Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts

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.

Phase recovery technology of a dual-frame phase-shifting interferogram based on first-order norm vector normalization

The pursuit of high-precision and high-efficiency phase recovery methods has been a research focus of interferometric technology. We propose a dual-frame phase-shifting interferogram phase recovery technique based on normalization of the first-order norm. A set of sine and cosine components is constructed by the addition and subtraction of dual-frame interferograms. Then the first-order norm normalization method is employed to achieve vector orthogonality. The phase distribution is then obtained through the arctangent operation. State-of-the-art dual-frame phase recovery techniques are evaluated, and it shows that the first-order norm normalization method outperforms the second-order norm normalization method. Especially in terms of computational efficiency, the method using the first-order norm is at least 50% more efficient than other methods.

Robust weighted principal components analysis demodulation algorithm for phase-shifting interferometry

We present an asynchronous phase-shifting demodulation approach based on the principal component analysis demodulation method that is robust to typical problems as turbulence, vibrations, and temporal instabilities of the optical setup. The method brings together a two-step and a phase-shifting asynchronous demodulation method to share their benefits while reducing their intrinsic limitations. Thus, the proposed approach is based on a two-fold process. First, the modulating phase is estimated from a two-step demodulation approach. Second, this information is used to compute weights to each phase-shifted pattern of the interferogram sequence, which are used in a novel weighted principal component demodulation approach. The proposed technique has been tested with simulated and real interferograms affected by turbulence and vibrations providing very satisfactory results in challenging cases.

Alternative interpretation of color phase-shifting profilometry and a self-contained method for fringe analysis

This work aims to report a robust single-shot color fringe projection method based on phase-shifting profilometry, where, to calculate the phase, is it not necessary to have knowledge of color coupling or color imbalance calibration parameters of the projector-camera system; additionally, it does not require knowledge of the phase-shifting step, nor projecting equally stepped fringe patterns. Furthermore, it does not require a high density of carrier fringes; it only requires a composite single color image. Theoretical and experimental results are provided. Its suitability to simplify absolute unwrapping is also discussed.

Phase-shift extraction of multiple-frame randomly phase-shifted interferograms by analysis of the amplitude of the analytic signal

We present a powerful phase-shift extraction algorithm for multiple-frame random phase-shifting fringe patterns. The proposed method is based on changing the regularity of the amplitude of a demodulated analytic signal with respect to different phase shifts and a one-dimensional optimization method. Compared with the existing universal phase-reconstruction method, the proposed method is accurate, stable, and efficient. Both numerical simulations and experimental data demonstrate the high accuracy and efficiency of the proposed method.

Rapid and precise phase retrieval from two-frame tilt-shift based on Lissajous ellipse fitting and ellipse standardization

A rapid and precise phase-retrieval method based on Lissajous ellipse fitting and ellipse standardization is demonstrated. It only requires two interferograms without pre-filtering, which reduces its complexity and shortens the processing time. The elliptic coefficients obtained by ellipse fitting are used for ellipse standardization. After compensating phase-shift errors by ellipse standardization, the phase distribution is extracted with high precision. It is suitable for fluctuation, noise, tilt-shift, simple and complex fringes. This method is effective for the number of fringes less than 1. The reliability of the method is verified by simulations and experiments, indicating high accuracy and less time consumption.

Random phase shifting shadow moiré using a one-dimensional minimizer

The introduction of a random phase-shifting technique into a shadow moiré system, where an equal and known (or unknown) phase step is used to demodulate the phase of interest, is beneficial for the improvement of measurement accuracy. However, in spite of recent advances in optical metrology phase-shifting techniques, simultaneously estimating unequal and unknown phase shifts from three random phase-shifting fringe patterns remains a significant challenge. This paper presents a one-dimensional minimizer-based technique to address this ill-posed problem of phase demodulation from random phase-shifting patterns. In this method, two new sets of connected fringe patterns, without background illumination, are constructed through normalizing the secondary fringe patterns. Then, a generalized phase-shifting algorithm is developed by utilizing the character of the modulation factor's standard deviation distribution. Both numerical simulations and optical experiments are performed to demonstrate the high accuracy and robustness of the proposed method.

Universal phase reconstruction approach of self-calibrating phase-shifting interferometry

Self-calibrating phase-shifting interferometry (PSI) reconstructs the phase map from three-frame or more phase-shifting interferograms without the need for knowing accurate phase steps. The existing phase reconstruction methods for self-calibrating PSI still have many constraints in terms of the required number of interferograms, special usage preconditions, etc. In this Letter, a universal, accurate, and efficient phase reconstruction method for self-calibrating PSI is proposed. In this approach, we search the solution space of phase shifts to obtain the modulation amplitude of interferograms with the minimum coefficient of variation (CV). Then the phase is reconstructed through the searched phase shifts. Numerical and experimental studies demonstrate that this method can realize highly accurate phase reconstruction consistently, compared to the currently popular methods in their own usable ranges. We anticipate that this Letter may provide a universal and powerful solution for the phase reconstruction of self-calibrating PSI.

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.

Two-frame fringe pattern phase demodulation using Gram-Schmidt orthonormalization with least squares method

Gram-Schmidt (GS) orthogonal normalization is a fast and efficient two-frame fringe phase demodulation method. However, the precision of the GS method is limited due to the residual background terms and noise, as well as several approximation operations in the GS method. To obtain a phase map with higher accuracy, we propose an algorithm combining GS orthogonal normalization and least squares iterative (LSI) phase shift algorithm (GS&LSI). In our method, the phase was first obtained using GS method, and then a refinement operation using LSI was adopted to get the final wrapped phase map. Because of the LSI process, the demodulation result is greatly improved in many cases. Simulation and experimental result are presented to validate the potential of the proposed method.

Object wave retrieval using normalized holograms in three-step generalized phase-shifting digital holography

Phase-shifting methods using interferogram normalization are often applied to smooth objects, for which the requirements for the normalization approach, including zero-order term elimination and the norm approximation condition, are easily achieved. Here we propose a three-step generalized phase-shifting method using the normalization approach for diffuse objects. In the proposed method, the zero-order terms are sufficiently suppressed by mutual subtraction of the phase-shifted holograms. The norm approximation condition is satisfied, and the complex field of the object wave can be estimated by the normalization approach when the hologram satisfies the phase randomness condition. We present an object wave retrieval algorithm using three phase-shifted holograms, in which estimation of phase-shift values is unnecessary. The proposed method is verified through simulations and optical experiments.

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.

Dual-channel simultaneous spatial and temporal polarization phase-shifting interferometry

Without the limitations of fringe number and fringe shape, a dual-channel simultaneous spatial and temporal polarization (DC-SSTP) phase-shifting interferometry system is proposed to achieve rapid and accurate phase retrieval through only one-time phase-shifting procedure with unknown phase shifts. First, an arbitrary phase shifts is simultaneously introduced into two channels of DC-SSTP system by a spatial light modulator (SLM). Second, by performing the subtraction operation between each pair of phase-shifting interferograms captured in the same channel, the background deduction of interferogram can be achieved easily, so the accurate phase can be retrieved rapidly. Especially, it is found even if the fringe number in interferogram is less than one, the proposed DC-SSTP method still reveals high accuracy of phase retrieval. Both the simulation and experimental results demonstrate the outstanding performance of proposed DC-SSTP method in phase measurement.

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.

Random phase-shifting algorithm by constructing orthogonal phase-shifting fringe patterns

The intensity distribution of fringe patterns becomes nonsinusoidal in real testing environments. Thus, the performance of existing phase shift extraction algorithms, which usually compute the desired phase shift by arccosine function or arcsine function, may be affected. In the presented paper, we report an arctangent-function-based technique to solve this disturbance. First, two orthogonal fringe patterns are constructed through subtraction and addition of two background-removed images. Second, the unequal amplitude between two new fringe patterns is eliminated using a normalization process. Third, the phase shift is determined by computing the norms of the two new images. The proposed method is fast and can be implemented easily in many applications. We verify the algorithm performance and robustness using both simulated and experimental data, indicating the high accuracy of the presented method.

An advanced phase retrieval algorithm in N-step phase-shifting interferometry with unknown phase shifts

In phase-shifting interferometry with unknown phase shifts, a normalization and orthogonalization phase-shifting algorithm (NOPSA) is proposed to achieve phase retrieval. The background of interferogram is eliminated through using the orthogonality of complex sinusoidal function; and the influence of phase shifts deviation on accuracy of phase retrieval is avoided through both normalization and orthogonalization processing. Compared with the current algorithms with unknown phase shifts, the proposed algorithm reveals significantly faster computation speed, higher accuracy, better stability and non-sensitivity of phase shifts deviation.

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.

Gram-Schmidt orthonormalization for retrieval of amplitude images under sinusoidal patterns of illumination

Structured illumination using sinusoidal patterns has been used for optical imaging of biological tissues in biomedical research, and of horticultural products in food quality evaluation. Implementation of structured-illumination imaging relies on retrieval of amplitude images, which is conventionally achieved by a phase-shifting technique that requires collecting a minimum of three phase-shifted images. In this study, we have proposed Gram-Schmidt orthonormalization (GSO) to retrieve amplitude component (AC) images using only two phase-shifted images. We have proposed two forms of GSO implementation, and prior to GSO processing, we eliminated the direct component (DC) background by subtracting a DC image we recovered using a spiral phase function (SPF) in the Fourier space. We demonstrated the GSO methods through numerical simulations and application examples of detection of bruise defects in apples by structured-illumination reflectance imaging (SIRI). GSO performed comparably to conventional three-phase-based demodulation. It is simple, fast and effective for amplitude retrieval and requires no prior phase information, which could facilitate fast implementation of structured-illumination imaging.

Three-frame self-calibration phase shift algorithm using the Gram-Schmidt orthonormalization approach

Affected by the height dependent effects, the phase-shifting shadow moiré can only be implemented in an approximate way. In the technique, a fixed phase step around π/2 rad between two adjacent frames is usually introduced by a grating translation in its own plane. So the method is not flexible in some situations. Additionally, because the shadow moiré fringes have a complex intensity distribution, computing the introduced phase shift from the existing arccosine function or arcsine function-based phase shift extraction algorithm always exhibits instability. To solve it, we developed a Gram-Schmidt orthonormalization approach based on a three-frame self-calibration phase-shifting algorithm with equal but unknown phase steps. The proposed method using the arctangent function is fast and can be implemented robustly in many applications. We also do optical experiments to demonstrate the correction of the proposed method by referring to the result of the conventional five-step phase-shifting shadow moiré. The results show the correctness of the proposed method.

Precise phase retrieval under harsh conditions by constructing new connected interferograms

To date, no phase-shifting method can accurately retrieve the phase map from a small set of noisy interferograms with low phase-shifts. In this Letter, we develop a novel approach to resolve this limitation under such harsh conditions. The proposed new method is based on constructing a set of connected interferograms by means of simple subtraction and addition operations, in which all the subset of interferograms have the same phase-shift interval of π/2. According to this characteristic, this set of connected interferograms can be processed with conventional phase retrieval methods as PCA or AIA obtaining accurate results. The reduction in the RMS errors after using our method reaches as high as 93.7% and 89.3% respectively comparing with conventional PCA and AIA methods under harsh conditions. Both simulation and experiment results demonstrate that the new proposed method provides an effective way, with high precision and robustness against noise, for phase retrieval.

Demodulation of two-shot fringe patterns with random phase shifts by use of orthogonal polynomials and global optimization

We propose a simple and robust phase demodulation algorithm for two-shot fringe patterns with random phase shifts. Based on a smoothness assumption, the phase to be recovered is decomposed into a linear combination of finite terms of orthogonal polynomials, and the expansion coefficients and the phase shift are exhaustively searched through global optimization. The technique is insensitive to noise or defects, and is capable of retrieving phase from low fringe-number (less than one) or low-frequency interferograms. It can also cope with interferograms with very small phase shifts. The retrieved phase is continuous and no further phase unwrapping process is required. The method is expected to be promising to process interferograms with regular fringes, which are common in optical shop testing. Computer simulation and experimental results are presented to demonstrate the performance of the algorithm.

Phase stepping optical profilometry using fiber optic Lloyd's mirrors

A three-step phase stepping profilometry based on a fiber optic Lloyd's mirror assembly is employed in the optical profilometry for the first time to measure the shapes of 3D objects. Required π/2 phase shifts for interference fringe pattern are obtained by mechanically sliding the Lloyd assembly via an ordinary micrometer stage. The experimental setup is simple and low cost to construct, and is insensitive to the ambient temperature fluctuations and environmental vibrations that cause unwanted effects on the projected fringe pattern. Consecutive interferograms are captured by a CCD camera and are processed with an algorithm to accomplish 3D topographies.

Visual measurement of the evaporation process of a sessile droplet by dual-channel simultaneous phase-shifting interferometry

To perform the visual measurement of the evaporation process of a sessile droplet, a dual-channel simultaneous phase-shifting interferometry (DCSPSI) method is proposed. Based on polarization components to simultaneously generate a pair of orthogonal interferograms with the phase shifts of π/2, the real-time phase of a dynamic process can be retrieved with two-step phase-shifting algorithm. Using this proposed DCSPSI system, the transient mass (TM) of the evaporation process of a sessile droplet with different initial mass were presented through measuring the real-time 3D shape of a droplet. Moreover, the mass flux density (MFD) of the evaporating droplet and its regional distribution were also calculated and analyzed. The experimental results show that the proposed DCSPSI will supply a visual, accurate, noncontact, nondestructive, global tool for the real-time multi-parameter measurement of the droplet evaporation.

Advanced principal component analysis method for phase reconstruction

Focus on the phase reconstruction from three phase-shifting interferograms with unknown phase shifts, an advanced principal component analysis method is proposed. First, use a simple subtraction operation among interferograms, two intensity difference images are obtained easily. Second, set the center region of the data of intensity difference images to zero, and then construct a covariance matrix to obtain a transformation matrix. Third, two principal components of interferograms can be determined by the Hotelling transform and then phase can be calculated from the two normalized principal components by an arctangent function. By means of the simulation calculation and the experimental research, it is proved that the phase with high precision can be obtained rapidly by the proposed algorithm.

Two-shot fringe pattern phase-amplitude demodulation using Gram-Schmidt orthonormalization with Hilbert-Huang pre-filtering

Gram-Schmidt orthonormalization is a very fast and efficient method for the fringe pattern phase demodulation. It requires only two arbitrarily phase-shifted frames. Images are treated as vectors and upon orthogonal projection of one fringe vector onto another the quadrature fringe pattern pair is obtained. Orthonormalization process is very susceptible, however, to noise, uneven background and amplitude modulation fluctuations. The Hilbert-Huang transform based preprocessing is proposed to enhance fringe pattern phase demodulation by filtering out the spurious noise and background illumination and performing fringe normalization. The Gram-Schmidt orthonormalization process error analysis is provided and its filtering-expanded capabilities are corroborated analyzing DSPI fringes and performing amplitude demodulation of Bessel fringes. Synthetic and experimental fringe pattern analyses presented to validate the proposed technique show that it compares favorably with other pre-filtering schemes, i.e., Gaussian filtering and continuous wavelet transform.