Error-compensating algorithms in phase-shifting interferometry: a comparison by error analysis

In this paper, two families of phase-shifting algorithms with π/2 phase steps are studied. In family I, three new algorithms are derived by using the averaging technique based on the Surrel six-sample algorithm with phase shifts of π/2. Family II includes four well-known algorithms derived by the averaging technique based on the conventional four-sample algorithm with π/2 phase steps. A polynomial model of phase-shift errors used to describe general expressions for calculation of the correct object phase via the Fourier spectra analysing method as a function of the harmonic order in the fringe signal is presented. The error-compensating properties of the algorithms in families I and II are investigated by the Fourier spectra analysing method. It is found that the averaging technique, when used in any of the algorithm with π/2 phase steps, can improve the phase-shifting algorithm property: it is insensitive to phase-shift error when the fringe signal contains the first harmonic, but it can't be used to enhance the phase-shifting algorithm properties when the fringe signal contains higher order harmonics (n2). P–V (peak–valley) phase errors are calculated by the computer simulation and tables and plots are presented, from which the algorithms in families I and II are compared. It is shown that the algorithms in family I are more insensitive to phase-shift errors when the fringe signal contains the second harmonic and the algorithms in family II are more insensitive to phase-shift errors when the fringe signal is a sinusoidal waveform.     

一种基于移相误差估计的5步移相算法

移相误差是用移相法进行相位测量的主要误差。本文提出一种 5步移相算法 ,分两步进行相位计算 ,首先估计实际步进移相的线性移相误差 ,然后再利用此移相误差估计值计算相位分布。移相误差估计公式和相位计算公式简洁 ,算法简单易行 ,对线性移相误差和二次谐波的敏感度低 ,可基本消除线性移相误差对解调相位的影响。对本文提出的算法进行了仿真研究 ,同时给出了 Hariharan 5步算法、Surrel 6步最小算法的仿真结果。结果表明 :本算法明显优于以上两种算法 ,可基本消除线性移相误差引起的相位偏移。本算法适用于作等步移相的相位测量或移相的标定。     

Identification of nonlinear recording error in phase shifting interferometry

The phase shifting method for quantitative fringe pattern analysis provides high accuracy if stringent requirements on the component interferogram recording are met. In the paper the issue of detection and identification of error sources in the two-beam interferogram phase shifting experiment is discussed. The phase shift angle histogram and lattice-site representation are applied for that purpose. Special attention is paid to possible nonlinear recording of component interferograms in the presence of linear and nonlinear phase step errors. Four and five step phase shifting algorithms are considered. The superiority of the lattice-site representation is shown. In the case of phase steps equal to π/2, however, the lattice-site representation of shift angles for five frame algorithm does not allow to detect recording nonlinearity. The four frame counterpart shows to be very helpful in this respect. Its properties related to the fringe pattern profile under study, including a defocused Ronchi grating, are discussed.     

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.     

相移阴影莫尔条纹正交化解调技术

提出一种基于克莱姆正则化分析法的三帧自标定相移阴影莫尔三维轮廓技术.该技术首先采用移动光栅的方法获得相移条纹图,然后通过不同帧相移条纹图相减去除条纹图背景,进而结合克莱姆正交化法和最小二乘法,发展了一种相位解调方法,提取了测量相位.以五步Harlharan算法为参考,用不同算法对同一物体表面进行测量.结果表明,相对于典型的三步相移法和主量分析方法,提出的方法测量得到的相位误差最小(0.5rad),且简化了测量过程.     

A comparison of phase shifting and fourier methods in the analysis of discontinuous fringe patterns

Phase Shifting Interferometry is a highly accurate data acquisition technique that efficiently utilizes several frames of information for each measurement. In this work, the advantages of phase shifting have been applied to a conventional moiré interferometer, yielding a system capable of recording phase shifted fringe patterns for both in-plane displacement components. Using this method, the phase of a wavefront of interest can be determined at each detector location, so that the resolution of the phase measurements is limited primarily by the detector discrimination and geometry. Unlike traditional Fourier fringe analysis, the noise rejection of phase shift processing algorithms does not degrade image fidelity in the presence of edges and discontinuities. A general discussion of both the phase shifting technique and the Fourier fringe analysis method is included to provide insight into the problems of processing discontinuous fringe patterns.     

基于梯形与正弦相移算法的非线性误差校正北大核心CSCD

相移轮廓术是一种广泛使用的光学三维测量方法,其精度不仅受相位展开算法本身的影响,也受测量系统中投影仪和摄像机的非线性影响。理论上,投射更多的相移条纹可减弱非线性误差的影响,但是增加了测量时间。为了提高误差校正的效率,提出了一种基于梯形正弦相移的测量方法。该方法需要两组改进的梯形相移条纹和一幅正弦条纹。梯形条纹提供图像强度信息和条纹级次信息,图像强度信息用来求取系统的非线性响应曲线,进一步消除系统的非线性。正弦条纹经过希尔伯特变换可求得额外的条纹图像,用来计算截断相位信息。经过校正的截断相位信息,可进一步获取精度较高的三维信息。相较于先前的梯形与正弦误差校正方法,该方法的测量效率提高了28%。     

用于全场光学相干层析成像的无色散相移器研究

提出了基于旋转λ/2波片无色散相移器的理论模型,和实施该相移器功能的全场光学相干层析成像(Optical coherence tomography,OCT)系统,为实现快速、高分辨光学相干层析成像提供了一种有效方法。该相移器能在宽光谱范围内无色散地获得八倍于λ/2波片旋转角的相移量,避免了利用单色光相移算法提取信号时存在的系统误差。针对修正Carré算法和三步相移算法所要求的不同相移量,在240 nm波长范围内,对相移器的性能进行了数值分析。结果表明,在给定光谱范围内,相移量越大,相移误差的绝对值也越大,幅值比变化范围也越大。用于实施无色散相移器功能的全场光学相干层析成像系统结构,与现有结构相比,具有一些有益的特点。     

Multi-step phase-shifting algorithms insensitive to linear phase shift errors

We describe and analyse a group of multi-step phase calculation algorithms for evaluation of interferometric measurements using the phase-shifting technique. Phase-shifting algorithms are proposed, with a constant but arbitrary phase shift between captured frames of the irradiance of the interference field. The algorithms are similarly derived as so called Carré algorithm. The phase evaluation process is not dependent on linear phase shift errors. An advantage of the described algorithms is their ability to determine the phase shift value at every point of the detector plane. Moreover, a complex error analysis of proposed algorithms is performed and the algorithms are compared to several common error compensating phase stepping algorithms.     

一种基于傅里叶变换的分析载波条纹的新方法

针对传统傅里叶变换法处理光载波干涉条纹图时会有边缘效应产生的问题,提出了一种基于傅里叶变换法的外推延拓方法,并从理论上进行了数学推导。为了验证这种方法的正确性,分别对一维数字信号和二维空间载波条纹图进行了数值模拟,进一步分析了误差产生的原因,并与传统的傅里叶变换法对比。结果表明该法可以有效抑制传统傅里叶变换法处理光载波干涉条纹图时边缘效应所造成的较大误差,在基于空间域相位调制技术的波面干涉测量中,对空间载波条纹图进行处理,可以使相位的计算精度达到3.3 mrad。     

Blind phase error suppression for color-encoded digital fringe projection profilometry

Color-encoded digital fringe projection profilometry (CDFPP) has the advantage of fast speed, non-contact and full-field testing. It is one of the most important dynamic three-dimensional (3D) profile measurement techniques. However, due to factors such as color cross-talk and gamma distortion of electro-optical devices, phase errors arise when conventional phase-shifting algorithms with fixed phase shift values are utilized to retrieve phases. In this paper, a simple and effective blind phase error suppression approach based on isotropic n-dimensional fringe pattern normalization (INFPN) and carrier squeezing interferometry (CSI) is proposed. It does not require pre-calibration for the gamma and color-coupling coefficients or the phase shift values. Simulation and experimental works show that our proposed approach is able to effectively suppress phase errors and achieve accurate measurement results in CDFPP.     

Characteristic polynomial theory of two-stage phase shifting algorithms

Two-stage phase shifting algorithms make possible to directly recover the sum or the difference of the optical phase of two different fringe patterns. These algorithms can be built by combining the known phase shifting algorithms in a non-linear way. In this work, we associate a two-dimensional characteristic polynomial to each two-stage phase shifting algorithm. This enables us to qualitatively compare their behaviour against the main systematic error sources, by means of an analysis protocol like that used for phase shifting algorithms. We show that this tool allows to understand the propagation of properties from precursor phase shifting algorithms to new evaluation algorithms built from them. As an experimental application, a wavefront distortion evaluation in differential phase-shifting interferometry is presented.     

A novel three-dimensional shape measurement method based on a look-up table

Fringe projection profilometry is widely used for three-dimensional shape measurement. In an oblique-angle projection, the fringe cycle is broadened on the reference plane. Phase errors are mainly caused by the nonlinear gamma of the projector and fringe cycle broadening. This study describes a phase error compensation method to eliminate these phase errors. A look-up table that stores phase errors is constructed for phase error compensation. Based on it, a new height equation is proposed. The experimental results show that the proposed method can compensate for the phase errors of the fringe projection profilometry, thereby improving the measurement accuracy significantly.     

Comparative analysis of interferogram noise filtration using wavelet transform and spin filtering algorithms

The aim of this paper is to analyze 2D fringe pattern denoising performed by two chosen methods based on quasi-1D two-arm spin filter and 2D discrete wavelet transform (DWT) signal decomposition and thresholding. The ultimate aim of this comparison is to estimate which algorithm is better suited for high-accuracy measurements by phase shifting interferometry (PSI) with the phase step evaluation using the lattice site approach. The spin filtering method proposed by Yu et al. (1994) was designed to minimize possible fringe blur and distortion. The 2D DWT also presents such features due to a lossless nature of the signal wavelet decomposition. To compare both methods, a special 2D histogram introduced by Gutman and Weber (1998) is used to evaluate intensity errors introduced by each of the presented algorithms.     

Reference phase shift determination in phase shifting interferometry

The phase shifting technique has recently seen application to many types of interferometers, including holographic, speckle and moiré systems for strain analysis. In these applications, close control of the phase shifts may be impossible due to the effect of mechanical vibrations, and this presents difficulties when extracting the phase information, since the reference phase shifts are generally assumed to be known. In this work, several approaches to the problem of determining the reference phase shifts in a perturbing environment are critically evaluated using computer simulations. The analyses provide a general relationship between precision and reference phase shift errors and demonstrate the performance of each method as certain idealised fringe pattern error parameters are varied. A practical example confirms the predictions of the numerical simulations and demonstrates the feasibility of using reference phase shift estimates to improve data reduction procedures.     

叠栅条纹信号细分误差的一种动态补偿方法

缺乏有效的误差补偿方法足制约长光栅测最精度提高的关键原因之一.提出一种动态的误差补偿方法.可以消除由直流漂移、两路信号不等幅和非止交导致的细分误差.其原理是跟踪光栅信号在一个周期上的8个特征值点(正余弦信号的过零点及绝对值交点),从特征值点的幅度值中首先分解出正弦信号的直流漂移误差.对其进行补偿;然后继续跟踪补偿后的信号.从中义能分解出余弦信号的直流漂移误差.再补偿.再跟踪.又能依次分解出不等幅误差和非正交误差.最多只需要3个光栅信号周期,就能对三种误差依次实现补偿.分析了谐波对该方法的影响并提出r改进措施.实验证实了该方法的有效性.     

用双图全息相移干涉法计算与时间有关的位相分布

提出一种双图全息相移干涉法．该方法可以用来计算与时间有关的位相分布，两幅全息干涉条纹图可以同时得到，并且有固定的９０°位相差．虽然两幅条纹图的位相可以随着时间变化，如环境影响，或载荷变化，但他们之间的位相差恒定．该方法利用二维傅氏变换，结合偏振相移法以及分光技术．     

几种任意步距步进相移算法的误差分析与对比

介绍了到目前为止的几种任意步距步进相移算法,并针对相移干涉仪的两种主要误差———移相误差和探测器非线性误差进行了计算仿真,进而比较分析了它们对这这些误差的抑制能力,其结果可为实际应用合理地选择算法提供理论依据。     

相位编码条纹投影轮廓术的相位展开误差校正方法

相位编码法是条纹投影技术中常用的一种相位展开方法。然而,受到随机噪声、图像采样、系统离焦等因素的影响,条纹阶次与包裹相位的对应关系容易被破坏,从而引入相位展开误差。根据相位展开误差的分布特点,提出一种附加二值条纹的相位展开误差校正方法。引入的二值条纹与相位编码条纹的码字存在半周期错位,使得计算出的2组条纹阶次值具有互补性,利用两者的互补性可以有效地消除相位展开误差。仿真和真实实验均表明,该方法能够准确地恢复出被测物体的绝对相位,具有较高的可靠性。