Fast blind extraction of arbitrary unknown phase shifts by an iterative tangent approach in generalized phase-shifting interferometry

Joint least-squares algorithm correcting phase-shift errors and detector nonlinearity simultaneously in phase-shifting interferometry

Phase-shifting interferometry may suffer from the errors caused by the miscalibration of the phase shifter and the nonlinearity of the detector simultaneously. These errors are not easy to eliminate because they are generally coupled with each other in interferograms. For solving this issue, we suggest a joint least-squares phase-shifting algorithm. It allows one to decouple these errors through an alternate least-squares fitting procedure, thus accurately estimating phases, phase shifts, and coefficients of the detector response simultaneously. The converging condition of this algorithm, associated with the uniqueness of the equation solution and anti-aliasing phase shifting, is discussed. Experimental results demonstrate that this proposed algorithm is helpful for improving phase-measuring accuracy in phase-shifting interferometry.

pyDHM: A Python library for applications in digital holographic microscopy

pyDHM is an open-source Python library aimed at Digital Holographic Microscopy (DHM) applications. The pyDHM is a user-friendly library written in the robust programming language of Python that provides a set of numerical processing algorithms for reconstructing amplitude and phase images for a broad range of optical DHM configurations. The pyDHM implements phase-shifting approaches for in-line and slightly off-axis systems and enables phase compensation for telecentric and non-telecentric systems. In addition, pyDHM includes three propagation algorithms for numerical focusing complex amplitude distributions in DHM and digital holography (DH) setups. We have validated the library using numerical and experimental holograms.

Phase-shifting digital holographic microscopy with an iterative blind reconstruction algorithm

In phase-shifting digital holographic microscopy (PS-DHM), the reconstructed phase map is obtained after processing several holograms of the same scene with a phase shift between them. Most of the reconstruction algorithms in PS-DHM require an accurate and known phase shift between the recorded holograms. This requirement limits the applicability of the method. To ease the use of PS-DHM, this paper presents an iterative-blind phase shift extraction method based on demodulation of the different components of the recorded holograms. The method uses a DHM system operating in a slightly off-axis architecture. The proposed method uses three-frame holograms with arbitrary and unequal phase shifts between them and therefore eases the use of the PS-DHM. We believe both simulated and experimental results demonstrate the goodness and feasibility of the proposed technique.

Dynamic phase-deforming interferometry: suppression of errors from vibration and air turbulence

In this Letter, dynamic phase-deforming interferometry (PDI) is proposed to solve the impact of environmental vibration and air turbulence on the measurement accuracy. Two interference passes, namely, the measurement pass (MP) and carrier-frequency pass (CP), are constructed, utilizing one detector to acquire the fringe patterns of the two passes simultaneously. The relative deformation phases of the CP fringe patterns are calculated with the Fourier transform method; then the phase extraction of the MP is performed via the least squares method to obtain the final phase distribution. The optical layout and phase-extraction algorithm of the PDI method are investigated, and the principle simulation and measurement experiments are performed. The experimental results show that the PDI can provide not only high-precision phase measurement under the influence of environmental vibration and air turbulence, but also has low system complexity and fast measurement speed.

Accurate and fast two-step phase shifting algorithm based on principle component analysis and Lissajous ellipse fitting with random phase shift and no pre-filtering

To achieve high measurement accuracy with less computational time-in-phase shifting interferometry, a random phase-shifting algorithm based on principal component analysis and Lissajous ellipse fitting (PCA&LEF) is proposed. It doesn't need pre-filtering and can obtain relatively accurate phase distribution with only two phase shifted interferograms and less computational time and is suitable for different background intensity, modulation amplitude distributions and noises. Moreover, it can obtain absolutely accurate result when the background intensity and modulation amplitude are perfect and can partly suppress the effect of imperfect background intensity and modulation amplitude. Last but not least, it removes the restriction that PCA needs more than three interferograms with well-distributed phase shifts to subtract relatively accurate mean. The simulations and experiments verify the correctness and feasibility of PCA&LEF.

Three-step random phase retrieval approach based on difference map normalization and diamond diagonal vector normalization

To overcome the phase shift error in phase shifting interferometry, a three-step random phase retrieval approach based on difference map normalization and diamond diagonal vector normalization (DN&DDVN) is proposed. It does not need pre-filtering for the interferograms and can obtain relatively accurate phase distribution with a simple process and less computational time. This simulation and experiment verify the correctness and feasibility of DN&DDVN.

Random two-step phase shifting interferometry based on Lissajous ellipse fitting and least squares technologies

To accurately obtain the phase distribution of an optical surface under test, the accurate phase extraction algorithm is essential. To overcome the phase shift error, a random two-step phase shifting algorithm, which can be used in the fluctuating and non-uniform background intensity and modulation amplitude, Lissajous ellipse fitting, and least squares iterative phase shifting algorithm (LEF&LSI PSA), is proposed; pre-filtering interferograms are not necessary, but they can get relatively accurate phase distribution and unknown phase shift value. The simulation and experiment verify the correctness and feasibility of the LEF & LSI PSA.

Spatial dual-orthogonal (SDO) phase-shifting algorithm by pre-recomposing the interference fringe

In the case that the phase distribution of interferogram is nonuniform and the background/modulation amplitude change rapidly, the current self-calibration algorithms with better performance like principal components analysis (PCA) and advanced iterative algorithm (AIA) cannot work well. In this study, from three or more phase-shifting interferograms with unknown phase-shifts, we propose a spatial dual-orthogonal (SDO) phase-shifting algorithm with high accuracy through using the spatial orthogonal property of interference fringe, in which a new sequence of fringe patterns with uniform phase distribution can be constructed by pre-recomposing original interferograms to determine their corresponding optimum combination coefficients, which are directly related with the phase shifts. Both simulation and experimental results show that using the proposed SDO algorithm, we can achieve accurate phase from the phase-shifting interferograms with nonuniform phase distribution, non-constant background and arbitrary phase shifts. Specially, it is found that the accuracy of phase retrieval with the proposed SDO algorithm is insensitive to the variation of fringe pattern, and this will supply a guarantee for high accuracy phase measurement and application.

Estimating phase shifts from three fringe patterns by use of cross spectrum

In phase-shifting technique, using self-calibrating algorithms allows us to determine phases and phase shifts simultaneously, thus eliminating the errors caused by miscalibrations of phase shifters. However, it is difficult to estimate phase shifts when only three fringe patterns are available, because in this case the problem becomes underdetermined. In this paper, we analyze the effects of phase-shift errors on the calculated phases, and find that the phase-shift errors introduce correlations between different frequency components of the calculated phases. We measure these correlations by calculating the cross spectrum of cis functions between the calculated phases and their trebles, and further define a single-valued objective function. A gradient-guided search strategy is used for minimizing this objective function, so that the phase shifts are estimated from three fringe patterns. The simulation and experimental results demonstrate that the newly proposed algorithm, in comparison with the existing correlation-based algorithms, has several advantages, such as being insensitive to the nonuniformities of the background intensities and the modulations, having a high stability, and offering improved computational efficiency.

Phase shifting interferometry from two normalized interferograms with random tilt phase-shift

We propose a novel phase shifting interferometry from two normalized interferograms with random tilt phase-shift. The determination of tilt phase-shift is performed by extracting the tilted phase-shift plane from the phase difference of two normalized interferograms, and with the calculated tilt phase-shift value the phase distribution can be retrieved from the two normalized frames. By analyzing the distribution of phase difference and utilizing special points fitting method, the tilted phase-shift plane is extracted in three different cases, which relate to different magnitudes of tilts. Proposed method has been applied to simulations and experiments successfully and the satisfactory results manifest that proposed method is of high accuracy and high speed compared with the three step iterative method. Additionally, both open and closed fringe can be analyzed with proposed method. What's more, it cannot only eliminate the small tilt-shift error caused by slight vibration in phase-shifting interferometry, but also detect the large tilt phase-shift in phase-tilting interferometry. Thus, it will relaxes the requirements on the accuracy of phase shifter, and the costly phase shifter may even be useless by applying proposed method in high amplitude vibrated circumstance to achieve high-precision analysis.

Statistical generalized phase-shifting digital holography with a continuous fringe-scanning scheme

We propose a novel statistical generalized phase-shifting digital holography using a continuous fringe-scanning scheme. In this method, the continuous fringe-scanning scheme is implemented using a PC-based measurement system without any synchronous circuit between the digital camera and the phase shifter. Thus, nonuniformly phase-shifted interference fringes are captured sequentially because of the fluctuation of the image-capturing interval. To cope with the nonuniform phase shifts, we employ a statistical generalized phase-shifting approach. Since the algorithm is designed to use an arbitrary phase shift, the nonuniform phase shifts do not obstruct object wave retrieval. Simulations and experiments demonstrate that the proposed method can be used to implement a practical and accurate digital holography system.

Phase recovery from interferograms under high amplitude vibrations

A phase recovery procedure using interferograms acquired in highly noisy environments as severe vibrations is described. This method may be implemented when disturbances do not allow obtaining equidistant phase shifts between consecutive interferograms due to tilt-shift and nonlinearity errors introduced by the vibrating conditions. If the amount of the tilt-shift is greater than π radians, it will lead a sign change in the phase estimation. This situation cannot be handled correctly by algorithms that consider small errors or non-equidistant phase shifts during the phase shifting process under moderate disturbances. In experimental applications, it is observed that the tilt-shift is often the most dominant error in phase differences that one must deal with. In this work, a Fourier technique is used for the processing and recovering of the cosine of the phase differences. Once the phase differences are obtained, the phase encoded in the interferograms is determined. The proposed algorithm is tested in two sets of interferograms obtained from the analysis of an optical component, finding an rms error in the phase reconstructions of 0.1388 rad.

Improving the accuracy performance of phase-shifting profilometry for the measurement of objects in motion

This Letter presents a new approach to reducing the errors associated with the shape measurement of a rigid object in motion by means of phase-shifting profilometry. While the work previously reported is only valid for the case of two-dimensional (2-D) movement, the proposed method is effective for a situation in which the object moves in a three-dimensional (3-D) space. First, a new model is proposed to describe the fringe patterns reflected from the object, which is subject to 3-D movement. Then, an iterative least-squares algorithm is presented to estimate the phase map. Experiments show that, in contrast to conventional phase-shifting profilometry, the proposed method is capable of significantly reducing the error caused by the 3-D movement of the object.

Robust phase-shift estimation method for statistical generalized phase-shifting digital holography

We propose a robust phase-shift estimation method for statistical generalized phase-shifting digital holography using a slightly off-axis optical configuration. The phase randomness condition in the Fresnel diffraction field of an object can be sufficiently established by the linear phase factor of the oblique incident reference wave. Signed phase-shift values can be estimated with a statistical approach regardless of the statistical properties of the Fresnel diffraction field of the object. We present computer simulations and optical experiments to verify the proposed method.

Robust adaptive phase-shifting demodulation for testing moving wavefronts

Optical interferometers are very sensitive when environment perturbations affect its optical path. The wavefront under test is not static at all. In this paper, it is proposed a novel and robust phase-shifting demodulation method. This method estimates the interferogram's phase-shifting locally, reducing detuning errors due to environment perturbations like vibrations and/or miscalibrations of the Phase-Shifting Interferometry setup. As we know, phase-shifting demodulation methods assume that the wavefront under test is static and there is a global phase-shifting for all pixels. The demodulation method presented here is based on local weighted least-squares, letting each pixel have its own phase-shifting. This is a different and better approach, considering that all previous works assume a global phase-shifting for all pixels of interferograms. Seeing this method like a black box, it receives an interferogram sequence of at least 3 interferograms and returns the modulating phase or wavefront under test. Here it is not necessary to know the phase shifts between the interferograms. It does not assume a global phase-shifting for the interferograms, is robust to the movements of the wavefront under test and tolerates miscalibrations of the optical setup.

Phase shift estimation from variances of fringe pattern differences

This paper presents a simple algorithm for estimating phase shifts from only three interferograms. In it, the fringe pattern differences are computed first in order to remove the background component, and then the variances and further the standard deviations (SDs) of fringe pattern differences are calculated. The phase shifts are estimated, by using the law of cosines, from a triangle whose lengths of sides are the SDs just calculated. This algorithm offers several advantages over others, e.g., being efficient, easy to implement, accurate, and less sensitive to noise. Numerical simulations and an experiment are performed to demonstrate its validity.

Iterative phase-shifting algorithm immune to random phase shifts and tilts

An iterative phase-shifting algorithm based on the least-squares principle is developed to overcome the random piston and tilt wavefront errors generated from the phase shifter. The algorithm iteratively calculates the phase distribution and the phase-shifting map to minimize the sum of squared errors in the interferograms. The performance of the algorithm is evaluated via computer simulations and validated by the Fizeau interferometer measurements. The results show that the proposed algorithm has a fast convergence rate and satisfactory phase-estimation accuracy, improving the measurement precision of the phase-shifting interferometers with significant phase-shifter errors.

Phase determination method in statistical generalized phase-shifting digital holography

A simple estimation method of the relative phase shift for generalized phase-shifting digital holography based on a statistical method is proposed. This method consists of a selection procedure of an optimum cost function and a simple root-finding procedure. The value and sign of the relative phase shift are determined using the coefficient and the solution of the optimum cost function. The complex field of an object wave is obtained using the estimated relative phase shift. The proposed method lifts the typical restriction on the range of the phase shift due to the phase ambiguity problem. Computer simulations and optical experiments are performed to verify the proposed method.

Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method

A method combining the principal component analysis (PCA) and the least squares method (LSM) is proposed to extract the phase from interferograms with random phase shifts. The method estimates the initial phase by PCA, and then determines the correct global phase sign and reduces the residual phase error by LSM. Some factors that may influence the performance of the proposed method are analyzed and discussed, such as the number of frames used, the number of fringes in interferogram and the amplitude of random phase shifts. Numerical simulations and optical experiments are implemented to verify the effectiveness of this method. The proposed method is suitable for randomly phase-shifted interferograms.

Principal component analysis of multiple-beam Fizeau interferograms with random phase shifts

A non-iterative method based on principal component analysis (PCA) is presented to directly extract the phase from multiple-beam Fizeau interferograms with random phase shifts. The PCA method is the approach that decomposes the multiple-beam Fizeau interferograms into many uncorrelated quadrature signals and then applies principal component analysis algorithm to extract the measured phase without any prior guess about the phase shifts. Some factors that affect the performance of the proposed method are analyzed and discussed. Numerical simulations and experiments demonstrate that the proposed method extracts phase fast and exhibits high precision. The method can be applied in high precision interferometry.

Blind self-calibrating algorithm for phase-shifting interferometry by use of cross-bispectrum

A blind self-calibrating algorithm for phase-shifting interferometry is presented, with which the nonlinear interaction introduced by phase shift errors, between the reconstructed phases and the reconstructed amplitudes of the reference wave, is measured with cross-bispectrum. Minimizing an objective function based on this cross-bispectrum allows accurately estimating the true phase shifts from only three interferograms in the absence of any supplementary assumptions and knowledge about these interferograms.

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-shift extraction for generalized phase-shifting interferometry

A simple algorithm for blind extraction of phase shifts is proposed for generalized phase-shifting interferometry from only three interferograms. Based on the statistical property of the object wave, the algorithm calculates approximately the involved phase shifts as initial values. The extraction is further improved by an iterative method, considering the fact that the closer the phase shifts approach their real values, the more uniform the reconstructed reference wave will become. The feasibility of this algorithm is demonstrated by both simulation and experiment.

Algorithm for multiple-beam Fizeau interferograms with arbitrary phase shifts

The objective of this paper is to describe a novel method for phase extraction from multiple-beam Fizeau interferograms with arbitrary phase shifts. The approach begins with applying FFT method to estimate the phase shifts and then utilizes least-squares iterative algorithm to extract phase and phase shifts simultaneously. If the spatial carrier frequency of the fringes is high enough to separate the phase of the first-order maximum in the Fourier domain, the proposed method requires only two iterative cycles to accurately extract phase information from seven multiple-beam Fizeau interferograms with arbitrary phase shifts. Numerical simulations and experiments demonstrate the effectiveness of the proposed algorithm. A comprehensive analysis of the influences of systematic errors (spatial carrier frequency, reflectivity coefficient, and random noise) on the evaluation of phase shifts and phase is presented. The method has applications in high precision interferometry.

Generalized in-line digital holographic technique based on intensity measurements at two different planes

In-line digital holography based on two-intensity measurements [Zhang Opt. Lett. 29, 1787 (2004)], is modified by introducing a pi shifting in the reference phase. Such an improvement avoids the assumption that the object beam must be much weaker than the reference beam in strength and results in a simplified experimental implementation. Computer simulations and optical experiments are carried out to validate the method, which we refer to as position-phase-shifting digital holography.

Accurate phase-shifting digital interferometry

In phase-shifting interferometry experiments, the accuracy of the phase shift is a major issue. Many experimental and data analyses are done to cancel phase-shift errors inherent to the modulation techniques used. We propose to remove most of the phase-shift error by recourse to a frequency-shifting method. This approach can be applied to both holography and interferometry. We validate the idea with a holographic experiment.