Applications of windowed Fourier fringe analysis in optical measurement: A review

The applications of windowed Fourier fringe analysis in the past decade are reviewed. Because fringe patterns from different optical measurement systems are similar, the reviewed applications are classified according to the functions of the windowed Fourier transform being used in fringe pattern analysis: denosing exponential phase fields, demodulating carrier fringe patterns, getting phase derivatives, and utilizing local properties. From these applications, the windowed Fourier transform is shown to be effective and versatile for fringe pattern analysis.     

On window size selection in the windowed Fourier ridges algorithm: Addendum

The windowed Fourier transform is a useful technique for fringe pattern analysis. It has been shown that the proper selection of the window size is a balance between the linear phase approximation error and the influence of noise [Q. Kemao, On widow size selection in windowed Fourier ridges algorithm. Opt Lasers Eng, 2007, accepted for publication]. Since the fringe intensity and noise level usually vary spatially, the window size should also be spatially adaptive in order to reach a good balance for each pixel of the fringe pattern. This addendum first shows that the windowed Fourier transform with a spatially fixed window size (SFWS) is still practically useful and then discusses the window size competition strategies for the windowed Fourier transform with a spatially adaptive window size (SAWS). The windowed Fourier transform with a SAWS is theoretically better than that with a SFWS but it is also more challenging in use. The windowed Fourier ridges algorithm is used for analysis throughout this paper. This analysis is also applicable to the windowed Fourier filtering algorithm.     

Two-dimensional multiscale windowed Fourier transform based on two-dimensional wavelet transform for fringe pattern demodulation

A two-dimensional multiscale windowed Fourier transform (2D-MWFT), based on two-dimensional Gabor wavelet transform (2D-GWT), for the phase extraction from a spatial fringe pattern in fringe projection profilometry is presented. First, the instantaneous frequencies on x and y direction of the modulated fringe pattern are determined by 2D-GWT, and then the local stationary lengths are obtained. The 2D-MWFT with different two-dimensional Gaussian windows whose width is set according to the local stationary length is preformed for each section of the modulated fringe pattern to achieve multiresolution analysis and phase demodulation. Comparing the result of the phase demodulated by 2D-GWT and two-dimensional windowed Fourier transform (2D-WFT) with that by 2D-MWFT in a numerical simulation, we show that the 2D-MWFT method is superior to these methods, especially for the local non-stationary signal with low frequency. The theory and the results of a simulation and experiment are shown.     

三维测量中一种新的自适应窗口傅里叶相位提取法

针对多尺度窗口傅里叶变换中,窗口尺寸的自适应选取及提取基频时的频谱混叠等问题,提出基于希尔伯特-黄变换(HHT)的自适应窗口傅里叶相位提取法。对变形条纹信号进行HHT后,通过谱分析,自适应确定能够准确描述条纹信号变化情况的瞬时频率及条纹图的背景分量。根据所得的瞬时频率,给出自适应定位条纹信号局部平稳区域的步骤,进而确定窗口尺寸。不需额外计算,可有效去除背景分量以减少基频提取过程中零频频谱的干扰。与现有的用最大脊法确定窗口尺寸的方法相比,本方法不受被测相位必须线性逼近且变化缓慢的前提约束。实验证明本方法有效、可行,且对测量携带陡峭边缘或面形复杂的物体也能进行较为精确有效的测量。     

Method for eliminating zero spectrum in Fourier transform profilometry

Windowed Fourier transform (FT) is introduced into Fourier transform profilometry (FTP) to extract zero component included in deformed fringe pattern without capturing two fringe patterns with π phase difference. First, we extract zero component of deformed fringe image using windowed FT and normalize it, then calculate the normalized Fourier spectra of deformed fringe image, finally subtract the zero component from Fourier spectra of deformed fringe image. We give the theoretical analysis of the proposed method. The experiment is carried out to demonstrate that the method can eliminate zero component and extend the measurable slope of height variation to nearly three times of the conventional FTP.     

An improved windowed Fourier transform for fringe demodulation

A modified algorithm of windowed Fourier transform (WFT) for phase retrieval from electronic speckle-shearing fringe patterns with carriers is proposed. The algorithm is based on the introduction of a fast Fourier transform (FFT) in WFT to reduce computation time for fringe demodulation. Since boundary effects in FFT will influence the accuracy of phase retrieval, the Gerchberg method is employed to extrapolate the fringe pattern at the boundaries to reduce boundary effects. A theoretical analysis of the algorithm is presented. Both simulated and experimental results show that the proposed method has reduced the computation time significantly compared with the convolution method of WFT without sacrificing measurement accuracy.     

窗口傅里叶变换轮廓术中窗口尺度选取的改进

多尺度窗口傅里叶变换法根据条纹信号的瞬时频率梯度来确定信号的局域平稳长度,再由局域平稳长度来控制窗口的尺度,即窗口的尺度和局域平稳长度成正比。使用多尺度窗口傅里叶变换法来使条纹信号的频谱局域化,可以在条纹信号的频率分辨率和空间分辨率之间达到一种较佳的调和。针对多尺度窗口傅里叶变换三维形貌测量技术中局域平稳长度提取算法的不足进行了改进,使窗口尺度的选取更具合理性,对变形光栅基频提取更精确,进一步减小了位相测量的误差。给出了理论分析、计算机模拟以及实验结果。     

Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB environment

In the process of measurements such as optical interferometry and fringe projection, an important stage is fringe pattern analysis. Many advanced fringe analysis algorithms have been proposed including regularized phase tracking (RPT), partial differential equation based methods, wavelet transform, Wigner–Ville distribution, and windowed Fourier transform. However, most of those algorithms are computationally expensive. MATLAB^® is a general algorithm development environment with powerful image processing and other supporting toolboxes. It is also commonly used in photomechanical data analysis. With rapid development of multicore CPU technique, using multicore computer and MATLAB^® is an intuitive and simple way to speed up the algorithms for fringe pattern analysis. The paper introduces two acceleration approaches for fringe pattern processing. The first approach is task parallelism using multicore computer and MATLAB^® parallel computing toolbox. Since some algorithms are embarrassing problems, our first approach makes use of this characteristic to parallelize these algorithms. For this approach, parallelized windowed Fourier filtering (WFF) algorithm serves as an example to show how parallel computing toolbox accelerates the algorithm. Second, data parallelism using multicore computer and MATLAB^® parallel computing toolbox is proposed. A high level parallel wrapping structure is designed, which can be used for speeding up any local processing algorithms. WFF, windowed Fourier ridges (WFR), and median filter are used as examples to illustrate the speedup. At last, the results show that the parallel versions of former sequential algorithm with simple modifications achieve the speedup up to 6.6 times.     

Phase retrieval of speckle fringe pattern with carriers using 2D wavelet transform

A novel two-dimensional (2D) Gabor continuous wavelet transform (CWT) for phase retrieval and fringe filtering of speckle fringe patterns with spatial carriers is proposed. Theoretical analysis of 2D Gabor CWT is presented and results are compared with advanced fan 2D CWT using both the computer simulated and experimental speckle fringe patterns. It is shown that noise reduction by 2D Gabor CWT demonstrates better results than that of the advanced fan 2D CWT. Two-dimensional Gabor CWT is also compared with 2D Fourier transform and results show that 2D Gabor CWT algorithm has better noise immunity.     

Characterizing Young's fringes’ orientation and spacing by Fourier transform and Radon transform

Orientation and spacing are important features of fringe patterns, especially in speckle photography when the displacement information of the object is determined by pointwisely filtering the double-exposure specklegram. The Fourier transform and Radon transform are both simple and effective tools for solving this problem, which extract the information in spectral domain and spatial domain, respectively. A hybrid method combining Radon transform and Fourier transform is also possible. The results can be further improved by enhanced Fourier transform and enhanced Radon transform. The theories, as well as real applications, are given in this paper.     

分数傅里叶变换啁啾滤波

针对常规傅里叶变换所不能解决的啁啾噪声滤除问题,利用Wigner分布函数分析分数傅里叶变换的空域和频域特性,提出在分数傅里叶变换域进行啁啾滤波的方法。并将该方法应用到图像处理中,对分数傅里叶变换滤除一维和二维图像的啁啾噪声进行了计算机仿真,获得了满意的效果,结果表明该方法在图像处理中的有效性。     

Fringe pattern analysis by S-transform

S-transform proposed in 1996 by Stockwell R.G is a simple and popular technique for the time–frequency analysis. It has been introduced in optical three-dimensional shape measurement, recently. In this paper, a study about applications of S-transform in the demodulation of deformed fringe patterns is performed. We focus on discussing not only the S-transform spectrum filtering technique, the S-transform ridge technique and the phase gradient calculation method based on S-transform used in fringe pattern demodulation, but also the phase unwrapping technique. In addition, a generalized S-transform was introduced to analyze fringe patterns, which is helpful to improve the measurement accuracy and flexibility of the method based on S-transform. The reconstruction results based on S-transform were compared with that on wavelet transform and windowed Fourier transform in fringe analysis.     

基于窗口傅里叶变换剪切干涉法波前检测

提出了一种利用二维窗口傅里叶变换从径向剪切干涉条纹中准确得到波前的重建技术。首先对剪切干涉条纹做二维窗口傅里叶变换,设置阈值和频率积分范围后,进行二维窗口傅里叶逆变换,然后对包裹相位做去载频和相位展开处理得到相位差分布,最后使用波前迭代算法从相位差中复原实际波前。模拟计算表明,使用该方法最大相位复原误差为0.82%,均方根值为0.020 9 rad,实验结果验证了该方法的有效性。同时也对窗口傅里叶变换的关键参数,如窗函数的选择、窗口大小的确定以及阈值的选取等进行了简要讨论。与传统傅里叶变换法(FFT)相比,基于窗口傅里叶变换的剪切干涉波前检测法有更高的精度和稳定性,为波前检测提供一种新的处理方法。     

Evaluation of spectral phase in spectrally resolved white-light interferometry: Comparative study of single-frame techniques

Spectral phase in a white-light interferogram contains information about the absolute optical path difference (OPD) in the interferometer. Evaluation of spectral phase is therefore important in applications such as profilometry with white light. In spectrally resolved white-light interferometry (SRWLI) the white-light interferogram is spectrally decomposed by a spectrometer in order to determine this phase. Several single-frame techniques in SRWLI have been proposed for the evaluation of the phase including Fourier transform, Hilbert transform, spatial phase shifting, windowed Fourier transform and wavelet transform. In this paper we present a comparative study of these techniques in this application in relation to the temporal phase-shifting technique which is a multi- frame method. Further, we also propose a modified method to remove the influence of source spectrum modulation in Hilbert transform procedure.     

Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations

Fringe patterns from optical metrology systems need to be demodulated to get the desired parameters. Two-dimensional windowed Fourier transform is chosen for the determination of phase and phase derivatives. Two algorithms, one based on filtering and the other based on similarity measure, are developed. Some applications based on these two algorithms are explored, including strain determination, phase unwrapping, phase-shifter calibration, fault detection, edge detection and fringe segmentation. Various examples are given to demonstrate the ideas. Finally implementations of these algorithms are addressed. Most of the work has appeared in various papers and its originality is not claimed. Instead, this paper gives an overview and more insights of our work on windowed Fourier transform.     

Fractional Gabor transform

A fractional Gabor transform (FRGT) is proposed. This new transform is a generalization of the conventional Gabor transform (GT) based on the Fourier transform to the windowed fractional Fourier transform (FRFT). The FRGT provides analyses of signals in both the real space and the FRFT frequency domain simultaneously. The space-FRFT frequency pattern can be rotated as the fractional order changes. The FRGT has an additional freedom, compared with the conventional GT, i.e., the transform order. The FRGT may offer a useful tool for guiding optimal filter design in the FRFT domain in signal processing.     

基于S变换的改进窗口傅里叶三维测量法

讨论了一维S变换在处理条纹图中的应用,推导了S变换表达式,提出一种基于S变换脊的自适应窗口傅里叶三维测量法。利用S变换对于频率的敏感性以及S变换脊思想,求得任意点的瞬时频率。推导了在相位跳变剧烈区域相位二阶导φ″(b)对S变换脊的影响,并求得去除此影响后更为准确的瞬时频率。然后将瞬时频率倒数作为经典窗口傅里叶变换(WFT)的窗口大小,解出条纹图的精确相位。最后对去除相位二阶导影响前后基于S脊的WFT进行对比仿真;实验验证其与传统基于小波脊的WFT在抗噪声、解相位精度上的优势。     

Windowed Fourier assisted two-dimensional Hilbert transform for fringes phase extraction

In this article, two-dimensional windowed Fourier assisted Hilbert transform for fringes phase extraction has been proposed. The mathematical derivations of the filtered fringes image and its Hilbert transformations are presented using windowed Fourier transform. The proposed method has been verified by calculating the wrapped and unwrapped phase of the experimental moire fringes. In comparison to the conventional fringes phase extraction methods, the proposed method is found to be effective.     

High-speed fringe analysis by using stair-shaped virtual grating demodulation

This article describes a novel fringe analysis technique using stair-shaped virtual grating demodulation algorithm for extracting the phase information from a single fringe pattern. In contrast to the Fourier transform technique and quadrature multiplicative moire method, the new method processes a fringe pattern in the integer signal domain instead of the frequency domain and real-signal domain. The spatial carrier of fringe image is not limited to any particular frequency as long as it fulfills the sampling theorem. The errors analysis of this technique is conducted by computer simulation and experiments.     

基于希尔伯特变换的干涉条纹相位解调新算法

提出了一种新的应用希尔伯特变换解调干涉条纹相位的算法,可以从单幅干涉条纹图中解调出全场相位分布.在实际应用中,常借助傅里叶变换实现希尔伯特变换算法,但是会忽略负频率成分,造成相位信息的丢失.对于相位分布非单调变化的干涉条纹,提出了一种判断函数,用来计算相位信息零频率点的分布.利用相位的零频率点分布构造了一个二元模板,使用该模板对本文提出的两次希尔伯特变换法产生的包裹相位图进行修正.对修正后的包裹相位图进行解包裹处理,可以得到连续的全场相位分布.对该方法用计算机模拟进行了验证.