An improved stair phase encoding method for absolute phase retrieval

An improved phase unwrapping method is proposed to reduce the projection fringes in three-dimensional (3D) surface measurement. Color fringe patterns are generated by encoding with sinusoidal fringe and stair phase fringe patterns in red and blue channels. These color fringe patterns are projected onto the tested objects and then captured by a color CCD camera. The recorded fringe patterns are separated into their RGB components. Two groups of four-step phase-shifting fringe patterns are obtained. One group of the stripes are four sinusoidal patterns, which are used to determine the wrapped phase. The other group of stripes are four sinusoidal patterns with the codeword embedded into stair phase, whose stair changes are perfectly aligned with the 2π discontinuities of sinusoidal fringe phase, which are used to determine the fringe order for the phase unwrapping. The experimental results are analyzed and compared with those of the method in Zheng and Da (2012. Opt Express 20(22):24139–24150). The results show that the proposed method needs only four fringe patterns while having less error. It can effectively reduce the number of projection fringes and improve the measuring speed.     

单周期条纹双四步相移投影仪的标定方法

针对投影仪标定方法中存在畸变及倾斜投影引起条纹周期、条纹级数变化的问题,提出一种单周期条纹双四步相移投影仪的标定方法.设计生成横向和纵向各两组单周期条纹图像,经投影仪投影到带有圆形标识的标定板上,相机同步采集标定板图像,叠加由双四步相移获得的两幅相位主值图,对叠加相位主值图相位展开,利用展开的绝对相位值计算投影仪像素坐标值,最终将投影仪标定转换为成熟的相机标定.实验结果表明:仿真投影仪标定实验准确度的最大重投影误差约为0.4pixel,均方根误差为0.132 96pixel;实际投影仪标定实验准确度的最大反投影误差约为0.46pixel,均方根误差为0.143 12pixel;实验结果与仿真结果的最大反投影误差相差15%,均方根误差相差7.6%.与现有的采用三频相位展开进行投影仪标定的方法相比,投影光栅图像数可减少8幅.该方法改善了现有投影仪标定方法的不足,标定准确度和标定效率均得到提高.     

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.     

The fluctuating phase error analysis in the digital grating phase-shifting profilometry

The nonlinear response of the experimental system and the saturation of fringe patterns can induce the fluctuating phase error in the projection grating phase-shifting profilometry. Two major factors of the fluctuating phase error are discussed by simulation. The fluctuating phase error caused by the nonlinear response of the system is four times the frequency of the fringe pattern when the conventional four-frame phase extracting algorithm is used. However, such error can be decreased by five-frame algorithm. On the other hand, the fluctuating phase error caused by the fringe saturation is five times the frequency of the fringe pattern by using conventional five-frame phase extracting algorithm. A novel phase recovering algorithm is used to decrease the phase error caused by the saturation. Furthermore, the applicability range of the proposed phase recovering algorithm is analyzed by simulation and experiments with different saturation degree of the fringe pattern and nonlinearity of the measurement system.     

Study on a novel phase-recovering algorithm for partial intensity saturation in digital projection grating phase-shifting profilometry

The phase error and its periodic behavior caused by the partial intensity saturation of fringe patterns in the digital projection grating phase-shifting profilometry are studied. A saturation coefficient K is defined to describe the saturation degree of the fringe patterns projected on a measured object. The distribution of the phase error is analyzed through a simulation method. Moreover, a novel phase-recovering algorithm is studied to resolve the phase error issue introduced by the partial saturated fringe patterns. The real phase can be recovered by the unsaturated intensity values. A series of results corresponding to different degrees of saturation defined by K are given to prove the validity of the proposed algorithm.     

Effect of intensity-correlated error due to quantization and noise on phase-shifting method

A method to investigate the effect of the interaction of the noise and the quantization on the computed phase in phase-shifting measurement (PSM) is proposed. The statistical relationship between the measured phase and the fringe intensity-correlated error is formulated for the standard N-sample algorithm. The phase error variance is evaluated following a nonlinear stochastic analysis.     

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.     

A novel three-dimensional shape measurement method based on a look-up table

Fringe projection profilometry is widely used for three-dimensional shape measurement. In an oblique-angle projection, the fringe cycle is broadened on the reference plane. Phase errors are mainly caused by the nonlinear gamma of the projector and fringe cycle broadening. This study describes a phase error compensation method to eliminate these phase errors. A look-up table that stores phase errors is constructed for phase error compensation. Based on it, a new height equation is proposed. The experimental results show that the proposed method can compensate for the phase errors of the fringe projection profilometry, thereby improving the measurement accuracy significantly.     

Complex object 3D measurement based on phase-shifting and a neural network

An accurate phase-height mapping algorithm based on phase-shifting and a neural network is proposed to improve the performance of the structured light system with digital fringe projection. As phase-height mapping is nonlinear, it is difficult to find the best camera model for the system. In order to achieve high accuracy, a trained three-layer back propagation neural network is employed to obtain the complicated transformation. The phase error caused by the non-sinusoidal attribute of the fringe image is analyzed. During the phase calculation process, a pre-calibrated phase error look-up-table is used to reduce the phase error. The detailed procedures of the sample data collection are described. By training the network, the relationship between the image coordinates and the 3D coordinates of the object can be obtained. Experimental results demonstrate that the proposed method is not sensitive to the non-sinusoidal attribute of the fringe image and it can recover complex free-form objects with high accuracy.     

Three-dimensional profile measurement of the blade based on surface structured light

As the blade must have precise size and accurate shape, three-dimensional profile measurement of the blade is very important. In this paper, three-dimensional profile measurement method of the blade based on surface structured light is proposed. A new phase unwrapping method is also proposed. This encoding method combined coding fringe pattern with the four-step phase-shifting method, the coding information is embedded into the phase, which can be used to determine fringe order for absolute phase retrieval. The encoding method is simple and easy to implement, so the phase unwrapping of the blade can be quickly perform. The experimental and simulation results show that the method can achieve a high precision, high speed and low cost three-dimensional profile measurement of the blade.     

Optical measurement on dynamic buckling behavior of stiffened composite panels under in-plane shear

The buckling behavior and failure mode of a composite panel stiffened by I-shaped stringers under in-plane shear is studied using digital fringe projection profilometry. The basic principles of the dynamic phase-shifting technique, multi-frequency phase-unwrapping technique and inverse-phase technique for nonlinear error compensation are introduced. Multi-frequency fringe projection profilometry was used to monitor and measure the change in the morphology of a discontinuous surface of the stiffened composite panel during in-plane shearing. Meanwhile, the strain history of multiple points on the skin was obtained using strain rosettes. The buckling mode and deflection of the panel at different moments were analyzed and compared with those obtained using the finite element method. The experimental results validated the FEM analysis.     

Three-dimensional profile measurement of the blade based on multi-value coding

As the blade must have precise size and accurate shape, three-dimensional (3D) profile measurement of the blade is very important. 3D profile measurement method based on multi-value coding is proposed. This method designs a multi-value coding stripe pattern combined with the four-step phase-shifting method. Two kinds of fringe patterns are projected onto the object respectively, one is sinusoidal intensity distribution used for wrapped phase, the other is multi-value coding fringe pattern for phase unwrapping. Because this encoding method is simple, and easy to implement, the absolute phase can be quickly implemented. Experimental results demonstrated that the proposed method can achieve a high precision, high speed and low cost 3D profile measurement of the blade.     

基于正弦和三角波条纹投影的三维测量方法

在现有的针对复杂物体表面形貌的三维测量方法中, 为了完成绝对相位的测量, 通常需要处理至少6幅条纹图像, 限制了测量速度。提出了采用2幅正弦条纹和2幅三角波条纹图来获得物体三维形貌的方法。利用两步相移正弦条纹和两步相移三角条纹得到截断相位, 再利用两步相移三角波条纹得到条纹级次, 减少了投影条纹幅数, 提高测量速度。在得到条纹级次时, 计算三角波条纹强度调制和强度对比度, 与计算相位相比, 可以减少数据处理的时间, 进一步提高测量速度, 同时能减小物体表面反射率的影响, 提高了测量精度。测量最大高度为39 mm的阶梯状标准块, 得到的最大绝对误差和最大的RMS误差分别为0.045 mm和0.041 mm。验证了该方法的有效性和实用性, 在高速实时的复杂形貌三维测量中有广泛的应用前景。     

Phase-Measuring Algorithms to Suppress Spatially Nonuniform Phase Modulation in a Two-Beam Interferometer

Phase modulation of presently used phase-shifting interferometers is assumed to be spatially uniform across the observing aperture. However, calibration errors or the configuration of an interferometer can cause a spatial nonuniformity in the phase modulation. Spatial nonuniformity causes a significant error in the measured phase when the phase modulator has nonlinear sensitivity. An even-order nonlinearity in the phase modulation in particular contributes to the errors. Lowest-order errors can be suppressed by adding a new symmetry to the sampling functions of the phase-shifting algorithm, however the algorithm suffers from large random noise. The random noise is shown to be decreased substantially by applying one more sampled frame to the algorithm. We derive new seven-sample and eight-sample algorithms that can compensate for a nonuniform phase shift and has much less random noise than the previous algorithm we proposed.     

双步相移光栅投影测量轮廓术

双三步相移算法证明可以较大地减少数字光栅投影测量轮廓术的测量误差,基于理论分析与实验验证,针对常用的四步、五步相移算法,提出了相应的双四步、双五步相移算法。通过两次传统相移算法得到两幅主值相位图,直接融合两幅主值相位图即可获得测量所需的相位信息,与已有的针对两幅展开相位进行相位融合方法相比,此方法实现简单且更加有效。相较于双三步相移算法,双四步和双五步相移算法实现简单且能够极大地减少测量误差,仅需通过投影2倍数目传统相移算法所需的投影光栅,且可保持常用三步、四步及五步相移算法固有的优点。     

Automatic evaluation of photoelastic fringe constant by the nonlinear least-squares method

A nonlinear least-squares method was applied to determine the photoelastic fringe constant from isochromatic phase field obtained by digital phase-shifting photoelasticity. To perform automatic evaluation, not only the photoelastic fringe constant but the center coordinates of a disk under diametral compression are also determined simultaneously using the nonlinear least-squares method. A set of over-determined nonlinear equations is obtained to estimate the unknown parameters by an iterative procedure based on Newton–Raphson method. Simulation and experimental results show that the parameters are evaluated accurately from the isochromatic phase field obtained from the phase-shifting photoelasticity. The photoelastic fringe constant and the disk center coordinates can be evaluated easily, simply and automatically by the technique.     

Phase-shifting interferometry in the presence of nonlinear phase steps, harmonics, and noise

A phase-shifting piezo device commonly employed in phase-shifting interferometry exhibits a nonlinear response to applied voltage. Hence, a method for estimation of phase distribution in the presence of nonlinear phase steps is presented. The proposed method compensates for the harmonics present in the intensity fringe, allows the use of arbitrary phase-step values between 0 and tau rad, and does not impose constraints on the selection of particular phase-step values for minimizing nonlinearity and compensating for the harmonics. The comparison of the proposed method with other well-known benchmarking algorithms shows that our method is highly efficient and also works well in the presence of noise.     

Phase deviation analysis and phase retrieval for partial intensity saturation in phase-shifting projected fringe profilometry

Intensity saturation may take place as a sinusoidal fringe pattern is projected onto an object which has a relative high reflective index on some regions of the surface. If a phase-shifting method is used, the illuming light intensities at the same point of the object may have different values for each projected phase-shifting fringe pattern. Therefore, when the intensity at a point of the object exceeds the saturation level for a certain phase step of projected fringe pattern, other intensities obtained at the same point with different phase steps of projected fringe patterns do not exceed the saturation level. This kind of saturation is defined as partial intensity saturation. In the case of partial intensity saturation, a relative larger phase error is introduced when a conventional phase reconstruction algorithm is used in the phase-shifting projected fringe profilometry (PSPFP). To get rid of such disadvantage, an improved algorithm for phase reconstruction is proposed in this paper. By using the new algorithm, the effect of partial intensity saturation for the phase reconstruction can be decreased greatly and a good quality of reconstructed phase map can be obtained. Phase deviation of the reconstructed phase is also analyzed. Finally, an experimental result with PSPFP is presented to validate the feasibility of the proposed algorithm.     

相移相位测量的全息再现算法及测量误差分析

用全息原理和方法研究相移相位测量,得到了N步整周期相移再现物光波复振幅同步叠加函数(N步相移函数),同时提出一种新的相移相位测量误差分析和最大误差估计方法。N步相移干涉图是以理想平行光为参考光的无衍射同轴全息图,将其与对应的相移参考光相乘后求和得到N步相移函数;在理想情况下,这是一种复振幅分离、测量和物光波复振幅函数同步叠加方法,存在误差时计算出的相位是最小二乘方法的最佳期望结果。利用N步相移函数得到的N 1步相移函数,说明非理想N步相移函数是理想N步相移函数与误差函数之和,可以把相位型误差转化为与振幅和强度相对误差同等的误差来对待,降低了相位测量中误差估计的难度,给出了N步相移算法最大误差的估计方法和公式。     

Further study of the phase-recovering algorithm for saturated fringe patterns with a larger saturation coefficient in the projection grating phase-shifting profilometry

The phase error will occur when captured fringe patterns are saturated in the projection grating phase-shifting profilometry. The phase-recovering algorithm corresponding to seven-frame phase-shifting method is deduced. The applicability ranges of phase-recovering algorithms corresponding to different phase-shifting methods are studied with different intensity saturation coefficients by a simulative method. Simulative results indicate that the phase error caused by the intensity saturation can be effectively decreased by the phase-recovering algorithm when the saturation coefficient of fringe patterns is within the applicability range of the corresponding phase-recovering algorithm. Furthermore, the applicability range of the phase-recovering algorithm will be extended with the increase of phase-shifting steps. An experimental result is presented to prove the availability of the phase-recovering algorithm.