Retrieving ESPI Map of Discontinuous Objects via a Novel Phase Unwrapping Algorithm

Abstract:  This work presents an effective scheme for the phase unwrapping of electronic speckle pattern interferometry (ESPI) map of discontinuous objects. ESPI is a highly effective measurement approach for industry and academia. However, due to the speckle noise, its unwrapping job is quite difficult, especially, when treating ESPI maps obtained from the deformation field of an object containing height discontinuities. The minimum L^p norm method developed by Ghiglia and Romero can treat the aforementioned problem with acceptable accuracy, but takes a long time to run. Therefore, this study presents a novel method based on a hybrid of the regional algorithm proposed by Gierloff, the branch cut method presented by Goldstein et al . and the (spatially) parallel unwrapping method with region-referenced algorithm developed by Huang and He. The proposed algorithm can retrieve ESPI maps with height discontinuities within acceptable accuracy and effectiveness.     

Double three-step phase-shifting algorithm

We describe what we believe is a new phase-shifting algorithm called a double three-step algorithm developed to reduce the measurement error of a three-dimensional shape-measurement system, which is based on digital fringe-projection and phase-shifting techniques. After comparing the performance of different existing phase-shifting algorithms, we present the new double three-step algorithm based on the error analysis of the standard three-step algorithm. In this algorithm, three-step phase shifting is done twice with an initial phase offset of 60 degrees between them, and the two obtained phase maps are averaged to generate the final phase map. Both theoretical and experimental results showed that this new algorithm worked well in significantly reducing the measurement error.     

Statistical phase-shifting step estimation algorithm based on the continuous wavelet transform for high-resolution interferometry metrology

We propose a statistical phase-shifting estimation algorithm for temporal phase-shifting interferometry (PSI) based on the continuous wavelet transform (CWT). The proposed algorithm explores spatial information redundancy in the intraframe interferogram dataset using the phase recovery property on the power ridge of the CWT. Despite the errors introduced by the noise of the interferogram, the statistical part of the algorithm is utilized to give a sound estimation of the phase-shifting step. It also introduces the usage of directional statistics as the statistical model, which was validated, so as to offer a better estimation compared with other statistical models. The algorithm is implemented in computer codes, and the validations of the algorithm were performed on numerical simulated signals and actual phase-shifted moiré interferograms. The major advantage of the proposed algorithm is that it imposes weaker conditions on the presumptions in the temporal PSI, which, under most circumstances, requires uniform and precalibrated phase-shifting steps. Compared with other existing deterministic estimation algorithms, the proposed algorithm estimates the phase-shifting step statistically. The proposed algorithm allows the temporal PSI to operate under dynamic loading conditions and arbitrary phase steps and also without precalibration of the phase shifter. The proposed method can serve as a benchmark method for comparing the accuracy of the different phase-step estimation methods.     

Two-wavelength phase-shifting interferometry insensitive to the intensity modulation of dual laser diodes

We have constructed two-wavelength phase-shifting interferometry that is insensitive to the intensity changes in interferograms associated with the current variations in two laser-diode (LD) sources by using a newly developed phase-extraction algorithm. The tested phase at a synthetic wavelength can be measured from six interferograms with different phase shifts. The algorithm becomes a simple form for seven interferograms and reduces to a minimum of five phase-shifted data in the proper conditions. We shifted the phases equally in opposite directions to one another by separately varying the stepwise currents in dual LD's on an unbalanced interferometer. The measurement accuracy has been improved compared with that of the two-wavelength four-step method. The phase error caused by the power changes in the dual LD's has been investigated theoretically and experimentally. The experimental results are shown to measure a step object with a 4.6-μm synthetic wavelength.     

Dynamic phase-shifting photoelasticity

The application of phase-shifting photoelasticity to a real-time dynamic event involves simultaneous recording of the four phase-shifted images. Here an instrument, believed to be novel, is developed and described for this purpose. Use of a Multispec Imager is introduced into digital photoelasticity for the first time to our knowledge. This device enables splitting the optical energy of an object into four identical paths, thus permitting recording of the required four phase-shifted images. Experimental demonstration is provided for validation.     

A new phase retrieval algorithm with pure generalized three-step phase-shifting under matrix norm processing

Phase-shifting interferometry (PSI) is one of the most effective techniques in imaging a phase specimen, in which the phase retrieval is a basic and significant process. A new phase retrieval method based on the matrix norm algorithm in PSI is proposed in this paper. In this algorithm, the value of phase shift can be determined by three different matrix norms of the intensity difference between two phase-shifted interferograms, and then the phase can be retrieved. Neither the iterative calculation nor the extra measurements of other parameters are necessary on account of this algorithm which only requires three phase-shifted interferograms. The feasibility and accuracy of this algorithm are demonstrated by the simulated results. Experimentally, the generalized phase shift can be realized by a simple device which adjusting the angle of glass accurately. It is found that this algorithm has a good application prospect in the field of dynamic imaging.     

Generating fringe-free images from phase-shifted interferometry data

Weighted averaging of a sequence of phase-shifted interference patterns yields a fringe-free intensity image that can be useful for machine vision, lateral metrology, defect detection, and other supplementary tasks in a surface-profiling interferometer. Coefficients for effective fringe-removal algorithms follow from a Fourier analysis of phase-shifting errors. Theoretical and experimental examples illustrate the substantially improved performance of a well-designed weighted average over a simple linear sum of data frames.     

High-speed dynamic speckle interferometry: phase errors due to intensity, velocity, and speckle decorrelation

The recently developed technique of high-speed phase-shifting speckle interferometry combined with temporal phase unwrapping allows dynamic displacement fields to be measured, even for objects containing global discontinuities such as cracks or boundaries. However, when local speckle averaging is included, small phase errors introduced at each time step are accumulated along the time axis, yielding total phase values that depend strongly on the speckle rereference rate. We present an analysis of the errors introduced in the phase evaluation by three sources: intensity errors, velocity errors, and speckle decorrelation. These errors are analyzed when they act both independently and together, for the most commonly used phase-shifting algorithms, with computer-generated speckle patterns. It is shown that, in a controlled out-of-plane geometry, errors in the unwrapped phase map that are due to speckle decorrelation rise as the time between rereferencing events is increased, whereas those due to intensity and velocity errors are reduced. It is also shown that speckle decorrelation errors are typically more important than the intensity and velocity errors. These results provide guidance as to the optimal speckle rereferencing rate in practical applications of the technique.     

Vibration-induced phase errors in high-speed phase-shifting speckle-pattern interferometry

We present results from numerical simulations of a dynamic phase-shifting speckle interferometer used in the presence of mechanical vibrations. The simulation is based on a detailed mathematical model of the system, which is used to predict the expected frequency response of the rms measurement error, in the time-varying phase difference maps, as a result of vibration. The performance of different phase-shifting algorithms is studied over a range of vibrational frequencies. Phase-difference evaluation is performed by means of temporal phase shifting and temporal phase unwrapping. It is demonstrated that longer sampling windows and higher framing rates are preferred in order to reduce the phase-change error that is due to vibration. A numerical criterion for an upper limit on the length of time window for the phase-shifting algorithm is also proposed. The numerical results are finally compared with experimental data, acquired with a phase-shifting speckle interferometer of 1000 frames/s.     

Evaluation of two-dimensional displacement components of symmetrical deformation by phase-shifting electronic speckle pattern interferometry

A method for the isolation of two-dimensional (2D) displacement components by using one phase map obtained by phase-shifting electronic speckle pattern interferometry (ESPI) is presented. When the typical ESPI is used for displacement measurement, a mixed phase distribution of deformation is measured. If the deformation of the object is symmetrical, two components of deformation can be separated from each other by using the mixed phase distribution. We turn over the mixed phase map first to obtain the second phase map, and then overlap them. Two displacement components can be separated from each other by boundary alignment and algebraic calculation between the two phase maps. This method has been proved feasible by a typical three-point bending experiment. Some experimental results are offered and compared with the results obtained by a dual-beam symmetrical illuminations experiment. This technique presented provides an alternative approach to 2D deformation measurement.     

Phase-shifted dynamic speckle pattern interferometry at 1 kHz

We describe a phase-shifting out-of-plane speckle interferometer operating at 1 kHz for studying dynamic events. The system is based on a Pockels cell that is synchronized to a high-speed video camera to ensure that the phase shifting occurs between frames. Phase extraction is performed by use of a standard four-frame algorithm, and temporal phase unwrapping allows sequences of several hundred absolute (rather than relative) displacement maps to be obtained fully automatically. The maximum theoretical surface velocity of 67 mum s(-1) is a factor of 40 greater than can be achieved with a speckle interferometer based on a conventional video camera. We test the system using a target that is displaced with constant speed in a direction normal to its surface by means of a piezoelectric transducer. The system's performance in a practical situation is illustrated with measurements on a thin plate undergoing out-of-plane deformation.     

Double-exposure phase calculation method in electronic speckle pattern interferometry based on holographic object illumination

Multiple-exposure phase calculation procedures are widely used in electronic speckle pattern interferometry to calculate phase maps of displacements. We developed a double-exposure process based on holographic illumination of the object and the idea of the spatial carrier phase-shifting method to examine transient displacements. In our work, computer-generated holograms and a spatial light modulator were used to generate proper coherent illuminating masks. In this adjustment all phase-shifted states were at our disposal from one recorded speckle image for phase calculation. This technique can be used in the large scale of transient measurements. In this paper we illustrate the principle through several examples.     

Optimized pulse width modulation pattern strategy for three-dimensional profilometry with projector defocusing

Three-dimensional profilometry by sinusoidal fringe projection using phase-shifting algorithms is usually distorted by the nonlinear intensity response of commercial video projectors. To overcome this problem, several methods including sinusoidal pulse width modulation (SPWM) were proposed to generate sinusoidal fringe patterns with binary ones by defocusing the project to some certain extent. However, the residual errors are usually nonnegligible for highly accurate measurement fields, especially when the defocusing level is insufficient. In this work, we propose two novel methods to further improve the defocusing technique. We find that by properly optimizing SPWM patterns according to some criteria, and combining SPWM technique with four-step phase-shifting algorithm, the dominant undesired harmonics will have no impact on the phase obtained. We also propose a new sinusoidal fringe generation technique called tripolar SPWM, which can generate ideal sinusoidal fringe patterns with a very small degree of defocusing. Simulations and experiments are presented to verify the performance of these two proposed techniques.     

Spatial carrier phase-shifting algorithm based on least-squares iteration

An advanced spatial carrier phase-shifting (SCPS) algorithm based on least-squares iteration is proposed to extract the phase distribution from a single spatial carrier interferogram. The proposed algorithm divides the spatial carrier interferogram into four phase-shifted interferograms. By compensating for the effects of the variations of phase shifts between pixels and the variations of background and contrast, the proposed algorithm determines the local phase shifts and phase distribution simultaneously and accurately. Numerical simulations show that the accuracy of the proposed algorithm is obviously improved by compensating for the effects of background and contrast variations. The peak to valley of the residual phase error remains less than 0.002 rad when the magnitude of spatial carrier is in the range from pi/5 to pi/2 and the direction of the spatial carrier is in the range from 25 degrees to 65 degrees. Numerical simulations and experiments demonstrate that the proposed algorithm exhibits higher precision than the existing SCPS algorithms. The proposed algorithm is sensitive to random noise, but the error can be reduced by N times if N measurements are taken and averaged.     

复合式结构光编码法测量3D图像

为了更加有效地采集物体的三维表面信息,使用结构光技术构建新的三维表面数字化检测平台.结合格雷和相移编码各自所具有的优势,提出了一种新的复合编码法,首先将图片分成一定的区域,然后在各个区域单独投影结构光条纹,之后每幅相移图依次向右移动一定像素.由于该法对投影的简化使编码周期明显缩短,减少了图像处理的数据量,从而提高了扫描速度,尤其适合于实现快速的三维模型重建.在自行研制的三维表面信息采集实验平台上进行了实验,新方法的有效性得到了证明.     

Compensation algorithm for the phase-shift error of polarization-based parallel two-step phase-shifting digital holography

We propose an algorithm for compensating the phase-shift error of polarization-based parallel two-step phase-shifting digital holography, which is a technique for recording a spatial two-step phase-shifted hologram. Although a polarization-based system of the technique has been experimentally demonstrated, there had been the problem that the phase difference of two phase-shifted holograms had been changed by the extinction ratio of the micropolarizer array attached to the image sensor used in the system. To improve the performance of the system, we established and formulated an algorithm for compensating the phase-shift error. Accurate spatial phase-shifting interferometry in the system can be conducted by the algorithm regardless of phase-shift error due to the extinction ratio. By the numerical simulation, the proposed algorithm was capable of reducing the root mean square errors of the reconstructed image by 1/4 and 1/5 in amplitude and phase, respectively. Also, the algorithm was experimentally demonstrated, and the experimental results showed that the system employing the proposed algorithm suppressed the conjugate image, which slightly appeared in the image reconstructed by the system not employing the algorithm, even when the extinction ratio was 10:1. Thus, the effectiveness of the proposed algorithm was numerically and experimentally verified.     

Extended-range temporal electronic speckle pattern interferometry

In recent years the availability of high-speed digital video cameras has motivated the study of electronic speckle pattern interferometry (ESPI) in the time domain. To this end a properly sampled temporal sequence of N-fringe patterns is used to analyze the temporal experiment. Samples of temporal speckle images must fulfill the Nyquist criteria over the time axis. When the transient phenomena under study are too fast, the required sampling frequency over time may not be fulfilled. In that case one needs to extend the measuring range of the algorithm used to extract the modulating phase. We analyze how to use short laser pulses or short video acquisition times with fairly long temporal separation among them to estimate the modulating phase of a dynamic ESPI experiment. The only requirement is that the modulating phase being estimated be properly sampled in the spatial domain.     

Precipitation of Cu in Fe–Cu alloys by high-speed deformation

The differences between defect structures in Fe–Cu alloys deformed at the high (4.3×10⁵ s⁻¹) and the low strain rate (67 s⁻¹) were studied. Positron lifetime and coincidence Doppler broadening (CDB) measurements were carried out to investigate the formation of vacancy clusters and Cu precipitates. Both the size of vacancy clusters and the total amount of vacancy-type defects were larger after high-speed deformation at room temperature. Cu precipitation in the specimen deformed at the high-speed stopped for 10 h after annealing at 400 °C, while that in the specimen deformed at the low-speed continued for 100 h. Transmission electron microscopy (TEM) observations showed a heterogeneous distribution of dislocations in the case of low-speed deformation but a homogeneous distribution in the case of high-speed deformation. These results suggested that the sink efficiency for defects was higher in the specimen deformed at the high-speed.     

Dynamic range of Ronchi test with a phase-shifted sinusoidal grating

The dynamic range of a Ronchi test with a phase-shifted sinusoidal grating was investigated theoretically and experimentally. As the number of fringes in a Ronchi interferogram increases, the fringe visibility decreases, which results in a decrease of phase-measurement resolution. It is shown that in order to optimize the dynamic range the effective wavelength of the interferogram should be tuned to the characteristic wavelength of the object wave front. The maximum dynamic range achievable is estimated to be 16 times larger than that of a Fizeau interferometer. Suppressing higher-order diffraction components has achieved sheared interferograms with a signal-to-noise ratio in excess of 60:1. The effects of nonsinusoidal transmittance of the grating and the phase-shift errors were minimized by a seven-sample phase-shifting algorithm, and a phase measurement uncertainty of less than 1/700 has been achieved.     

Multiple-step triangular-pattern phase shifting and the influence of number of steps and pitch on measurement accuracy

We present new extensions of the two-step, triangular-pattern phase-shifting method for different numbers of phase-shifting steps to increase measurement accuracy and to analyze the influence of the number of phase-shifting steps and pitch of the projected triangular intensity-profile pattern on the measurement accuracy. Phase-shifting algorithms to generate the intensity ratio, essential for surface reconstruction, were developed for each measurement method. Experiments determined that higher measurement accuracy can be obtained with a greater number of phase-shifting steps and a lower value of pitch, as long as the pitch is appropriately selected to be divisible by the number of phase-shifting steps and not below an optimal value, where intensity-ratio unwrapping failure would occur.