Four-step Phase-shifting Algorithm Research Articles

Automatic optimization of projection intensity for high dynamic range 3D surface measurement

Stripe projection profilers are widely used in many fields because of their high speed, high accuracy, and clear surface textures. In the structured light projection method, the complex reflectance distribution on the surface of the high-dynamic-range objective is an important factor that affects the measurement accuracy, as it can result in image over- and under-exposure. In this study, a method is proposed to adaptively determine the number of stripe patterns and the corresponding number of light intensities of the stripe patterns based on the complex reflectance of the surface of the object being measured. A multi threshold Otsu algorithm is used to classify the histogram of the surface reflectance distribution of the measured object, and several sets of stripe patterns with optimal light intensity are generated according to the light intensity response function. The original stripe patterns acquired at different light intensities are synthesized pixel by pixel into a high dynamic range (HDR) stripe image, and a four-step phase-shifting algorithm is used to obtain the unwrapped phase from the synthesized image. Experimental results show that this method can accurately measure the target of a surface-reflectance-transformed HDR.

High-Speed Phase Structured Light Integrated Architecture on FPGA

Phase-shifting fringe projection profilometry (FPP) in three-dimension (3D) structured light measurement is widely applied in high-accuracy 3D shape measurement scenarios. But motion objects or other dynamic scenarios make the traditional phase-shifting FPP hard to deal with due to its sizable CPU-based execution time. Therefore, an embedded high-speed and high-accuracy phase-shifting FPP system is designed and implemented on FPGAs. The system implements a three-frequency four-step phase-shifting hardware algorithm with RTL-level caching, multi-frame image ping-pong buffering, and data handling, etc. The system exhibits fully-parallel and pipeline processing while behaving with lower resource and power consumption. Measurement experiments indicate that the proposed method can reach high-accuracy real-time 3D shape measurement with a 3D decoding speed of 40.15 fps when the clock frequency is 200MHz, and the image size is 2432×2048.

Parallel Phase-Shifting Digital Holographic Phase Imaging of Micro-Optical Elements with a Polarization Camera

In this paper, we propose a measurement method of micro-optical elements with parallel phase-shifting digital holographic phase imaging. This method can record four phase-shifting holograms with a phase difference of π/2 in a single shot and correct the pixel mismatch error of the polarization camera using a bilinear interpolation algorithm, thereby producing high-resolution four-step phase-shifting holograms. This method reconstructs the real phase information of the object to be measured through a four-step phase-shifting algorithm. The reproduced image eliminates the interference of zero-order images and conjugate images, overcoming the problem that traditional phase-shifting digital holography cannot be measured in real time. A simulation analysis showed that the relative error of this measurement method could reach 0.0051%. The accurate surface topography information of the object was reconstructed from an experimental measurement through a microlens array. Multiple measurements yielded a mean absolute error and a mean relative error for the vertical height of the microlens array down to 5.9500 nm and 0.0461%, respectively.

Distortion-Correction Method Based on Angular Spectrum Algorithm for Holographic Display

The reconstruction quality of computer-generated holographic display is subject to distortions, which are caused by manufacture defects and misalignments of optical elements. Distortion correction based on precision optics is costly and time-consuming. In this paper, a numerical distortion-correction method based on the angular spectrum algorithm is proposed. A wavefront acquisition system using a camera and the four-step phase-shifting algorithm is employed for calculating the distorted distributions. A wavefront correction system using the spatial light modulator is developed for eliminating distorted distributions. Effective distance, calculation time and iterations, and limitations of the method are discussed. The proposed method is robust in reducing distortions in computer-generated holographic display.

Three-dimensional refractive index measurement of special optical fiber based on optical vortex phase-shifting digital holographic microscopy

We propose an approach for three-dimensional (3-D) refractive index (RI) measurement of special optical fiber by phase-shifting digital holographic microscopy based on optical vortex. The experimental system consists of a modified Mach–Zehnder interferometer and related lab-developed software. The phase shifts are obtained by loading different computer-generated spiral phase holograms on a liquid-crystal spatial light modulator with digital blazed grating, which is more accurate than the traditional phase-shift methods. The optical vortex generated in +1 diffraction order of the diffraction field is utilized as the reference beam. Eight digital holograms with the phase-shift increments of 0, π / 2, π, and 3π / 2 are recorded by a CCD before and after the tested optical fiber is placed in the experimental system. Then, a four-step phase-shifting algorithm and phase unwrapping algorithm are adopted to extract the phase distribution from the holograms. Based on a filtered backprojection algorithm, the 3-D RI distribution of the tested fiber is obtained at high accuracy. Experimental results demonstrate the feasibility of the proposed method for 3-D RI measurement of special optical fiber.

Correction of errors due to uneven intensity distribution and inaccurate phase shifts in prism-based simultaneous phase-shifting interferometry.

Simultaneous phase-shifting interferometry (SPSI) is a suitable technique for dynamic surface measurement due to its rapid measurement capability. Based on whether there are obvious multiple phase-shifting channels in the setup, it can be divided into the multi-channel type and the pixelated phase-mask type. For the former type, two important error sources are the uneven intensity distribution among different phase-shifting channels and inaccurate phase shifts in each phase-shifting channel. In order to diminish the influence of these two error sources, several methods have been proposed. However, in some of these methods, only one error source of the two was considered and corrected. In the methods where the two error sources were both considered, some assumptions such as the uniform background intensity and fringe contrast are needed, so a complete and universal suppression method for both error sources is still needed in actual applications. In this paper, for a prism-based SPSI, we proposed such an error-correction method that mainly contains the following contributions. First, the intrinsic parameters of the system, including intensity distribution coefficients and relative phase shifts among phase-shifting channels, are calibrated in advance. We also checked the uniformity of phase shifts in each phase-shifting channel. Second, based on the obtained parameters, a corrected four-step phase-shifting algorithm is deduced to recover the more accurate measured phase. Results of both simulation and experiment verify the effectiveness of the proposed method.

A novel phase retrieval method in fringe projection based on phase-shifting algorithm

In order to reduce nonlinear phase error in fringe projection based on phase-shifting algorithm, a novel phase retrieval method is proposed. Firstly, on the basis of expression for wrapped phase and corresponding phase in N-step phase-shifting algorithm, the nonlinear phase error and its simple expression is deduced. Secondly, based on the four-step phase-shifting algorithm, the wrapped phase expression of actual measured object is obtained. The corresponding nonlinear phase error is derived in a simple form, and then the new phase error value is obtained by offsetting π/4 from the nonlinear phase value. The mean of the new and original phase error values yields the final phase error; hence, the nonlinear phase errors are significantly reduced. Finally, the results of numerical simulations and experiments show that the proposed method is effective to reduce the nonlinear phase error due to system gamma nonlinearity in phase-shifting algorithm.

Volume moiré tomography based on projection extraction by spatial phase shifting of double crossed gratings

To realize volume moiré tomography (VMT) for the real three-dimensional (3D) diagnosis of combustion fields, according to 3D filtered back projection (FBP) reconstruction algorithm, the radial derivatives of the projected phase should be measured firstly. In this paper, a simple spatial phase-shifting moiré deflectometry with double cross gratings is presented to measure the radial first-order derivative of the projected phase. Based on scalar diffraction theory, the explicit analytical intensity distributions of moiré patterns on different diffracted orders are derived, and the spatial shifting characteristics are analyzed. The results indicate that the first-order derivatives of the projected phase in two mutually perpendicular directions are involved in moiré patterns, which can be combined to compute the radial first-order derivative. And multiple spatial phase-shifted moiré patterns can be simultaneously obtained; the phase-shifted values are determined by the parameters of the system. A four-step phase-shifting algorithm is proposed for phase extraction, and its accuracy is proved by numerical simulations. Finally, the moiré deflectometry is used to measure the radial first-order derivative of projected phase of a propane flame with plane incident wave, and the 3D temperature distribution is reconstructed.

Full-Field Calibration of Color Camera Chromatic Aberration using Absolute Phase Maps.

The refractive index of a lens varies for different wavelengths of light, and thus the same incident light with different wavelengths has different outgoing light. This characteristic of lenses causes images captured by a color camera to display chromatic aberration (CA), which seriously reduces image quality. Based on an analysis of the distribution of CA, a full-field calibration method based on absolute phase maps is proposed in this paper. Red, green, and blue closed sinusoidal fringe patterns are generated, consecutively displayed on an LCD (liquid crystal display), and captured by a color camera from the front viewpoint. The phase information of each color fringe is obtained using a four-step phase-shifting algorithm and optimum fringe number selection method. CA causes the unwrapped phase of the three channels to differ. These pixel deviations can be computed by comparing the unwrapped phase data of the red, blue, and green channels in polar coordinates. CA calibration is accomplished in Cartesian coordinates. The systematic errors introduced by the LCD are analyzed and corrected. Simulated results show the validity of the proposed method and experimental results demonstrate that the proposed full-field calibration method based on absolute phase maps will be useful for practical software-based CA calibration.

Enhanced high dynamic range 3D shape measurement based on generalized phase-shifting algorithm

Measuring objects with large reflectivity variations across their surface is one of the open challenges in phase measurement profilometry (PMP). Saturated or dark pixels in the deformed fringe patterns captured by the camera will lead to phase fluctuations and errors. Jiang et al. proposed a high dynamic range real-time three-dimensional (3D) shape measurement method (Jiang et al., 2016) [17] that does not require changing camera exposures. Three inverted phase-shifted fringe patterns are used to complement three regular phase-shifted fringe patterns for phase retrieval whenever any of the regular fringe patterns are saturated. Nonetheless, Jiang's method has some drawbacks: (1) the phases of saturated pixels are estimated by different formulas on a case by case basis; in other words, the method lacks a universal formula; (2) it cannot be extended to the four-step phase-shifting algorithm, because inverted fringe patterns are the repetition of regular fringe patterns; (3) for every pixel in the fringe patterns, only three unsaturated intensity values can be chosen for phase demodulation, leaving the other unsaturated ones idle. We propose a method to enhance high dynamic range 3D shape measurement based on a generalized phase-shifting algorithm, which combines the complementary techniques of inverted and regular fringe patterns with a generalized phase-shifting algorithm. Firstly, two sets of complementary phase-shifted fringe patterns, namely the regular and the inverted fringe patterns, are projected and collected. Then, all unsaturated intensity values at the same camera pixel from two sets of fringe patterns are selected and employed to retrieve the phase using a generalized phase-shifting algorithm. Finally, simulations and experiments are conducted to prove the validity of the proposed method. The results are analyzed and compared with those of Jiang's method, demonstrating that our method not only expands the scope of Jiang's method, but also improves measurement accuracy.

Phase extraction algorithm considering high-order harmonics in fringe image processing.

A phase calculation method using discrete Fourier series (DFS) is proposed to eliminate the effects of nonsinusoidal characteristics. In this method, the fundamental coefficients are extracted from continuous N samples in one cycle by DFS, with which four images with π/2 intervals are reconstructed, and then more accurate phase distribution can be further obtained. This method is applicable for improving the precision of the traditional phase-shifting algorithm. Its effectiveness and accuracy are verified by computer simulations and moiré fringe and projecting fringe experiments with about 85% of the phase error reduced compared with a four-step phase-shifting algorithm, about 70% reduction compared with a 16-step phase-shifting algorithm.

Frequency domain phase extraction algorithm for phase-shifting interferometry with random phase-shifting amount and low-sampling rate

To extract the exact phase of the measured object in the environmental conditions with random noise, a novel kind of frequency domain phase extraction method for phase-shifting interferometry with random phase-shifting amount and low-sampling rate is proposed, in which the time-varied intensity signal of an arbitrary pixel of phase-shifting interferograms is cut into a lot of sub-sections intensity signals with the same length, and then the frequency domain weighted stack operation is performed for the corresponding Fourier spectrum of these sub-section intensity signals. Importantly, the phase obtained from the proposed method coincides well with the conventional four-step phase-shifting algorithm, but the requirement for the environmental conditions in the former are greatly lowered than that in the latter, moreover, while the same number of phase-shifting interferograms is employed to perform the phase extraction, the phase measuring accuracy in the proposed method is greatly higher than that in the time domain Fourier transform (TFT) method.

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.

Four-step spatial phase-shifting shearing interferometry from moiré configuration by triple gratings

A spatial phase-shifting shearing interferometry is presented in this paper. The whole optical configuration is simple and consists of three Ronchi gratings. Four phase-shifted shearing interferograms can be obtained simultaneously. The explicit intensity distributions of shearing interferograms are given and a corresponding four-step spatial phase-shifting algorithm is proposed to extract phase information from the new interferometry. This spatial phase-shifting configuration is applied to extract phase projection of a propane flame and a mathematical error analysis is presented.

Continuous-wave terahertz phase imaging using a far-infrared laser interferometer

Terahertz phase imaging can reveal the depth information of an optically opaque object and provide much better contrast for weak-absorption materials. We demonstrate a continuous-wave terahertz interferometric imaging method in which a far-infrared laser interferometer is used to measure the phase distribution with diffraction-limited lateral resolution and subwavelength axial resolution. An improved four-step phase-shifting algorithm is introduced to retrieve the phase map with very high accuracy and low distortion. The relative depth profiles of two transparent samples are successfully extracted by using this method. Experimental results verify that terahertz interferometric imaging in combination with the phase-shifting technique enables effective reconstruction of the phase image of the object under test.

A novel phase-shifting method based on strip marker

A new phase-shifting method to obtain rapid phase measurement is proposed. Different from the traditional four-step phase-shifting algorithm, a strip marker is added in the projected fringe, which gives the indication for strip order. By the presented phase algorithm, not only the primary phase map but also the strip order can be obtained from the phase-shifting images, and thus the final phase map can be obtained. Using this method, no additional fringe pattern is required, that makes the time cost on fringe projection highly reduced. Experimental results show that the new algorithm has more efficiency in phase measurement.

Windowed Fourier transform for fringe pattern analysis: theoretical analyses

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.

Whole-field strain measurement in a pin-loaded plate by electronic speckle pattern interferometry and the finite element method

The strain field in an epoxy plate loaded in tension through a steel pin is determined using electronic speckle pattern interferometry (ESPI) and the finite element method (FEM). In a dual-beam illumination speckle interferometer, the in-plane component of the displacement at the plate's surface is accurately measured using a four-step phase-shifting algorithm. Digital image processing algorithms have been developed for noise reduction and strain calculation directly in the computer from the phase map with a strain gage length of about 0.4 mm. A whole-field strain map is obtained, as well as distributions of strain concentration factor, in critical regions near the hole of the plate. FEM is used to perform a nonlinear contact analysis accounting for friction effects at the pin/hole interface. The agreement between experimental results and numerical predictions is good. In terms of speed, accuracy and ease of use, dual-beam ESPI appears to be a superior method of whole-field strain analysis.