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

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

High-resolution 3D profilometry with binary phase-shifting methods

This paper presents a novel pixel-level resolution 3D profilometry technique that only needs binary phase-shifted structured patterns. This technique uses four sets of three phase-shifted binary patterns to achieve the phase error of less than 0.2%, and only requires two sets to reach similar quality if the projector is slightly defocused. Theoretical analysis, simulations, and experiments will be presented to verify the performance of the proposed technique.     

Systematic errors in small deformations measured by use of shadow-moiré topography

Phase-shift shadow-moiré topography is a noncontact optical technique for measuring the shapes of surfaces. Artifactual bands resembling isoheight surface contours are observed during measurement of small changes in shape by use of this technique. The shape-reconstruction algorithm used in shadow-moiré topography is based on a mathematical model of the fringe patterns generated on the surface to be measured. We hypothesize that the observed bands reflect systematic errors caused by ignoring height-dependent terms in the mathematical model of the fringe patterns. We test the assumption by simulating the fringe patterns for a virtual test surface by using a model that contains height-dependent terms and one term that is idealized by ignoring these terms. Small systematic errors in shape are observed only when the surface is reconstructed from fringe patterns simulated with a model containing the height-dependent terms. Shape-error curves are computed as a function of the surface height by the subtraction of the reconstructed shape from the known shape. Simulated shape-error curves agree with experimental measurements in that they show an increase in error with surface height, and both the experimental and the simulated shape-error curves contain ripples. Although the errors are small in comparison with the dimensions of the surface and are negligible in shape measurements and in most deformation measurements, they may show up as noticeable bands in images of small deformations.     

Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector

A structured light system using a digital video projector is widely used for 3D shape measurement. However, the nonlinear gamma of the projector causes the projected fringe patterns to be nonsinusoidal, which results in phase error and therefore measurement error. It has been shown that, by using a small look-up table (LUT), this type of phase error can be reduced significantly for a three-step phase-shifting algorithm. We prove that this algorithm is generic for any phase-shifting algorithm. Moreover, we propose a new LUT generation method by analyzing the captured fringe image of a flat board directly. Experiments show that this error compensation algorithm can reduce the phase error to at least 13 times smaller.     

Comparison of the squared binary, sinusoidal pulse width modulation, and optimal pulse width modulation methods for three-dimensional shape measurement with projector defocusing

This paper presents a comparative study on three sinusoidal fringe pattern generation techniques with projector defocusing: the squared binary defocusing method (SBM), the sinusoidal pulse width modulation (SPWM) technique, and the optimal pulse width modulation (OPWM) technique. Because the phase error will directly affect the measurement accuracy, the comparisons are all performed in the phase domain. We found that the OPWM almost always performs the best, and SPWM outperforms SBM to a great extent, while these three methods generate similar results under certain conditions. We will briefly explain the principle of each technique, describe the optimization procedures for each technique, and finally compare their performances through simulations and experiments.     

Flexible error-reduction method for shape measurement by temporal phase unwrapping: phase averaging method

Temporal phase unwrapping is an important method for shape measurement in structured light projection. Its measurement errors mainly come from both the camera noise and nonlinearity. Analysis found that least-squares fitting cannot completely eliminate nonlinear errors, though it can significantly reduce the random errors. To further reduce the measurement errors of current temporal phase unwrapping algorithms, in this paper, we proposed a phase averaging method (PAM) in which an additional fringe sequence at the highest fringe density is employed in the process of data processing and the phase offset of each set of the four frames is carefully chosen according to the period of the phase nonlinear errors, based on fast classical temporal phase unwrapping algorithms. This method can decrease both the random errors and the systematic errors with statistical averaging. In addition, the length of the additional fringe sequence can be changed flexibly according to the precision of the measurement. Theoretical analysis and simulation experiment results showed the validity of the proposed method.     

A general gamma nonlinearity compensation method for structured light measurement with off-the-shelf projector based on unique multi-step phase-shift technology

ABSTRACT

Nonlinearities in the application of fringe projection metrology make it very difficult to acquire perfect 3D data. This paper describes a six-step phase-shifting technique for a structured light measurement system with an off-the-shelf projector. First, the phase error is analysed and a gamma model is established by deriving the relative expression between the wrapped phase and input images. This is then expressed in matrix form to derive a unique solution, which is used for the gamma solver. The complex gamma calibration and projector error compensation can be removed once the gamma value of the off-the-shelf projector has been determined. The ideal model reconstruction results are obtained through simulations and experiments, and the standard deviation of the phase error is found to be only 0.0039 radians. Hence, the proposed method eliminates the nonlinear errors associated with fringe projection technology using existing projectors and improves the overall image reconstruction quality.     

Quantization error of CCD cameras and their influence on phase calculation in fringe pattern analysis

We present an investigation into the phase errors that occur in fringe pattern analysis that are caused by quantization effects. When acquisition devices with a limited value of camera bit depth are used, there are a limited number of quantization levels available to record the signal. This may adversely affect the recorded signal and adds a potential source of instrumental error to the measurement system. Quantization effects also determine the accuracy that may be achieved by acquisition devices in a measurement system. We used the Fourier fringe analysis measurement technique. However, the principles can be applied equally well for other phase measuring techniques to yield a phase error distribution that is caused by the camera bit depth.     

Phase Error Compensation of Three-Dimensional Reconstruction Combined with Hilbert Transform

Nonlinear response is an important factor affecting the accuracy of three-dimensional image measurement based on the fringe structured light method. A phase compensation algorithm combined with a Hilbert transform is proposed to reduce the phase error caused by the nonlinear response of a digital projector in the three-dimensional measurement system of fringe structured light. According to the analysis of the influence of Gamma distortion on the phase calculation, the algorithm establishes the relationship model between phase error and harmonic coefficient, introduces phase shift to the signal, and keeps the signal amplitude constant while filtering out the DC component. The phase error is converted to the transform domain, and compared with the numeric value in the space domain. The algorithm is combined with a spiral phase function to optimize the Hilbert transform, so as to eliminate external noise, enhance the image quality, and get an accurate phase value. Experimental results show that the proposed method can effectively improve the accuracy and speed of phase measurement. By performing phase error compensation for free-form surface objects, the phase error is reduced by about 26%, and about 27% of the image reconstruction time is saved, which further demonstrates the feasibility and effectiveness of the method.     

相移条纹投影三维形貌测量技术综述

结构光三维形貌测量系统目前得到了越来越广泛的应用和研究,相移条纹投影三维形貌精密测量技术是其重要的发展方向。对结构光相移条纹投影三维形貌测量系统的应用发展、工作过程、不同系统构成方式、相移条纹的各种形式及特点、相位误差校正方法、不同相位解包裹算法及其优缺点和适用场合、测量系统数学模型的实现方法及其相应的优缺点、高动态范围测量技术等进行了详细的分析。对相移条纹投影系统的工作流程、实现方法、关键技术的发展及其存在问题等进行了比较全面的梳理,为三维形貌精密测量技术进一步满足先进制造中更高精度的要求指出了后续的研究方向。     

位相测量轮廓术中相移误差和最佳相移次数的研究

位相测量轮廓术是三维面形测量的一种重要方法,它采用的离散相移技术要求精确的相移,在实际系统中不可避免地存在着相移误差,它将导致计算位相和重建面形的误差,本文利用已建立的三维面形仿真系统,定量研究了不同线性相移误差下所引起的位相误差和面形误差大小,本文还讨论了应如何选取最佳的相移次数,本文的工作可为实际测量的校准提供理论数据,对实际测量工作具有指导意义。     

Intensity range extension method for three-dimensional shape measurement in phase-measuring profilometry using a digital micromirror device camera

Phase-measuring profilometry is an accurate and effective technique for performing three-dimensional (3D) shape and deformation measurements of diffuse objects by fringe projection. However, phase analysis cannot be performed in underexposed or overexposed areas of the detector when an object with wide reflectance is measured. A novel intensity range extension method using a digital micromirror device (DMD) camera is proposed. In the optics of the DMD camera, each pixel of the CCD corresponds exactly to each mirror of the DMD. The phase-shifted fringe patterns with high contrast can be easily captured by programming an inverse intensity pattern that depends on the reflectance of the object. Our method can provide a wider intensity range and higher accuracy for 3D shape measurement than other conventional methods in both underexposed and overexposed areas. The measurements of a replica of a metallic art object and a flat plane are analyzed experimentally to verify the effectiveness of our method. In the experiment, the percentage of invalid points due to underexposure and overexposure can be reduced from 20% to 1%.     

Uneven fringe projection for efficient calibration in high-resolution 3D shape metrology

A novel uneven fringe projection technique is presented whereby nonuniformly spaced fringes are generated at a digital video projector to give evenly spaced fringes in the measurement volume. The proposed technique simplifies the relation between the measured phase and the object's depth independent of pixel position. This method needs just one coefficient set for calibration and depth calculation. With uneven fringe projection the shape data are referenced to a virtual plane instead of a physical reference plane, so an improved measurement with lower uncertainty is achieved. Further, the method can be combined with a radial lens distortion model. The theoretical foundation of the method is presented and experimentally validated to demonstrate the advantages of the uneven fringe projection approach compared with existing methods. Measurement results on a National Physical Laboratory (UK) "step standard" confirm the measurement uncertainty using the proposed method.     

Optimized two-frequency phase-measuring-profilometry light-sensor temporal-noise sensitivity

Temporal frame-to-frame noise in multipattern structured light projection can significantly corrupt depth measurement repeatability. We present a rigorous stochastic analysis of phase-measuring-profilometry temporal noise as a function of the pattern parameters and the reconstruction coefficients. The analysis is used to optimize the two-frequency phase measurement technique. In phase-measuring profilometry, a sequence of phase-shifted sine-wave patterns is projected onto a surface. In two-frequency phase measurement, two sets of pattern sequences are used. The first, low-frequency set establishes a nonambiguous depth estimate, and the second, high-frequency set is unwrapped, based on the low-frequency estimate, to obtain an accurate depth estimate. If the second frequency is too low, then depth error is caused directly by temporal noise in the phase measurement. If the second frequency is too high, temporal noise triggers ambiguous unwrapping, resulting in depth measurement error. We present a solution for finding the second frequency, where intensity noise variance is at its minimum.     

The International Commission for Optics

An apparatus is described that automatically measures the course of the accommodation of the human eye with infra-red light. The principle of measurement is based on the spreading of the image of a test target on the retina when the eye is defocused. The measuring technique is an autocollimating one. Each single measurement tests the state of accommodation over a range from ?7 to +2 D. Targets with a complicated structure such as gratings can be used to get a better criterion for defocusing. The apparatus gives 50 measurements of accommodation per second. Compared to the slow changes of accommodation this is an effectively continuous registration. The lag of accommodation and the transfer function of accommodation computed from the response to a sudden shift in target distance were measured with the apparatus.     

Effects of process errors on the diffraction characteristics of binary dielectric gratings

The effects of fabrication errors on the predicted performance of surface-relief phase gratings are analyzed with a rigorous vector diffraction technique. For binary elements, errors in the dimensions of the profile [depth, linewidth (fill factor), and grating period], as well as errors in the shape of the profile, are investigated. It is shown that the dimension errors do not have a significant effect on grating performance when the grating is designed for either maximum or minimum diffraction efficiency. A trapezoid is used to model the shape error of the profile. For the first time, design rules that significantly reduce the effects of any shape error are presented.     

Measurement and calibration of interferometric imaging aberrations

Phase-shifting interferometry is the standard method for testing figure error on optical surfaces. Instruments measuring spheres and flats are readily available, but the accurate measurement of aspheres requires null correction. One problem with the general (nonull) testing of aspheres is the loss of common path. Systematic errors are introduced into the measurement by the fringe imaging optics. The sources and types of error are reviewed, as well as their effect on a wave-front measurement. These nonnull errors are predicted generally, with third-order analytic expressions derived for a tilted or a defocused test surface. An interferometer is built to test the expressions. The imaging system is a single lens, nominally image telecentric. Measurements are performed on a test surface defocused from -5 to 5 mm. The resulting measurement bias is shown to be in good agreement with third-order aberration theory predictions.     

Model-based inversion of speckle interferometer fringe patterns

Micrometer-scale rigid-body translations are determined fromelectronic speckle interferometric fringe patterns. An iterativeminimum error procedure employs the relative fringe order of pickedpositions of fringe maxima and minima within a single interferogram tocalculate the displacement field directly. The method does notcalculate the displacement at a single point but relies on theassumption that the character, but not the magnitudes or directions, ofthe displacements over the viewing area of the interferogram isknown. That is, a model of the displacements exists. Onperfect, noise-free forward modeled fringe patterns calculated for an 8.0-mum displacement, the phase error is less than 2 x10(-6) fringe orders (1.3 x 10(-5) rad)and probably results only from numerical noise in the inversion. Onreal fringe patterns obtained in electronic speckle interferometricexperiments, mean phase errors are generally less than 5 x10(-5) fringe orders (3.2 x 10(-4)rad), suggesting that the technique is robust despite errorsresulting from speckle noise, lack of accuracy in positioning ofexperimental components, and image-distortion corrections.     

基于改进Sierra 抖动算法的微离焦投影三维轮廓测量技术研究

通过分析传统频率调制的相位轮廓测量技术,提出基于蛇形扫描改进Sierra抖动算法,结合微离焦投影,可减小正弦光栅二值化的量化误差,同时抑制非对称纹理,能较大地改善离焦后光栅的正弦性。将该抖动算法生成的离焦光栅用于三维轮廓测量技术,与Bayer有序抖动算法、Sierra抖动算法和Floyd-steinberg误差扩散抖动算法进行了比较,实验结果表明:改进Sierra抖动算法具备更好的适应性,能够较大程度地降低相位误差,运算速度快,生成光栅准确度较高,改善了相位质量,适用于高精度三维轮廓测量。