Discuss the measuring range of dual-frequency wavelet transform profilometry

We discuss the structure condition and sampling condition of wavelet transform profilometry (WTP) based on dual-frequency fringe pattern in this work. In the mentioned method a grating fringe with dual-frequency components is projected onto an object. And two wavelet ridge lines can be extracted by means of wavelet analysis, from which we can calculate two groups of wrapped phase information. Afterwards the retrieved phase with higher precision can be obtained through phase unwrapping process. However, it should be noted that the spectral aliasing of the deformed fringe pattern must be avoided in order to restore the correct phase information. And the two fringe carrier frequencies have to obey some rules as well. In this paper, the structure condition and sampling condition of the proposed method is deduced from the point of view of frequency analysis. It is proven that there would be no frequency overlapping in the deformed fringe pattern only when both of the two conditions mentioned previously are fulfilled. The results of computer simulations and experiments verify the validity of our theory.     

小波变换轮廓术的测量范围研究

利用小波“脊”处的小波系数来提取变形条纹中的相位信息可以在很大程度上抑制条纹图中有用的基频分量与零频和其它谐波频率分量的混叠,弥补了傅里叶变换轮廓术的不足。从离散信号频域分析角度,推导了变形条纹小波变换的频谱描述形式,讨论了其测量范围,包括结构条件和抽样条件。结果表明,只有在无周期内瞬时频谱混叠,即任意位置处物体瞬时高度变化满足h/xx=b<1/3条件时,和不存在抽样引起的周期间瞬时频谱混叠的抽样条件下(即一个周期内的抽样点数m≥4时),小波变换轮廓术才能正确恢复被测物体的三维面型。计算机模拟和实验验证了该结论。     

单幅条纹图相位解调的小波分析方法

相位解调是条纹相位分析的关键问题.本文提出一种应用小波频率估计联合频率导数对变形条纹进行瞬时频率分析,从中提取参考基频,从而依靠单一变形条纹实现相位解调的方法.首先,理论上证明了当变形条纹瞬时频率空间导数等于零,该空间点的瞬时频率等于参考基频频率;其次,引入Gabor小波提取变形条纹的瞬时频率空间分布,利用变形条纹瞬时频率的空间导数分布识别提取参考基频,从而实现相位解调.利用该方法进行了三维形貌测量的实验,结果表明该方法在实现相位解调中效果良好.     

单幅条纹图相位解调的小波分析方法

相位解调是条纹相位分析的关键问题.本文提出一种应用小波频率估计联合频率导数对变形条纹进行瞬时频率分析,从中提取参考基频,从而依靠单一变形条纹实现相位解调的方法.首先,理论上证明了当变形条纹瞬时频率空间导数等于零,该空间点的瞬时频率等于参考基频频率;其次,引入Gabor小波提取变形条纹的瞬时频率空间分布,利用变形条纹瞬时频率的空间导数分布识别提取参考基频,从而实现相位解调.利用该方法进行了三维形貌测量的实验,结果表明该方法在实现相位解调中效果良好.     

A new wavelet transform for reliability-guided phase unwrapping of optical fringe patterns

A new daughter wavelet definition for reliability-guided phase unwrapping of the optical fringe pattern is introduced. The new daughter wavelet definition is given by normalized mother wavelet with a new factor including a Gaussian function. The modulus of the wavelet transform coefficients, obtained by using daughter wavelet under this new daughter wavelet definition, includes not only modulation information but also local frequency information of the deformed fringe pattern. Therefore, it can be treated as a good parameter which represents the reliability of the reconstructed phase data. Mathematical demonstration of this parameter is given. Both the computer simulation and experiment reveal the validity of the proposed method.     

小波相位分析测量成像径向畸变

为了测量光学成像像面各个像素的径向畸变大小,提出将小波变换载频条纹相位分析应用于径向畸变测量。采用正弦载波条纹作为测量模板,把径向畸变转化为径向调制相位。应用条纹相位分析导出径向调制相位和径向畸变的转化关系。采用小波频率估计和相位估计提取变形条纹的相位,由于变形条纹中心点是零畸变,中心点的瞬时频率和相位可以计算参考条纹的基频相位。两种基频相位之差就是与所有像素径向位置畸变分布对应的三维调制相位——称为径向畸变分布。利用校正公式和立方卷积插值算法对彩色畸变图像进行校正,给出详细的理论分析和实验结果。     

Shifted-phase calibration for a 3-D shape measurement system based on orthogonal composite grating projection

This paper proposes a novel method for reducing measurement error caused by spectrum overlapping in orthogonal-composite-grating-based 3-D measurement method. For 3-D measurement systems based on orthogonal composite grating projection, spectrum overlapping causes phase of each deformed phase-shifting fringe changed differently, which violates the principle that the shifted phases between adjacent deformed fringes must be equivalent to 2π/3, and therefore results in phase measurement error. The proposed shifted-phase calibration method is based on that phase variation of each deformed fringe is independent of height and reflectivity of the measured object. Three composite gratings are projected on the reference plane, and each carrier channel includes three phase-shifting gratings needed in phase measuring profilometry (PMP). Because the adjacent phase-shifting fringes demodulated from the same carrier channel have the phase difference of 2π/3, we can respectively calculate the reference plane's phases of three carrier channels by the phase algorithm of PMP method, and the shifted phases between them are obtained. When an object is measured, the shifted phases between deformed phase-shifting fringes can be calibrated. A new 3-D measurement mathematical model is set to reconstruct object. Our experiments prove that the proposed method can effectively restrain the effect of spectrum overlapping and improve measurement accuracy almost one times.     

Analysis on fringe pattern demodulation by use of 2-D CWT

We discuss the wavelet transform profilometry based on the continuous wavelet transform technique as viewed from frequency analysis. We deduce the expression of one-dimensional (1-D) and two-dimensional (2-D) wavelet transform in frequency domain and analyze their characteristics in the application of demodulating the fringe patterns. We also compare 1-D CWT and 2-D CWT in demodulating the oblique fringe patterns with dual carrier frequency components. When oblique fringe patterns are processed, the direction normal to the grating line and x axis or y axis is not identical. By 1-D CWT, in which wavelet transform is carried out row by row, we cannot obtain the most similarity between local signal and the wavelet functions with different dilation values. While a fan 2-D continuous wavelet transformation can deal with the fringe pattern as a 2-D unit as well as has multi-directions, its advantage is that it can be used to exact the information in the spatial direction. However, its spatial localization ability is not very good, which leads that it is not suitable for demodulating the fringe patterns with high phase variation. Computer simulations and experiments have verified our analysis.     

BP neural network applied to 3D object measurement based on fringe pattern projection

In the fringe projection profilometry, the traditional triangle method, such as Fourier-transform profilometry (FTP), is difficult to recover the stepped shape object from the deformed fringe pattern. In order to solve this problem, the neural network is introduced to deal with this kind of fringe patterns and gain the three-dimensional (3D) information of the measured object. By training the network, the relationship between the deformed fringe pattern and the height of the object can be obtained, and thereby the height of the object can be obtained. Furthermore, the object can be reconstructed perfectly without knowing the optical parameters of the experiment system. An obvious merit of this network method is that it can recover the 3D object in a short time and only need one deformed fringe pattern. Computer simulations and experiment validate the feasibility of the method.     

Comparison of results of interferometric phase extraction algorithms for three-dimensional flow-field tomography

A comparison of the results of interferometric phase information retrieval and subsequent tomographic reconstruction from deformed wavefront projections by phase unwrapping is presented. In interferogram processing, conventional fringe tracking and Fourier transform methods have been utilized for comparison. With these methods by injecting carrier fringes, the projection data of any axial cross section can be extracted in all projection directions to implement three-dimensional tomographic reconstruction. The results of experiments of a simulated temperature field prove that the phase extraction based on the Fourier transform method produces tomographic reconstruction much superior to the conventional fringe tracking technique.     

On window size selection in the windowed Fourier ridges algorithm

The windowed Fourier ridges (WFR) algorithm can be used to extract the frequency and phase information from an exponential or carrier fringe pattern. The selection of the window size for this algorithm is investigated in this paper. For exponential phase fringe patterns, the window size does not need to be adjusted according to the signal frequency. Hence the scaling strategy of the wavelet transform is not necessary. Instead, the window size should be selected according to the balance between the linear phase approximation error and the noise level. For carrier fringe patterns, the influence of the conjugate component should also be considered. However it can be ignored when the lowest frequency of the fringe pattern is higher than 1.5/σ (σ is used to represent the window size).     

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.     

On-line phase measuring profilometry based on phase matching

On-line phase measuring profilometry based on phase matching is proposed. While just one fixed sinusoidal fringe is projected on a measured object moving with the pipeline, deformed patterns modulated by the object moving at the same distance moment are captured synchronously by the CCD camera. The phase information of the object in those captured deformed patterns can be predicted using FTP method to assist the pixel matching so as to realize the point-to-point correspondence of the object in the captured deformed patterns. Meanwhile, the equivalent phase-shifting deformed patterns can be extracted. So the three dimensional shape of the object can be reconstructed successfully with an equal phase-shifting algorithm. Numerical simulation and experiments show feasibility and validity of the proposed method.     

Neural network applied to reconstruction of complex objects based on fringe projection

The neural network has been introduced into the reconstruction of the complex object based on fringe projection. In this method, the neural network with powerful property of approximation is used to get the continuous approximate function of a discrete fringe pattern captured by an image frame grabber. The depth-related phase of the measured object modulated into the fringe pattern can be demodulated by dealing with the approximate function. Compared with the Fourier transform profilometry (FTP), in the network method, one deformed fringe pattern is needed to reconstruct the tested object, and a high spatial resolution is maintained for no filtering process. Therefore, this method performs better than FTP in the measurement of the complex object. Moreover, the network method is capable of demodulating more depth-related phase even in the case that the local shadow exists in the fringe pattern. Computer simulations and experiments validate the feasibility of this method.     

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.     

傅里叶变换轮廓术中相位失真的预矫正方法

介绍了一种基于傅里叶变换轮廓术的三维面形测量系统中相位失真的预矫正方法。由于投影系统和成像系统的空间三角位置关系、投影仪的发散照明和两套系统蕴含的光学畸变,投影一幅相位与空间坐标成理想线性关系的标准正弦光场,拍摄到的条纹相位和空间坐标不再呈线性分布,引起相位失真,甚至会影响系统测量精度。该方法借鉴反向条纹投影的思想,计算拍摄光场的非线性相位分布与理想的线性相位分布之间的关系,预先矫正,反算出一个新的待投影光场。实验结果表明这种方法能有效地减小该类相位失真所导致的测量误差,获得了更好的测量结果．     

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.     

基于傅里叶变换解相法的像增强管多参数测量

 提出一种基于傅里叶变换解相法的像增强管成像质量多参数同时测量的新方法。将一正弦光栅图输入到像增强管,如果像增强管成像质量存在问题,将使图像输出发生改变,形成一变形条纹,采用傅里叶变换解相法对输出的变形条纹进行解相,可同时获得放大率、放大率不均匀度、枕形畸变、桶形畸变、蛇形畸变、剪切畸变等相关信息,对测量系统进行标定,即可实现对上述参数的同时测量。实验测得中心放大率与设定放大率的差值到了10-3量级,测得的畸变和畸变真值两者差异在0.4 pixel以内,结果证明了方法的有效性。     

小波相位解调轮廓术

针对傅里叶变换轮廓术因混频难以准确提取基频的问题,可将小波变换用于三维形貌相位直接解调。对其原理进行了研究,通过编程实现了小波相位解调。取Morlet复小波作为小波母函数,对调制栅线图逐行进行连续小波变换,从各位置的沿尺度方向的小波变换系数幅值的极值中可直接求取对应的相位数据。由于小波函数具有空域_频率两域的局部化特性,因此它对变形栅具有很强的自适应能力。为验证新方法,对其进行了仿真分析,同时还对石膏半球模型和化妆品瓶进行了实际测量,并分别用傅里叶变换和小波变换进行了处理。结果表明,新方法有效地克服了混频的问题,改善了相位解调效果,提高了测量精度,特别适合于复杂物体的形貌测量。     

Study on Fourier transforms profilometry based on bi-color projecting

Projecting a bicolor sinusoidal fringe pattern consisting of two interlaced RGB format base color fringe patterns with π phase difference onto an object thought digital light projector, we can capture a deformed color pattern by color digital camera, then decode two individual sinusoidal fringe patterns with π phase difference by color-separating technique. Accessing these two fringe patterns, not only are zero-order spectra eliminated, but mask function is also built to mark valid unwrapping area in FTP, automatically.Moreover, because the wrapped phase just inside the valid areas is needed unwrapping, we can mark these areas with mask function, which avoids the error transferring resulting from unwrapping the invalid areas and shortens the unwrapping time. Furthermore, in Fourier transform processing, the full-field deformed fringe pattern generally needed to guarantee measurement precision can be formed by expanding non-full-field fringe pattern captured using the mask function.