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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.

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.

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

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.

Two-wavelength fringe projection profilometry (FPP) unwraps a phase with the unambiguous phase range (UPR) of the least common multiple (LCM) of the two wavelengths. It is accurate, convenient, and robust, and thus plays an important role in shape measurement. However, when two non-coprime wavelengths are used, only a small UPR can be generated, and the unwrapping performance is compromised. In this Letter, a spatial pattern-shifting method (SPSM) is proposed to generate the maximum UPR (i.e., the product of the two wavelengths) from two non-coprime wavelengths. For the first time, to the best of our knowledge, the SPSM breaks the constraint of wavelength selection and enables a complete (i.e., either coprime or non-coprime) two-wavelength FPP. The SPSM, on the other hand, only requires spatially shift of the low-frequency pattern with the designed amounts and accordingly adjusting the fringe order determination, which is extremely convenient in implementation. Both numerical and experimental analyses verify its flexibility and correctness.

Phase-shifting techniques are extremely important in modern optical metrology. The advanced iterative algorithm (AIA) is an elegant, flexible and effective phase-shifting algorithm that can extract phase from fringe patterns with arbitrary unknown phase-shifts. However, comparing it with traditional phase-shifting algorithms, AIA has not been sufficiently investigated on (i) its applicability to different types of fringe patterns; (ii) its performance with respect to different phase-shifts, frame numbers and noise levels and thus the possibility of further improvement; and (iii) the predictability of its accuracy. To solve these problems, a series of innovations are proposed in this paper. First, condition numbers are introduced to characterize the least squares matrices used in AIA, and subsequently a fringe density requirement is suggested for the success of AIA. Second, the performance of AIA regarding different phase-shifts, frame numbers and noise levels is thoroughly evaluated by simulations, based on which, an overall phase error model is established. With such understanding, three individual improvements of AIA, i.e., controlling phase-shifts, controlling frame numbers and suppressing noise, are proposed for better performance of AIA. Third, practical methods for estimating the overall phase errors are developed to make the AIA performance predictable even before AIA is executed. We then integrate all these three innovations into an enhanced AIA (eAIA), which solves all the problems we mentioned earlier. The significant contributions of eAIA include the insurability of the convergence, the controllability of the performance, and achievability of a desired accuracy. An experiment is carried out to demonstrate the effectiveness of eAIA. Economic Development Board (EDB) Published version Economic Development Board - Singapore (S17-1579-IPP-II).

Accurately extracting phase or phase derivative is the most important requirement in optical metrology. However, in practice, there are many error sources, among which nonlinear distortion in fringe patterns is often encountered. Several techniques have been proposed over time to remove the nonlinearity error. Among these techniques, the windowed Fourier ridges (WFR) algorithm has been shown to be an effective solution insensitive to nonlinearity, but it lacks a theoretical justification. In this paper, we theoretically analyze the local frequency estimation error and phase extraction error, which not only proves the mentioned insensitivity, but also supports the performance prediction and error control, and thus is very important and useful in optical measurement. The theoretical results have been verified by computer simulations. Other error sources such as model error and noise are also compared and discussed.

The applications of windowed Fourier fringe analysis in the past decade are reviewed. Because fringe patterns from different optical measurement systems are similar, the reviewed applications are classified according to the functions of the windowed Fourier transform being used in fringe pattern analysis: denosing exponential phase fields, demodulating carrier fringe patterns, getting phase derivatives, and utilizing local properties. From these applications, the windowed Fourier transform is shown to be effective and versatile for fringe pattern analysis.

Phase-shifting profilometry combined with Gray-code patterns projection has been widely used for 3D measurement. In this technique, a phase-shifting algorithm is used to calculate the wrapped phase, and a set of Gray-code binary patterns is used to determine the unwrapped phase. In the real measurement, the captured Gray-code patterns are no longer binary, resulting in phase unwrapping errors at a large number of erroneous pixels. Although this problem has been attended and well resolved by a few methods, it remains challenging when a measured object has step-heights and the captured patterns contain invalid pixels. To effectively remove unwrapping errors and simultaneously preserve step-heights, in this paper, an effective method using an adaptive median filter is proposed. Both simulations and experiments can demonstrate its effectiveness.

Optical interferometric techniques offer non-contact, high accuracy and full field measurements, which are very attractive in various research and application fields. Single fringe pattern processing is often needed when measuring fast phenomenon. However, several difficulties are encountered in phase retrieval, among which the discontinuity problem of the fringe pattern is challenging and requires attention due to the increasing complexity of manufactured pieces. In this paper, we propose a complete flowchart for discontinuous single fringe pattern processing, which uses segmentation as universal pre-processing for all discontinuous fringe pattern problems. Within the flowchart, we also propose a systematic way to introduce boundary-awareness into demodulation methods by a masking function to improve demodulation accuracy and a quality-guided scanning strategy with a novel composite quality map to improve demodulation robustness. To the best of our knowledge, this is the first time a complete solution is proposed for single and discontinuous fringe pattern processing. Three typical demodulation methods, the frequency-guided regularized phase tracker with quadratic phase matching, the windowed Fourier ridges, and the spiral phase quadrature transform, are used to demonstrate how demodulation methods can be made boundary-aware. The proposed methods are verified by successful demodulation results from both simulated and experimental fringe patterns. Published version

In this Letter, for the first time, to the best of our knowledge, we propose a digital holographic reconstruction method with a one-to-two deep learning framework (Y-Net). Perfectly fitting the holographic reconstruction process, the Y-Net can simultaneously reconstruct intensity and phase information from a single digital hologram. As a result, this compact network with reduced parameters brings higher performance than typical network variants. The experimental results of the mouse phagocytes demonstrate the advantages of the proposed Y-Net.

Phase shifting profilometry (PSP) using binary fringe patterns with projector defocusing is promising for high-speed 3D shape measurement. To obtain a high-quality phase, the projector usually requires a high defocusing level, which leads to a drastic fall in fringe contrast. Due to its convenience and high speed, PSP using squared binary patterns with small phase shifting algorithms and slight defocusing is highly desirable. In this paper, the phase accuracies of the classical phase shifting algorithms are analyzed theoretically, and then compared using both simulation and experiment. We also adapt two algorithms for PSP using squared binary patterns, which include a Hilbert three-step PSP and a double three-step PSP. Both algorithms can increase phase accuracy, with the latter featuring additional invalid point detection. The adapted algorithms are also compared with the classical algorithms. Based on our analysis and comparison results, proper algorithm selection can be easily made according to the practical requirement.

Extracting phase distribution from arbitrary phase-shifted fringe patterns, if possible, is very useful in phase-shifting interferometry. The advanced iterative algorithm (AIA) is introduced and the windowed Fourier ridges and least squares fitting (WFRLSF) is proposed. Both algorithms are sensitive to noise, which limits their applications to almost perfect fringe patterns. The windowed Fourier filtering (WFF) algorithm is proposed for both pre-filtering and post-filtering to suppress the noise. Simulation results show that with the effective noise suppression, the phase error is reduced to less than 0.1 rad. Experimental examples are also given for verification. The almost identical results produced by the AIA and the WFRLSF suggest that both algorithms can be used for phase extraction with cross-validation.

Fringe projection profilometry is widely used for three-dimensional (3-D) surface shape measurement using phase-shifting (PS) methods with multiple images or transform methods with single projected fringe pattern. In this paper, phase extraction methods from a single fringe pattern using different transform methods are compared using both simulations and experiments. The principles of Fourier transform (FT), windowed Fourier transform (WFT) and wavelet transform (WT) methods for fringe pattern processing are introduced. Implementation of 1-D and 2-D algorithms and phase compensation are discussed. Noisy and non-sinusoidal waveforms are involved into this comparison. The merits and limitations of each of these processing methods are indicated.

A simple phase unwrapping approach based on windowed Fourier filtering was proposed recently [K. Qian et al. A simple phase unwrapping approach based on filtering by windowed Fourier transform. Opt Laser Technol 2005;37:458–62]. The windowed Fourier filtering algorithm is an essential ingredient that suppresses the noise effectively and makes the phase unwrapping trivial. This paper adds a note on the threshold selection in the windowed Fourier filtering algorithm. A large interval can be selected as the threshold to obtain almost optimal filtering results. Once the selected threshold is suitable, it is almost optimal. This makes the threshold selection in the windowed Fourier filtering algorithm extremely easy.

A simple phase unwrapping approach based on windowed Fourier filtering (WFF) was proposed recently. In this paper, the effects of the WFF on edges are further investigated, which show that the phase near an edge remains unchanged and no special consideration is needed for the edges. This useful property will be theoretically proven and numerically verified.

The images used in phase-shifting interferometry are usually noisy and smoothing filters are frequently used. Filtering can be done on the fringe patterns before phase retrieving or on the phase map after phase retrieving. Filtering fringe patterns—a form of preprocessing, requires the same processing to be done on at least three patterns. Filtering phase map, on the other hand, needs processing of only one pattern. Preprocessing prevents noise from being carried through while post-processing might clip parts of the signal. Thus there is a need to evaluate which one is better. A theoretical analysis was carried out for this basic problem, followed by computer simulations and a real experiment for verification. The comparison was also extended to phase difference measurement which uses similar but slightly different phase retrieval algorithms.

Orientation and spacing are important features of fringe patterns, especially in speckle photography when the displacement information of the object is determined by pointwisely filtering the double-exposure specklegram. The Fourier transform and Radon transform are both simple and effective tools for solving this problem, which extract the information in spectral domain and spatial domain, respectively. A hybrid method combining Radon transform and Fourier transform is also possible. The results can be further improved by enhanced Fourier transform and enhanced Radon transform. The theories, as well as real applications, are given in this paper.

Fringe-based optical measurement techniques require reliable fringe analysis methods, where empirical mode decomposition (EMD) is an outstanding one due to its ability of analyzing complex signals and the merit of being data-driven. However, two challenging issues hinder the application of EMD in practical measurement. One is the tricky mode mixing problem (MMP), making the decomposed intrinsic mode functions (IMFs) have equivocal physical meaning; the other is the automatic and accurate extraction of the sinusoidal fringe from the IMFs when unpredictable and unavoidable background and noise exist in real measurements. Accordingly, in this paper, a novel bidimensional sinusoids-assisted EMD (BSEMD) is proposed to decompose a fringe pattern into mono-component bidimensional IMFs (BIMFs), with the MMP solved; properties of the resulted BIMFs are then analyzed to recognize and enhance the useful fringe component. The decomposition and the fringe recognition are integrated and the latter provides a feedback to the former, helping to automatically stop the decomposition to make the algorithm simpler and more reliable. A series of experiments show that the proposed method is accurate, efficient and robust to various fringe patterns even with poor quality, rendering it a potential tool for practical use.

By using a special Ronchi phase grating and polarization phase shifting method, four phase shifted patterns can be captured simultaneously, and then the dynamic deformation can be measured. The analysis and experiments are given.

The windowed Fourier transform is a useful technique for fringe pattern analysis. It has been shown that the proper selection of the window size is a balance between the linear phase approximation error and the influence of noise [Q. Kemao, On widow size selection in windowed Fourier ridges algorithm. Opt Lasers Eng, 2007, accepted for publication]. Since the fringe intensity and noise level usually vary spatially, the window size should also be spatially adaptive in order to reach a good balance for each pixel of the fringe pattern. This addendum first shows that the windowed Fourier transform with a spatially fixed window size (SFWS) is still practically useful and then discusses the window size competition strategies for the windowed Fourier transform with a spatially adaptive window size (SAWS). The windowed Fourier transform with a SAWS is theoretically better than that with a SFWS but it is also more challenging in use. The windowed Fourier ridges algorithm is used for analysis throughout this paper. This analysis is also applicable to the windowed Fourier filtering algorithm.

In Chapters 2 and 3, the WFRn and WFFn are developed for exponential phase fields (T1 fringe patterns). In Chapter 4, wrapped phase maps (T2) are filtered and unwrapped. In this short chapter, demodulation of carrier fringe patterns (T3) is discussed. Due to the presence of a carrier frequency, the demodulation is not difficult. The Fourier transform (FT) is a widely used technique that converts a carrier fringe pattern to an EPF, from which phase can be obtained. The WFRn/WFFn can be used with or without the FT technique—these two options will be compared in Section 5.1. Fringe projection profilometry provides carrier fringe patterns containing profile information. They often come with undesired nonlinear intensities. In Section 5.2, The WFRn/WFFn and FT will again be compared when the nonlinearity occurs. We will focus on 2D fringe patterns and 2D transforms because they can be easily adapted to 1D and higher dimensions.

Windowed Fourier ridges algorithm can provide a quality map to assist the quality-guided phase unwrapping. Its performance for discontinuous phase maps is investigated in this paper, where the influence of window size in the algorithm is examined. Three discontinuous phase boundaries, straight, curved, and fused, are tested for both noiseless and noisy situations. Encouraging results are observed. Small window size can be used for higher boundary detection accuracy and can be enlarged if noise is heavy and/or the discontinuities are not obvious in a small area.

Fringe patterns with noise and discontinuity are often encountered but difficult to analyze. Discontinuity-detectable and boundary-aware processing techniques are demanded. A local orientation coherence based fringe segmentation (LOCS) method and its cooperation with boundary-aware coherence enhancing diffusion (BCED) for discontinuous fringe pattern denoising are proposed in this paper. The LOCS method has three steps. First, as orientation coherence indicated by structure tensors is informative to describe fringe structures, it is selected for discontinuity recognition. Due to the complexity of the discontinuity problem, the detected boundary often has missing parts and is not very accurate. Boundary completion by cubic splines and boundary refinement based on partial structure tensors are further performed as the second and third steps, respectively. Subsequently, the BCED method is developed to adapt the original CED to fringe segments with irregular boundaries. Simulated and experimental fringe patterns are tested and successful results have been obtained.

The regularized phase tracker (RPT) is one of the most powerful approaches for demodulation of a single fringe pattern. However, two disadvantages limit the applications of the RPT in practice. One is the necessity of a normalized fringe pattern as input and the other is the sensitivity to critical points. To overcome these two disadvantages, a generalized regularized phase tracker (GRPT) is presented. The GRPT is characterized by two novel improvements. First, a general local fringe model that includes a linear background, a linear modulation and a quadratic phase is adopted in the proposed enhanced cost function. Second, the number of iterations in the optimization process is proposed as a comprehensive measure of fringe quality and used to guide the demodulation path. With these two improvements, the GRPT can directly demodulate a single fringe pattern without any pre-processing and post-processing and successfully get rid of the problem of the sensitivity to critical points. Simulation and experimental results are presented to demonstrate the effectiveness and robustness of the GRPT. Published version

An algorithm for dynamic phase retrieval in temporal speckle pattern interferometry using least squares method and windowed Fourier filtering is proposed. The least squares method is used to evaluate the phase change between two speckle patterns provided that the phase of either one speckle pattern has been estimated. The windowed Fourier filtering is used to eliminate the noise in the phase change. Based on these two techniques, the proposed algorithm determines the phase of the initial speckle pattern by phase shifting method at first, then the phase of the rest speckle patterns are retrieved by sequentially evaluating the phase changes between every two consecutive speckle patterns. The algorithm solves the problem of speckle decorrelation by refreshing the reference image frame by frame, and also avoids the problem of error accumulation during the reference image refreshing process by the windowed Fourier filtering. Two experimental results are presented to demonstrate the effectiveness and robustness of the proposed algorithm.

Fringe patterns produced by various optical interferometric techniques encode information such as shape, deformation, and refractive index. Noise affects further processing of the fringe patterns. Denoising is often needed before fringe pattern demodulation. Filtering along the fringe orientation is an effective option. Such filters include coherence enhancing diffusion, spin filtering with curve windows, second-order oriented partial-differential equations, and the regularized quadratic cost function for oriented fringe pattern filtering. These filters are analyzed to establish the relationships among them. Theoretical analysis shows that the four filters are largely equivalent to each other. Quantitative results are given on simulated fringe patterns to validate the theoretical analysis and to compare the performance of these filters.

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 windowed Fourier filtered and quality guided (WFF-QG) phase unwrapping algorithm was proposed recently [Appl. Opt.47, 5420-5428 (2008)10.1364/AO.47.005420APOPAI0003-6935], based on the windowed Fourier transform [Appl. Opt.47, 5408-5419 (2008)APOPAI0003-693510.1364/AO.47.005408] where the phase is assumed to be locally quadric. We consider a locally higher order polynomial phase. After the phase is filtered and unwrapped by the WFF-QG, it is postprocessed by a congruence operation (CO), so that the unwrapped phase is congruent to the original wrapped phase. The unwrapped phase can now be assumed to be a locally high-order polynomial, and, consequently, least squares fitting (LSF) is proposed to suppress the noise. This postprocessing algorithm is abbreviated as CO-LSF. The CO-LSF is theoretically a reasonable choice to improve the WFF-QG results, especially when the noise is severe. This is because for severe noise, a large window is necessary for reliable phase extraction in the WFF-QG. However, this large window makes the quadric phase assumption less reasonable and leads to a large phase error. The CO-LSF thus helps to reduce the phase error by more reasonably assuming that the phase is a high-order polynomial. The polynomial order of 4 is suggested for the CO-LSF, as higher order polynomials do not give significant improvement to the WFF-QG. One disadvantage of the CO-LSF is that it is more sensitive to phase discontinuities than the WFF-QG.

In optical interferometers, fringe projection systems, and synthetic aperture radars, fringe patterns are common outcomes and usually degraded by unavoidable noises. The presence of noises makes the phase extraction and phase unwrapping challenging. Windowed Fourier transform (WFT) based algorithms have been proven to be effective for fringe pattern analysis to various applications. However, the WFT-based algorithms are computationally expensive, prohibiting them from real-time applications. In this paper, we propose a fast parallel WFT-based library using graphics processing units and computer unified device architecture. Real-time WFT-based algorithms are achieved with 4 frames per second in processing 256×256 fringe patterns. Up to 132× speedup is obtained for WFT-based algorithms using NVIDIA GTX295 graphics card than sequential C in quad-core 2.5GHz Intel(R)Xeon(R) CPU E5420.

The nonlinear intensity response of a digital fringe projection profilometry (FPP) system causes the captured fringe patterns to be nonsinusoidal waveforms and leads to an additional phase measurement error for commonly used three- and four-step phase-shifting algorithms. We perform theoretical analysis of the phase error owing to the nonsinusoidal waveforms. Based on the derived theoretical model, a novel and simple iterative phase compensation algorithm is proposed to compensate the nonsinusoidal phase error. Experiments show that the proposed algorithm can be used for effective phase error compensation in practical phase-shifting FPP.

Electronic speckle pattern interferometry is one of the methods measuring the displacement on object surfaces in which fringe patterns need to be evaluated. Noise is one of the key problems affecting further processing and reducing measurement quality. We propose an application of coherence-enhancing diffusion to fringe-pattern denoising. It smoothes a fringe pattern along directions both parallel and perpendicular to fringe orientation with suitable diffusion speeds to more effectively reduce noise and improve fringe-pattern quality. It is a generalized work of Tang'semet al.'s/em[Opt. Lett.33, 2179 (2008)] model that only smoothes a fringe pattern along fringe orientation. Since our model diffuses a fringe pattern with an additional direction, it is able to denoise low-density fringes as well as improve denoising effectiveness for high-density fringes. Theoretical analysis as well as simulation and experimental verifications are addressed.

Some effective filtering methods for wrapped phase maps, a regularized phase-tracking method (RPT) without the regularization term, a multiple-parameter least-square method (MPLS), a windowed Fourier ridges method (WFR), an autocorrelation function method (ACF), and a sine/cosine average filter (SCAF), are analyzed in order to establish their transversal relationship. The analysis shows that principles of the RPT, MPLS, WFR, and ACF are equivalent and the SCAF also leads to the WFR by some extension, which elegantly unifies all these methods for filtering unwrapped phase maps.

Quality-guided phase unwrapping (QGPU) has been proven as an effective phase unwrapping method. As the QGPU assumes that the true phase map is continuous, it suffers from the discontinuity problem. To solve this problem, the windowed Fourier transform (WFT) based methods have been adopted to obtain effective quality measures. Nevertheless, in some special cases, using the quality maps from the WFT-based methods could not lead to a correct result. In this paper, we propose a method that incorporates the GVF snake model to assist the unwrapping process. This new method achieves better results than using the WFT-based methods directly in some discontinuous examples.

Fringe projection profilometry has become one of the most popular 3D information acquisition techniques being developed over the past three decades. However, the general and practical issues on valid point detection, including object segmentation, error correction and noisy point removal, have not been studied thoroughly. Furthermore, existing valid point detection techniques require multiple case-dependent thresholds which increase processing inconvenience. In this paper, we proposed a new valid point detection framework, which includes the k-means clustering for automatic background segmentation, unwrapping error correction based on theoretical analysis, and noisy point detection in both temporal and spatial directions with automatic threshold setting. Experimental results are given to validate the proposed framework.

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.

Novel approaches based on windowed Fourier transform for demodulation of fringe patterns were previously presented [Appl. Opt. 43, 2695-2702 (2004)], where extraction of phase and phase derivatives from either phase-shifted fringe patterns or a single-carrier fringe pattern was the main focus. I show that the same methods can be applied to process a single closed-fringe pattern in either noise reduction or phase approximation, which adds to the versatility of the windowed Fourier-transform method for fringe pattern analysis.

A phase Ronchi grating-based real-time polarization phase-shifting method can be efficiently used for dynamic phase measurement in optical interferometry. A thorough error analysis is required for exhibiting how error sources influence phase-measurement results. We analyze the phase-measurement errors that are induced by the retardation error and azimuth angle error of the quarter-wave plate, the azimuth angle error of polarizers, the phase and intensity aberrations of diffractive wave fronts, and pixel mismatch of the interferometric patterns. The results will also be useful for evaluating the phase-measurement accuracy of other similar systems.

Pixelated micropolarizer arrays (PMAs) have recently been used as key components to achieve real-time phase measurement. PMA fabrication by electron beam lithography and inductively coupled plasma-reactive ion etching is proposed in this work. A 320 × 240 aluminum PMA with 7.4 μm pitch is successfully fabricated by the proposed technique. The period of the grating is 140 nm, and the polarization directions of each of the 2 × 2 units are 0°, 45°, 90°, and 135°. The scanning electron microscopy and optical microscopy results show that the PMA has a good surface characteristic and polarization performances. When the PMA is applied to phase-shifting interferometry, four fringe patterns of different polarization directions are obtained from only one single frame image, and then the object wave phase is calculated in real time.

Quality-guided phase unwrapping (QGPU) is a widely used technique, and an adjoin list plays a very important role in the QGPU process. Indexed interwoven linked list (I2L2) is a data structure for implementing the adjoin list. In this paper, we propose three improvements on the I2L2. The first improvement is resumed searching, which records the highest nonempty level in the I2L2 and reduces the computational redundancy; the second is an adaptive mapping between the quality values and the I2L2 levels, which reduces the effect of concentrated quality value distribution. Last, I2L2-H, a new variant of the I2L2 combining the advantages of both the I2L2 and heap, is developed. With these three improvements, the improved I2L2 is over 6 times faster than the original one in the best cases, and it can process large phase maps in almost real time. Published version

Fringe patterns produced by various optical interferometric techniques encode information such as shape, deformation, and refractive index. Denoising and demodulation are two important procedures to retrieve information from a single closed fringe pattern. Various existing denoising and demodulation techniques require fringe direction/orientation during the processing. Fringe orientation is often easier to obtain but fringe direction is needed in some demodulation techniques. A quality-guided orientation unwrapping scheme is proposed to estimate direction from orientation. Two techniques, one based on windowed Fourier ridges and the other based on fringe gradient, are proposed for the quality-guided orientation unwrapping scheme. The direction qualities are compared for both simulated and experimental fringe patterns. Their application to demodulation technique is also given. Published version

Quality-guided phase unwrapping is a widely used technique with different quality definitions and guiding strategies reported. It is thus necessary to do a detailed comparison of these approaches to choose the optimal quality map and guiding strategy. For quality maps, in the presence of noise, transform-based methods are found to be the best choice. However in the presence of discontinuities, phase unwrapping is itself unresolved and hence quality-guided phase unwrapping is not sufficient. For guiding strategies, classical, two-section, and stack-chain guiding strategies are chosen for comparison. If accuracy is the foremost criterion then the classical guiding strategy with a data structure of indexed interwoven linked list is best. If speed is of essence then the stack-chain guiding strategy is the one to use.

Phase demodulation from a single fringe pattern is a challenging task but of interest. A frequency-guided regularized phase tracker and a frequency-guided sequential demodulation method with Levenberg-Marquardt optimization are proposed to demodulate a single fringe pattern. Demodulation path guided by the local frequency from the highest to the lowest is applied in both methods. Since critical points have low local frequency values, they are processed last so that the spurious sign problem caused by these points is avoided. These two methods can be considered as alternatives to the effective fringe follower regularized phase tracker. Demodulation results from one computer-simulated and two experimental fringe patterns using the proposed methods will be demonstrated.

A windowed Fourier ridges (WFR) algorithm and a windowed Fourier filtering (WFF) algorithm have been proposed for fringe pattern analysis and have been demonstrated to be versatile and effective. Theoretical analyses of their performances are of interest. Local frequency and phase extraction errors by the WFR and WFF algorithms are analyzed in this paper. Effectiveness of the WFR and WFF algorithms will thus be theoretically proven. Consider four phase-shifted fringe patterns with local quadric phase [c(20)=c(02)=0.005 rad/(pixel)(2)], and assume that the noise in these fringe patterns have mean values of zero and standard deviations the same as the fringe amplitude. If the phase is directly obtained using the four-step phase-shifting algorithm, the phase error has a mean of zero and a standard deviation of 0.7 rad. However, when using the WFR algorithm with a window size of sigma(x)=sigma(y)=10 pixels, the local frequency extraction error has a mean of zero and a standard deviation of less than 0.01 rad/pixel and the phase extraction error in the WFR algorithm has a mean of zero and a standard deviation of about 0.02 rad. When using the WFF algorithm with the same window size, the phase extraction error has a mean of zero and a standard deviation of less than 0.04 rad and the local frequency extraction error also has a mean of zero and a standard deviation of less than 0.01 rad/pixel. Thus, an unbiased estimation with very low standard deviation is achievable for local frequencies and phase distributions through windowed Fourier transforms. Algorithms applied to different fringe patterns, different noise models, and different dimensions are discussed. The theoretical analyses are verified by numerical simulations.

Moire interferometry is an effective experimental technique for measurement of in-plane deformation. However, it is information on the derivatives of the deformation, i.e., strains, that is usually desired in experimental mechanics. It is shown that the desired strains are the instantaneous frequencies of the fringe pattern and that either an energy operator or wavelet ridges can be used to extract the instantaneous frequencies from a single fringe pattern. The energy operator is a pixelwise processor; thus the strain extraction can be done on the fly, but it is sensitive to noise. The wavelet ridges extract the local features in the fringe pattern. The strain extraction is thus insensitive to noise, and good results are obtainable at the cost of longer computation time. The two methods can thus be chosen for different needs in strain analysis. The properties of the two methods as well as their applications to a real fringe pattern are given. The effectiveness of the proposed methods is illustrated by their comparison with traditional methods.