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.     

Phase retrieval of optical fringe patterns from the ridge of a wavelet transform

A new method for phase retrieval of optical fringe patterns is presented. This method is based on a wavelet transform and is capable of extracting the full 2D phase distribution from a single fringe pattern. An important conclusion that the phase of the optical fringe pattern is equal to the phase of its wavelet transform on the ridge of the wavelet transform is theoretically clarified. The method is compared with the Fourier transform and the integration methods. A numerical simulation and an experimental example of phase retrieval are shown.     

基于结构光投影的二维S变换轮廓术

S变换是一种集合了窗口傅里叶变换和小波变换优点的时频分析技术,目前一维S变换已成功应用于结构光投影的条纹相位解调中。由于二维S变换可以对图像在两个方向上进行时频分析,具有更优于一维S变换的分析和处理能力。为了完善S变换的条纹相位解调理论,将二维S变换方法引入到基于结构光投影的三维光学测量中,研究了二维S变换在条纹相位解调中的原理及应用,给出了详尽的理论分析,并同一维S变换结果进行了比较。模拟和实验都表明,在条纹图解相中,二维S变换比一维S变换提取的相位精度更高,即使在存在较严重噪声污染的情况下也表现出良好的可靠性,体现出二维S变换提取相位的优势。     

Determination of interferometer phase distributions by use of wavelets

A new technique for directly extracting phase gradients from two-dimensional (2-D) interferometer fringe data is presented. One finds the gradients by tracking the maximum modulus of the continuous wavelet transform of the fringe data and the phase distribution that is obtained, with a small error, by integration. Problems associated with phase unwrapping are thereby avoided. The technique is compared with standard methods, and excellent agreement is found. In common with Fourier-transform methods, the technique is capable of extracting the full 2-D phase distribution from a single image.     

A numerical spatial carrier for single fringe pattern analysis algorithm

Hilbert transform and continuous wavelet transform are combined to form an algorithm for the analysis of a single fringe pattern with open or closed fringes. Only one recorded fringe pattern is needed to extract the phase information. A second π/2 phase shifted fringe pattern should first be generated from the other using Hilbert transform to superpose numerically the spatial carrier. Several phase extraction techniques are conceived to demodulate fringe patterns. In this paper, the wavelet technique is used to obtain the required phase distribution. The algorithm offers an advantage that it is appropriate when the spatial carrier is impossible to be added experimentally. Finally, a simulation was carried out to validate the algorithm, giving good results.     

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.     

Phase recovery from fringe patterns using the continuous wavelet transform

Interferometry is well established as an optical technique in which a measurand is encoded as the phase of a periodically varying intensity pattern. In view of the inherent accuracy of interferometry, many methods have been developed to retrieve the phase from images of the fringe pattern. Our focus in this paper is one such technique—the continuous wavelet transform (CWT). We begin by reviewing the CWT and the space–spatial–frequency localisation properties of wavelets. We show that a path which follows the maximum modulus of the CWT (the wavelet ridge) gives the instantaneous fringe frequency as a function of spatial displacement. The phase is automatically and trivially obtained, without discontinuities, by integration. Examples of practical wavelets are given and algorithms to isolate the wavelet ridge reviewed.     

Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform

In this paper, a filtering technique based upon two-dimensional continuous wavelet transform (2D-CWT) is used to eliminate the low frequency components of fringe patterns. The filtered fringe patterns are subsequently demodulated using a standard Fourier transform profilometry (FTP) algorithm. This image pre-filtering stage improves the noise performance of the FTP algorithm and enables the FTP method to demodulate fringe patterns with larger bandwidths. Also, the 2D-CWT technique reduces speckle noise significantly. Moreover, only a single fringe pattern is required in this technique. The 2D-CWT algorithm is capable of separating low frequency terms from the high frequency terms that contain phase-modulated fringe information, even when both interfere, greatly, in the frequency domain. The proposed algorithm is tested, both via computer simulation and using real fringe patterns. This revealed the robustness of this algorithm and also demonstrably enables the demodulation of a wider range of fringe patterns using the FTP technique.     

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.     

基于时域小波变换相位提取的三维形貌测量

提出基于时域小波变换的连续振动物体相位提取方法,恢复连续振动物体的瞬时三维形貌。将小波变换用于序列影栅云纹相位分析中,对连续振动物体的序列影栅图像进行处理,利用小波变换的多分辨力特点,对各点的灰度变化进行连续复小波变换,通过提取小波脊所对应的相位,得到各点相对于基准光栅的完整相位调制信息,从而获取运动物体的瞬时三维形貌。利用该方法对连续振动悬臂梁进行了分析,得到连续振动悬臂梁的瞬时速度及瞬时三维形貌,再现了悬臂梁的连续振动过程。为研究动态物体瞬时特性提供了一种新的方法。     

基于指数尺度间隔连续小波变换的相位提取算法

只需要一幅调制图像的光栅投影测量方法主要有傅里叶变换轮廓术(FTP)、小波变换轮廓术(WTP)等。采用基于指数尺度间隔的连续小波变换与重构方法,提取调制图像的瞬时相位。针对指数尺度间隔连续小波变换,指出了足够大的噪声能够改变小波变换脊的位置,并且该脊向上移动的概率最大。因此,为了重构载频信号,选择脊及其紧邻的较大的那个尺度所对应的小波系数,采用灰度图像阈值分割中最大类间方差法(OTSU),剔除掉幅值较小的系数;针对斑点噪声的影响,对OTSU算法的结果进行了修正;使用修正后的系数集合重构载频信号,并计算该信号的瞬时相位。理论分析和实验结果表明算法有效且具有稳健性。     

A fast,accurate phase unwrapping method for wavelet-transform profilometry

A novel phase unwrapping method used in wavelet-transform profilometry is presented to improve the accuracy and speed of the unwrapping process. The wrapped phase of the fringe image is extracted by using wavelet transform. How to establish quality map using scale factor at wavelet ridge is discussed firstly. The proposed method can reflect the reliability of the pixels in fringe image effectively. As the consuming time of traditional flood-fill phase unwrapping algorithm is too long, an improved quality-guided method is used in phase unwrapping process. We divide the wrapped phase map into two levels following the established quality map, and process these two levels using different algorithms. Compared with the traditional flood-fill algorithm, we can get the accurate result while the consuming time of the phase unwrapping process is much less by using the proposed method. Simulation and experiment results verify that the 3D information of the measured object can be obtained rapidly and accurately by using the proposed method. Also the dynamic object can be measured too.     

Automatic interferogram analysis techniques applied to quasi-heterodyne holography and ESPI

This paper describes significant developments in methods for the automatic, quantitative analysis of interferograms. All areas of analysis have been considered: fringe field generation, pre-processing, and phase unwrapping.

A new quasi-heterodyne holographic technique is described in which the image is reconstructed using a single beam. The errors in the reconstructed fringe field are mainly linear in form, and an error compensation scheme is proposed. The final error in the phase measurement using automatic analysis is λ/40.

The process of image smoothing by an averaging filter is considered to reduce the effects of random noise. It is shown that by measuring the signal-to-noise ratio of the fringe field an optimum degree of smoothing may be applied. This is demonstrated on holographic and electronic speckle pattern interferometry (ESPI) data.

Two methods for cosinusoidal fringe image combination are compared, using three or four fields. It is shown that an automatic analysis can be achieved using four phase stepped images.

A new algorithm to automatically unwrap the phase of complex fringe patterns is described. The fringe field is segmented into small rectangular areas, called tiles. This allows local data to be obtained on fringe consistency and density. A confidence tree can then be formed to produce an optimal solution for the whole field. Results are presented and discussed for both holographic and ESPI data.     

Computer aided evaluation of fringe patterns

The crucial step in computer aided evaluation of interferometric fringe patterns is the determination of the interference phase distribution from the recorded and stored intensity pattern. Methods for determination of the interference phase distributions from fringe patterns are presented and the most important methods, which are skeletonising, temporary heterodyning, phase shifting and Fourier transform evaluation are compared with regard to experimental requirements, achievable resolution and precision, as well as inherent noise suppression and image enhancement. The comparison shows that whenever phase shifting is possible, it is the best choice. If only one interference pattern is offered, Fourier transform evaluation is recommendable. For demodulation of the wrapped phase a path-independent algorithm is presented. Practical examples are given from holographic interferometric measurements.     

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.     

Conversion from phase map to coordinate: Comparison among spatial carrier, Fourier transform, and phase shifting methods

For full-field phase measurement methods, many algorithms have been developed to extract a phase map from fringe image(s). Both phase wrapping and unwrapping algorithms have been extensively investigated by many researchers, but few papers can be found on how to calculate the coordinates of surface points from a phase map. This paper focuses on algorithms that show how a phase map can be used to calculate coordinates. Details are given for single image methods such as Fourier transform, spatial carrier methods, and multiple image methods like traditional phase-shifting methods. Algorithms that can be used to convert a phase map to coordinates and some issues related to these conversion algorithms are discussed. An artifact is measured using these phase measurement methods. The results show that using the correct algorithm to convert a phase map to coordinates is a key to obtaining accurate measurement results.     

Single fringe projection profilometry based on sinusoidal intensity normalization and subpixel fitting

A novel fringe projection profilometry using a single sinusoidal fringe pattern projected is proposed. Computer-generated sinusoidal fringe and uniform intensity patterns are firstly projected on a testing object by a liquid crystal display projector. The variable reflection intensity of a fringe pattern is then roughly normalized by division operation applied to the grabbed fringe and uniform intensity patterns projected. Fringe intensity is further normalized by employing an interpolation algorithm. The deformed sinusoidal pattern encoding object shape is converted to a wrapped phase map without using phase-shifting or Fourier transform. Computer simulation and experimental performance are evaluated to demonstrate the validity of the proposed method. The experimental results compared with those of the four-step phase-shifting and fast Fourier transform methods are also presented.     

散斑干涉相位条纹图的频域滤波处理

为了解决在数字散斑干涉技术测量时,散斑干涉相位条纹图像中大量噪声对相位解包裹结果和精度产生严重影响的问题,介绍了一种条纹正余弦分解和频域低通滤波结合的方法,实现了散斑干涉相位条纹图的高精度滤波。该方法的基本思路是在对相位图像进行滤波处理前,先将相位图通过正余弦函数进行映射转换成两幅图,分别经过频域滤波,然后再合成为相位图。这种分解频域滤波方法可以在滤波的同时,有效保留相位跳变信息。实验结果表明：与传统的图像降噪方法相比,该方法能够在保留图像“尖峰”信息的基础上,较好地滤除图像中的散斑噪声,方法简单有效,有效解决了传统滤波方法应用在相位条纹图中,相图灰度信息丢失10%~40%的问题。     

Speckle correlation fringes denoising using stationary wavelet transform. Application in the wavelet phase evaluation technique

This paper presents a stationary wavelet transform (SWT) method for speckle noise reduction in digital speckle pattern interferometry fringes. The main advantage of SWT is its translation invariance, which makes it important in statistical image processing applications. This method was used to denoise a simulated speckle fringe patterns, a good fidelity value was obtained. Applied to the wavelet phase evaluation, it has provided a phase distribution with a good accuracy.     

Wavelet transform as a processing tool in white-light interferometry

Results of the application of wavelet transform for signal processing in white-light interferometry are reported. The mother wavelet frequency is chosen to be the light-source correlogram 1's, and accurate phase measurements are obtained from simple correlation computations. The fringe envelope is also addressed and permits a complete analysis of coherence-limited fringe patterns. Miscalibrations of phase shift and mean wavelength are also considered.