A simple phase unwrapping approach based on filtering by windowed Fourier transform: The phase near edges

A simple phase unwrapping approach based on windowed Fourier filtering (WFF) was proposed recently. In this paper, the effects of the WFF on edges are further investigated, which show that the phase near an edge remains unchanged and no special consideration is needed for the edges. This useful property will be theoretically proven and numerically verified.     

A simple phase unwrapping approach based on filtering by windowed Fourier transform: A note on the threshold selection

A simple phase unwrapping approach based on windowed Fourier filtering was proposed recently [K. Qian et al. A simple phase unwrapping approach based on filtering by windowed Fourier transform. Opt Laser Technol 2005;37:458–62]. The windowed Fourier filtering algorithm is an essential ingredient that suppresses the noise effectively and makes the phase unwrapping trivial. This paper adds a note on the threshold selection in the windowed Fourier filtering algorithm. A large interval can be selected as the threshold to obtain almost optimal filtering results. Once the selected threshold is suitable, it is almost optimal. This makes the threshold selection in the windowed Fourier filtering algorithm extremely easy.     

基于快速傅里叶变换的四种相位解包裹算法

为了快速准确地对含有噪声的包裹相位图进行相位展开,采用理论分析与计算机模拟及实验验证相结合的方法,对基于快速傅里叶变换（FFT）的四种典型算法四次FFT算法(4-FFT)、二次FFT算法（2-FFT）、四次离散余弦变换算法(4-DCT)及横向剪切干涉与FFT相结合的算法(LS-FFT)作了对比研究。结果表明：2-FFT算法运行速度最快,4-FFT算法次之,LS-FFT算法速度最慢;4-FFT算法对含有较强噪声和轻微欠采样的实验数据的处理效果是最好的;LS-FFT算法对强噪声数据的处理效果最差。     

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.     

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.     

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.     

基于快速傅里叶变换的四种相位解包裹算法

为了快速准确地对含有噪声的包裹相位图进行相位展开,采用理论分析与计算机模拟及实验验证相结合的方法,对基于快速傅里叶变换（FFT）的四种典型算法四次FFT算法(4-FFT)、二次FFT算法（2-FFT）、四次离散余弦变换算法(4-DCT)及横向剪切干涉与FFT相结合的算法(LS-FFT)作了对比研究。结果表明:2-FFT算法运行速度最快,4-FFT算法次之,LS-FFT算法速度最慢;4-FFT算法对含有较强噪声和轻微欠采样的实验数据的处理效果是最好的;LS-FFT算法对强噪声数据的处理效果最差。     

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.     

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。     

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。     

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).     

Measurement of sub-surface delaminations in carbon fibre composites using high-speed phase-shifted speckle interferometry and temporal phase unwrapping

A high-speed phase-shifted speckle interferometer has been developed recently for studying dynamic events. Speckle interferograms are continuously recorded by a CCD camera operating at 1 kHz with temporal phase shifting carried out by a Pockels cell running at the same frequency. Temporal phase unwrapping through sequences of more than 1000 frames allows the determination of time-varying absolute displacement maps. This paper presents the application of this speckle interferometry system to the detection and measurement of sub-surface delamination defects in carbon fibre specimens. The influence of re-referencing the temporal phase unwrapping algorithm after different time intervals is analysed to reduce the random phase errors produced by speckle decorrelation and vibration. The performance of several phase-shifting algorithms to minimize the influence of the vibration noise caused by the vacuum pump used to load the specimen is also investigated.     

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.     

基于小波变换的最小二乘相位解缠算法

最小二乘法是求解二维相位解缠问题最稳健的方法之一,其本质是在最小二乘意义下使缠绕相位的离散偏导数与解缠相位的偏导数整体偏差最小,并等效为可求解一大型的稀疏线性方程系统。由于系统矩阵结构的稀疏性,在采用迭代法求解时收敛速度非常慢。为了改善收敛特性,提出一种基于多分辨率表示的离散小波变换相位解缠算法。利用小波变换将原线性系统转化成具有较好收敛条件的等价新系统。仿真实验表明,该方法能够很好的恢复真实相位,其解缠效果优于Gauss-Seidel松弛迭代和多重网格法。     

S变换引导的窗口傅里叶变换相位提取

提出了一种基于S变换引导的自适应窗口傅里叶变换相位提取方法.通过连续S变换,得到局部条纹的最佳变换窗口,可以保证窗口尺寸随变形条纹的频率变化而自动调整,适合复杂物体的面形测量.计算机模拟和实验验证表明,该方法克服了传统窗口傅里叶变换窗口大小固定的缺点,相位提取的误差由-2～8rad降低为-1～1rad,而且还可以得到被...     

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.     

Adaptive filter for unwrapping noisy phase image in phase-stepping interferometry

An adaptive filter for preprocessing wrapped phase image with high noise content has been developed. The window size of this median filter is adjustable according to its location in the noisy phase image. This size-adjustable median filter has not only high noise-rejection ability, but also both good phase-jump identification and image-detail preservation properties. Phase unwrapping can then be done to recover the real phase image without the hampering influence of noise, especially the noise near the phase jumps. Both numerical simulation and experimental results used to examine the effectiveness of this adaptive filter are presented.     

Comparison of eight unwrapping algorithms applied to Fourier-transform profilometry

Phase unwrapping is a task common to many applications like interferometry imaging, medical magnetic resonance imaging, solid-state physics, etc. Fourier transform profilometry (FTP) values the height distribution of object, elaborating the interference between a plane reference grating and a deformed object grating. Since the height information is extracted from the phase of a complex function, the phase unwrapping is a critical step of the process. Several unwrapping algorithms are proposed in literature, but applied to measurement technologies different from FTP. The purpose of this paper is to define the performances of eight different unwrapping algorithms applied to FTP optical scan method and to define the best one. The algorithms chosen are: Goldstein's algorithm, quality guided path following method, Mask cut method, Flynn's method, multi-grid method, weighted multi-grid method, preconditioned conjugate gradient method and minimum L_p-norm method. The methods were tested on real images acquired by a FTP scanner developed and calibrated for these experiments. The objects used vary from simple geometries, like planes and cylinders, to complex shapes of common use objects. Algorithms were qualified considering the phase unwrapping errors, execution time and accuracy of the shape of objects obtained from the scan method in comparison with real ones. The results show that quality guided algorithm best fits in FTP application.     

Fractional Fourier transform dual random phase encoding of time-varying signals

Optical techniques have shown great potential in the field of information security to encode high-security images. Among several established methods, a double-random phase encryption technique (DRPE) for encoding a primary image into stationary white noise was developed by using the analogy between Fresnel diffraction patterns and the fractional Fourier transform (FrFT-DRPE). In this case, additional keys are obtained through the knowledge of the fractional orders of the FrFTs. In this work we propose an encoding setup for time-varying signals, mainly for short-haul fiber optics link applications, that can be considered as the temporal analogue of the spatial FrFT-DRPE. The behavior of the proposed method is analyzed by employing the Wigner distribution function. As a result, general expressions for both signal time width and spectral bandwidth, at all encryption stages are obtained. Numerical simulations have been carried out to illustrate the system performance. The obtained results indicate that this encryption method could be a good alternative to other well-established methods.     

Single phase encoding method based on the fractional Fourier transform

We propose a single phase encoding scheme for encrypting image by using fractional Fourier transform. Single phase mask is designed in order to be symmetrical about certain direction, which can be used in the process of both encryption and decryption. A conjugate mask is not required in the image decryption process, which is very convenient for the practical application in optics. Moreover, the optical implementation of the image encryption and decryption is given. The implementing structure is composed of lens and spherical mirror. Numerical simulations have demonstrated the validity and security of the encryption algorithm.