Fringe harmonics elimination in multi-frequency phase-shifting fringe projection profilometry

Orthogonal Spatial Binary Coding Method for High-Speed 3D Measurement

Temporal phase unwrapping based on single auxiliary binary coded pattern has been proven to be effective for high-speed 3D measurement. However, in traditional spatial binary coding, it often leads to an imbalance between the number of periodic divisions and codewords. To meet this challenge, a large codewords orthogonal spatial binary coding method is proposed in this paper. By expanding spatial multiplexing from 1D to 2D orthogonal direction, it goes beyond the traditional 8 codewords to 27 codewords at three-level periodic division. In addition, a novel full-period connected domain segmentation technique based on local localization is proposed to avoid the time-consuming global iterative erosion and complex anomaly detection in traditional methods. For the decoding process, a purely spatial codewords recognition and a spatial-temporal hybrid codewords recognition methods are established to better suppress the percentage offset caused by static defocusing and dynamic motion, respectively. Obviating the need for intricate symbol recognition, the decoding process in our proposed method encompasses a straightforward analysis of statistical distribution. Building upon the development of special spatial binary coding, we have achieved a well-balance between low periodic division and large codewords for the first time. The experimental results verify the feasibility and validity of our proposed whole image processing method in both static and dynamic measurements.

Phase correction strategy based on structured light fringe projection profilometry

Fringe projection profilometry based on structured light has been widely used in 3-D vision due to its advantages of simple structure, good robustness, and high speed. The principle of this technique is to project multiple orders of stripes on the object, and the camera captures the deformed stripe map. Phase unwrapping and depth map calculation are important steps. Still, in actual situations, phase ambiguity is prone to occur at the edges of the object. In this paper, an adaptive phase segmentation and correction (APSC) method after phase unwrapping is proposed. In order to effectively distinguish the stable area and unstable area of the phase, a boundary identification method is proposed to obtain the structural mask of the phase. A phase compensation method is proposed to improve the phase accuracy. Finally, we obtain the 3-D reconstruction result based on the corrected phase. Specific experimental results verify the feasibility and effectiveness of this method.

Nonlinear error reduction for phase-shifting profilometry considering periodicity and symmetry of a phase histogram

Phase-shifting profilometry is extensively utilized for three-dimensional (3D) measurement. However, because of gamma nonlinearity, the image intensities of the captured fringe patterns are regrettably distorted. An effective nonlinear error reduction method without requiring parameter estimation is presented in this paper. Differing from the traditional whole-period phase histogram equalization (PHE) method, our method takes into account not only the periodicity but also the symmetry of the phase histogram. Taking a three-step phase-shifting algorithm as an example, the phase error frequency triples the fringe frequency; thus, we first propose a 1/3-period PHE method. Moreover, since the phase error distribution is sinusoidal with symmetry, we further propose a 1/6-period PHE method. Simulations and experiments both indicate that the 1/6-period PHE method, compared with the whole-period PHE and 1/3-period PHE methods, can further reduce the nonlinear error.

Harmonics elimination in phase-shifting fringe projection profilometry by use of a non-filtering algorithm in frequency domain

In phase-shifting fringe projection profilometry, fringe harmonics caused by device nonlinearities as well as other factors may badly ruin the measurement results. Generally, the used phase-shifting algorithm enables restraint of effects of harmonics below a certain order depending on the number of phase shifts. When reducing the number of phase shifts for efficiency, high order harmonics will affect the phase-measuring results because of aliasing caused by insufficient sampling rate. To overcome this issue, this paper suggests a non-filtering technique operating in frequency domain, that enables improvement of measurement accuracy by eliminating effects of high order harmonics. With this technique, the phase-shifting algorithm is restated as a process of retrieving the fundamental complex fringes from the phase-shifted fringe patterns. Implementing a Fourier transform to this calculated complex fringe pattern, the actual fundamental signals and the aliased harmonics have their own lobes with separated peaks in the frequency domain. We reconstruct each order of the aliased harmonics by exploiting their relations with the fundamental signals and then estimate their magnitudes by using the spectral peaks. Instead of directly filtering the fringe spectrum, we subtract spectra of the harmonics from Fourier transform of the just calculated complex fringes, so that the Fourier spectrum of the fundamental fringes without harmonics is recovered through an iterative operation. Further, the phase map is measured accurately. Simulation and experimental results confirm that this proposed method can significantly suppress effects of fringe harmonics. Meanwhile, by taking advantage of non-filtering, it effectively preserves the edges and details of the measured surfaces from being blurred.

Mask information-based gamma correction in fringe projection profilometry

For fringe projection profilometry (FPP), the gamma effect of the camera and projector will cause non-sinusoidal distortion of the fringe patterns, leading to periodic phase errors and ultimately affecting the reconstruction accuracy. This paper presents a gamma correction method based on mask information. Since the gamma effect will introduce higher-order harmonics into the fringe patterns, on top of projecting two sequences of phase-shifting fringe patterns having different frequencies, a mask image is projected to provide enough information to determine the coefficients of higher-order fringe harmonics using the least-squares method. The true phase is then calculated using Gaussian Newton iteration to compensate for the phase error due to the gamma effect. It does not require projecting a large number of images, and only 2 × 3 phase shift patterns and 1 mask pattern minimum are required. Simulation and experimental results demonstrate that the method can effectively correct the errors caused by the gamma effect.

Super-resolution phase retrieval network for single-pattern structured light 3D imaging

Structured light 3D imaging is often used for obtaining accurate 3D information via phase retrieval. Single-pattern structured light 3D imaging is much faster than multi-pattern versions. Current phase retrieval methods for single-pattern structured light 3D imaging are however not accurate enough. Besides, the projector resolution in a structured light 3D imaging system is expensive to improve due to hardware costs. To address the issues of low accuracy and low resolution of single-pattern structured light 3D imaging, this work proposes a super-resolution phase retrieval network (SRPRNet). Specifically, a phase-shifting module is proposed to extract multi-scale features with different phase shifts, and a refinement and super-resolution module is proposed to obtain refined and super-resolution phase components. After phase demodulation and unwrapping, high-resolution absolute phase is obtained. A sine shifting loss and a cosine shifting loss are also introduced to form the regularization term of the loss function. As far as can be ascertained, the proposed SRPRNet is the first network for super-resolution phase retrieval by using a single pattern, and it can also be used for standard-resolution phase retrieval. Experimental results on three datasets show that SRPRNet achieves state-of-the-art performance on 1×, 2×, and 4× super-resolution phase retrieval tasks.

Fast combined-frequency phase extraction for phase shifting profilometry

Due to the nonlinearity in phase shifting profilometry (PSP) system, the captured images are often distorted with fringe harmonics, resulting in inaccurate phase map and measurement. Considering the fact that the phase error can be significantly reduced by modeling high-order fringe harmonics, this work formulates the phase extraction problem - with different frequency images and high-order fringe harmonic model - as a maximum likelihood estimation (MLE). To optimize it efficiently, we thus propose a combined-frequency phase extraction (CFPE) solution by introducing a latent phase map and incorporating the famous expectation-maximization (EM) framework. As a result, our CFPE method only needs ∼5% execution time of a high-order baseline, whilst keeps the high-order accuracy. Tested on synthetic images as well as practical measurements, our CFPE method demonstrated its performance improvement of efficiency and accuracy. In addition, our detailed implementation with experimental arrangement is also provided for interested researchers.

Robust gamma correction based on chord distribution coding considering projector defocusing

In phase-measurement profilometry (PMP), the gamma effect can cause severe nonlinear distortion of the phase pattern (i.e., water ripples on the surface profile). Gamma correction is an effective method to eliminate the gamma effects of commercial projectors. However, projector defocusing on the suppression of higher harmonics inevitably results in an estimated gamma deviation from the true value. In this study, gamma mapping is constructed using the duty ratio (DR) to code the chord distribution of the simulated distorted phase while considering projector defocusing. With the known gamma mapping, the accurate gamma is calculated by DR coding of the actual distorted phase under projector defocusing. Simulated experiments verified that the relative errors of the gamma calculated by the proposed method under different degrees of defocus were less than 3.5%. Furthermore, the experimental results demonstrate that the proposed gamma calculation method is robust to the defocus effect of the projector and that a smoother surface can be reconstructed after gamma correction.

Anti-aliasing phase reconstruction via a non-uniform phase-shifting technique

The conventional phase-shifting techniques commonly suffer from frequency aliasing because of their number of phase shifts below the critical sampling rate. As a result, fringe harmonics induce ripple-like artifacts in their reconstructed phase maps. For solving this issue, this paper presents an anti-aliasing phase-measuring technique. Theoretical analysis shows that, with phase-shifting, the harmonics aliased with the fundamental frequency component of a fringe signal depend on the greatest common divisor (GCD) of the used phase shifts. This fact implies a possibility of removing such aliasing effects by selecting non-uniform phase shifts that together with 2π have no common divisors. However, even if we do so, it remains challenging to separate harmonics from the fundamental fringe signals, because the systems of equations available from the captured fringe patterns are generally under-determined, especially when the number of phase shifts is very few. To overcome this difficulty, we practically presume that all the points over the fringe patterns have an identical characteristic of harmonics. Under this constraint, using an alternate iterative least-squares fitting procedure allows us to estimate the fringe phases and the harmonic coefficients accurately. Simulation and experimental results demonstrate that this proposed method enables separating high order harmonics from as few as 4 fringe patterns having non-uniform phase shifts, thus significantly suppressing the ripple-like phase errors caused by the frequency aliasing.

Signal-enhanced and bi-directional interferometric Rayleigh scattering velocimetry using an asymmetry cavity

Interferometric Rayleigh scattering technique is commonly employed to measure single-point velocity fluctuation and its standard deviation in a high-speed flow due to many benefits, such as high accuracy, easy data interpretation, and high sampling rate. However, this technique suffers from a severe problem often referred to as the weak Rayleigh scattering signal, especially in the supersonic and hypersonic flow with an extremely low gas molecule density. An asymmetry cavity structure that could cost-effectively improve the Rayleigh scattering (RS) signal of interest is designed and used in the interferemetric Rayleigh scattering technique. The ZEMAX simulations suggest that the parallel beam can be repeatedly reflected in the resonant cavity and can be focused in a measurement region with the order of 0.67 mm×1.31 mm. The number of propagating rays inside the cavity can reach about 50. The fidelity of this proposed cavity is then verified by the Rayleigh scattering imaging experiments. Results show that this cavity allows the laser beam to reflect several times in the resonant cavity, and the RS signal intensity in the major axis can be 10.4 times larger than that of the incident laser. The cavity is finally employed under realistic supersonic flow velocity measurements, where the results conclusively illustrate that the Rayleigh scattering signal of interest in a single direction can be improved by a factor of 4∼5. In addition, the bi-directional (both the axial and radial directions) velocity parameters can also be obtained simultaneously. The axial velocity and its standard deviation are similar to conventional single-line ones.

Improved computer-generated moiré profilometry with flat image calibration

An improved computer-generated moiré profilometry (CGMP) with flat image calibration is proposed. In CGMP, the purification of the AC component plays a decisive role. While a composite grating modulated with both the sinusoidal grating and its background light substitutes for the sinusoidal grating itself, the sinusoidal deformed pattern and flat image can be demodulated from the captured pattern. It is found that the sinusoidal deformed pattern and flat image may deviate, which is caused by ambient light. So flat image calibration is conducted to obtain a purer AC component that can effectively suppress the influence of ambient light and ensure the measurement accuracy, even if spectrum aliasing exists. Experimental results show the feasibility and validity of the proposed method.

Spatial pattern-shifting method for complete two-wavelength fringe projection profilometry

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.