Lissajous Ellipse Research Articles

Two-step random phase retrieval approach based on Gram-Schmidt orthonormalization and Lissajous ellipse fitting method.

To overcome the phase shift error in phase shifting interferometry, a two-step random phase retrieval approach based on Gram-Schmidt (GS) orthonormalization and Lissajous ellipse fitting (LEF) method (GS&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. It is suitable for different background intensity, modulation amplitude distributions and noises. Last but not least, this method is effective for circular, straight or complex fringes. The simulations and experiments verify the correctness and feasibility of GS&LEF.

An accurate phase shift extraction algorithm for phase shifting interferometry

Abstract The accuracy of phase shift extraction has a significant influence on measurement results in surface micro-topography interferometry. Phase shifting errors are mainly caused by nonlinearity of the employed phase shifter, environmental turbulence, camera imperfection and so on. In this paper, a general algorithm based on Lissajous figures and ellipse fitting is proposed for extracting the phase distribution from a set of phase-shifting interferograms with random noise. Two sets of pixels with π ∕ 2 phase difference in all the investigated interferograms are selected and used for ellipse fitting. Both numerical simulations and optical experiments have proven the validity, rapidity, and accuracy of the proposed method. Experiments show that the proposed method is a general phase extraction method, which can work for straight fringe patterns, circle fringes patterns and other anomalous features.

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.

Precise phase demodulation of single carrier-frequency interferogram by pixel-level Lissajous figure and ellipse fitting

Phase demodulation from a single carrier-frequency fringe pattern is becoming increasingly important particularly in areas of optical metrology such as dynamic interferometry, deflectometry and profilometry. The Fourier transform (FT) method and the spatial-carrier phase-shifting technique (SCPS) are two popular and well-established approaches to demodulation. However FT has the drawback of significant edge errors because of the Gibbs effect, whilst detuning errors for the local phase shift occur when SCPS is applied. A novel demodulation method based on pixel-level Lissajous figure and ellipse fitting (PLEF) is presented in this paper. Local demodulation in the spatial domain makes PLEF more flexible than the FT method, without spectral leakage. Based on a more adaptable approach, account is taken of variations in illumination and phase distribution over a few neighboring pixels. The mathematic demodulation model is of interest and has been demonstrated via simulation. Theoretical phase extraction error is as low as 10−4 rad. Experiments further corroborate the effectiveness of the proposed method. In conclusion, various influencing factors, e.g. variations of background/modulation, phase amplitude, carrier frequency, additive noise that may affect the precision of PLEF are discussed in detail.

Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering

This paper presents a novel method to extract the phase shift and phase distribution from two interferograms simultaneously. By employing Hilbert–Huang transform based prefiltering, the background intensities and modulation amplitudes of the two interferograms are suppressed and normalized respectively. With the addition and subtraction operation of the two prefiltered interferograms, two parametric equations are achieved which can be regarded as the complex harmonic motion of the Lissajous figure. The phase of the Lissajous figure can be directly demodulated by the ellipse fitting algorithm. Apart from the advantages of other well-known two-step phase demodulation algorithms, i.e., high accuracy and efficiency of the Gram-Schmidt orthonormalization (GS) method and the less stringent requirement concerning the fringe number in the extreme value of interference (EVI) method, proposed Lissajous figure and ellipse fitting (LEF) approach has another bonus related to its robustness to the fluctuations of the fringe patterns noise, background intensity and modulation amplitude. Simulations demonstrate the outstanding performance of the proposed method, and experiments further corroborate its effectiveness.

Are simultaneous postural adjustments (SPA) programmed as a function of pointing velocity?

This paper deals with the influence of velocity on the postural adjustments that occur during the course of a voluntary movement, that is to say, simultaneous postural adjustments (SPA). To this aim, a pointing task performed at different velocities (V) was considered. Upper limb kinematics and body kinetics were recorded. Using a 2-DOF model, the body was divided into two parts: the right upper limb (termed the "focal" chain) and the rest of the body (termed the "postural" chain). This model allowed us to calculate the kinetics of both subsystems (-F x and [Formula: see text]), with one corresponding to the resultant action on the shoulder (AoSh: -F x) and the other to the resultant reaction of the shoulder (RoSh: [Formula: see text]). The influence of pointing velocity on peak amplitudes and durations was evaluated, as was their instantaneous relationship ("Lissajous ellipse"). The results showed that RoSh and AoSh display similar diphasic profiles, whose amplitude and duration vary with movement velocity. In addition, RoSh is in phase advance of AoSh, the advance being all the shorter as the focal movement velocity becomes faster. Finally, SPA appears to play a dual role, which includes a propulsive action during upper limb acceleration and body stabilization during deceleration. These new findings strengthen the hypothesis that the postural chain is programmed according to task velocity in the same way as the focal chain and that both are coping in order to make the task more efficient.

Generalized phase shifting interferometry based on Lissajous calibration technology

The feasibility and limitation of directly using the Lissajous figure and ellipse fitting technology to correct the phase extraction error in generalized data reduction algorithm (GDRA) for phase extraction of randomly phase-shifted interferograms are analyzed and discussed. By combining Lissajous calibration technology, which represents the transformative process of Lissajous ellipse to circle (ETC), with advanced iterative algorithm (AIA) we propose a novel generalized phase shifting algorithm (GPSA), and here it is abbreviated as ETCI method. The phase distribution and phase shifts that extracted from randomly phase shifted interferograms by use of ETCI are more accurate and the whole process is far faster than AIA. Additionally, proposed method is less sensitive to non-uniform background intensity and modulation amplitude. Numerical simulations are conducted to evaluate the performance of ETCI, and some influential factors are elaborated. The experimental results further indicate proposed method is suitable for truly random phase shifted interferograms.

Correction of phase extraction error in phase-shifting interferometry based on Lissajous figure and ellipse fitting technology.

The accuracy of phase-shifting interferometers (PSI) is crippled by nonlinearity of the phase shifter and instability of the environment such as vibration and air turbulence. A general algorithm, utilizing Lissajous figures and ellipse fitting, of correcting the phase extraction error in the phase shifting interferometry is described in this paper. By plotting N against D, where N and D represent the numerator and denominator terms of the phase extraction function (i.e. an arctangent function) respectively, a Lissajous ellipse is created. Once the parameters of the ellipse are determined by ellipse fitting, one can transform the ellipse to a unit circle (ETC). Through this process the phase extraction error caused by random phase shift errors can be corrected successfully. Proposed method is non-iterated, adapts to all phase shifting algorithms (PSAs), and has high accuracy. Some factors that may affect the performance of proposed method are discussed in numerical simulations. Optical experiments are implemented to validate the effectiveness of proposed algorithm.

Simultaneous Postural Adjustments (SPA) scrutinized using the Lissajous method

The goal of this research was to study the postural adjustments that occur during the course of a voluntary movement (Simultaneous Postural Adjustments: SPA). A pointing task performed at maximal velocity was considered and upper limb kinematics and body kinetics were recorded. A 2-DOF model was elaborated that distinguishes between the body segments that are mobilized in order to perform the pointing movement. These segments are the right upper limb (termed the “focal” component) and the rest of the body (termed the “postural” component). This model allowed for the calculation of both sub-systems׳ kinetics and a comparison of the resultant reaction (RoSh) with the corresponding action (AoSh) at the shoulder level. The analysis was based on the ellipsoidal shape of their relationship. The ellipse computation (“Lissajous ellipse”) allowed the time lag to be estimated. The results showed that the kinetics of the postural component preceded that of the focal ones and that the time lag during the SPA was not statistically different from the APA duration (dAPA). In addition, the kinetics of the postural component were found to be opposed to the perturbation induced by the pointing movement, but only during part of the SPA time interval. It was concluded that the postural component plays a dual role during the movement, which consists of postural stabilization and propulsive action, with one prevailing over the other depending on the time-instant of movement evolution. This new evidence in healthy subjects is helpful to further specify differences associated with motor impairments.

Phase step measurement and variable step algorithms in phase-shifting interferometry

A novel method, utilizing Lissajous figures and ellipse fitting, of calculating the phase difference between interferograms obtained from a phase shifting interferometer is described. Intensity bias and intensity modulation of interferograms are also calculated using this technique. Two new phase extraction algorithms are presented which use intensity and phase step information to calculate a surface from interferograms acquired with uneven phase steps. One algorithm allows surface phase to be calculated from two interferograms. Preliminary results from the Lissajous figure technique and the presented algorithms are discussed.

An investigation of two‐second microseisms associated with cold fronts and a new method for tracking the cold front center

Two tripartite stations have been set up six miles apart. Each consists of three two‐second horizontal seismographs recording at a drum speed of 600 mm/min.In addition, three stations equipped with two‐second vertical and horizontal instruments have been set up on a triangle with sides of 8.3, 6.75 and 6.0 miles.Results to date are somewhat confusing. There is evidence of a microseismic source southwest of New York. There is also evidence of wave trains identifiable between two stations six miles apart indicating a velocity of microseismic waves of less than one mile per second. A new method is presented of recording microseisms on an oscilloscope and reading the time differences of each pair of stations along the axis of the observed Lissajous ellipse.