Four-step Phase-shifting Method Research Articles

A three-dimensional measurement method based on reusing equally spaced binary stripes

A three-dimensional measurement method based on reusing equally spaced binary stripes

Few-fringe-based phase-shifting profilometry employing hilbert transform

In optical measurement techniques, when the projector works at a constant projection rate, reducing the number of projected fringe patterns is an effective way to reduce the projection time. In this paper, we propose a few-fringe-based phase-shifting profilometry employing Hilbert transform. We project only two fringe patterns for each frequency of fringes, with the phase shift between the fringes designed to be π. The Hilbert transform makes these two captured fringes phase shifted by π/2, thus transforming the original two fringes into four, and the wrapped phase can be obtained through these four fringes with a phase difference of π/2. To improve the accuracy and robustness of the method, we adopt three frequencies of fringes for phase unwrapping. Further, since different phase unwrapping methods can be used for the different frequencies of the fringes, we propose a heterodyne 2H+2 M+2 L method and a hierarchical 2H+2 M+2 L respectively. Multiple experimental results have confirmed the feasibility and effectiveness of the two 2H+2 M+2 L methods, and the different characteristics of the reconstructed shapes obtained by the two 2H+2 M+2 L methods are compared to provide useful guidance in the selection of the required method for different application scenarios. The experimental results demonstrate that when using the heterodyne and hierarchical methods for phase unwrapping, the RMSE of the 2H+2 M+2 L method is only higher than that of the three-frequency four-step phase-shifting method by 0.0056 and 0.0024 respectively, but this 2H+2 M+2 L method improves the efficiency of the three-frequency four-step phase-shifting method by 50%.

Phase Deflectometry for Defect Detection of High Reflection Objects.

A method for detecting the surface defects of high reflection objects using phase deflection is proposed. The abrupt change in the surface gradient at the defect leads to the change in the fringe phase. Therefore, Gray code combined with a four-step phase-shift method was employed to obtain the surface gradients to characterize the defects. Then, through the double surface illumination model, the relationship between illumination intensity and phase was established. The causes of periodic error interference were analyzed, and the method of adjusting the fringe width to eliminate it was proposed. Finally, experimental results showed the effectiveness of the proposed method.

A novel experimental procedure for lock-in thermography on solar cells

<abstract> <p>The occurrence of defects in solar cells is intrinsically related to a reduction in the efficiency and reliability of these devices. Therefore, monitoring techniques, such as lock-in thermography, electroluminescence and the I-V characteristic curve are adopted in order to evaluate the integrity of the solar cells. In the present work, a novel experimental procedure for the lock-in thermography of solar cells is proposed, aiming to improve the detection capability of the assay. Conventional techniques use pulse width modulation to operate the cell at a fixed point on the I-V curve. Instead, we propose a methodology based on a sinusoidal electric current excitation in order to extend the range of operational points that are close to the maximum power point as the cell operates in the field. Some traditional image processing techniques (principal component analysis, the fast Fourier transform and the four-step phase-shifting method) have been used to analyze the thermal images captured by an infrared camera during steady-state operation mode of the solar cells using both sinusoidal electric current signal and standard pulse width modulation procedures. Comparison between the results of both procedures found that this novel approach provides smoother and clearer delimitation of the defects. Furthermore, the contrast of the phase images was found to exhibit significant changes between the defective and non-defective regions for different modulation frequencies and types of defects. From the achieved results, it was possible to obtain a satisfactory characterization of the existing defects.</p> </abstract>

Azimuth measurement based on OAM phase spectrum of optical vortices

The optical vortex is a new type of structured light field with a helical wavefront. Because of carrying orbital angular momentum (OAM), the optical vortex can be used to measure the shape and angular orientation of a static object in optical remote sensing. These measurements take advantage of variations in the intensity or phase of the OAM mode. In this paper, we propose a method of target azimuth measurement based on OAM phase spectrum of optical vortices. The OAM phase spectrum consists of helical phase terms in different OAM modes, where the helical phase is obtained by a four-step phase-shifting method. By using the approximate duality relationship between the OAM phase spectrum and the azimuth, we can accurately measure the azimuth of a single target or multiple targets. The resolution of the azimuth measurement is inversely proportional to the OAM scanning range. Experimentally, the target azimuth measurement by employing OAM phase spectrum analysis is realized and the measurement error is less than 2.5°. On this basis, the inverse relationship between the OAM scanning range and azimuth resolution is verified, where the error between experimental and theoretical azimuth resolution is less than 0.3°.

Research on partially coherent spatial light interference microscopy.

Based on partial coherence theory, this study rigorously deduces the principle of spatial light interference microscopy (SLIM) and improves the calculation method of SLIM. The main problem we found with SLIM is that it simply defaults the phase of the direct light to 0. To address this problem, we propose and experimentally demonstrate a double four-step phase shift method. Simulation results show that this method can reduce the relative error of oil-immersed microsphere reconstruction to about 3.7%, and for red blood cell reconstruction, the relative error can be reduced to about 13%.

Analysis of the influence of simultaneous phase-shifting shearing interference polarizer

The aspheric surface testing system based on the shearing interference principle is used to better analyze how the installation and fabrication errors of linear polarizer affects the precision of the aspheric surface. The effects of the angle error of the linear polarizer and the phase-shifting error of the wave plate on the reconstruction of the aspheric surface are studied. Constructing the model of the Jones matrix with the error term system, the reconstruction surface and residual after the corresponding error system are obtained using the four-step phase-shifting and differential Zernike method. The polarizer array of an angle of θ was varied from ( θ − 1 deg ) to ( θ + 2 deg ) every 0.2 deg, and the changes of the fitting surface and the residual surface were simulated. The phase-shifting error of the simulated one-fourth wave plate was <λ / 300. Then, the change of the fitting surface and residual surface when these two errors exist at the same time were simulated. The simulation results show that the influence of the phase-shifting error of the wave plate on the fitting surface is much greater than the angle error of the polarizer array. When assembling the interference system, the phase-shifting error of the wave plate should be <λ / 300 when the light transmission angle error of the polarizer array is between −0.2 deg and 0.2 deg, which will ensure the testing precision of the aspheric surface testing system.

Vectorial Manipulation of High-Resolution Focusing Optical Field through a Scattering Medium

The manipulation of the polarization states of the light transmitted through a scattering medium has become an emerging field due to the novel fundamental physics interest and potential applications. Here, the manipulation of the polarization states in the focusing high-resolution optical field (points and vector beams) after passing a scattering medium is theoretically and experimentally demonstrated. The vector transmission matrix (VTM) of a scattering medium is measured with the vector basis of orthogonally circular polarizations by the two-dimensional (2D) holographic grating combined with the four-step phase-shifting method. The incident wavefronts for the creation of desired high-resolution optical fields through a scattering medium are modulated according to the calculation with the VTM of the medium. The theoretical and experimental results show that the constructed high-resolution optical field with spatially variant states of polarization can be realized through frosted glass. These results provide a new way to vectorially manipulate the constructed high-resolution optical field by passing through a scattering medium.

Analysis of phase errors introduced by single-shot color phase-shifting profilometry on colored objects

We aim to report an error analysis on single-shot profilometry when projecting a composite colored fringe pattern onto a surface containing different colors. Several studies have analyzed white surfaces using profilometry; however, when color is incorporated, new considerations must be taken into account. In this work, it is found theoretically and corroborated experimentally that each color introduces local phase variations that render in phase errors with a severity that is related to the crosstalk matrix of the projector–camera system. It is also shown that the phase errors are phase-shifting dependent and could be reduced with appropriate phase recovery algorithms. Experimental results are presented and compared with those obtained using a standard temporal four-step phase-shifting method.

The analysis of Influencing Factors of Morie Fringe based on 3D Imaging Simulation and experiment measurement

We propose a three-dimension imaging construction scheme based on Moire fringe and investigate the influencing factors for the fringe generation. Specifically, the principle of the Moire fringe is introduced and a corresponding proof-of-principle experiment is designed and accrued out accompanying with corresponding programming codes subsequently. Four-step phase-shift method and Fourier transform are applied in the experiment to obtain the phase information. Afterwards, the Fourier transformation is added up to a model that shows the surface of the object in terms of the unwrap the phase package. According to the results, the ability of Moire fringe to image is confirmed, which pave a path for various applications, e.g., medical treatment and cryptology.

Analysis and Compensation of Phase Error Caused by Noise and Nonlinear Response of 3-D Measurement System

For static objects where the fringes are not saturated, the phase accuracy is mainly affected by the unavoidable factors of the introduction of fringe noise and the nonlinear response of the measurement system. The existing noise-induced phase error model is based on a Gaussian noise model, but in fact the noise of the fringe pattern is random. In this work, we establish a new evaluation method that directly converts the noise variance into phase variance, and we compare the anti-noise performance of the three-step and four-step phase-shift methods according to the phase error coefficient. Theoretical and experimental results confirm that the anti-noise performance of the four-step phase-shifting method is 25% higher than that of the three-step phase-shifting method. Subsequently, we also established a phase error model for the nonlinear response of the three-dimensional (3D) measurement system. Since the low measurement efficiency of the existing double <i>N</i>-step phase-shifting (DNPS) method, an improved double <i>N</i>-step phase-shifting (IDNPS) method is proposed. The RMSE of the compensated phase with IDNPS is approximately equal to that with DNPS, but the number of additional fringes with IDNPS is reduced to half that with DNPS. The total time cost of using IDNPS is reduced to 68.5% of using DNPS. Simulations and experiments are conducted to verify the validity of the theoretical analysis results.

3D reconstruction of structured light based on infrared MEMS

The traditional 3D reconstruction process of structured light is based on a DLP visible light grating projection method; the DLP method has many flaws, including great bulk, and harmful to human eyes caused by visible light and so on. In this article, we propose a method based on infrared micro electro mechanical systems (MEMS) instead of DLP to generate programmable coded structured light. First, MEMS emits a laser beam by using a point-beam laser. Then, the laser beam is modulated by the optical system and transformed into a laser line, which is reflected by MEMS and projected onto the object to be measured. The grating pattern can be a square wave or a sinusoidal wave for different needs. Due to the speckle characteristics of the infrared laser, the phase error obtained by the traditional four-step phase-shifting method is very large. Therefore, an eight-step phase-shifting method based on the Gaussian filter is proposed to improve the decoding accuracy of the grating. An improved calibration method based on phase space is used to calibrate the monocular system. Finally, the 3D data of the object are obtained according to this method, and the 3D reconstruction accuracy is better than 0.1 mm.

Fast two-step phase-shifting method for measuring the three-dimensional contour of objects

A two-step phase-shifting method is proposed. The method realized the rapid measurement of the three-dimensional (3D) contour of some objects using mean envelope denoising based on the striped structured light. First, Butterworth low-pass pre-filtering in the frequency domain was performed on the two deformed fringe images containing the height information of the objects. It reduced the influence of higher harmonics. Then the peak and valley values of fringes with sub-pixel accuracy were obtained through numerical interpolation and point-by-point comparison. Furthermore, the mean envelope was accurately obtained and removed. After removing the mean envelope, the rectangular low-pass filtering was used to thoroughly filter out the remaining background signal. Finally, the fast reconstructions of the 3D contour of the objects were realized. The 3D contour reconstructions of the objects were simulated and experimentally measured using the method proposed. The calculation speed and accuracy were compared with the traditional four-step phase-shifting method and other two-step phase-shifting methods. The method has the advantages of fast calculation speed and high calculation accuracy. It is a valuable method for quickly measuring the 3D contour of some objects.

High-efficiency single-pixel imaging using discrete Hartley transform

Single-pixel imaging technology is popular with invisible wavelengths and low light environments. However, the time-consuming steps hindered the development of single-pixel imaging technology. To improve imaging efficiency, a high-efficiency one-step single-pixel imaging method based on the discrete Hartley transform is proposed. The proposed method does not require a large number of fringe patterns and only requires a real-number calculation. The number of fringe patterns required for the proposed method is only half of that required for the four-step phase-shift Fourier method at the same sampling rate. Although a one-step method, it also uses the idea of differential measurements and adds upsampling processing strategies, which simultaneously improve the signal-to-noise ratio of the recovered image. The simulation shows that the peak signal-to-noise ratio and structural similarity index of the recovered target scene exceed 20 dB and 80%, respectively, when the sampling rate is 30%. Only 20 164 patterns are needed to reconstruct a (256 × 256)-pixel image. After defocusing the gray stripe pattern into a binary pattern, it only takes milliseconds to project these patterns into the target. It can be seen that the experimental results of the proposed method are significantly better than those of the two-step phase-shift method under dramatical noise interference. With the rapid development of advanced equipment, this method will represent significant progress in the real-time reconstruction of single-pixel imaging.

The sensitivity and the measuring range of the typical differential optical flow method for displacement measurement using the fringe pattern

Emerging from motion analysis with the benefits of non-contact, high sensitivity, high precision, and simple operation, and the optical flow method can be utilized to measure micro-displacement. Because only one shot of the deformed fringe pattern is required in measurement, it is suitable for dynamic measurement. As a new tool for fringe analysis, some of its basic characteristics have not been found out, such as the measuring range and sensitivity, which are essential to displacement measurement. In this work, these basic characteristics of typical optical flow algorithms, the global H–S algorithm, and the local L–K algorithm, are discussed by using numerical simulation. The results prove that the minimum of the measurable phase change by both the H–S algorithm and the L–K algorithm is10−13πwith the relative error less than 0.4% under noiseless. The corresponding displacement on the image plane is 1.6×10−12 pixels, which is as good as that of the four-step phase-shifting method. The results also show that when the relative error is less than 2% and the Root-Mean-Square (RMS) error is less than 3%, the measurement range of the H–S and the L–K algorithm is about π/6 and π/2, respectively. But the minimum of the measurable phase change is decreased to 0.01π, which is 0.16 pixels displacement on the image plane, with the relative error less than 0.77% in the H–S algorithm and 1.71% in the L–K algorithm when the fringe patterns are polluted by the Gaussian noise of 40 dB. With increasing noise, the measurable maximum of the displacement will decrease while the minimum increase and the measuring range will lessen.

Three-dimensional reconstruction Based on Tri-frequency Heterodyne Principle

Abstract The traditional 3d reconstruction method of stereo vision is based on image feature point extraction, but it is very inconvenient to extract feature points in more complex cases, and the reconstructed 3D information is not comprehensive. In this paper, a 3d reconstruction method based on three-frequency heterodyne four-step phase shift method is proposed. System parameters is obtained by binocular calibration algorithm first, and then using digital grating projector to the object under test of three groups of different frequency phase shift fringe and collecting images, using the four step phase-shift and three phase frequency heterodyne method to get the main value and get the absolute phase of continuous distribution, combining with the polar correction and phase matching reconstruction of 3 d point cloud data of the object to be tested, finally complete 3 d measurement in 3 d software. The experimental results show that the system can realize 3d dimension measurement of heterogeneous castings with high detection efficiency.

A Multifrequency Heterodyne Phase Error Compensation Method for 3D Reconstruction

In view of the problem of “jumping points” in the phase unwrapping of the multifrequency heterodyne principle, this paper proposes a novel method to improve the multifrequency heterodyne. By solving the root-mean-square error of the original frequency function, it includes the relationship between the error and the adjacent phase in the condition of constrained phase unwrapping, and it compensates phase ±2π of the skip point. To ensure the accuracy of phase unwrapping, the function with a “jump point” after each phase unwrapping and the absolute phase curve of the principal value function are used to establish the threshold judgment model of the least square method, and the initial phase unwrapping of the principal value function with different frequencies is carried out continuously. The simulation analysis of phase compensation with the four-step phase-shifting method shows that the error is reduced to 36% under the set environment. The experimental result of 3D reconstruction by measuring the flatness of the plate shows that the error decreases by 41% after phase compensation compared with before phase compensation. The three-dimensional reconstruction experiment of pitch measurement with a nut shows that the nut after phase compensation is smooth without noise, and the pitch error is 0.033 mm, which verified the method is workable and effective.

Filter construction using Ronchi masks and Legendre polynomials to analyze the noise in aberrations by applying the irradiance transport equation.

We tested different optical elements placed in three different positions by applying the irradiance transport equation (ITE), obtaining the wavefront $[ W(x,y) ]$[W(x,y)] and aberration surface [$ \textit{AS}(r,\theta ) $AS(r,θ)]. The existing noise in the captures $ I $I as well as in the $ W $W and $ AS $AS were analyzed applying several filters: first a filter based on Legendre polynomials (LP), generating the most probable points increasing the data resolution; second, a filter based on a 50 deg 2D-LP was used as a multilineal fit (multiple linear regression); and third, an ideal bandpass filter in the Fourier space after inducing a periodicity using Ronchi simulated masks with periods in $ x,y,xy $x,y,xy was used to perform data scanning (similar to the four-step phase-shifting method). Signal-to-noise ratio values were obtained for each proposed filter along with the most probable image free from noise, determined from a linear combination of the original data and the applied filters.

Accurate blind extraction of arbitrary unknown phase shifts by an improved quantum-behaved particle swarm optimization in generalized phase-shifting interferometry

An optimized method of accurately extracting the arbitrary unknown and unequal phase steps in phase-shifting interferometry is proposed. The approximate phase steps are first calculated based on the statistical nature of the diffraction field, and the mutated adaptive quantum-behaved particle swarm optimization is used to further extract the arbitrary unknown and unequal phase steps. We improve the mutated adaptive quantum-behaved particle swarm optimization by adding mutation operator with Gaussian probability distribution, thereby increasing the population diversity. The developed method here is fast, highly accurate, and can effectively overcomes the “sawtooth-solution phenomenon” often encountered in traditional direct search approach. We demonstrate the speed and quality of the solution by measuring the transmissive object with the four-step phase-shifting interference method, as is verified by both the computer simulations and optical experiments.

Digital four-step phase-shifting technique from a single fringe pattern using Riesz transform.

A digital four-step phase-shifting method for obtaining the optical phase distribution from a single fringe pattern is proposed in this Letter. By computing the first-, second-, and third-order Riesz transform components for a given fringe pattern, three π/2 phase-shifted fringe patterns are generated from the obtained Riesz components and, finally, the wrapped phase map is extracted. The validity of the proposed method is demonstrated on both the simulated and experimentally obtained fringe patterns. The performance of the proposed method is evaluated by using the image quality index and edge preservation index. Further, the performance of the proposed method is tested on speckled correlation fringes obtained from digital speckle pattern interferometry, and the resulting phase from the proposed method is compared with the phase obtained from three experimentally recorded phase-shifted fringe patterns. The obtained results reveal that the proposed method provides a simple and robust solution for optical phase extraction from a single fringe pattern with good accuracy and, therefore, make it suitable for real-time measurement applications.