Two-step Phase-shifting Algorithm Research Articles

Quasi-pointwise two-step phase-shifting profilometry with the fringe parameters estimated statistically.

Fringe projection profilometry is a popularly used three-dimensional measurement technique in which phase-measuring algorithms based on two-step phase shifting are usually used because of their best tradeoff between measurement resolution and speed. Most two-step phase-shifting algorithms involve neighboring or other spatial operations, thus having degraded accuracies at edges and discontinuities of the measured object surface. Pointwise two-step algorithms enable overcoming this issue. With them, however, the offsets of the dynamic ranges of the projector and camera are usually improperly overlooked or inaccurately estimated, thus inducing errors in their measurement results. For solving this problem, this paper suggests a quasi-pointwise two-step phase-shifting algorithm for fringe projection profilometry. This algorithm models the captured fringe patterns practically by taking the offsets of the dynamic ranges of the projector and camera into account, and estimates the fringe parameters from the statistics of fringe intensities. As a result, we can calculate fringe phases in a pointwise way from two fringe patterns having a phase difference of π/2 radians. The simulation and experimental results demonstrate that the proposed method has a relatively low level of errors in measuring object surfaces having isolated regions and discontinuities.

Two-step gradient-assisted phase-shifting demodulation algorithm for fast 3D reconstruction

The conventional multi-frequency heterodyne method is one of the most widely used methods in non-contact 3D measurement. However, it needs to project different phase-shifting patterns with different frequencies, so a large number of patterns are required. For most conventional methods, the fringe period number of the projected patterns is usually small due to its limited noise tolerance, though a larger fringe period number always means higher accuracy. We propose a two-step phase-shifting demodulation algorithm based on an intensitygradient. In this method, only two patterns for each frequency are required. With the intensity gradient of the two patterns, we obtain the wrapped phase of each frequency. Next, the absolute phase is retrieved from the three wrapped phases with the heterodyne algorithm. Because only two patterns are required for each frequency, the proposed method is more robust and has higher measuring speed compared with the traditional three-frequency four-step heterodyne method. Simulations and experiments prove the feasibility and effectiveness of the method, and demonstrate that the proposed method extends the noise tolerance and achieves highprecision with only half of the patterns required by the traditional three-frequency four-step method.

Spatial phase-shifting profilometry by use of polarization for measuring 3D shapes of metal objects.

In this paper, we present a polarization spatial phase-shifting method for fringe projection profilometry. It enables us to measure the three-dimensional shape of a metal object in a fast way requiring only a single-shot implementation. With this method, a couple of projectors are equipped, in front of their lens, with linear polarization filters having orthogonal polarization directions, so that they can simultaneously cast two sinusoidal fringe patterns having different phase shifts onto the measured metal surfaces without mixture. To register the two projected patterns, we suggest a fringe alignment method based on the epipolar geometry between the projectors. By taking advantage of the property of metal surfaces in maintaining polarization state of incident light, the deformed fringe patterns on the measured surfaces are captured by using two coaxially-arranged polarization cameras. As a result, the fringe phases are calculated by using a two-step phase-shifting algorithm and further the 3D shapes of the measured surfaces are reconstructed. Experimental results demonstrate the proposed method to be valid and efficient in measuring metal objects.

A two-step phase-shifting algorithm dedicated to fringe projection profilometry

Improving the time efficiency of fringe projection profilometry (FPP) is an attractive problem. For FPP using phase-shifting, it is desired to improve the efficiency by reducing the step number for phase retrieval. This paper proposes a two-step phase-shifting algorithm dedicated to FPP. Considering the physical process of FPP, the captured fringe image is formulated with two variables, i.e. surface reflectance and phase value. And a phase shift is introduced to get the two equations, which lead to the close-form solution for phase calculation. Then the phase error due to ambient light is analyzed via a line-circle model, and an algorithm of refining the phase calculation is proposed based on the estimation of the actual fringe contrast. The validity of the proposed approach is demonstrated with experiments.

Research and application of dual-camera dynamic in-line digital holography using a two-step phase-shifting cepstrum technique.

Due to limits in the properties of digital cameras, in-line digital holography is commonly used to take full advantage of the sampling space of the camera. To realize the dynamic high-resolution measurement of in-line digital holography, dual-camera dynamic in-line digital holography is proposed. By means of a two-step phase-shifting cepstrum algorithm and a dual-camera parallel phase-shifting recording optical path, the complex amplitude of the object wave can be reconstructed without its zero-order and conjugate terms. Meanwhile, a novel spherical wave interference calibration method is also developed for the dual-camera recording system, and image correction is carried out via rotation, translation, and diffraction, with an average error of phase correction of 0.1107 rad. Finally, the feasibility and effectiveness of the proposed technique is well demonstrated by a practical application of dynamic temperature field measurement in a transparent medium.

Analysis of the synchronous phase-shifting method in a white-light spectral interferometer.

For white-light spectral interferometry, the phase information is usually retrieved via the Fourier transform method and the temporal phase-shifting method. In comparison, the synchronous phase-shifting method can be used to synchronously acquire interferometric signals with good accuracy and reduced noise. Therefore, it has potential for online measurement and is suitable for application in precision industries and for ultrahigh-speed measurement. In this work, a white-light spectral interferometer for synchronous phase shifting based on polarization interference was built, and the two-step phase-shifting algorithm was used to retrieve phase information. A variety of spectral interferometric signals were simulated based on the mathematical model of the two-step phase-shifting algorithm to illustrate the effects of differences in intensity and envelope shape, random noise, and phase-shift error on measurement of the absolute distance. Measurements of the absolute distance were conducted, and they indicated that the system had high accuracy.

Single-shot 3D shape measurement of discontinuous objects based on a coaxial fringe projection system.

Fringe projection profilometry has been widely used in high-speed three-dimensional (3D) shape measurement. To improve the speed without loss of accuracy, we present a novel single-shot 3D shape measuring system that utilizes a coaxial fringe projection system and a 2CCD camera. The coaxial fringe projection system, comprising a visible light (red, green, and blue) projector and an infrared (IR) light projector, can simultaneously project red, green, blue, and IR fringe patterns. The 2CCD camera, as the name suggests, has two CCD chips that can acquire visible and IR fringe patterns at the same time. Combining the two-step phase-shifting algorithm, Fourier transform profilometry, and the optimum three-frequency selection method, 3D shape measurement of complex surfaces such as large slopes or discontinuous objects can be obtained from single-shot acquisition. A virtual fringe projection measurement system has been established to generate pre-deformed fringe patterns to correct positional deviations of the coaxial fringe projection system. This method has been applied to simulations and experiments on static and dynamic objects with promising results.

Fast phase-shifting technique for 3-D surface micro-topography measurement

This paper presents a two-step phase-shifting interferometry technique for fast inspection of uniformity of optoelectronic samples. The in-line configuration was applied to make full use of the space-bandwidth product (SBP) of the camera and hence can resolve finer spatial details of the sample. Two-step phase-shifting (TSPS) algorithm was used to reconstruct the object with only two in-line interferograms generated when a quarter wave plate (QWP) is placed in the light path of one arm of the Mach-Zehnder interferometer. Experimental results show that the extracted phases calculated by the TSPS technique match with the phases extracted by the three-step phase-shifting (THSPS) technique.

Dynamic Phase Measurement Based on Two-Step Spatial Carrier-Frequency Phase-Shifting Interferometry

A dynamic phase measurement method based on a two-step spatial carrier-frequency phase-shifting (TS-SCFPS) technique is proposed. First, by the subtraction operation between a pair spatial carrier-frequency interferograms with a certain phase shifts, which are captured by dual-channel interferometry system, we can achieve one-frame background-removed carrier-frequency interferogram. Second, from two-frame phase-shifting subinterferograms splitting from the above background-removed carrier-frequency interferogram, we can rapidly achieve the accurate phase through using a two-step orthogonal and normalization algorithm. Compared with the existing SCFPS methods, the proposed TS-SCFPS method reveals good background deductions and antinoise performance. Moreover, by using a two-step phase-shifting algorithm, without any special requirement for the phase shifts, the proposed TS-SCFPS method can rapidly achieve high-accuracy phase retrieval relative to other SCFPS methods. Especially, the achieved experimental results demonstrate the outstanding performance of the proposed TS-SCFPS method in both static and dynamic phase measurements.

Phase retrieval in moiré volume computed tomography based on spatial phase shifting by triple-crossed gratings.

Due to its advantages of nonintrusiveness, wide measurement range, and insensitivity to variation, Moiré deflectometry is a powerful tool for quantitative measurement of a flow field's physical parameters such as density and temperature. Moiré volume computed tomography (MVCT), combining the moiré deflectometry and volume optical computed tomography (VOCT), can realize real three-dimensional parameters reconstruction, in which the radial derivatives extraction of the projected phase is of great importance. In this paper, a spatial phase-shifting-interferometry-based MVCT system was proposed to extract the radial shearing phase distribution. The system is simple and compact, and consists of three crossed gratings and lenses, with no wave plates or polarizers introduced. Via using a 4-f system, the optical path was shortened, and four spatial phase-shifting grid moiré projections can be obtained simultaneously. Each grid interferogram was filtered and separated into two linear interferograms in two orthogonal directions. Moreover, a two-step spatial phase-shifting algorithm was applied to obtain the first-order derivative phase in two mutually perpendicular directions, respectively. Simulations were implemented to verify the feasibility and accuracy of the proposed phase retrieval method. The measured results for the radial first-order partial derivative of the phase projection of a propane flame in the experimental VOCT system are presented.

一种两步相移相位解算方法

一种两步相移相位解算方法

Parallel-quadrature on-axis phase-shifting common-path interferometer using a polarizing beam splitter.

A common-path parallel-quadrature on-axis phase-shifting interferometry using a modified Michelson configuration with a polarizing cube beam splitter is proposed for quantitative phase measurement. The frequency spectrum of the circularly polarized object beam is split into two beams using a beam splitter. One beam is converted to a 45° linearly polarized beam to act as the object beam, and the other beam is low-filtered by a pinhole mirror to act as the reference beam. Two interferograms with quadrature phase shift can be simultaneously captured by combining the 45° linearly polarized object beam with the circularly polarized reference beam through a 45° tilted polarizing cube beam splitter, and the phase of a specimen can be then retrieved through a two-step phase-shifting algorithm. Experiments are carried out to demonstrate the validity and stability of the proposed method.

Visual measurement of the evaporation process of a sessile droplet by dual-channel simultaneous phase-shifting interferometry.

To perform the visual measurement of the evaporation process of a sessile droplet, a dual-channel simultaneous phase-shifting interferometry (DCSPSI) method is proposed. Based on polarization components to simultaneously generate a pair of orthogonal interferograms with the phase shifts of π/2, the real-time phase of a dynamic process can be retrieved with two-step phase-shifting algorithm. Using this proposed DCSPSI system, the transient mass (TM) of the evaporation process of a sessile droplet with different initial mass were presented through measuring the real-time 3D shape of a droplet. Moreover, the mass flux density (MFD) of the evaporating droplet and its regional distribution were also calculated and analyzed. The experimental results show that the proposed DCSPSI will supply a visual, accurate, noncontact, nondestructive, global tool for the real-time multi-parameter measurement of the droplet evaporation.

Two-step spatial phase-shifting lateral shearing interferometry by triple gratings

To acquire phase projection information for moiré tomography, a new spatial phase-shifting lateral shearing interferometry is presented in this paper. The system is very simple and contains only three linear gratings and two filters. No wave plates or polarization elements are introduced. Moreover, via using a 4-f system, the optical path is greatly shortened and two complete spatial phase-shifted lateral shearing interferograms can be obtained simultaneously. A corresponding two-step phase-shifting algorithm is used for phase retrieval and the interferometry is used to extract the first-order partial derivative of the spherical wave. The results show that the proposed method is not only feasible but also has high accuracy. Propane flame phase information is measured by the optical system. Finally, error analysis of phase projection extracted by two-step spatial phase-shifting method is mathematically conducted.

采用相位半角的两步相移法

采用相位半角的两步相移法

Two-step spatial phase-shifting radial shearing interferometery with circular gratings

This Letter presents a simple radial shearing interferometery with circular gratings to measure the radial first-order derivative of arbitrary wavefront phase. Two spatial phase-shifted radial shearing interferograms can be simultaneously obtained by the system, and a two-step phase-shifting algorithm with arbitrary phase-shift value is proposed for phase retrieval. The measurement for spherical wave has shown the feasibility and validity, and the optical system is used to measure the radial first-order derivative of projection wavefront phase of a propane flame with plane incident wave.

Two-step phase-shift interferometry with known but arbitrary reference waves: a graphical interpretation

There are many applications in biology and metrology where it is important to be able to measure both the amplitude and phase of an optical wave field. There are several different techniques for making this type of measurement, including digital holography and phase retrieval methods. In this paper we propose an analytical generalization of this two-step phase-shifting algorithm. We investigate how to reconstruct the object signal if both reference waves are different in phase and amplitude. The resulting equations produce two different solutions and hence an ambiguity remains as to the correct solution. Because of the complexity of the generalized analytical expressions we propose a graphical-vectorial method for solution of this ambiguity problem. Combining our graphical method with a constraint on the amplitude of the object field we can unambiguously determine the correct result. The results of the simulation are presented and discussed.

Two-step phase retrieval method with unknown phase shift on non-absorbtion grating X-ray differential phase contrast imaging system

Conventional phase retrieval methods for X-ray differential phase-contrast imaging are carried out with at least three interferograms. In this contribution, we present a novel two-step phase-shifting algorithm with an arbitrary unknown phase shift between 0 and π. Based on the detecting model of a grating interferometer, the unknown phase shifts are extracted by a noniterative algorithm. Then the phase distribution of the detected sample can be further obtained from the retrieval phase shift. This method is experimentally validated based on non-absorbtion grating imaging system and the result verified its effectiveness.

Optimized two-step phase-shifting algorithm applied to image encryption

The two-step phase-shifting algorithm proposed previously is optimized, the original object wave field can be reconstructed by only one phase shift value in (0, π) and two interferograms, with the removal (or suppression) of background intensity (or dc term), and the additional measurements such as the object wave intensity, reference wave intensity, etc., are no longer required. Together with double random phase encoding technique in the Fresnel domain, the optimized two-step phase-shifting algorithm is then applied to image encryption system. The feasibility of the proposed scheme is verified by computer simulation. Furthermore, the sensitivity of geometrical keys has also been tested and analyzed.

Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities

A wavefront reconstruction and three-dimensional (3-D) shape measurement method by a two-step phase-shifting algorithm with arbitrary phase shift in (0,pi) is proposed. In this method, only two phase-shifted intensities, with the removal of the dc term by an averaging technique in spatial domain or low-pass filter operation in the frequency domain, are needed, and the other additional measurements are no longer required. The simulation for an irregular wavefront has shown the feasibility, and the optical experiment for a 3-D face mask in the case of a sinusoidal fringe projection system has illustrated the validity of the proposed method.