Carrier frequency removal in phase measuring deflectometry with non-telecentric imaging

Fringe projection profilometry (FPP) and phase measuring deflectometry (PMD) are the currently popular technologies for measuring diffuse and specular reflective surfaces respectively. Both methodologies demonstrate superior performance for single-type reflective surfaces. However, with the rapid development of advanced manufacturing and intelligent manufacturing technologies, products have seen significant improvements in complexity, diversity, and personalization, often exhibiting composite reflective characteristics on their surfaces. The aforementioned single technology cannot meet the measurement requirements for composite reflective surfaces. To address this challenge, this paper presents a composite reflective surfaces measurement method combining FPP with PMD techniques, and an optimized system calibration method based on binocular stereo vision is proposed to enhance the system's measurement accuracy. To improve the measurement efficiency, a strategy in which the LCD screen and projector project different colorful fringe patterns simultaneously is adopted. To mitigate stray light interference from the projector on specular surface profilometry, a partition projection method is presented. Instead of establishing the pixel mapping relationship between the projector and diffused surface aeras, the coordinate mapping relationship between the specular reflective region and the projector pixel is established according to the phase-height relationship to realize the accurate projection of the projector to the diffused surface. Experiments have been performed to validate the proposed method, and the results demonstrate that it can reconstruct the composite surfaces with higher accuracy. In particular, the reconstruction accuracy of specular surface can reach several tens of nanometres.

Phase measuring deflectometry (PMD) is a competitive method for specular surface measurement that offers the advantages of a high dynamic range, non-contact process, and full field measurement; furthermore, it can also achieve high accuracy. Camera calibration is a crucial step for PMD. As a result, a method based on the calibration of the entrance pupil center is introduced in this paper. Then, our proposed approach is compared with the most popular photogrammetric method based on Zhang’s technique (PM) and Huang’s modal phase measuring deflectometry (MPMD). The calibration procedures of these three methods are described, and the measurement errors introduced by the perturbations of degrees of freedom in the PMD system are analyzed using a ray tracing technique. In the experiment, a planar window glass and an optical planar element are separately measured, and the measurement results of the use of the three methods are compared. The experimental results for the optical planar element (removing the first 6 terms of the Zernike polynomial) show that our method’s measurement accuracy reached 13.71 nm RMS and 80.50 nm PV, which is comparable to accuracy values for the interferometer.

This paper proposes a new shearing approach to eliminate the systematic errors in phase measuring deflectometry (PMD). Besides a normal measurement using PMD, another three measurements are taken when the test surfaces are translated and rotated with an amount of known magnitude in our proposed method. The four measurements give three sheared wavefronts, and then the difference Zernike polynomial fitting and a least squares algorithm are proposed to reconstruct the true test surface figure. The proposed method is demonstrated and verified through computer simulation and experimental results. The error propagation when the sheared wavefronts include random noise and the introduced errors due to the incorrect shearing values are both analyzed.

Phase Measuring Deflectometry (PMD) and Structured-Light Modulation Analysis Technique (SMAT) perform effectively in shape and defect measurements of specular objects, but the difficulty of giving consideration to accuracy and speed has also restricted the further development and application of them. Inspired by recent successes of deep learning techniques for computational imaging, we demonstrate for the first time that deep learning techniques can be used to recover high-precision modulation distributions of specular surfaces from a single-frame fringe pattern under SMAT, enabling fast and high-quality defect detection of specular surfaces. This method can also be applied to recover higher-precision phase distributions of specular surfaces from a single-frame fringe pattern under PMD, so as to realize the 3D shape measurement. In this paper, we combine depthwise separable convolution, residual structure and U-Net to build an improved U-Net network. The experimental results prove that the method has excellent performance in the phase and modulation retrieval of specular surfaces, which almost reach the accuracy of the results obtained by ten-step phase-shifting method.

In fringe projection profilometry, by taking the projector as a camera, depth measurement has the similar principle with binocular vision. The matching of feature points carrying depth information plays an important role in measurement. Due to the existence of difficultly-corrected projector errors, the measurement results of using a fixed point in camera corresponding to the points in the projector or of conversely using a fixed point in projector corresponding to the points in the camera is not exactly same. According to fringe projection system, a function representing the relationship between the depths and the corresponding fringe phases is introduced for measurement. This traditional method usually requires three or more known reference planes to calibrate parameters determined by the measurement system. However, another way is also feasible. Based on the measurement system, a generic function representing the relationship between the depths and the corresponding pixel shifts are also presented in this paper. Using these two functions allows us to recover the depth information of the object. Theoretical analyses are carried out to investigate the performances of these two methods. Comparative experiments are implemented, and their results demonstrate that the depth measurement method from pixel shifts is immune to the influence of high order harmonics as a little expense of efficiency, being better than the method from fringe phases having high measurement efficiency regardless of the low accuracy.

Profilometry by fringe projection is an optical metrology technique implemented projecting a sinusoidal grating onto a three-dimensional surface, where the resulting deformed grating image represents the object's height dis- tribution modulated in phase. Fringe projection can be implemented as a phase shifting process or as a one-frame method. In this work, both approaches are compared using a correlation and a Fourier transform based algorithm for phase extraction. As a first step of the surface recovering process, a wrapped phase image is obtained. Therefore, a correction is applied by means of a phase un- wrapping algorithm, obtaining a continuous phase distribution. The results show that the phase shifting approach returns a phase distribution with higher resolution and less noise than the one-frame approach.

A predistortion-based 3-D phase mapping method with linear epipolar constraints for fringe projection system calibration and measurement is reported. Hybrid system calibration methods have been developed recently to provide accurate measurement, high speed, and flexibility for fringe projection profilometry (FPP). However, these methods still have significant limitations. With projector distortion, for example, rigorous post-undistortion often results in low efficiency in computation. Conversely, simple, approximated undistortion is only effective for low levels of distortion. As a result, projector undistortion is usually not prioritized in practice. Aiming to achieve high-accuracy 3-D reconstruction with FPP, an easily implementable projector undistortion method is proposed in this article. Theoretical analyses and simulations confirm that the proposed method is rigorous. Experiments validate that the pixelwise root-mean-square reconstruction error is 0.042 mm, a 24% reduction from that of a simple post-undistortion method. The proposed method is also efficient computationally. Finally, a 3-D phase mapping lookup table is built. The results show that the proposed method is effective for achieving high accuracy and speed in 3-D surface measurement.

Single-shot high-accuracy reconstruction of three-dimensional (3D) objects plays an essential role in many fields. Wavelet-based fringe projection profilometry (WFPP) is a popular single-shot imaging method due to its multiple-resolution analysis and non-contact property. However, the construction of mother wavelets for high-accuracy WFPP is still challenging. In this paper, a general framework of two-dimensional (2D) compact support complex wavelet employing the dual-tree concept combined with the one-dimensional (1D) asymmetric wavelet function is proposed. The constructed wavelet has an asymmetric frequency envelope in the radial direction and concentrated energy in a single quadrant of the frequency domain. A method that the constructed wavelet applied in dual-angle rotation-based 2D WFPP is presented for higher accuracy. The theoretical analysis of the proposed method is presented, and a comparison of frequency characteristics between the constructed wavelet and other popular wavelets is conducted. Computer simulations and experiments have demonstrated that the proposed method provides a better reconstruction of 3D shapes in single-shot WFPP.

A combined linear and nonlinear ultrasound time-domain approach for impact damage detection in composite structures using a constructive nonlinear array technique

This paper forecast/predicted the one-year time charter weekly freight rates earned by a 65 000 dwt bulk carrier using 996 weeks of data from 1989 to 2008. First, the need and the importance, but also the futility, of forecasting is discussed in shipping. This is a volatile industry that can be easily likened to the roulette. The introduction is followed by a literature review that has examined the principal recent works in this area and presented a critique of earlier works. Most of the research studied dealt with the shipping industry per se. Since the methods used are considered as a departure from the classical Random Walk, a comprehensive section of the paper is devoted to the methodology of nonlinear, chaotic and deterministic methods. The relevant time series have been transformed into stationary ones, as this is the proper practice (using first logarithmic differences). The time series were tested for randomness (identically and independently distributed) and for long-term correlation using BDS statistic. The methods used were: Rescaled Range Analysis and the related Hurst Exponent; Power Spectrum Analysis; V-statistic and BDS Statistic (using software MATLAB 5.3 and NLTSA V.2.0/2000). The analysis of the data was presented in three separate sections. The relevant ‘attractor’ of the system has been graphically shown. System's dimension has been calculated, which was found to be non-integer, fractal and equal to 3.95. This finding permitted us to proceed to forecasting, as this is a case of a low dimensional chaos (3.95 < 10 dimensions). In order for the predictions to be robust, the prediction horizon allowed was found equal to 8.24 weeks, as indicated by the positive maximum Lyapunov exponent (0.12 rounded). Then NLTSA software was used to make prediction inside- and forecasting outside- the sample, using by selection nonlinear Principal Components and Kernel Density Estimation methods.

The measurement and data analysis used to evaluate highway roughness

In 3D shape measurement, because deformed fringes often contain low-frequency information degraded with random noise and background intensity information, a new fringe-projection profilometry is proposed based on 2D empirical mode decomposition (2D-EMD). The fringe pattern is first decomposed into numbers of intrinsic mode functions by 2D-EMD. Because the method has partial noise reduction, the background components can be removed to obtain the fundamental components needed to perform Hilbert transformation to retrieve the phase information. The 2D-EMD can effectively extract the modulation phase of a single direction fringe and an inclined fringe pattern because it is a full 2D analysis method and considers the relationship between adjacent lines of a fringe patterns. In addition, as the method does not add noise repeatedly, as does ensemble EMD, the data processing time is shortened. Computer simulations and experiments prove the feasibility of this method.

The phase diversity technique is studied as a wave-front sensor to be implemented with widely extended sources. The wave-front phase expanded on the Zernike polynomials is estimated from a pair of images (in focus and out of focus) by use of a maximum-likelihood approach. The propagation of the photon noise in the images on the estimated phase is derived from a theoretical analysis. The covariance matrix of the phase estimator is calculated, and the optimal distance between the observation planes that minimizes the noise propagation is determined. The phase error is inversely proportional to the number of photons in the images. The noise variance on the Zernike polynomials increases with the order of the polynomial. These results are confirmed with both numerical and experimental validations. The influence of the spectral bandwidth on the phase estimator is also studied with simulations.

The deformation of large- aperture optical mirror is induced by gravity. It is very important to analyze the support structure and format in grounding and testing. By finite element software, a &Phi;620mm mirror is created and its theoretical analysis about three comparative testing programs of vertical support is proposed. Zernike polynomials are used to fit optical surface and separate corresponding aberrations. By reducing the aberrations with great impact on mirror, at the same time, RMS excel 0.025&lambda; (&lambda;=632.8nm) has been achieved. Experiments of three kinds of support verify that the entirely bound by strip is the best way in all supports. It affords a more scientific and rational support to large-diameter optical mirror reflection. Through research and analysis of support on large-diameter mirror by finite element, it provides a theory of design and guidance for the development and test of reflection mirror in future.

Carrier removal has been typically associated with wrap reduction, and its application has strongly been recommended as a previous step to phase unwrapping. In general, the carrier signal contributes to an increase of the number of wraps, and its removal leads to a reduction of the number of phase inconsistencies, having a positive impact on the subsequent unwrapping operation. In this paper, we show that phase wrap reduction can be better tackled by other alternative and more effective methods that exploit local information and attempt to also remove the signal component associated with the object’s surface. The results are analyzed on a series of real fringe patterns and confirm the better performance of the proposed method, which leads to signals with a lesser number of wraps than by just removing the carrier.