A simple arbitrary phase-step digital holographic reconstruction approach without blurring using two holograms

This work presents a simple approach for arbitrary phase-step digital holography using two holograms without ambiguity. Using this approach, a magnitude-contrast image is reconstructed without dc term and twin-image blurring, but the phase-contrast image is filled with phase distortion. Computer simulations are carried out to verify the proposed approach, and optical experiments are performed to validate it. The advantages, benefits and limitations of the proposed method compared to conventional two-step phase-shifting methods are also discussed herein.     

Complex surface three-dimensional shape measurement method based on defocused Gray code plus phase-shifting

Binary pattern defocused projection method can overcome the nonlinear gamma of the projector, as well as filter high harmonics and high-frequency noise. However, high-accuracy three-dimensional (3-D) shape measurement of complex surface using it still remains a challenge. Therefore, a novel Gray code plus phase-shifting method based on defocusing is proposed to solve the problem. The edges of Gray code patterns become blurred owing to defocus, which makes the recovery of accurate Gray code patterns difficult. To solve this problem, the positive and inverse Gray code patterns are projected to obtain threshold values, which are used to achieve the binarization of Gray code patterns. This method is robust and suitable for different defocus levels. Compared with the traditional Gray code plus phase-shifting method, the experimental results prove the effectiveness and feasibility of the proposed method.     

Experimental requirement of the error-compensating five-frame interferogram-collecting sequence used in phase-shifting interferometry

Digital phase-shifting interferometry (PSI), a technique widely used in optical testing, requires interferograms collected at optical phase differences separated by a definite phase step. The five-frame interferogram-collecting sequence suggested by Hariharan et al. [Appl. Opt. 26, 2504 (1987)] is extremely effective in significantly reducing the errors in height profiles derived using PSI that are caused by phase-step errors. In this Letter, we report on a class of five-frame sequence that, owing to its mathematical equivalence with the one suggested by Hariharan et al. and its ease of execution, is more commonly used but is much less effective in reducing the height profile errors caused by phase-step errors.     

Chirp estimation in phase-shifting interferometry

We propose a new approach for estimating the phase in the presence of a nonlinear response of a phase-shifting device: a piezoelectric transducer (PZT). The method is complemented well by the high resolution and the maximum likelihood estimation techniques in the estimation of the phase step and the nonlinear coefficient. The advantage of the proposed method is that it can be extended to the extraction of multiple phases in configurations involving multiple PZTs in the presence of nonlinearity. Symmetricity in the phase steps is not required in this method. Hence hysteresis of the PZT does not have any influence on the accuracy of the phase estimation. The effectiveness of the method is shown by experimental results.     

A novel approach for characterizing the nonlinear phase steps of the PZT in interferometry

We propose a novel approach to estimate the phase distribution in the presence of nonlinear response of the PZT. The proposed method is an outcome of two different concepts based on the discrete chirp Fourier transform (DCFT) and the maximum-likelihood estimator for the estimation of nonlinear phase step. The robustness of the proposed method is tested for various magnitudes of nonlinearity and harmonics. Comparison of our proposed method with the bench marking algorithm by Hibino et al.'s six and nine sample algorithm in the presence of noise shows that our method is highly efficient.     

Multi-frequency inverse-phase fringe projection profilometry for nonlinear phase error compensation

A new multi-frequency inverse-phase method was proposed to compensate for nonlinear phase errors in fringe projection profilometry and to measure the three-dimensional shape of discontinuous objects. After introducing a phase offset of π/4 into the multi-frequency four-step phase-shifting method the corresponding nonlinear phase error reversed its sign, which allowed the addition of unwrapped phases before and after the phase-offset operation to compensate for the error. For the four-step phase-shifting method, simulation analysis showed that the nonlinear phase error had quadrupled the fringe frequency. Moreover, experimental results verified the feasibility and applicability of the proposed method.     

An effective method in calibrating nonlinear errors for phase-shifting interferometry

In phase measurement or digital holography for phase-shifting interferometry, the key role is the variation of reference light wave and recover algorithm based on interferograms and reference phase, so the calculation result is directly affected by phase-shift accuracy. However, because of the errors of nonlinear and other random factors, it is difficult to control the actual phase-shifting amount accurately. In this paper, we aim to propose an efficient method for phase-shifting interferometry which does not require accurate initial estimation of phase-shift amounts, only a few pixels with several randomly shifted interferograms are sufficient for accurate extraction of phase information. This method has reduced the dependence of reference phase, and can obtain phase-shifting amount directly without using complex revised algorithm for correcting phase-shifting nonlinear errors.     

Effect of intensity-correlated error due to quantization and noise on phase-shifting method

A method to investigate the effect of the interaction of the noise and the quantization on the computed phase in phase-shifting measurement (PSM) is proposed. The statistical relationship between the measured phase and the fringe intensity-correlated error is formulated for the standard N-sample algorithm. The phase error variance is evaluated following a nonlinear stochastic analysis.     

相移相位测量的全息再现算法及测量误差分析

用全息原理和方法研究相移相位测量,得到了N步整周期相移再现物光波复振幅同步叠加函数(N步相移函数),同时提出一种新的相移相位测量误差分析和最大误差估计方法。N步相移干涉图是以理想平行光为参考光的无衍射同轴全息图,将其与对应的相移参考光相乘后求和得到N步相移函数;在理想情况下,这是一种复振幅分离、测量和物光波复振幅函数同步叠加方法,存在误差时计算出的相位是最小二乘方法的最佳期望结果。利用N步相移函数得到的N 1步相移函数,说明非理想N步相移函数是理想N步相移函数与误差函数之和,可以把相位型误差转化为与振幅和强度相对误差同等的误差来对待,降低了相位测量中误差估计的难度,给出了N步相移算法最大误差的估计方法和公式。     

Modified temporal-phase-unwrapping method for measuring in real time the out-of-plane displacements of the tympanic membrane of Mongolian Gerbil

A technique for measuring in real-time continuous out-of-plane displacements of delicate objects is proposed, and demonstrated on the tympanic membrane of Mongolian Gerbil. The technique is based on the combination of two methods: the spatial phase shifting (SPS) and the modified temporal phase unwrapping (TPU). The combination allows to obtain, in several steps, the phase values of the points that undergo out-of-plane displacement as the object is deformed. The technique reduces the frame acquisition time of the standard TPU used in moiré interferometry by a factor of 4, which is important to diminish post-mortem artifacts during in-vitro experiments and to reduce motion artifacts in in-vivo tests. The proposed technique is robust against problems associated with the temporal phase-shifting method, such as nonlinear phase shift and noise. The advantages and disadvantages are discussed.     

Rapid phase extraction method based on the difference image between interferograms with unknown phase-shifting amount

In phase-shifting interferometry, based on the difference image between interferograms with unknown phase-shifting amount, a rapid phase extraction method is proposed. In this method, first, by means of the simple subtraction operation between interferograms captured in one phase-shifting period, a sequence of difference images are generated easily; second, to decrease the random phase-shifting errors induced during the phase-shifting procedure, the summation operation is performed for these difference images; third, based on one time phase recovery operation and one time phase-unwrapping operation, the measured phase with high precision can be obtained rapidly. Moreover, in the proposed method, by means of a simple phase summation operation for all pixel-phases in one interferogram, the corresponding phase-shifting amount of interferogram can be determined conveniently.     

Generalized linear prediction method in phase-shifting interferometry in the presence of noise

The effectiveness of phase-shifting interferometry (PSI) techniques employing piezoelectric device PZT in the estimation of phase depends largely on the accuracy with which the phase shifts are imparted to the device and the noise influencing the measurement. Several effective algorithms have been proposed to compute the phase shifts imparted to the device and subsequently obtain the phase using least-squares estimation technique. In this paper, we propose a generalized approach, which accurately estimates the phase shifts in the presence of noise. The method is based on the idea of linear prediction and explores the fact that sampling more data frames yields a reliable phase step estimate in a least-squares sense. We also compare our method with a commonly used generalized phase-shifting method based on histogram analysis and show that our proposed approach is highly effective. We also present simulation and experimental validations of our proposed method.     

Phase-shifting mask design for interference exposure of chirp blazed grating

An interference exposure system using a phase-shifting mask was proposed for generating periodic sawtooth optical intensity profile with chirp in period. Multiple space-harmonic waves are launched from the mask by diffraction, and the sawtooth optical intensity is formed by interference based on Fourier synthesis after propagation through an air gap. The design concept merely utilizing positive diffraction-order harmonics with tilt illumination of an exposure light and a design algorithm based on time-reversed configuration for calculating required phase shift amount were proposed and discussed. A phase-shifting mask for a blazed grating of very high chirp rate was designed to demonstrate its potential and feasibility. The grating period changes from 2.2 to 3.1 μm gradually within 0.3 mm length. Sawtooth-like optical intensity profile was confirmed by theoretical simulation as well as preliminary experimental results.     

基于光学涡旋相移技术的离面位移测量

设计了基于光学涡旋相移技术的离面位移测量实验方案,实现了电子散斑干涉中相移的数字控制.该方法利用输入液晶空间光调制器中的叉形光栅产生涡旋光束,通过涡旋光束绕轴的旋转产生相移;同时,产生的涡旋光束又作为参考光与物光干涉.实验中,在物体发生离面位移前后依次输入四幅叉形光栅,产生相移步长为π/2的涡旋光束,利用CCD获得涡旋光与物光的干涉光场,从而获得离面位移场的包裹相位;再通过解包裹,获得物体离面变形的相位变化.光学涡旋相移法可应用于离面位移测量.     

Phase deviation analysis and phase retrieval for partial intensity saturation in phase-shifting projected fringe profilometry

Intensity saturation may take place as a sinusoidal fringe pattern is projected onto an object which has a relative high reflective index on some regions of the surface. If a phase-shifting method is used, the illuming light intensities at the same point of the object may have different values for each projected phase-shifting fringe pattern. Therefore, when the intensity at a point of the object exceeds the saturation level for a certain phase step of projected fringe pattern, other intensities obtained at the same point with different phase steps of projected fringe patterns do not exceed the saturation level. This kind of saturation is defined as partial intensity saturation. In the case of partial intensity saturation, a relative larger phase error is introduced when a conventional phase reconstruction algorithm is used in the phase-shifting projected fringe profilometry (PSPFP). To get rid of such disadvantage, an improved algorithm for phase reconstruction is proposed in this paper. By using the new algorithm, the effect of partial intensity saturation for the phase reconstruction can be decreased greatly and a good quality of reconstructed phase map can be obtained. Phase deviation of the reconstructed phase is also analyzed. Finally, an experimental result with PSPFP is presented to validate the feasibility of the proposed algorithm.     

Automatic evaluation of photoelastic fringe constant by the nonlinear least-squares method

A nonlinear least-squares method was applied to determine the photoelastic fringe constant from isochromatic phase field obtained by digital phase-shifting photoelasticity. To perform automatic evaluation, not only the photoelastic fringe constant but the center coordinates of a disk under diametral compression are also determined simultaneously using the nonlinear least-squares method. A set of over-determined nonlinear equations is obtained to estimate the unknown parameters by an iterative procedure based on Newton–Raphson method. Simulation and experimental results show that the parameters are evaluated accurately from the isochromatic phase field obtained from the phase-shifting photoelasticity. The photoelastic fringe constant and the disk center coordinates can be evaluated easily, simply and automatically by the technique.     

基于主元分析的随机相移算法

将统计理论中的主元分析应用到干涉条纹分析中,提出一种基于主元分析的随机相移算法。该算法先将多帧条纹数据分解成不相关的主元成分,然后从主元成分中提取2维相位信息。数值模拟结果表明:该算法不需要迭代运算就能从4帧完全随机相移干涉图中有效提取相位信息,运算时间少,精度高。实验结果表明主元分析相移算法比现有迭代相移算法更适合于随机相移干涉图分析。     

基于波长移相剪切干涉的准直波前重构技术

为了实现干涉仪出射准直波前的重构,提出了基于波长调谐移相的横向剪切干涉技术。干涉仪出射波前分别经楔板的前后表面反射,通过角锥棱镜返回后在干涉仪CCD上形成剪切干涉条纹。采用波长移相方法提取剪切干涉条纹的相位信息从而实现准直波前重构。分析相对剪切比对波面重构精度的影响,推导相对剪切比和其影响因素间的关系公式,给出波长移相中光程差常数分量的估算方法。测量干涉仪的三组出射波前,波前的峰谷值分别为3.22λ、2.10λ、0.83λ。该方法简化了传统测量干涉仪准直波前的横向剪切干涉装置,提高了测量精度,特别适合于测量波长移相干涉仪的出射波前。     

四步相移数字全息中随机误差的抑制方法

提出一种在四步相移数字全息中消除随机相移误差及CCD噪声的方法。通过多次重复四步相移数字全息的实验过程,使得每一步的相移都有多幅全息图像与之对应,把每一步对应的多幅全息图像进行数学平均就会得到4幅全息均值图像。理论分析表明,使用这4幅全息均值图像进行数字全息的重建,可以得到精确的物光波的分布,有效地消除随机相移误差及CCD噪声的影响。计算机模拟结果表明:该方法有助于提高数字全息重建光场质量。     

Analysis of phase distortion in phase-shifted fringe projection

This paper describes the analysis of phase distortion in phase-shifted fringe projection method. A phase distortion occurs when the phase shifting technique is applied to extract the phase values from projected fringe patterns in surface contouring. The phase distortion will induce measurement errors especially in the measurement of micro-components. The cause of such phase distortion is investigated and the influence of phase distortion on the measurement of micro-components is discussed. To eliminate the phase distortion, a continuous wavelet transform (CWT) is employed to extract phase values from object surface modulated fringe patterns. Principle of the proposed CWT phase extraction method is described and experiments are conducted to verify the proposed method. It is shown that by the use of CWT phase extraction method phase distortion induced in conventional phase-shifting technique can be completely eliminated.