Lissajous Ellipse Research Articles

Dynamic range extension method for sinusoidal phase modulating interferometer based on in-phase component signal reconstruction

Abstract The continuous increase of the amplitude of the measured vibration will cause spectrum aliasing in the interference signal of the sinusoidal phase modulating interferometer (SPMI), thereby constraining its measurement dynamic range (DR). To address this issue, a dynamic range extension method of SPMI based on the reconstruction of in-phase component signal is proposed. On the basis of conventional SPMI, an additional reference signal is introduced for estimating the phase modulation depth (PMD) and the major to minor axis ratio Re of the Lissajous ellipse formed by orthogonal interference signal pair and a phase demodulation algorithm named as PGC-ICR-DCM which integrates PGC-DCM algorithm and the in-phase component signal reconstruction (ICR) is proposed. Initially, the PMD is regulated to approximately 1.8412 rad, ensuring that the quadrature component signal can be accurately extracted from the mixed signal of the interference and carrier signals, even in the presence of spectrum aliasing in the original interference signal. Then, the undistorted quadrature component signal is used to reconstruct an in-phase component signal with the estimated value of Re. Finally, the reconstructed in-phase component signal and quadrature component signal are normalized, and a differential-and-cross-multiplying algorithm (DCM) is applied to obtain the measured vibration. A SPMI measurement system based on Michelson interferometer structure is constructed to experimentally validate the proposed method. The experimental results demonstrate that the proposed method can effectively extend the DR of the SPMI system in the presence of spectrum aliasing. The DR of the system is increased from 97.52 dB @ 55 Hz to 102.6 dB @ 55 Hz, with the maximum measurable amplitude increasing from 7728.75 nm @ 55Hz to 13705.16 nm @ 55 Hz.

Accurate two-step random phase retrieval approach without pre-filtering based on hyper ellipse fitting

In this work, we propose a hyper ellipse fitting-based high-precision random two-frame phase shifting algorithm to improve the accuracy of phase retrieval. This method includes a process of Gram-Schmidt orthonormalization, followed by a hyper ellipse fitting procedure. The Gram-Schmidt orthonormalization algorithm constructs a quadrature fringe pattern relative to the original fringe pattern. These two quadrature fringe patterns are then fed into the hyper ellipse fitting procedure, which reconstructs the phase map and refines the background light to produce the final accurate phase of interest. Due to the hyper ellipse fitting procedure, the demodulation results are significantly improved in many cases. This method allows us to design a two-shot phase reconstruction algorithm without the need for least squares iteration or pre-filtering, effectively mitigating residual background to the greatest extent. It combines the advantages of both the Gram-Schmidt orthonormalization method and the Lissajous ellipse fitting method, making our hyper ellipse fitting approach a simple, flexible, and accurate phase retrieval algorithm. Experiments show that by using the weighted least squares method and adjusting the weights, this method can prioritize data points with more significant information or higher reliability, ensuring more accurate estimation of the ellipse parameters.

A PGC demodulation scheme based on the Lissajous ellipse fitting for homodyne interferometer without using second harmonic carrier signal

This paper proposes a phase generated carrier (PGC) demodulation scheme for sinusoidal phase modulation interferometer (SPMI) that does not require a second harmonic carrier signal. In this scheme, an in-phase interference signal component with direct current (DC) bias is obtained by low-pass filtering the original interference signal, and a quadrature interference signal component is obtained by conventional means. The Lissajous figure formed by these two quadrature interference signal components is an ellipse whose center deviates from the origin of the I-Q coordinates. By fitting this ellipse, the major and minor axes and the center coordinates can be obtained. These elliptic parameters are then used to eliminate DC bias and normalize the amplitude of the quadrature signal components. Finally, the phase demodulation of the interference signal can be realized by using arctangent algorithm or differential cross multiplication (DCM) algorithm. The results of simulation and experiment have verified the validity and accuracy of the proposed PGC demodulation scheme, and show that it is insensitive to the phase modulation depth C. Compared with conventional PGC demodulation, this scheme requires lower depth of carrier phase modulation. The vibration measurement results indicate that the average signal-to-noise and distortion Ratio (SINAD) and total harmonic distortion (THD) values of demodulation results in the frequency range of 130 Hz to 250 Hz are 34 dB and -33 dB respectively. The proposed PGC demodulation scheme shows good performance in displacement measurement using SPMI.

Detection of Water Surface Acoustic Waves Using Sinusoidal Phase Modulation Interferometer and Prenormalized PGC-Arctan Algorithm

A sinusoidal phase modulation laser interferometer is proposed to detect water surface acoustic waves excited by underwater acoustic radiation, and an improved PGC-Arctan demodulation algorithm that combines prenormalization and Lissajous ellipse fitting is proposed to demodulate detection signals. In this paper, the effects of phase modulation depth, carrier phase delay, and interference signal visibility on the Lissajous figure formed by quadrature interference components are analyzed. The demodulation algorithm first uses the amplitudes of multiple Fourier spectral components of an interference signal to calculate the phase modulation depth C, and calculation of the carrier phase delay Vc is achieved through the introduction of a quadrature carrier signal. Then, certain coefficients regarding C and Vc are constructed for prenormalization of the two quadrature interference signal components to eliminate the local nonuniform widening phenomenon of Lissajous ellipse. Next, the outer and the inner contours are extracted from a uniformly widened Lissajous ellipse resulting from light intensity disturbance, and the axial ratio of the ellipse is obtained, which is used to correct the ratio of the quadrature interference signal to eliminate the effect of filter gain coefficients. At last, through the combination of an Arctan algorithm and a phase-unwrapping algorithm, high-precision demodulation of the interference signal is realized. A sinusoidal phase modulation interferometer was set up to detect water surface acoustic waves, and a series of detection experiments were carried out. The experiment results show that the detection method and demodulation algorithm described in this paper can accurately realize the measurement of weak water surface acoustic waves. The proposed algorithm shows less distortion in demodulation results, and its signal-to-noise distortion ratio is less than 20 dB at 500 Hz, which is significantly better than traditional algorithms. The experimental results demonstrate the effectiveness and accuracy of water surface acoustic wave detection using sinusoidal phase modulation interferometer.

Phase retrieval from single interferogram without carrier using Lissajous ellipse fitting technology

Phase extraction from single interferogram is of high significance and increasingly interest in optical metrology. In this contribute we propose an advanced Pixel-level Lissajous Ellipse Fitting (APLEF) method to extract the phase from single interferogram without carrier. At each pixel, a Lissajous figure is created by plotting N against D, where N and D are subtractions and additions of intensities of adjacent pixels in a small window. The so created Lissajous figure is already in phase quadrature because of the subtraction and addition process, and the Lissajous Figure is forced to be closed by taking the opposite values of N and D, i.e. –N and -D into account. The closed and in phase quadrature Lissajous Figure is the key point for APLEF to demodulate the single inteferogram without carrier in theoretically. The simulation shows its higher accuracy than existed SPT and Garbusi’s method and the experiments finally corroborate its effectiveness.

Two-step orthogonalization phase demodulation method based on a single differential interferogram.

To reduce the acquisition time of interferogram and provide a dynamic phase retrieval method with arbitrary phase shift using a dual-channel simultaneous polarization phase-shifting system, a two-step orthogonalization phase demodulation method (TOPD) based on a single differential interferogram is proposed in this paper. In this method, the differential interferogram obtained by subtracting two phase-shifting interferograms and one of the Gaussian filtered based-interferograms are used to normalize and orthogonalize, and then the phase related parameters are solved by the Lissajous ellipse fitting method. Finally, the measured phase is obtained with high accuracy. The proposed method further reduces the deviation caused by the filtering operation performed in the two-step phase demodulation method. At the same time, combined it with the Lissajou ellipse fitting method reduces the limitation associated with the approximation conditions of the orthogonalization and normalization method. The experimental and simulation results demonstrate that this method provides a solution with high accuracy, high stability, strong practicability, and few restrictions for phase extraction in quantitative phase imaging.

Normalized and orthogonal phase retrieval algorithm based on Lissajous ellipse fitting

A normalized and orthogonal phase retrieval algorithm based on Lissajous ellipse fitting (NOLEF) is proposed in this study, which can extract phase distributions under random phase shifts with high accuracy. In this algorithm, the parameters related to the phase can be calculated via Lissajous ellipse fitting after the normalization and orthogonalization processes. Compared with other high-accuracy phase-shifting algorithms, the proposed method has a shorter calculation time and provides stable and accurate phase extraction with random phase-shifting interferograms, in which the accuracy is not affected by the number of fringes and phase shifts; it is also suitable for interferograms with different backgrounds and modulations. This method's feasibility and performance at different number of fringes and noise levels are verified via experimental and simulation analyses.

Phase-shifted demodulation technique with additional modulation based on a 3 × 3 coupler and EFA for the interrogation of fiber-optic interferometric sensors.

A phase-shifted demodulation technique with a 3×3 coupler and ellipse fitting algorithm (EFA) for the interrogation of interferometric sensors is proposed. To reduce the error of the EFA as to measure small phase signals, additional phase modulation is introduced. The additional modulation provides a walk of the operating point along the Lissajous ellipse large enough to permit calculation of the ellipse parameters at every moment. Experimental result shows that this technique demonstrates high accuracy and stability for measuring small phase signals. The setting of this technology expands the application of the EFA in fiber-optic phase demodulation technology.

A precise three-step phase retrieval algorithm suitable for the interferograms with less than one fringes numbers

In order to improve the accuracy and expand the range of the phase-shifting algorithm application, a precise three-step phase retrieval algorithm suitable for the interferograms with less than one fringes numbers is proposed. The proposed method improves the accuracy, on the one hand because normalization and subtraction are introduced to reduce background intensity fluctuation, on the other hand because Lissajous ellipse fitting (LEF) method is introduced to reconstruct phase maps from the normalized and Gram-Schmidt (GS) orthonormalized maps. Because the approximation operations for cumulative sum aren’t used in the GS method, the proposed algorithm is suitable for not only interferograms that the fringes numbers are more than one but also interferograms that fringes numbers are less than one. Moreover, the proposed method can be applied to simple and complex patterns. Thus, the proposed algorithm has a wider range of applications. The proposed algorithm’s reliability and performance are confirmed by simulations and experiments.

Single frame phase estimation based on Hilbert transform and Lissajous Ellipse Fitting method in fringe projection technology

Fringe projection three-dimensional (3D) measurement technique is an effective technique to measure 3D shape of objects, in which phase retrieval is an important procedure. To overcome the need of pre-filtering process in the existing single-frame methods, this paper proposes a novel method to extract phase distribution solely from one-frame fringe pattern without pre-filtering, which reduces its complexity and retains more phase details. The proposed method consists of two stages. Firstly, the Hilbert transform is applied to one-frame fringe pattern to generate another frame of fringe pattern. Secondly, the phase distribution is calculated based on Lissajous Ellipse Fitting (LEF) technique by using original fringe pattern and the Hilbert transformed fringe pattern. The results provides a practically efficient methodology for single frame phase reconstruction, which shows potential useful means for reconstructing object in rapid measurement.

Two-step phase shifting profilometry based on Lissajous ellipse fitting technique

<sec> Phase shifting profilometry (PSP) is an effective technique to reconstruct the three-dimensional shape of object. In general, PSP needs three or more fringe patterns with phase-shifting accurately known to extract the phase distribution of objects. Therefore, the scene and the test objects should remain stationary during capturing the fringe patterns. However, the phase shifts may be unknown in an actual PSP measurement system, especially when measuring the moving object, that is, the phase-shift error may be introduced during the obtaining of the phase-shifting fringe patterns of moving object. In the dynamic measurement scenario, the use of fewer fringe patterns can realize the faster measurement speed and suppress the phase shift error introduced by the moving object. In this paper, a two-step PSP algorithm is proposed based on Lissajous ellipse fitting (LEF). The proposed method uses only two fringe patterns to extract the phase distribution of the object and can suppress the phase shift error caused by the moving object. </sec><sec> However, in a practical PSP system, the spatiotemporally varying background intensity and modulation also significantly affect the phase accuracy extracted by LEF, and thus three error-suppressing methods are proposed to reduce the phase error caused by the non-uniform background intensity and modulation. In order to verify the effectiveness of the three error-suppressing methods, we analyze and compare their performances of error suppression under different background intensities and modulations. The advantages of three error-suppressing methods can be summarized as follows. 1) The mean and modulation correction technique has greater advantage than the other two when the background intensity and modulation vary with time. 2) When the background intensity and modulation are relevant to pixel position and the number of fringe patterns, the empirical mode decomposition normalization can more effectively suppress the influence of the non-uniform background intensity and modulation. </sec><sec> In experiment, a two-step phase-shifting dynamic measurement based on LEF is conducted. Compared with the traditional PSP which needs at least three fringe patterns, the two-step PSP algorithm successfully extracts the phase with only two fringe patterns and suppresses the phase shift error caused by the motion of the object. Compared with Fourier transform profilometry (FTP), the two-step PSP algorithm can obtain very accurate phase distribution and retain many phase details. </sec>

Fringe analysis with a two-step phase shifting method based on local Lissajous ellipse fitting

This work aims to report an algorithm for performing fringe analysis with two phase-stepped fringe patterns. By performing a particular Least Squares Fitting (LSF) it is intended to overcome the restrictions to fringe patterns of having constant background or being high-pass filtered when using Lissajous ellipse fitting. With the new method, the phase step and the fringes background are represented by single coefficients that depend on image position, avoiding the utilization of additional arithmetic operations for its calculation, as done with previous methods, increasing its accuracy and range of applicability. Experimental and simulations are presented to elucidate our proposal.

Modelling and correction for polarization errors of a 600 mm aperture dynamic Fizeau interferometer.

The polarization errors of large aperture dynamic interferometers based on the polarization phase shifting method are mainly coming from the effects of imperfect polarized elements and birefringence of large elements. Using the Lissajous ellipse fitting algorithm to correct the influence of the polarized device can effectively eliminate single and double frequency print through errors. We develop a wave plate model for analyzing the birefringence effect, and on this basis, we establish the relationship between the calculated phase and the ideal phase distribution. Experiments are carried out on a 600mm aperture Fizeau interferometer and then compared with the result acquired through the wavelength tuning method. The difference between PV is only 0.002λ.

The role(s) of “Simultaneous Postural Adjustments” (SPA) during Single Step revealed with the Lissajous method

Dynamical phenomena in the postural chain occur before, during and after the voluntary movement. These phenomena correspond to anticipatory (APA), simultaneous (SPA), and consecutive (CPA) postural adjustments, respectively. APA and, more recently, CPA, have been extensively investigated in the literature. SPA have surprisingly received much less attention. The aim of the present study was to examine the role(s) of SPA associated with a single step task (SST). Ten healthy young adults performed series of SST on a force-plate. A 2-DOF mechanical model was used to separate the dynamics of the swing leg and the dynamics of the rest of the body, corresponding to the focal and the postural component of the SST, respectively. The postural component was plotted against the focal one during SPA (from heel-off to foot-contact), and this plot was modelled as a Lissajous ellipse. Result showed that this ellipse systematically ran through the same three quadrants of the diagram. For each of these quadrants, the role of the postural component in regards to the focal one was interpreted according to the relative orientation of the postural and focal dynamics. Results thus showed that SPA ensured the following successive roles: counter-perturbation of swing leg dynamics following heel-off, propulsion of swing leg, counter-perturbation of swing leg dynamics again, and then braking swing leg movement. These new findings contribute to a better knowledge of postural adjustments properties, and may provide new insights for understanding balance troubles with aging and in neurological patients (e.g. people with Parkinson’s disease).

Phase extraction from two randomly phase shifted interferograms by combining advanced principal component analysis and Lissajous ellipse fitting

In order to achieve the high accuracy and high speed in phase shifting interferometry, a phase extraction approach by combining advanced principal component analysis and Lissajous ellipse fitting (APCA-LEF) is proposed. It can obtain accurate phase distribution with only two no pre-filtering phase shifted interferograms, and it costs less time simultaneously. It removes the restriction that PCA needs more than three interferograms with well distributed phase shifts to subtract relatively accurate mean. Moreover, adjacent pixels taken part in the APCA process increases the accuracy by suppressing the effect of noise, also, it is suitable for different levels of noises. Last but not least, if the high accuracy is required, the phase shift would be best to far away from 0 rad and π rad. The simulations and experiments verify the correctness and feasibility of APCA-LEF.

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.

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

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.

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.

Random three-step phase retrieval approach based on difference map Gram–Schmidt orthonormalization and Lissajous ellipse fitting method

To achieve high measurement accuracy with less computational time in phase-shifting interferometry, a random phase retrieval approach based on difference map Gram–Schmidt othonormalization and Lissajous ellipse fitting method (DGS-LEF) is proposed, it easy to implement and only needs three phase shifted interferograms, it doesn't need pre-filtering, moreover, the phase shift can be random except for too small relative phase shift between two adjacent interferograms, last but not least, this method is effective for the symmetrical and asymmetrical fringes. The simulations and experiments verify the correctness and feasibility of DGS-LEF.

Single-shot deflectometry for dynamic 3D surface profile measurement by modified spatial-carrier frequency phase-shifting method

We propose a new concept of single-shot deflectometry for real-time measurement of three-dimensional surface profile using a single composite pattern. To retrieve an accurate phase from one-frame composite pattern, we adapt the Fourier Transform (FT) method and the spatial carrier-frequency phase-shifting (SCPS) technique to our proposed deflectometry. Based on Lissajous figure and ellipse fitting method, we also correct the phase extraction error in SCPS technique by reducing the effect of background and modulation variations. The proposed technique is verified by comparing our measurement results with phase-shifting deflectometry, and the maximum difference between two measurement results is less than 30 nm rms. We also test the robustness to vibration and the measurement capability for dynamic object.