Fringe Phase 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.

Three-dimensional measurement for specular reflection object based on adaptive fringe projection and phase measuring profilometry

ObjectiveUsing phase measuring profilometry (PMP) to measure the object with specular reflections is a very challenging task, a large deviation between the measured result and the actual data is caused by the highlights of the measured object surface. Therefore, a 3D measurement method for the object with a specular reflection based on adaptive fringe pattern projection (AFPP) is proposed in this paper. MethodsFirstly, a set of uniform grayscale images are projected to the measured object by a DLP projector. The gray level gradually decreases from 255 and stops when the number of saturated pixels in the camera image reaches 0. The masks of saturated pixels at each gray level are obtained. Secondly, the masks corresponding to two adjacent gray levels are compared respectively, and the optimal projection intensity of the camera image is obtained. Thirdly, the coordinates of the highlight pixels in the camera image are mapped to the projector image coordinate system, and the new fringe images are generated using the optimal projection intensity. Fourthly, the new fringe images are projected onto the measured object for 3D measurement. ResultsThe measurement experiments were implemented on the measured objects made of metal, plastic, and ceramic, respectively. Experiment results show that the proposed method can reduce the measurement deviation caused by the highlights and make the 3D reconstruction model more complete without additional cameras and hardware.

Axial Resolution Enhancement of Optical Sectioning Structured Illumination Microscopy Based on Three-Beam Interference

As a branch of 3D microscopy, optical sectioning structured illumination microscopy (OS-SIM) has the advantages of fast imaging speed, weak photobleaching and phototoxicity, and flexible and compatible configuration. Although the method of using the one-dimensional periodic fringe pattern projected on the sample can remove the out-of-focus background from the in-focus signal, the axial resolution of the final reconstructed 3D image is not improved. Here, we propose a three-beam interference OS-SIM, namely TBOS, instead of the common-used dual-beam interference OS-SIM (DBOS). The three-beam interference scheme has been adopted in 3D super-resolution SIM (3D-SR-SIM), where the fringe phase shifting needs to be along each of the three orientations. In contrast, TBOS applies phase shifting only in one arbitrary direction. We built a TBOS SIM microscope and performed the 3D imaging experiments with 46 nm diameter fluorescent microspheres and a mouse kidney section. The axial resolution of the 3D image obtained with TBOS was enhanced by a factor of 1.36 compared to the DBOS method, consistent with the theoretical analysis and simulation. The OS-SIM with enhanced axial resolution for 3D imaging may find a wide range of applications in the biomedical field.

Fringe-pattern analysis with ensemble deep learning

In recent years, there has been tremendous progress in the development of deep-learning-based approaches for optical metrology, which introduce various deep neural networks (DNNs) for many optical metrology tasks, such as fringe analysis, phase unwrapping, and digital image correlation. However, since different DNN models have their own strengths and limitations, it is difficult for a single DNN to make reliable predictions under all possible scenarios. In this work, we introduce ensemble learning into optical metrology, which combines the predictions of multiple DNNs to significantly enhance the accuracy and reduce the generalization error for the task of fringe-pattern analysis. First, several state-of-the-art base models of different architectures are selected. A K-fold average ensemble strategy is developed to train each base model multiple times with different data and calculate the mean prediction within each base model. Next, an adaptive ensemble strategy is presented to further combine the base models by building an extra DNN to fuse the features extracted from these mean predictions in an adaptive and fully automatic way. Experimental results demonstrate that ensemble learning could attain superior performance over state-of-the-art solutions, including both classic and conventional single-DNN-based methods. Our work suggests that by resorting to collective wisdom, ensemble learning offers a simple and effective solution for overcoming generalization challenges and boosts the performance of data-driven optical metrology methods.

Digital phase-shift method based on distance mapping for phase recovery of an ESPI fringe pattern.

In view of the limitation of the traditional method to recover the phase of the single fringe pattern, we propose a digital phase-shift method based on distance mapping for phase recovery of an electronic speckle pattern interferometry fringe pattern. First, the direction of each pixel point and the centerline of the dark fringe are extracted. Secondly, the normal curve of the fringe is calculated according to the fringe orientation to obtain the fringe moving direction. Thirdly, the distance between each pixel point and the next pixel point in the same phase is calculated by a distance mapping method according to the adjacent centerlines; then the moving distance of the fringes is obtained. Next, combining the moving direction and moving distance, the fringe pattern after the digital phase shift is obtained by full-field interpolation. Finally, the full-field phase corresponding to the original fringe pattern is recovered by four-step phase shifting. The method can extract the fringe phase from a single fringe pattern through digital image processing technology. The experiments show that the proposed method can effectively improve the phase recovery accuracy of a single fringe pattern.

CF3DNet: A learning-based approach for single-shot 3D reconstruction from circular fringes

3D reconstruction by fringe projection is an ill-posed problem that is of significance in many practical applications. Conventional algorithms employ multiple steps like fringe analysis, phase unwrapping, and calibration to reconstruct the 3D profile. Recent deep learning (DL) methods circumvent this multi-step approach by directly mapping the deformations in linear fringes caused by objects to the absolute phase shifts. However, this mapping is not one-to-one for the discontinuous objects due to 2π periodicity of linear fringes, and hence it is impossible to reconstruct using a single deformed fringe. In this work, we propose Circular Fringe-to-3D reconstruction Network (CF3DNet), a learning-based approach for 3D reconstruction from circular fringes, that can establish a one-to-one mapping between the phase deformations and the absolute phase shifts. This is viable because the radially symmetric circular fringes can distinctly record the lateral shifts caused by objects, and the learning-based approach can uniquely map them to absolute phase shifts. Thus, the proposed CF3DNet can reconstruct discontinuous object profiles, which is infeasible with linear fringe-based methods. Further, to evade the tedious process of real data collection and to bring generalization, the model is trained with synthetically generated smooth, discontinuous and Computer-Aided Design (CAD) object profiles. The results demonstrate that the proposed method led to tenfold improved performance over linear fringe-based methods in 3D reconstruction of various synthetic objects, and accurately estimate the real object profile compared to the other state-of-art methods. The extensive analysis presented in this work shows that the proposed method can work with carrier frequencies as low as 10 cycles/row to reconstruct dynamic ranges up to 100 radians under high noise, paving a way for the development of a cost-effective fringe projection system.

A Targetless Method for Simultaneously Measuring Three-Degree-of-Freedom Angular Motion Errors with Digital Speckle Pattern Interferometry.

As a guide rail is the basic motion unit of precision equipment, the measurement of and compensation for its motion errors are important preconditions for precision machining and manufacturing. A targetless and simultaneous measurement method of three-degree-of-freedom (3-DOF) angular motion errors using digital speckle pattern interferometry (DSPI) is introduced in this paper. Based on the analysis of the sensitivity mechanism of DSPI to DOF errors and the formation mechanism of the phase fringes, the relationship between the angular motion errors and the distribution of the interferometric phases was established, and a new simultaneous measurement model of 3-DOF angular motion errors was further proposed. An optical setup based on a three-dimensional spatial-carrier DSPI with a right-angle symmetrical layout was used in the measurement system. Furthermore, repetitive tests, noise tests, and precision analysis were carried out to verify the performance of the system. The test results showed that the measurement resolution of the system was <1 μrad, which is capable of measuring the pitch angle, yaw angle, and roll angle at the submicron arc level simultaneously without target mirrors. The method has the advantages of no need to install cooperative targets and high measurement resolution, showing broad application prospects in many fields, including mechanical manufacturing, laser detection, aerospace, etc.

Complex amplitude domain filtering for phase measurement in speckle interferometric optics

Phase map filtering is considered an indispensable process in the measurement delivered by speckle interferometric optics. For recent measurements, quite a lot of scenarios require an optimized time consumption. A notable feature of the raw phase map is the 2π modulo of the phase distribution, as retrieved by the inverse tangent operation. Instead of employing the filtering method directly on the phase fringe pattern, the wrapped phase is converted into the sine–cosine domain with its sine and cosine components, and then the filtering method is applied to them. However, it is the nature of the sine–cosine filtering framework that consumes approximately 70% calculation resources for the recalculation of the intermediate phase map and the sine/cosine components at every loop, which limits the temporal property of related measurement. To address the issues of efficiency and quality at the same time, a new filtering domain is proposed, namely the complex amplitude domain filtering framework. The filtering is conducted at the stage of recovering the complex amplitude without calculating the intermediate phase map at every loop. The merit of complex amplitude domain filtering is that it has a good filtering efficiency to remove the speckle noise, preserves the details of the sharp edge in the phase-wrapped area at the same time, and no additional phase shift is generated during the filtering. Most of the existing filters and strategies used in the sine–cosine filtering domain can be directly applied in the proposed complex amplitude domain. Preliminary experiments show that, compared to the sine–cosine domain filtering, the proposed complex amplitude domain filtering accelerates the filtering process by at least 40% at no cost of filtering quality, and particularly shows a better filtering performance with the partial differential equations (PDE) noise reduction method.

Phase Deflectometry for Defect Detection of High Reflection Objects.

A method for detecting the surface defects of high reflection objects using phase deflection is proposed. The abrupt change in the surface gradient at the defect leads to the change in the fringe phase. Therefore, Gray code combined with a four-step phase-shift method was employed to obtain the surface gradients to characterize the defects. Then, through the double surface illumination model, the relationship between illumination intensity and phase was established. The causes of periodic error interference were analyzed, and the method of adjusting the fringe width to eliminate it was proposed. Finally, experimental results showed the effectiveness of the proposed method.

Analysis of influence of imaging resolution of relay lens on image reconstruction quality in pixel-wise coded exposure imaging technology

Pixel-wise coded exposure (PCE) imaging based on digital micromirror device (DMD) is an advanced high-speed imaging technology, which can realize the high-speed imaging by using a low-frame-rate camera. During exposure time, the multi-frame image information of a dynamic object can be integrated into one encoded image, and then the multi-frame sub-exposure images can be extracted by the post-processing algorithm. Therefore, the accurate pixel-to-pixel alignment between the DMD and the camera is the key step to realize PCE imaging, which has drawn much attention from researchers. So their studies mainly focused on how to achieve accurate pixel matching. However, the resolution of the relay imaging lens, as another important influence factor of PCE imaging, also has a significant influence on the imaging results, but few people have studied and analyzed it. To solve this problem, in this work, we theoretically analyze the influence of the resolution of the relay imaging system on the reconstructed decoded images, and verifies the theoretical analysis through simulation and imaging experiments. On this basis, a PCE imaging system is built, and a point spread function (PSF) estimation method of relay lens based on the fringe phase is proposed. Furthermore, a Richard-Lucy deconvolution algorithm is introduced into the reconstruction process of coded image to effectively improve the quality of PCE imaging, which is of great significance in developing the PCE imaging technology.

Single-shot structured light projection profilometry with SwinConvUNet

Structured light profilometry (SLP) is now widely utilized in noncontact three-dimensional (3D) reconstruction due to its convenience in dynamic measurements. Compared with classic fringe projection profilometries, multiple deep neural networks are proposed to demodulate or unwrap the fringe phase, and these networks utilize convolution layers to extract local features while omitting global characteristics. In this paper, we propose SwinConvUNet, a deep neural network for single-shot SLP that can extract local and global features simultaneously. In the network structure design, convolution layers are applied in shallow layers to extract local features, whereas transformer layers extract global features in deep layers, and an improved loss function by combining gradient-based structural similarity is employed to improve reconstruction details. The experimental results demonstrate that SwinConvUNet is more effective than the U-net model at decreasing learnable parameters while maintaining 3D reconstruction accuracy.

Achromatic design of a photonic tricoupler and phase shifter for broadband nulling interferometry

Nulling interferometry is one of the most promising technologies for imaging exoplanets within stellar habitable zones. The use of photonics for carrying out nulling interferometry enables the contrast and separation required for exoplanet detection. So far, two key issues limiting current-generation photonic nullers have been identified: phase variations and chromaticity within the beam combiner. The use of tricouplers addresses both limitations, delivering a broadband, achromatic null together with phase measurements for fringe tracking. Here, we present a derivation of the transfer matrix of the tricoupler, including its chromatic behaviour, and our 3D design of a fully symmetric tricoupler, built upon a previous design proposed for the GLINT instrument. It enables a broadband null with symmetric, baseline-phase-dependent splitting into a pair of bright channels when inputs are in anti-phase. Within some design trade space, either the science signal or the fringe tracking ability can be prioritised. We also present a tapered-waveguide $180^\circ$-phase shifter with a phase variation of $0.6^\circ$ in the $1.4-1.7~\mu$m band, producing a near-achromatic differential phase between beams{ for optimal operation of the tricoupler nulling stage}. Both devices can be integrated to deliver a deep, broadband null together with a real-time fringe phase metrology signal.

Phase Target-Based Calibration of Projector Radial Chromatic Aberration for Color Fringe 3D Measurement Systems.

The camera and projector are indispensable hardware parts of a color fringe projection 3D measurement system. Chromatic aberration between different color channels of the projector and camera has an impact on the measurement accuracy of the color fringe projection 3D profile measurement. There are many studies on camera calibration, but the chromatic aberration of the projector remains a question deserving of further investigation. In view of the complex system architecture and theoretical derivation of the traditional projector radial chromatic aberration method, a phase target based on projector radial chromatic aberration measurement and the correction method are proposed in this paper. This method uses a liquid crystal display with a holographic projection film as the phase target. The liquid crystal display sequentially displays red, green, and blue horizontal and vertical sinusoidal fringe images. The projector projects red, green, and blue horizontal and vertical sinusoidal fringe images to the phase target in turn, and calculates the absolute phases of the display fringes and reflection fringes, respectively. Taking the green channel as the reference channel, a phase coordinate system is established based on the phases of the vertical and horizontal directions displayed on the display screen, using the phase of the reflection fringes on the display screen as the ideal phase value of the phase point. Then, the phase coordinate system of the red and blue channels is transferred to the green phase coordinate system to calculate the chromatic aberration of the red-green channels and the blue-green channels, and pre-compensation is conducted. Experimental results prove that this method can measure and calibrate the radial chromatic aberration of the projector without being affected by the image quality of the camera. The correction effect of this method is that the maximum chromatic aberration of the red-green channel decreases from 1.9591/pixel to 0.5759/pixel, and the average chromatic aberration decreases from 0.2555/pixel to 0.1865/pixel. In addition, blue-green channel maximum chromatic aberration decreased from 1.8906/pixel to 0.5938/pixel, and the average chromatic aberration decreased from 0.2347/pixel to 0.1907/pixel. This method can improve the projection quality for fringe projection 3D profile measurement technology.

Deep Learning Network for Speckle De-Noising in Severe Conditions.

Digital holography is well adapted to measure any modifications related to any objects. The method refers to digital holographic interferometry where the phase change between two states of the object is of interest. However, the phase images are corrupted by the speckle decorrelation noise. In this paper, we address the question of de-noising in holographic interferometry when phase data are polluted with speckle noise. We present a new database of phase fringe images for the evaluation of de-noising algorithms in digital holography. In this database, the simulated phase maps present characteristics such as the size of the speckle grains and the noise level of the fringes, which can be controlled by the generation process. Deep neural network architectures are trained with sets of phase maps having differentiated parameters according to the features. The performances of the new models are evaluated with a set of test fringe patterns whose characteristics are representative of severe conditions in terms of input SNR and speckle grain size. For this, four metrics are considered, which are the PSNR, the phase error, the perceived quality index and the peak-to-valley ratio. Results demonstrate that the models trained with phase maps with a diversity of noise characteristics lead to improving their efficiency, their robustness and their generality on phase maps with severe noise.

Moiré Effects in Silicon Photonic Nanowires

Photonic moiré lattices offer an attractive platform for manipulating the flow and confinement of light from remarkably simple device geometries. This emerging field draws inspiration from the rapid research progress observed in twisted bilayer van der Waals materials or “twistronics,” instead of applying moiré physics to photon propagation in wavelength-scale optical media. However, to date, only a limited number of experimental studies have been performed in this area, and there is strong interest in understanding how moiré effects can be tailored in compact and scalable optical technologies such as an integrated photonics platform. In this work, we map the moiré effects of one-dimensional (1D) photonic moiré lattices composed of width-modulated silicon nanowires, including the construction of a 1D experiment analogous to the twisting of a two-dimensional (2D) lattice. Although the twist angle Δθ and/or lattice mismatch ΔΛ are the sole defining parameters for infinite moiré crystals, we demonstrate how the crystal size, symmetry, and moiré fringe phase Δϕ also serve as important degrees of freedom. Through tailoring these parameters, we map a wide range of behaviors including the formation of moiré photonic crystal cavities, the onset of miniband formation and operation as a coupled resonator optical waveguide (CROW), widely tunable Q-factors and group velocities, suppression of grating sidebands, and persistent vs extinguishable tunneling. These results provide insight into the moiré physics of 1D optical systems and highlight various operating regimes relevant to the design of finite photonic moiré lattices and devices.

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.

Alternate Iterative Least-Squares Algorithm Based on Nonuniform Phase Shifting for Suppressing Nonlinearity Errors in Fringe Projection Profilometry

In fringe projection technique, nonlinearities of used devices are recognized as major factors inducing phase errors, especially when the used camera and projector simultaneously have nonlinear features. In this case, ripple artifacts appear to be unevenly distributed over the reconstructed surface depending on the local reflectivity. For suppressing these errors, this paper suggests a phase-measuring technique which is characterized by two critical steps including anti-aliasing phase-shifting and iterative least-squares phase estimation. Anti-aliasing phase-shifting requires the selected phase shifts to be nonuniform and to have no common divisors. It provides a possibility of accurately recovering phases from few to four fringe patterns and eliminating influence of high order harmonics on the measurement results. By expanding the fringe signals from projector into harmonics and modeling the camera response as a polynomial, we derive a least-squares algorithm based on alternate iterations to calculate the fringe phase map and the related nonlinearity coefficients simultaneously. Through practical experiments, it has been demonstrated that this technique allows us to decrease artifacts caused by the simultaneous nonlinearities of the camera and the projector, thus improving measurement accuracy significantly.

Step Surface Profile Measurement Based on Fringe Projection Phase-Shifting Using Selective Sampling

Fringe projection is a non-contact optical method that is widely used in the optical precision measurement of complex stepped surfaces. However, the accuracy of the fringe phase extraction employed has a direct impact on the measurement precision of the surface shape. Where phase-shifting measurement is used, the classical equal step phase extraction algorithm can only be used to measure simple and smooth surfaces, and leads to measurement errors on complex stepped surfaces, which affects the accuracy of the phase extraction. In addition, the iterative process lasts for a long time, resulting in a low efficiency. This paper proposes a step-by-step phase-shifting extraction algorithm based on selective sampling to measure the contour of the stepped surface. Firstly, the fringe pattern is sampled at equal intervals to reduce the iterative calculation time. Finally, the accurate measurement phase is calculated by the alternating iteration method. The phase extraction accuracy and iteration times are compared in experimental measurements between classical iterative algorithms such as four-step phase-shifting algorithms and the variable phase shift phase interpolation algorithm based on selective sampling. It is shown that the variable frequency phase-shifting extraction algorithm based on selective sampling has a shorter operation time, smaller error, and higher accuracy than the traditional iterative algorithm in fringe projection measuring complex stepped surfaces.

Deep-learning-based fringe-pattern analysis with uncertainty estimation

Deep learning has gained increasing attention in the field of optical metrology and demonstrated great potential in solving a variety of optical metrology tasks, such as fringe analysis and phase unwrapping. However, deep neural networks cannot always produce a provably correct solution, and the prediction error cannot be easily detected and evaluated unless the ground-truth is available. This issue is critical for optical metrology, as the reliability and repeatability of the measurement are of major importance for high-stakes scenarios. In this paper, for the first time to our knowledge, we demonstrate that a Bayesian convolutional neural network (BNN) can be trained to not only retrieve the phase from a single fringe pattern but also produce uncertainty maps depicting the pixel-wise confidence measure of the estimated phase. Experimental results show that the proposed BNN can quantify the reliability of phase predictions under conditions of various training dataset sizes and never-before-experienced inputs. Our work allows for making better decisions in deep learning solutions, paving a new way to reliable and practical learning-based optical metrology.

Defocused binary fringe phase error modeling and compensation using depth-discrete Fourier series fitting.

The binary defocus fringe projection is a widely adopted way to increase the speed of fringe projection profilemotry. However, the projected patterns may deviate from ideal ones at some depths. We propose a theoretical model and a corresponding compensation method to explain and calibrate the phase error of defocus-projected patterns. We first low-pass filter the projected patterns at different depths to obtain corresponding ideal ones. Then, we calibrate the model coefficients based on the errors between the original and ideal fringe patterns. The calibrated phase error model can be used to compensate the phase error at arbitrary depths within the calibration volume. Experiments are conducted to verify the feasibility and performance of the proposed method.