phase step and nonlinearity. The method can be extended to the cases of extraction of multiple phases in the configurations involving multiple PZTs such as holographic moiré in the presence of nonlinearity. The robustness of the proposed method is verified by comparing with Cramér–Rao lower bound. Experimental results prove the method’s feasibility.

An integral approach to phase shifting interferometry using a super-resolution frequency estimation method

The objective of this paper is to describe an integral approach - based on the use of a super-resolution frequency estimation method - to phase shifting interferometry, starting from phase step estimation to phase evaluation at each point on the object surface. Denoising is also taken into consideration for the case of a signal contaminated with white Gaussian noise. The other significant features of the proposal are that it caters to the presence of multiple PZTs in an optical configuration, is capable of determining An integral approach to phase shifting the harmonic content in the signal and effectively eliminating their influence on measurement, is insensitive to errors arising from PZT miscalibration, is applicable to spherical beams, and is a robust performer even in the presence of white Gaussian intensity noise.