Phase-shifting interferometry in the presence of nonlinear phase steps, harmonics, and noise

A phase-shifting piezo device commonly employed in phase-shifting interferometry exhibits a nonlinear response to applied voltage. Hence, a method for estimation of phase distribution in the presence of nonlinear phase steps is presented. The proposed method compensates for the harmonics present in the intensity fringe, allows the use of arbitrary phase-step values between 0 and tau rad, and does not impose constraints on the selection of particular phase-step values for minimizing nonlinearity and compensating for the harmonics. The comparison of the proposed method with other well-known benchmarking algorithms shows that our method is highly efficient and also works well in the presence of noise.     

A hybrid strategy to control uncertain nonlinear chaotic system

In this paper, a new method, based on firefly algorithm(FA) and extreme learning machine(ELM), is proposed to control chaos in nonlinear system. ELM is an efficient predicted and classified tool, and can match and fit nonlinear systems efficiently. Hence, mathematical model of uncertain nonlinear system is obtained indirectly. For higher fitting accuracy, a novel swarm intelligence algorithm FA is drawn in our proposed way. The main advantage is that our proposed method can remove the limitation that mathematical model must be known clearly and can be applied to unknown nonlinear chaotic system.     

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.     

阴影叠栅相移非线性误差补偿算法研究

相移阴影叠栅干涉场的相位(高度)存在非线性关系,而传统的相移阴影叠栅技术往往忽略了相位与高度的非线性关系,从而在测量系统中引入测量误差。对此提出了一种基于迭代相位解调自调算法相移阴影叠栅技术,该方法利用最小二乘技术获得相移量估算值,利用该估算值通过迭代算法消除相移阴影叠栅的全场相位误差,从而得到正确的相位分布。模拟计算表明该方法可以有效解决相移不均产生的相位测量误差问题,且可实现光栅移动量的精确估算,其误差不超过3.4%。对比实验进一步说明了所提出方法的正确性和优越性。     

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.     

High-resolution frequency estimation technique for recovering phase distribution in interferometers

An integral approach to phase measurement is presented. First, the use of a high-resolution technique for the pixelwise detection of phase steps is proposed. Next, the robustness of the algorithm that is developed is improved by incorporation of a denoising procedure during spectral estimation. The pixelwise knowledge of phase steps is then applied to the Vandermonde system of equations for retrieval of phase values at each pixel point. Conceptually, our proposal involves the design of an annihilating filter that has zeros at the frequencies associated with the polynomial that describes the fringe intensity. The parametric estimation of this annihilating filter yields the desired spectral information embedded in the signal, which in our case represents the phase steps. The proposed method offers the advantage of extracting the interference phase of nonsinusoidal waveforms in the presence of miscalibration error of the piezoelectric transducer. In addition, in contrast to previously reported methods, this method does not require the application of selective phase steps between data frames for nonsinusoidal waveforms.     

The fluctuating phase error analysis in the digital grating phase-shifting profilometry

The nonlinear response of the experimental system and the saturation of fringe patterns can induce the fluctuating phase error in the projection grating phase-shifting profilometry. Two major factors of the fluctuating phase error are discussed by simulation. The fluctuating phase error caused by the nonlinear response of the system is four times the frequency of the fringe pattern when the conventional four-frame phase extracting algorithm is used. However, such error can be decreased by five-frame algorithm. On the other hand, the fluctuating phase error caused by the fringe saturation is five times the frequency of the fringe pattern by using conventional five-frame phase extracting algorithm. A novel phase recovering algorithm is used to decrease the phase error caused by the saturation. Furthermore, the applicability range of the proposed phase recovering algorithm is analyzed by simulation and experiments with different saturation degree of the fringe pattern and nonlinearity of the measurement system.     

Phase shifting interferometry using a robust parameter estimation method

The paper proposes a maximum-likelihood (ML) method based on spectral estimation theory for the estimation of phase distribution in interferometry in the presence of nonsinusoidal waveforms, noise, and the miscalibration of the piezoelectric device. The proposed method also allows the use of arbitrary phase steps. ML estimators are asymptotically efficient for large number of data samples. The method is complemented well by the incorporation of global search algorithm known as Probabilistic Global Search Lausanne for minimizing the ML function. The performance of the proposed method is studied in the presence of noise.     

Polarization shifting interferometric profilometer

A phase sensitive Michelson interferometer based on interference microscope configuration with a polarization adjustment approach is proposed to determine the two-dimensional (2-D) surface profile of optical grating with real time capability. In the proposed method, nonlinear behavior of a PZT phase shifter is avoided by use of polarization stepping and a phase map is developed with the four-bucket algorithm. The phase map is unwrapped to give the true surface profile of the sample. A close agreement of measurements is found between the measured result determined by the proposed method and that determined by an atomic force microscope (AFM). We also analyzed the estimated uncertainty of measurement in the nanometer range for the random fluctuations of the experimental parameters.     

Three-dimensional image authentication using binarized images in double random phase integral imaging

We proposed a three-dimensional(3D) image authentication method using binarized phase images in double random phase integral imaging(InI). Two-dimensional(2D) element images obtained from InI are encoded using a double random phase encryption(DRPE) algorithm. Only part of the phase information is used in the proposed method rather than using all of the amplitude and phase information, which can make the final data sparse and beneficial to data compression, storage, and transmission. Experimental results verified the method and successfully proved the developed 3D authentication process using a nonlinear cross correlation method.     

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.     

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%。     

Fringe projection profilometry with nonparallel illumination: a least-squares approach

Under a nonparallel illumination condition, fringe patterns projected on an object have unequal fringe spacing that would introduce a nonlinear carrier phase component. This Letter describes a nonlinear carrier removal technique based on a least-squares approach. In contrast with conventional methods, the proposed algorithm would not magnify phase measurement uncertainty, nor does it require direct estimation of system geometrical parameters. The theoretical expression of the carrier phase function on the reference is derived and expanded in a power series. The unknown coefficients in the series are determined by a least-squares method. By subtracting the calculated carrier phase function from the unwrapped phase map, the phase distribution of the object profile is obtained.     

一种混沌海杂波背景下的微弱信号检测方法

基于复杂非线性系统的相空间重构理论,提出了一种基于遗传算法的支持向量机预测方法.利用改进的自相关法和饱和关联维数法确定混沌信号的时间延迟和嵌入维,从而实现相空间重构.通过遗传算法优化支持向量机中的惩罚系数和核函数参数,并结合支持向量机建立混沌序列的单步预测模型,从预测误差中检测出淹没在混沌背景中的微弱信号(包括瞬态信号和周期信号).以Lorenz系统和加拿大McMaster大学利用IPIX雷达实测得到的海杂波数据作为混沌背景噪声进行仿真实验,结果表明该方法能够有效地从混沌背景噪声中检测出微弱目标信号,所得的均方根误差为0.00049521(信噪比为-89.7704 dB),这比传统支持向量机方法的均方根误差(0.049,信噪比为-54.60 dB)降低了两个数量级.     

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.     

利用颜色编码辅助计算绝对相位的傅里叶轮廓术

针对传统去卷积算法时间需求的弊端,提出一种新的使用颜色编码辅助的绝对相位并行计算方法。该算法采用对光栅数目需求最少的傅里叶变换轮廓术（FTP）做为卷积相位求取的方法;颜色编码光栅被用来标识轮廓的序数。直接使用FTP计算出的卷积相位以及从彩色光栅中获得的轮廓序数,即可方便求出当前像素的绝对相位值;同时只用一副图像标识轮廓序数也比其他轮廓序数标识方法简单。本方法由于使用绝对相位计算方法,局部相位误差不会扩展。实验结果也证明了此算法对于多个分离物体以及复杂物体的有效性。     

基于旋转相位编码与照明光束匹配的叠层衍射成像算法研究

在传统的叠层成像算法中引入一种旋转相位编码与照明光束相匹配.采用分块均匀分布的旋转相位调制器对衍射光进行调制,在探测面获得了相应的衍射图样.这种照明光束与相位匹配的方式有效地增加了解的约束条件,具有收敛速度快、抗噪声及抗位置偏差能力强等优点.与随机相位调制相比,本算法所引入的旋转相位调制器构造简单、使用便捷,因而在实时显微成像、超分辨成像等领域将具有更高的实用价值.     

材料热物性与热源强度辨识的改进遗传算法

建立二维非线性导热逆问题(IHCP)的数学模型.通过对基本遗传算法逐步改进得到三种不同改进阶段的遗传算法,分别用于反演导热问题中材料的导热系数及其内部热源强度,并比较遗传算法在各改进阶段用于求解导热逆问题时的收敛速度与求解精度,寻求一种使导热逆问题求解效率与计算精度更高的遗传算法改进策略.结果表明:提出的遗传算法改进策略达到了预期目的.     

Adaptive higher-order split-step Fourier algorithm for simulating lightwave propagation in optical fiber

In this paper, we propose an adaptive step-size control algorithm for solving nonlinear Schrödinger’s equations. The proposed algorithm has a fourth-order local accuracy and is a system-independent rule for adjusting the step sizes. This algorithm has two potential advantages, an automatic step adjustment mechanism and higher-order accuracy. A test example shows that, by comparing to the fixed step-size method and the local-error method, our method decreases the computational time by about 10 and 70 times, respectively. The performance of the proposed method is validated and compared to commonly used step-size selection methods by simulating the evolution of a third-order soliton, the collision of a fundamental soliton pair, and the supercontinuum generation. Numerical simulations show that the proposed method can increase the computational efficiency by more than one and two orders of magnitude in comparison with the symmetrized split-step Fourier method. In addition, the computational efficiency is improved with the increase of the accuracy of solutions.