Fast frequency-guided sequential demodulation of a single fringe pattern

Fast frequency-guided sequential demodulation (FFSD) for demodulating a single closed-fringe pattern is proposed as an improvement of frequency-guided sequential demodulation (FSD). Instead of using optimization to estimate the local frequencies for determining the sign of the phase, the FFSD estimates the local frequencies by directly calculating the gradient of the obtained phase with an undetermined sign. This improvement considerably reduces the computational complexity of the FSD and leads to a faster and simpler method. Simulated and experimental fringe patterns are used to test the proposed method and show that the demodulation speed of FFSD is about 150 times faster than that of the FSD, while the robustness and accuracy remain almost the same.     

A derivative based simplified phase tracker for a single fringe pattern demodulation

In this paper, a novel fringe demodulation method for the estimation of phase and its first-order derivative from a closed-fringe interferogram is proposed. The proposed method determines the phase derivatives in both x&y directions from fringe orientation and density. The phase derivatives are subsequently used to determine phase values using a novel simplified phase tracker. In the phase tracking model, the complexity of the cost function is reduced using predetermined derivatives so computation time required for phase tracking is reduced considerably. The proposed model is more robust while dealing with saddle points in fringes than the conventional phase tracker model. Hence it does not require any specialized scanning strategy. The proposed method is validated with simulated and experimental fringe patterns (obtained using electronic speckle pattern interferometry and optical holographic interferometry) and a comparison study is carried out with conventional regularized phase tracker. The simulation results show that the proposed method has good accuracy and requires less computation time than existing phase-tracking algorithms. The experimental results demonstrate the robustness of the proposed method against speckle noise and its practical applicability for static and dynamic applications.     

Window fringe pattern demodulation by multi-functional fitting using a genetic algorithm

We present a new way to demodulate complicated fringe patterns containing closed fringes using a genetic algorithm (GA). The entire fringe pattern is divided into a set of partially overlapping smaller sub-image windows. Each of these has a lower dimensionality and as a consequence is faster and can be demodulated more reliably. The demodulation process proceeds row-by-row way passing from one sub-image in a neighborhood until the whole interferogram is processed. The modulating phase of each sub-image is modeled as a parametric analytic function whose parameters are optimized using a GA. The technique is demonstrated demodulating some normalized fringe patterns that have two main difficulties: closed fringes within the interferogram and regions of under-sampled fringes. These fringe images cannot be demodulated by techniques such as the regularized phase tracker (RPT).     

光载波条纹图的计算机辅助分析——阶梯形虚拟光栅解调法

本文提出一种两维光测条纹图的相位测量技术--阶梯形虚光栅解调算法。此法只需要获取一幅光载波条纹图,由计算机产生两个由N(整数)个象素构成、透射函数呈阶梯形分布、彼此间有一定相位差的两个虚光栅(参考信号),通过在整数域的运算实现对条纹图和虚光栅的相乘型叠加,低通滤波后得到两个相位差已知的叠栅函数,然后从两个相差为已知值的正弦信号中提取相位信息。理论分析和计算机模拟表明,具有相当高的测量精度,计算简便快捷。文中还给出了阶梯虚光栅解调法在投影光栅法三维形貌测量中的应用实例。     

Sequential demodulation of a single fringe pattern guided by local frequencies

A simple but effective approach for the demodulation of a single fringe pattern is proposed. The phase with an undetermined sign is directly obtained by taking the arccosine value of a preprocessed fringe pattern. The local frequencies, also with an undetermined sign, are then estimated by local matching. The sign ambiguity is then removed simply by forcing the continuity of the local frequencies. The priority of sign determination is guided by the value of total local frequency (fringe density) so that the critical points are processed last. The proposed approach is verified by successful demodulation of a simulated fringe pattern and two experimental fringe patterns.     

Fringe-pattern demodulation using an iterative linear digital phase locked loop algorithm

A novel technique called an iterative linear digital phase locked loop (DPLL) for fringe-pattern demodulation is presented. The proposed technique has better noise immunity than the basic one-dimensional DPLL algorithms (e.g., conventional and linear DPLLs) because it processes a fringe pattern in two dimensions. In the first iteration of the proposed algorithm, it uses a flat reference to estimate the phase of the fringe pattern. The estimated phase map resulting from the first iteration is smoothed using a moving average window and then it is used as a reference to estimate the phase of the fringe pattern in the second iteration. In the third iteration, the phase map resulting from the second iteration is smoothed and used as a reference to estimate the phase of the fringe pattern in the fourth iteration and so on until the demodulated phase map has a good signal-to-noise ratio.     

Phase locked loop profilometry

An innovative technique for obtaining automatic three-dimensional shape information of a diffuse surface is presented. The technique is based on a new approach of phase measuring from a fringe pattern. The target to be studied is first modulated by projecting a linear grating onto its surface. The linear fringes are deformed according to the surface shape. Demodulation is carried out by determining the phase of these deformed fringes using a novel digital phase locked loop demodulation algorithm. The technique has the main advantage that the traditional phase unwrapping process is not required. The phase is determined continuously as the algorithm scans the two-dimensional fringe pattern. Owing to its sequential nature, this phase detection technique can be implemented using a special purpose video processor for phase determination at video rates, making it a very fast algorithm. The algorithm along with experimental results of real surface profiles are presented.     

Comparison of phase recovery methods in spiral speckle pattern interferometry correlation fringes

Spiral interferometry can be used as a solution to the problem of sign ambiguity presented in the conventional speckle pattern interferometric technique when the optical phase needs to be reconstructed from a single closed fringe system. Depressions and elevations of the topography corresponding to the object deformation are distinguished by the direction of rotation of the local spiral fringe pattern. In this work, we implement and compare several methods for optical phase reconstruction by analyzing a single image composed of spiral speckle pattern interferometry correlation fringes. The implemented methods are based on contour line demodulation, center line demodulation, Spiral Phase Quadrature Transform and the 2D Riesz transform with multivector structure. Contour line and center line demodulation approaches are exclusively dedicated to images containing a fringe system with spiral structure. The others are based on the 2D Riesz transform, these being well known approaches in conventional interferometry. We examine simulated experiments and analyze some of the emerging drawbacks for solving the phase reconstruction problem by using different mean values of speckle size and background noise levels. We also discuss several numerical procedures that may well improve the efficiency and robustness of the presented numerical implementations. The performance of the implemented demodulation methods is evaluated by using a universal image quality index and therefore a quantitative comparison is also presented.     

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.     

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.     

Measurement of surface topography by RGB Shadow-Moiré with direct phase demodulation

In this paper we present the application of a direct demodulation method for the measurement of surface topography by means of Shadow-Moiré. In our set-up, we use three LEDs (with green, red and blue peak wavelengths) to illuminate the grating. Due to the different position of these light sources, a polychromatic Shadow-Moiré fringe pattern is produced, which can be described as the superposition of three monochromatic (red, green and blue) fringe patterns. Taking the image of this polychromatic fringe pattern with a RGB CCD camera, we get a monochromatic fringe pattern stored at each RGB channel of the CCD. The direct demodulation algorithm employed uses these fringe patterns to calculate the wrapped phase map. After unwrapping the phase map using a standard multi-grid technique, we implemented an automatic procedure to detect the area of interest of the phase map by removing low modulation zones and to calculate the absolute value of the phase. In this way it is possible to determine the topography of a surface with a single RGB snapshot maintaining a simple experimental set-up, which is an important feature, especially for the study of dynamic phenomena such as deformations. We present the experimental results obtained after measuring different objects with both smooth and rough surface textures.     

改进的次条纹积分相位分析技术

条纹花样的单帧相位解调算法在实时、动态测量技术中具有广泛应用。次条纹积分算法具有较高抗噪能力,但其采用载频近似估算形变条纹周期时存在频率失配的固有缺陷,提出一种改进的次条纹积分算法,对条纹的局域频率进行探测,利用局域频率进行次条纹积分,然后再计算相位。模拟实验表明,改进算法的最大误差约为原算法的1/3,并通过条纹投影轮廓术实验进一步证明了改进算法的有效性,为单帧条纹相位分析技术提供了一种可供选择的新算法。     

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.     

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.     

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.     

Robust profilometer for the measurement of 3-D object shapes based on a regularized phase tracker

A robust profilometer is proposed to measure 3-D object shapes based on a regularized phase tracker that is capable of demodulating fringe patterns with a high noise presence and broad bandwidth due to the object shape. As shown herein, the technique acts as an adaptive filter and is capable of giving the detected phase continuously so that no further unwrapping process is required. Experimental results of real surface profiles are presented.     

Interferometric Phase-Measurement Using a One-Dimensional Discrete Hilbert Transform

A phase demodulation scheme using a discrete Hilbert transform that can change the interferometric phase by /2 has been investigated. In-quadrature components of a fringe pattern are obtained from one captured interferogram using a one-dimensional (1-D) discrete Hilbert transform and a 1-D discrete high-pass filtering that are based on a digital signal processing technique. The phase distribution in the range of 15 (rad) can be demodulated with the proposed method. The 1-D discrete Hilbert transform can be extended to two-dimensional calculation with a raster scanning procedure. © 2005 The Optical Society of Japan     

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。     

Phase difference determination by fringe pattern matching

A method of phase difference determination in interferometry is presented. In this method, the phase difference between two interferograms is determined by fringe pattern matching with subpixel accuracy. The signal-to-noise ratio is significantly improved due to the region-based fringe pattern matching and its effect of averaging noise. The experiment shows that this method is useful for the determination of phase difference between two equi-spaced fringe patterns, and it has the advantages of high precision of measurement and high resistance to noise.