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The phase-shifting interferometry has been intensively studied for more than half a century, and is still actively investigated and improved for more demanding precision measurement requirements. A proper phase-shifting algorithm (PSA) for phase extraction should consider various error sources including (i) the phase-shift errors, (ii) the intensity harmonics, (iii) the non-uniform phase-shift distributions and (iv) the random additive intensity noise. Consequently, a large pool of PSAs has been developed, including those with known phase shifts (abbreviated as kPSA) and those with unknown phase shifts (abbreviated as uPSA). While numerous evaluation works have been done for the kPSAs, there are very few for the uPSAs, making the overall picture of the PSAs unclear. Specifically, there is a lack of (i) fringe pattern parameters' restriction analysis for the uPSAs and (ii) performance comparison within the uPSAs and between the uPSAs and the kPSAs. Thus, for the first time, we comprehensively evaluated the pre-requisites and performance of four representative uPSAs, the advanced iterative algorithm, the general iterative algorithm (GIA), the algorithm based on the principal component analysis and the algorithm based on VU factorization, and then compare the uPSAs with twelve benchmarking kPSAs. From this comparison, the demand for proper selection of a kPSA, and the restriction and attractive performance of the uPSAs are clearly depicted. Due to the outstanding performance of the GIA, a hybrid kPSA-GIA is proposed to boost the performance of a kPSA and relieve the fringe density restriction of the GIA. Economic Development Board (EDB) Ministry of Education (MOE) Published version Economic Development Board - Singapore (S17-1579-IPP-II); Ministry of Education - Singapore (MOET2EP20220-0008).

Fringe projector profilometry (FPP) is an important three-dimensional (3D) measurement technique, especially when high precision and speed are required. Thus, theoretical interrogation is critical to provide deep understanding and possible improvement of FPP. By dividing an FPP measurement process into four steps (system calibration, phase measurement, pixel correspondence, and 3D reconstruction), we give theoretical analysis on the entire process except for the extensively studied calibration step. Our study indeed reveals a series of important system properties, to the best of our knowledge, for the first time: (i) in phase measurement, the optimal and worst fringe angles are proven perpendicular and parallel to epipolar line, respectively, and can be considered as system parameters and can be directly made available during traditional calibration, highlighting the significance of the epipolar line; (ii) in correspondence, when two sets of fringes with different fringe orientations are projected, the highest correspondence precision can be achieved with arbitrary orientations as long as these two orientations are perpendicular to each other; (iii) in reconstruction, a higher reconstruction precision is given by the 4-equation methods, while we notice that the 3-equation methods are almost dominatingly used in literature. Based on these theoretical results, we propose a novel FPP measurement method which (i) only projects one set of fringes with optimal fringe angle to explicitly work together with the epipolar line for precise pixel correspondence; (ii) for the first time, the optimal fringe angle is determined directly from the calibration parameters, instead of being measured; (iii) uses 4 equations for precise 3D reconstruction but we can remove one equation which is equivalent to an epipolar line, making it the first algorithm that can use 3-equation solution to achieve 4-equation precision. Our method is efficient (only one set of fringe patterns is required in projection and the speed is doubled in reconstruction), precise (in both pixel correspondence and 3D reconstruction), and convenient (the computable optimal fringe angle and a closed-form 3-equation solution). We also believe that our work is insightful in revealing fundamental FPP properties, provides a more reasonable measurement for practice, and thus is beneficial to further FPP studies. Ministry of Education (MOE) Published version Funding: Ministry of Education - Singapore (MOE-T2EP20220-0008); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2019221); National Natural Science Foundation of China (11903036, 1803037, 61805243, 62127901); Bureau of International Cooperation, Chinese Academy of Sciences (181722KYSB20180015); Key Research Program of Frontier Science, Chinese Academy of Sciences (QYZDJ-SSW-JSC038).

Computer-Controlled Optical Surfacing (CCOS) has been greatly developed and widely used for precision optical fabrication in the past three decades. It relies on robust dwell time solutions to determine how long the polishing tools must dwell at certain points over the surfaces to achieve the expected forms. However, as dwell time calculations are modeled as ill-posed deconvolution, it is always non-trivial to reach a reliable solution that 1) is non-negative, since CCOS systems are not capable of adding materials, 2) minimizes the residual in the clear aperture 3) minimizes the total dwell time to guarantee the stability and efficiency of CCOS processes, 4) can be flexibly adapted to different tool paths, 5) the parameter tuning of the algorithm is simple, and 6) the computational cost is reasonable. In this study, we propose a novel Universal Dwell time Optimization (UDO) model that universally satisfies these criteria. First, the matrix-based discretization of the convolutional polishing model is employed so that dwell time can be flexibly calculated for arbitrary dwell points. Second, UDO simplifies the inverse deconvolution as a forward scalar optimization for the first time, which drastically increases the solution stability and the computational efficiency. Finally, the dwell time solution is improved by a robust iterative refinement and a total dwell time reduction scheme. The superiority and general applicability of the proposed algorithm are verified on the simulations of different CCOS processes. A real application of UDO in improving a synchrotron X-ray mirror using Ion Beam Figuring (IBF) is then demonstrated. The simulation indicates that the estimated residual in the 92.3 mm × 15.7 mm CA can be reduced from 6.32 nm Root Mean Square (RMS) to 0.20 nm RMS in 3.37 min. After one IBF process, the measured residual in the CA converges to 0.19 nm RMS, which coincides with the simulation.

Introducing a carrier into a fringe pattern makes the phase retrieval problem well-posed, and has become one of the pillar techniques in optical metrology. The carrier fringe pattern analysis for high-accuracy phase retrieval is hence important, for which, many well-known methods have been developed, such as the Fourier transform method, the windowed Fourier transform based methods (the windowed Fourier filtering and the windowed Fourier ridges), the spatial carrier phase-shifting method, and the sampling moire method. These methods were developed independently and seemingly different. The purpose of this paper is to reveal the close relationships between them, and to connect these methods into a tight-knit methodology family.

In this Letter, a deep learning solution (Y4-Net, four output channels network) to one-shot dual-wavelength digital holography is proposed to simultaneously reconstruct the complex amplitude information of both wavelengths from a single digital hologram with high efficiency. In the meantime, by using single-wavelength results as network ground truth to train the Y4-Net, the challenging spectral overlapping problem in common-path situations is solved with high accuracy.

An accurate and flexible real-time digital image correlation (RT-DIC) system utilizing a pipelined CPU and GPU parallel computing framework is proposed. First, the respective advantages of CPU and GPU in performing the fast Fourier transform-based cross-correlation (FFT-CC) algorithm and the inverse-compositional Gauss Newton (IC-GN) algorithm of the employed path-independent DIC (PI-DIC) method are elucidated. Second, based on the different properties and performances of CPU and GPU, a pipelined system framework unifying five Variants of combinations of CPU and GPU is proposed, which can be flexibly applied to various practical applications with different requirements of measurement scales and speeds. Last, both the accuracy and speed of the entire pipelined framework are verified by a PC implementation of the RT-DIC system integrating Variants 2–5. Variants 2 and 5 are also implemented on an iPhone 5S for the feasibility investigation of realizing a portable RT-DIC system on mobile devices using the same framework.

High-speed three-dimensional (3D) shape measurement techniques based on fringe projection profilometry (FPP) have undergone huge advances over the past two decades. However, accurate 3D displacement mapping and deformation analysis of dynamic scenes using FPP remains an unsolved problem. Because fringe patterns are projected rather than attached on the tested surfaces, the full-field point-to-point correspondence cannot be accurately established between any two 3D shape results. To deal with this challenge, a DIC-assisted FPP for high-speed 3D shape, displacement and deformation measurement of textured surfaces is proposed. Firstly, a high-speed 3D shape measurement system is adopted using our recently proposed robust and efficient Gray-coded coding strategy, which can accurately reconstruct full-field shape of discontinuous surfaces with rich texture information. Then, the modulation-based method is proposed to retrieve high-quality texture map from three phase-shifting fringe patterns, which can eliminate the adverse influence of the nonuniform and time-varying ambient light. By matching the retrieved surface texture images at difference states using DIC, accurate point tracking between the measured 3D shape data can be fulfilled, leading to precise 3D displacement and deformation measurement. Experiments have verified that the proposed method can achieve 3D shape, displacement and deformation measurement of dynamic scenes at a frame rate of 542 fps without any extra expense of hardware or calibration for the FPP system. The presented method is reliable and promising for further 3D displacement mapping and deformation analysis of dynamic scenes using FPP.

Phase unwrapping, either spatial or temporal, plays an important role in fringe projection profilometry (FPP) to recover a continuous phase map. Compared with the spatial phase unwrapping (SPU), the temporal phase unwrapping (TPU), including multi/two-frequency, phase coding, and gray code methods, is more widely used due to its ability to measure discontinuous objects. However, multiple patterns are required in TPU, which limits the measurement speed and the application of dynamic object measurement. To overcome this problem, different techniques, such as binary defocusing projection, two-plus-one phase-shifting algorithm, geometry/continuity constrained phase unwrapping, and ternary/quaternary gray code phase unwrapping have been proposed and actively studied recently, which either enhance the projection speed or reduce the number of projected patterns. However, there are very few studies on how these techniques affect the accuracy of TPU and which TPU method is most accurate under the scenario of high-speed measurement. This paper compares the accuracy of the two-frequency (TF), phase coding (PC), and gray code (GC) methods with different situations, including the traditional 8-bit focused FPP (aFPP), the high-speed binary defocused FPP (bFPP), and the geometry/continuity constrained binary defocused FPP (cFPP). We classify the phase unwrapping errors caused by system noises into uniformly and non-uniformly distributed errors, and analyze their distributions and rates in different TPU methods and different FPP systems. By comparative simulations and experiments, we find that, for low-frequency phase unwrapping, all the three TPU methods have a good result, while for high-frequency phase unwrapping which is desired in high-quality measurement, GC in aFPP and TF in cFPP provide higher accuracy. Thus, for measurement where accuracy is more concerned than the speed GC in aFPP is preferred, for dynamic measurement where extreme high speed is required, TF in cFPP is suggested.

A simple and efficient visible light positioning (VLP) system based on a single LED lamp is proposed and demonstrated. In our system, a common circular LED lamp with a point marker is used as a transmitter and a smartphone camera serves as a receiver. The LED image is no longer treated as a point as in the existing works, but as an image whose geometric features are exploited to determine the receiver's orientation and location relative to the reference LED lamp. The expressions for determining the receiver's orientation and location are derived in terms of the geometric parameters. Our algorithm enhances the system robustness by overcoming the possible interruptions in conventional VLP systems, which require multiple LEDs. Meanwhile, it avoids the usage of angular sensors, which are unable to provide reliable and accurate measurements on receiver's orientation in mobile indoor environment. We experimentally evaluate and analyze the performance of the proposed VLP system. A centimeter-level positioning accuracy is achieved in an area of 3 m × 3 m.

Size prediction and garment customization are two main goals of body measurement for garment design. Traditional body measurement, involving manual measurement and trying clothes in person, is time-consuming and not cost-efficient. With the help of 3D body scanner and neural networks, body measurement can be fast and accurate, thus reducing the cost. This paper introduces neural network models to predict body sizes and the measurements used to customize clothes from various body data. Three kinds of input data are used: raw 3D point clouds of human bodies, key body locations, and estimated body measurements. Raw point clouds are collected by scanning the participants' body, and key body locations and estimated measurements are automatically computed by existing software based on the point clouds. Then the manual measurement is applied to the participants to obtain the size labels and useful measurements for garment customization, which are used as the ground-truth values of output data. Different network structures are utilized for different kinds of input data. The results show that neural networks can achieve decent performance in predicting measurements for making clothes. While the results of the three models are comparable, the feed-forward network model with estimated measurements achieves the best result in numerical measurement prediction. In terms of size label prediction, the models using estimated measurements achieve similar results, while the CNN model with key body locations and the simplified PointNet model applied on raw point clouds are unable to achieve high accuracy. The initial attempts show the potential of using networks for body measurement. The models can be further improved with a larger amount of data, in order to make it production-ready.

Phase measurement techniques using a single-shot carrier fringe pattern play an important role in optical science and technology and have been widely used for various applications. In this paper, we focus on the comparative study of two major fringe analysis techniques, the sampling moire (SM) and the windowed Fourier transform (WFT). While SM converts a single-fringe pattern to multiple phase-shifted moire fringe patterns to extract the phase information in the spatial domain, WFT obtains the phase information in the windowed Fourier domain; thus, the two methods look entirely different. We evaluate the phase extraction errors of SM and windowed Fourier ridges (WFRs) as a typical WFT method for both linear and nonlinear phases with/without noise against the reference Fourier transform (FT) technique. For the simulated fringe patterns with linear or nonlinear phase and different random noise level, all the methods have high phase extraction accuracies. For a real experiment with more complicated phase and discontinuities, SM and WFR, both local methods, yield quite similar results and outperform FT.

Digital holography (DH) is one of the most promising quantitative phase measurement techniques and has been successfully used in 3D imaging and measurement. One of its attractive advantages is its excellent theoretical axial measurement accuracy of better than 1 nanometer. However, in practice, the axial accuracy has been quoted to be in the range of tens nanometers limited by the axial errors existing in DH system. In order to improve the axial measurement accuracy to approach the theoretical value, it is necessary to identify error sources and then reduce the errors according to their properties. In this paper, the space-variance effect of digital holography system is investigated and demonstrated to be an important systematic axial measurement error (SAME) source, especially for features with high frequency. The properties of the space-variant SAME are investigated through simulations and experiments. The object position, object height, object frequency content and object-CCD distance are found to be related to the space-variant SAME. Careful and appropriate placement of the object according to its features is thus necessary to reduce such SAME in a DH system. Based on the investigation, the guideline to appropriately position an object according to its properties is provided in this work.

Phase unwrapping is an important but challenging issue in phase measurement. Even with the research efforts of a few decades, unfortunately, the problem remains not well solved, especially when heavy noise and aliasing (undersampling) are present. We propose a database generation method for phase-type objects and a one-step deep learning phase unwrapping method. With a trained deep neural network, the unseen phase fields of living mouse osteoblasts and dynamic candle flame are successfully unwrapped, demonstrating that the complicated nonlinear phase unwrapping task can be directly fulfilled in one step by a single deep neural network. Excellent anti-noise and anti-aliasing performances outperforming classical methods are highlighted in this paper. Published version

The speed of a fringe projection profilometry system is of great importance for the measurement of dynamic objects. The previous quaternary gray-code phase unwrapping method uses only five patterns to recover an absolute phase which is suitable for high-speed measurement. However, this method requires a pre-knowledge of the minimum depth and it limits the object to move in a small range of depth. To extend the measurement range, we proposed an octal gray-code phase unwrapping method, in which five patterns are required. We apply this method to different objects and analysis the experiment results. This method can recover a good-quality phase map for sample objects, while its performance is decreased when the object is complex or with an improper defocusing level. Accepted version

Precision optics have been widely required in many advanced technological applications. X-ray mirrors, as an example, serve as the key optical components at synchrotron radiation and free electron laser facilities. They are rectangular silicon or glass substrates where a rectangular Clear Aperture (CA) needs to be polished to sub-nanometer Root Mean Squared (RMS) to keep the imaging capability of the incoming X-ray wavefront at the diffraction limit. The convolutional polishing model requires a CA to be extended with extra data, from which the dwell time is calculated via deconvolution. However, since deconvolution is very sensitive to boundary errors and noise, the existing surface extension methods can hardly fulfill the sub-nanometer requirement. On one hand, the figure errors in a CA were improperly modeled during the extension, leading to continuity issues along the boundary. On the other hand, uncorrectable high-frequency errors and noise were also extended. In this study, we propose a novel Robust Iterative Surface Extension (RISE) method that resolves these problems with a data fitting strategy. RISE models the figure errors in a CA with orthogonal polynomials and ensures that only correctable errors are fit and extended. Combined with boundary conditions, an iterative refinement of dwell time is then proposed to compensate the errors brought by the extension and deconvolution, which drastically reduces the estimated figure error residuals in a CA while the increase of total dwell time is negligible. To our best knowledge, RISE is the first data fitting-based surface extension method and is the first to optimize dwell time based on iterative extension. An experimental verification of RISE is given by fabricating two elliptic cylinders (10 mm × 80 mm CAs) starting from a sphere with a radius of curvature around 173 m using ion beam figuring. The figure errors in the two CAs greatly improved from 204.96 nm RMS and 190.28 nm RMS to 0.62 nm RMS and 0.71 nm RMS, respectively, which proves that RISE is an effective method for sub-nanometer level X-ray mirror fabrication.

Speckle projection profilometry (SPP), which establishes the global correspondences between stereo images by projecting only a single speckle pattern, has the advantage of single-shot 3D reconstruction. Nevertheless, SPP suffers from the low matching accuracy of traditional stereo matching algorithms, which fundamentally limits its 3D measurement accuracy. In this work, we propose a single-shot 3D shape measurement method using an end-to-end stereo matching network for SPP. To build a high-quality SPP dataset for training the network, by combining phase-shifting profilometry (PSP) and temporal phase unwrapping techniques, high-precision absolute phase maps can be obtained to generate accurate and dense disparity maps with high completeness as the ground truth by phase matching. For the architecture of the network, a multi-scale residual subnetwork is first leveraged to synchronously extract compact feature tensors with 1/4 resolution from speckle images for constructing the 4D cost volume. Considering that the cost filtering based on 3D convolution is computationally costly, a lightweight 3D U-net network is proposed to implement efficient 4D cost aggregation. In addition, because the disparity maps in the SPP dataset should have valid values only in the foreground, a simple and fast saliency detection network is integrated to avoid predicting the invalid pixels in the occlusions and background regions, thereby implicitly enhancing the matching accuracy for valid pixels. Experiment results demonstrated that the proposed method improves the matching accuracy by about 50% significantly compared with traditional stereo matching methods. Consequently, our method achieves fast and absolute 3D shape measurement with an accuracy of about 100µm through a single speckle pattern.

Phase-extraction is important in various fields of optical metrology, for which, many phase-shifting algorithms have been developed. Among them, the advanced iterative algorithm (AIA) can accurately extract phase from fringe patterns with random unknown phase-shifts by iteratively estimating the phase and phase-shifts. However, these iterations make the AIA much slower than traditional phase-shifting algorithms. This problem is severer when both the pixel number and the frame number are large for high resolution and accuracy, restricting AIA’s wide application. In this paper, based on the detailed analysis of the algorithm’s structure, a fully parallelized GPU-based AIA (gAIA) is proposed for the first time. Without scarifying the phase extraction accuracy, the gAIA achieves 500 × speedup comparing with the sequential implementation on a single-core-CPU, and 10 × speedup comparing with the state-of-the-art partial GPU implementation which has a potential convergence issue. Also, for the first time, the real-time phase extraction with AIA is achieved by using a normal NVIDIA RTX 2080 Ti GPU, i.e., the proposed gAIA only takes 26.55 ms to extract phase from 13 frames of fringe patterns with 2048 × 2048 pixels per frame. Finally, through the implementation and testing of the gAIA, it is discovered that increasing the frame number has little effect on the speed performance, which is against our intuition. As a consequence, more frames can be used for gAIA to increase the phase extraction accuracy with little influence on the speed.

The effectiveness of the renowned empirical mode decomposition (EMD) is affected by the mode-mixing problem (MMP) if a signal contains intermittent modes. The ensemble EMD (EEMD) and several modified and extended algorithms solve this problem by adding random white noises. However, the necessary large size of the ensemble and the inevitable manual intervention limits the application of EEMD. In this letter, a novel regenerated phase-shifted sinusoid-assisted EMD (RPSEMD) is proposed. Sinusoids with different scales are iteratively generated and added to cope with all possible MMPs in different intrinsic modes (IMs), where each sinusoid is designed adaptively and automatically. Furthermore, the sinusoids are shifted for better retaining the details of each IM and eliminating the added sinusoids. In the comparison experiments, the RPSEMD provides more reasonable results with less computation time. Accepted Version

The transport of intensity equation (TIE) is an ideal candidate for phase imaging with partially coherent illuminations. TIE has the advantages of simplicity in phase calculation due to its closed-form solution and no requirement for a reference beam and phase unwrapping due to its non-interferometric nature. However, TIE requires multiple through-focus intensity images, and is very sensitive to image boundaries and noise. Thus, in this paper, we combine deep learning with TIE, abbreviated as dTIE. After being trained by TIE phase results, the dTIE retains the advantages of TIE, and overcomes the shortcomings of TIE as follows: (i) only one de-focus intensity image is required for phase imaging while the result is very close to the TIE result with SSIM index reaches 0.95, enabling more efficient phase imaging; (ii) the boundary problem automatically disappears due to the translation invariance of the convolutional networks; (iii) it is insensitive to noise even with very heavy noise. All these enhancements are verified in the application of dTIE for phase imaging of real cells.

Although the 3D-DIC method has matured both theoretically and technically, the 2D-DIC method still plays an important role in in-plane deformation measurements. However, the accuracy of 2D-DIC is affected by out-of-plane motion (including out-of-plane translation and out-of-plane rotation) and non-perpendicular alignment. To tackle this problem, we propose to directly measure these unfavorable error sources by cross-shaped structured light (CSSL) and the optical triangulation method. Subsequently, pseudo-strains are calculated and compensated using an integrated mathematical model developed in this paper. To avoid mutual interference between the structured light strips and the speckle image, color coding is also proposed to use different color information for 2D-DIC processing and error compensation. Experiments with controlled out-of-the-plane motions show that the mean error after compensation can be as small as 50με. Uniaxial tension tests were also conducted to verify the feasibility of the proposed method in the real material parameter estimation experiment. This work was supported by the China Scholarship Council (grant nos. 201906340036).

We present an erratum and generalization to our Letter [Opt. Lett. 45, 3115 (2020)OPLEDP0146-959210.1364/OL.392102]. This erratum corrects an error in Eq. (12), and the generalization converts R h to k h for more general situations of wavelengths. Neither has any influence on the conclusions of the original Letter.