Unwrapping Method Research Articles

Few-fringe-based phase-shifting profilometry employing hilbert transform

In optical measurement techniques, when the projector works at a constant projection rate, reducing the number of projected fringe patterns is an effective way to reduce the projection time. In this paper, we propose a few-fringe-based phase-shifting profilometry employing Hilbert transform. We project only two fringe patterns for each frequency of fringes, with the phase shift between the fringes designed to be π. The Hilbert transform makes these two captured fringes phase shifted by π/2, thus transforming the original two fringes into four, and the wrapped phase can be obtained through these four fringes with a phase difference of π/2. To improve the accuracy and robustness of the method, we adopt three frequencies of fringes for phase unwrapping. Further, since different phase unwrapping methods can be used for the different frequencies of the fringes, we propose a heterodyne 2H+2 M+2 L method and a hierarchical 2H+2 M+2 L respectively. Multiple experimental results have confirmed the feasibility and effectiveness of the two 2H+2 M+2 L methods, and the different characteristics of the reconstructed shapes obtained by the two 2H+2 M+2 L methods are compared to provide useful guidance in the selection of the required method for different application scenarios. The experimental results demonstrate that when using the heterodyne and hierarchical methods for phase unwrapping, the RMSE of the 2H+2 M+2 L method is only higher than that of the three-frequency four-step phase-shifting method by 0.0056 and 0.0024 respectively, but this 2H+2 M+2 L method improves the efficiency of the three-frequency four-step phase-shifting method by 50%.

Local wavenumber imaging with low-sampled wavefield for delamination characterization in composite plate

In this paper, a phase unwrapping-based local wavenumber imaging method is proposed for delamination detection in the composite plate. Local wavenumber imaging has shown significant advantages in damage localization and quantification. However, a dense of spatial measurements and a high requirement of sampling intervals are crucial for conventional spatial Fourier transform methods. To address these challenges, a phase unwrapping method adopting low-sampled wavefields for local wavenumber estimation is proposed for damage imaging. Specifically, the wavefield sampling process can divide the detection area into various grids while making each grid consist of four neighbor measurements. Under the far-field assumption, the wave path distances of grid measurements can be approximately obtained by projecting their real distances along the wave vector direction. Meanwhile, by revealing the phase differences of grid measurement pairs with the low-sampled wavefield data, the frequency-related wavenumbers are estimated subsequently. Hence, the local wavenumber can be recovered robustly by averaging the estimated wavenumbers from all measurement pairs in the grid. On this basis, the structural damage can be geometrically imaged and quantitatively evaluated. Both the simulation study and the experimental investigation have shown the efficiency of the phase unwrapping-based method for local wavenumber imaging, which demonstrates the strong capability of damage localization and quantification in composite structures with low-sampled wavefield.

Temporal phase-unwrapping in fringe projection profilometry: Increasing the accuracy with equidistant long time-steps sampling

Temporal phase-unwrapping methods for fringe projection profilometry, mainly represented by the multi-frequency and multi-wavelength classes, are fundamental to recovering the 3D information from the continuous phase, also allowing measure discontinuities and isolated objects. In this paper, it is proposed a temporal phase unwrapping method for fringe projection profilometry. This method is specially designed to unwrap temporal phase sequences with equidistant long-time steps. As will be shown, this kind of temporal sampling allows increasing the accuracy of measurements. Simulated and real experimental results realized on a flat plate demonstrate the superiority of this proposal compared with leading already reported methods. Also, some experiments with complex objects are presented to demonstrate its applicability.

Deep learning-enabled invalid-point removal for spatial phase unwrapping of 3D measurement

The single-camera system projecting single-frequency patterns is the ideal option among all proposed Fringe Projection Profilometry (FPP) systems in terms of measurement speed and system cost. This system necessitates a robust spatial phase unwrapping algorithm. However, robust spatial phase unwrapping remains a challenge in complex scenes. Quality-guided spatial phase unwrapping needs more efficient ways to identify unreliable points in phase maps before unwrapping. End-to-end deep learning spatial phase unwrapping face generality and interpretability problems. This paper proposes a hybrid spatial phase unwrapping method combining deep learning-enabled invalid-point removal and traditional path-following. This hybrid method demonstrates better robustness than traditional quality-guided methods, better interpretability than end-to-end deep learning schemes, and generality on unseen data. Experiments on the real dataset of multiple illumination conditions and multiple FPP systems differing in image resolution, fringe number, fringe direction, and optics wavelength verify the effectiveness of the proposed method.

Dual-wavelength real-time simultaneous phase imaging based on off-axis interferometry

Dual-wavelength interferometry is widely used in optical measurement, including the of great significance phase imaging of transparent samples. Here, we present a dual wavelength real-time phase imaging system based on off-axis interferometry with two cameras for recording complex fields transmitted through transparent samples in a wide-field and non-intrusive manner. The method of phase unwrapping based on combing dual wavelength with Fourier transform was derived in detail. The advantage of the proposed system is, compared with single-shot dual wavelength holograms methods, that it can easily extract the complex fields at both wavelengths without consideration of spectrum overlap or needing extra optic device to create lateral shearing. For phase noise, we calculated and removed the phase signal without sample in Fourier domain, which can also ensure the measured phase accurately. Experimental results on standard sample (polystyrene microspheres) demonstrated the efficiency of this imaging system. Implementation in quantitative imaging of living cells showed the time of single-shot acquisition of samples was only 0.025s, which indicated that the system can be applied to real-time and label-free identification of phase objects.

Robust phase unwrapping via non-local regularization.

Phase unwrapping is an indispensable step in recovering the true phase from a modulo-2π phase. Conventional phase unwrapping methods suffer from error propagation under severe noise. In this Letter, we propose an iterative framework for robust phase unwrapping with high fidelity. The proposed method utilizes the transport-of-intensity equation to solve the phase unwrapping problem with high computational efficiency. To further improve reconstruction accuracy, we take advantage of non-local structural similarity using low-rank regularization. Meanwhile, we use an adaptive iteration strategy that dynamically and automatically updates the denoising parameter to avoid over-smoothing and preserve image details. A set of simulation and experimental results validates the proposed method, which can provide satisfying results under severe noise conditions, and outperform existing state-of-the-art phase unwrapping methods with at least 6 dB higher peak SNR (PSNR).

Centimeter-wave Free-space Neural Time-of-Flight Imaging

Depth sensors have emerged as a cornerstone sensor modality with diverse applications in personal hand-held devices, robotics, scientific imaging, autonomous vehicles, and more. In particular, correlation Time-of-Flight (ToF) sensors have found widespread adoption for meter-scale indoor applications such as object tracking and pose estimation. While they offer high depth resolution at competitive costs, the precision of these indirect ToF sensors is fundamentally limited by their modulation contrast, which is in turn limited by the effects of photo-conversion noise. In contrast, optical interferometric methods can leverage short illumination modulation wavelengths to achieve depth precision three orders of magnitude greater than ToF, but typically find their range is restricted to the sub-centimeter. In this work, we merge concepts from both correlation ToF design and interferometric imaging; a step towards bridging the gap between these methods. We propose a computational ToF imaging method that optically computes the GHz ToF correlation signal in free space before photo-conversion. To acquire a depth map, we scan a scene point-wise and computationally unwrap the collected correlation measurements. Specifically, we repurpose electro-optical modulators used in optical communication for ToF imaging with centimeter-wave signals, and achieve all-optical correlation at 7.15 GHz and 14.32 GHz modulation frequencies. While GHz modulation frequencies increase depth precision, these high modulation rates also pose a technical challenge. They result in dozens of wraps per meter which cannot be estimated robustly by existing phase unwrapping methods. We tackle this problem with a proposed segmentation-inspired phase unwrapping network , which exploits the correlation of adjacent GHz phase measurements to classify regions into their respective wrap counts. We validate this method in simulation and experimentally, and demonstrate precise depth sensing using centimeter wave modulation that is robust to surface texture and ambient light. Compared to existing analog demodulation methods, the proposed system outperforms all of them across all tested scenarios. CCS Concepts: • Computing methodologies ;

A novel phase unwrapping method for binocular structured light 3D reconstruction based on deep learning

Fringe projection has been prevalently adopted in the field of three-dimensional(3D) reconstruction. However, it is still a challenge to achieve high-precision and efficient measurement. To achieve real-time and efficient 3D shape measurement of the objects, we must extract the 3D coordinate data of the object from as few fringe patterns as possible. In the paper, we develop a new technique combining composite fringe coding and stair phase coding to solve the high-quality absolute phase and realize binocular 3D morphology reconstruction based on deep learning. This method only needs to project a composite fringe coding pattern to acquire the wrapped phase and a stair phase coding pattern to obtain the fringe order. Then we can recover the unwrapped phase maps of the captured fringe patterns. According to the calibration parameters of the binocular camera, the absolute phase is stereo matched to get the disparity map, and then converted into 3D spatial coordinates of the object. The influence of defocus and noise on the deviation between the wrapped phase and fringe order is solved by staggered phase unwrapping method. We demonstrated the validity and robustness of the novel method by experiments, which provides a promising research direction for 3D shape measurement based on binocular structured light.

A Parallel InSAR Phase Unwrapping Method Based on Separated Continuous Regions

Phase unwrapping is an imperative step in interferometry processing that has a significant influence on the quality of subsequent products. Many existing phase unwrapping algorithms have been designed to solve for the unwrapped phase under the assumption that noisy areas with discontinuities are small or that reliable continuity can be recovered there. They attempt to restore the unwrapped phase by using continuity and data quality measures, such as residues. However, when the observing field is divided into separate zones of continuous phase due to a large range of noise, such as those caused by rivers or mountains, it is difficult to use traditional phase unwrapping techniques to recover global continuity in these noisy areas. To address this challenge, we present a two-dimensional parallel phase unwrapping method that is designed to handle cases where the continuity of the phase is separated by closed noisy loops. Based on continuity distances, this method aims to identify continuous regions that are free of hidden phase discontinuities and restore phase continuity between the separated regions. A heterogeneous residual diffusion scheme is used to restore the unwrapped phase outside continuous regions. The parallel algorithm for extracting continuous regions, restoring continuity between the regions, and diffusing residuals was implemented on a GPU device to increase the processing efficiency. We applied our method to typical TanDEM-X data covering rivers, islands, and mountains and demonstrated that it is a promising solution for large-scale, heavily noisy phase unwrapping problems.

An error distribution-related function-trained two-dimensional InSAR phase unwrapping method via U-GauNet

An error distribution-related function-trained two-dimensional InSAR phase unwrapping method via U-GauNet

Inter-partition phase unwrapping method based on Gray code

格雷码因具有良好的鲁棒性和抗噪性，被广泛应用到结构光投影三维成像方法中。在三维测量过程中，由于设备以及其他环境噪声的影响，格雷码方法解码条纹级次边沿和截断相位边沿通常无法处于理想的对齐状况，使得展开的相位出现跳变现象。为了更好地避免级次跳变误差，使得边沿跳变区域的容错宽度更大，提出基于格雷码的分区间相位展开方法。在互补格雷码基础上增加一幅格雷码图像，利用所有格雷码解得附加码字，通过对附加码字进行不同位移量的条纹级次映射，得到2个辅助条纹级次。利用所有条纹级次，对截断相位进行分区间相位展开，在边沿跳变区域错误大于半个周期时，仍能获取到无跳变现象的展开相位。实验结果表明，当边沿错误区域宽度小于3/4个条纹周期宽度时，可以有效避免级次跳变产生的误差。

Pixel-Wise Phase Unwrapping With Adaptive Reference Phase Estimation for 3-D Shape Measurement

Phase unwrapping plays an important role in phase-based three-dimensional (3D) shape measurement. This paper proposes an effective pixel-wise phase unwrapping method without requiring additional fringe patterns. Firstly, compared with the geometric constraint-based (GCB) method which needs calibrate the fringe projection system and create a reference phase at the closest plane <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z_min</i> within the measurement volume, the proposed method can adaptively and directly estimate the reference phase from the wrapped phase without system calibration and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z_min</i> estimation. Secondly, based on the finding that the slope angle of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z_min</i> plane ignored by the GCB method has significant impact on the measurement range, thus the phase gradients that highly correlates with the optimal slope angle are involved into reference phase estimation. The wrapped phase can be pixel-by-pixel unwrapped by comparing with the reference phase. Both simulations and experiments have been conducted to demonstrate the efficiency of the proposed method. Compared with traditional two-frequency method and phase-coding method, the proposed method can improve the measuring speed by 2 times. Experiment results show that the proposed method can effectively remove the phase unwrapping errors and achieve the mean absolute phase error of 0.0137 rad, which has obvious advantages relative to traditional methods.

Semi-Global Matching Assisted Absolute Phase Unwrapping.

Measuring speed is a critical factor to reduce motion artifacts for dynamic scene capture. Phase-shifting methods have the advantage of providing high-accuracy and dense 3D point clouds, but the phase unwrapping process affects the measurement speed. This paper presents an absolute phase unwrapping method capable of using only three speckle-embedded phase-shifted patterns for high-speed three-dimensional (3D) shape measurement on a single-camera, single-projector structured light system. The proposed method obtains the wrapped phase of the object from the speckle-embedded three-step phase-shifted patterns. Next, it utilizes the Semi-Global Matching (SGM) algorithm to establish the coarse correspondence between the image of the object with the embedded speckle pattern and the pre-obtained image of a flat surface with the same embedded speckle pattern. Then, a computational framework uses the coarse correspondence information to determine the fringe order pixel by pixel. The experimental results demonstrated that the proposed method can achieve high-speed and high-quality 3D measurements of complex scenes.

Adaptive horizontal scaling method for speckle-assisted fringe projection profilometry.

Phase-shifting method is widely used in fringe projection profilometry to obtain high-precision wrapped phase maps. The wrapped phase map needs to be converted to an absolute phase map to recover 3D information. The speckle pattern based phase unwrapping method requires only one additional auxiliary pattern, showing great potential for fast 3D measurements. In this paper, a speckle assisted four-steps phase-shifting method was proposed for 3D measurements. This method requires five structured light patterns to complete 3D measurements, including four-steps phase-shifting fringe patterns and a speckle pattern which is used to remove phase ambiguity. The main challenge of speckle based phase unwrapping method is to overcome the mismatch problem which often occurs in some very steep surfaces. In order to improve the speckle matching accuracy, an adaptive horizontal scaling method was proposed. A voting strategy based on phase-connected regions was proposed to reduce the computational overhead. The experiments demonstrate its superior performance, and an accuracy of 0.21 mm was achieved.

Single-frequency and accurate phase unwrapping method using deep learning

Phase unwrapping is an important part of fringe projection profilometry(FPP), which greatly affects the efficiency and accuracy of reconstruction. Phase unwrapping methods with deep learning achieve single-frequency phase unwrapping without additional cameras. However, existing methods have low accuracy in the real complex scene, and can not process data whose resolution is greater than the resolution of training data. This paper introduces a neural convolutional network named as VRNet which achieves accurate and single-frequency phase unwrapping without extra cameras. VRNet with encoder-decoder structure gets multi-scale feature maps through feeding the wrapped phase map into the encoder, then fuses the feature maps recursively by using the proposed feature fusion module to accomplish precise prediction. In order to further improve the accuracy of phase unwrapping, this paper presents a phase correction method based on the distribution characteristics of the absolute phase. The method divides the cross-section of the absolute phase map into several curves and identifies a misclassified pixel by comparing its absolute phase value with the value of neighboring curves. In contrast to existing methods, the method is row-independent and does not require segmentation of image. Moreover, this paper accomplishes the prediction of high-resolution data through the phase stitching strategy and fine-tuning the phase correction method. Extensive experiments show that the proposed method is able to achieve high-accuracy and single-frequency phase unwrapping in real scenes which consist of at least one complex object, and is also effective for wrapped phase maps with a resolution larger than the training data.

An Improved Synthesis Phase Unwrapping Method Based on Three-Frequency Heterodyne

An improved three-frequency heterodyne synthesis phase unwrapping method is proposed to improve the measurement accuracy through phase difference and phase sum operations. This method can reduce the effect of noise and increase the equivalent phase frequency. According to the distribution found in the phase difference calculation process, the Otsu segmentation is introduced to judge the phase threshold. The equivalent frequency obtained from the phase sum is more than those of all projected fringe patterns. In addition, the appropriate period combinations are also studied. The simulations and related experiments demonstrate the feasibility of the proposed method and the ability to improve the accuracy of the measurement results further.

Extraction of the Electromagnetic Parameters of a Metamaterial Using the Nicolson–Ross–Weir Method: An Analysis Based on Global Analytic Functions and Riemann Surfaces

The characterization of electromagnetic metamaterials (MMs) plays a fundamental role in their engineering processes. To this end, the Nicolson–Ross–Weir (NRW) method is intensively used to recover the effective parameters of MMs, even though this is affected by the branch ambiguity problem. In this paper, we face this issue in the context of global analytic functions and Riemann surfaces. This point of view allows us to rigorously demonstrate the mathematical foundations of an algorithmic approach for avoiding the branch ambiguity problem, in which the phase unwrapping method is merged with K-K relations for recovering the effective parameters of an MM. In addition, exploiting the intimate relationship between the K-K relations and the Hilbert transform, a simple variant of the above algorithm is presented.

A Robust Phase Unwrapping Algorithm With High Measurement Range Based on Unscented Kalman Filter in Interferometric Optical Fiber Sensing

Phase unwrapping is the prerequisite to obtain the true phase in interferometric fiber sensing. This article proposes a method based on an unscented Kalman filter (UKF) for interferometric fiber sensors. This method can break the limitation that the traditional phase unwrapping methods fail with a high probability when the differences in phase change are more than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\pi $ </tex-math></inline-formula> radians according to the Itoh criterion. Then, the measurement range of the phase signal can be expanded. This method combines the true phase and the phase gradient into the state vector estimation for UKF. The feasibility of this method is verified theoretically and experimentally. In the specific experiment, the phase signal with a maximum absolute phase difference of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1.5\pi $ </tex-math></inline-formula> (about 4.705) between adjacent sampling points can be recovered perfectly. The time and frequency domain results using this method show better performance. This method can effectively improve the detection range of the system and reduce the sampling rate to a certain extent, which is suitable for large-scale array systems.

A new phase unwrapping method for cross-track interferometric synthetic aperture radar systems

One of the essential applications of interferometric synthetic aperture radar (InSAR) systems is to estimate the ground surface height. To this end, the altitude of each pixel in the region is estimated according to the phase difference of the two received signals by two sensors. However, the sensor has an ambiguity in the relative terrain altitude due to the 2π cyclic nature of the interferometric phase, which leads to a significant error in altitude estimation. In this paper, by focusing on the phase unwrapping issue in InSAR processing, an efficient iterative approach to minimize the phase difference error is presented. For this purpose, first, according to the InSAR imaging geometry, the interferometric phase relationships are described. Second, based on the least-squares approach, the phase unwrapping problem is expressed as an optimization issue with a specific cost function. Third, the cost function is converted to a submodular convex Markov random field (MRF) function. Finally, the resulting optimization problem is solved by using a fast and efficient approach to optimize submodular convex MRF functions. The proposed method's performance is examined using several numerical studies compared to its relevant competitors. The mean square error (MSE) of elevation estimation is considered the performance metric. Numerical simulations have confirmed the efficiency of the proposed algorithm in terms of MSE of elevation estimation. As seen, the outperformance of the proposed method with respect to its best competitor is at least 2 times (3 dB). The operation times of various methods are also compared as a measure of their convergence rate. Based on simulation results, the convergence rate of the proposed algorithm is more than 50 times better than that of its closest competitor.

A measurement framework using THz Time-Domain sensing for wood quality assessment across tree ring samples

To investigate the effectiveness of selective tree improvement work, an accurate and calibration-free framework based on THz time-domain sensing for intrinsic wood quality traits assessment was specifically designed for tree ring core sample analysis. This study contains a global phase unwrapping strategy, effective medium theory (EMT)-based measurement, combined measurement, and resolution enhancement, all of which aim to solve and discuss several challenges in THz time-domain sensing. Namely, the global phase unwrapping strategy eliminates the global jumps in phase difference by analysing the principle of jump generation; EMT-based THz measurement can assess properties of hundreds of consecutive sampling points on wood cores without model training; the combined measurement adjusts relative permittivity and dielectric loss relative to one other in a coordinate system; the resolution enhancement converts 3-D PSF into 1-D PSF, and extracts attenuation to achieve resolution enhancement of target properties. The results show that the global phase unwrapping strategy has the potential to deal with phase differences of both homogeneous and heterogeneous samples and can be considered a general unwrapping method. All R2 obtained by combined measurement are higher than that obtained by separate relative permittivity measurement. The resolution enhancement can achieve more accurate and faster measurement. The framework presented here offers significant potential in promoting the application of THz in wood quality traits assessment by providing a low-cost and high precision analysis system.