Robust Phase Unwrapping Research Articles

High-frequency average phase compensation method for gamma nonlinearity based on optimal-frequency strategy

The accuracy of fringe projection phase-shifting profilometry (PSP) is affected by gamma nonlinearity greatly, and the average phase compensation method is an effective technique to reduce the nonlinear error. However, double fringe patterns are commonly required, especially combined with the multi-frequency phase unwrapping method (MFPU), using 6 × 3 images in three-frequency method, which limits the measurement eiciency. To reduce the number of required images, this paper presents an efficient average phase compensation method using 6f h + 3f l + 3f u algorithm based on an optimal-frequency strategy. Six high-frequency standard and π/3 shifted 3-step phase-shifting fringe patterns are used together to generate high-accuracy wrapped phase. Three unit-frequency and three low-frequency fringe patterns are used to obtain coarse a unit-frequency wrapped phase and a coarse low-frequency wrapped phase, respectively. To ensure the robust phase unwrapping for high-frequency phase, the mathematical model of the optimal frequency is derived and determined by phase error amplitude calculation. Simulation and experimental results verified that only applying average phase compensation under the guidance of optimal-frequency selection strategy could achieve robust phase unwrapping and high-accurate measurement by reducing the nonlinear error substantially.

Three-Shot Dual-Frequency Fringe Scheme Based on Spatial Computer-Generated Moiré Fringe

A highly robust dual-frequency hierarchical temporal phase unwrapping (DHTPU) based on the novel spatial computer-generated Moiré profilometry (SCGMP) is proposed. The method requires only three patterns: a high-frequency fringe to provide robust surface information, a multi-period low-frequency fringe to eliminate the 2π-phase ambiguities, and a flat pattern to remove the average intensity of the two fringes. In decoding, different from traditional Moiré profilometries that rely on spectrum filters, SCGMP only employs spatial-domain calculations to extract the wrapped phase, thereby preserving more detailed information. Furthermore, we fully explore SCGMP’s capability to significantly alleviate phase ambiguity and provide an algorithm to determine the maximum measurable height range for a fixed system, enabling the direct extraction of the continuous basic phase from the multi-period low-frequency fringe. Consequently, the proposed basic phase exhibits an enhanced signal-to-noise ratio, compared to the traditional basic phase derived from the single-period fringes, effectively releasing the high-frequency restriction in the traditional DHTPU. The experimental results verify that the proposed DHTPU method has considerable accuracy and great potential for high-speed measurements, due to there being only three shots required.

A new 3D phase unwrapping method by region partitioning and local polynomial modeling in abdominal quantitative susceptibility mapping.

Accurate phase unwrapping is a critical prerequisite for successful applications in phase-related MRI, including quantitative susceptibility mapping (QSM) and susceptibility weighted imaging. However, many existing 3D phase unwrapping algorithms face challenges in the presence of severe noise, rapidly changing phase, and open-end cutline. In this study, we introduce a novel 3D phase unwrapping approach utilizing region partitioning and a local polynomial model. Initially, the method leverages phase partitioning to create initial regions. Noisy voxels connecting areas within these regions are excluded and grouped into residual voxels. The connected regions within the region of interest are then reidentified and categorized into blocks and residual voxels based on voxel count thresholds. Subsequently, the method sequentially performs inter-block and residual voxel phase unwrapping using the local polynomial model. The proposed method was evaluated on simulation and in vivo abdominal QSM data, and was compared with the classical Region-growing, Laplacian_based, Graph-cut, and PRELUDE methods. Simulation experiments, conducted under different signal-to-noise ratios and phase change levels, consistently demonstrate that the proposed method achieves accurate unwrapping results, with mean error ratios not exceeding 0.01%. In contrast, the error ratios of Region-growing (N/A, 84.47%), Laplacian_based (20.65%, N/A), Graph-cut (2.26%, 20.71%), and PRELUDE (4.28%, 10.33%) methods are all substantially higher than those of the proposed method. In vivo abdominal QSM experiments further confirm the effectiveness of the proposed method in unwrapping phase data and successfully reconstructing susceptibility maps, even in scenarios with significant noise, rapidly changing phase, and open-end cutline in a large field of view. The proposed method demonstrates robust and accurate phase unwrapping capabilities, positioning it as a promising option for abdominal QSM applications.

VDE-Net: a two-stage deep learning method for phase unwrapping.

Phase unwrapping is a critical step to obtaining a continuous phase distribution in optical phase measurements and coherent imaging techniques. Traditional phase-unwrapping methods are generally low performance due to significant noise or undersampling. This paper proposes a deep convolutional neural network (DCNN) with a weighted jump-edge attention mechanism, namely, VDE-Net, to realize effective and robust phase unwrapping. Experimental results revealed that the weighted jump-edge attention mechanism, which is first proposed and simple to calculate, is useful for phase unwrapping. The proposed algorithm outperformed other networks or common attention mechanisms. In addition, an unseen wrapped phase image of a living red blood cell (RBC) was successfully unwrapped by the trained VDE-Net, thereby demonstrating its strong generalization capability.

Experimental Results of Three-Dimensional Modeling and Mapping with Airborne Ka-Band Fixed-Baseline InSAR in Typical Topographies of China

Interferometric synthetic aperture radar (InSAR) has become a key technology for producing high-precision digital surface models (DSMs) and digital orthophoto maps (DOMs) in full time and all weathers. Airborne millimeter-wave InSAR, with large-scale and high-resolution imaging, is characterized by high spatial resolution, flexibility, and immunity to loss-of-correlation. This paper introduces our modeling experiments with airborne dual-antenna, Ka-band InSAR regarding typical topographies of China. Ka-band SAR data were acquired in designated experimental areas in flat (Heyang area in Shaanxi) and mountainous areas (Shibing area in Guizhou and Qionglai area in Sichuan). The key processing of the experimental data for DSMs and DOMs is demonstrated in the paper, especially the proposed robust and efficient phase unwrapping (PU) method for the interferometric data and block adjustment method of strip calibration. The results show that the proposed unwrapping method can provide reliable unwrapped phase results in undulating areas, and the block adjustment can carry out consistent calibration for strips with sparse ground control points (GCPs). The accuracy assessment of the DSM shows that the coordinate root mean square error (RSME) of the obtained DSM is less than 2 m in height, and 2.5 m horizontally, which meets the 1:5000 requirement for topographic mapping in difficult areas.

Robust and fast dual-wavelength phase unwrapping in quantitative phase imaging with region segmentation

Dual-wavelength digital holographic microscopy extends the unambiguous measurement range in quantitative phase imaging. However, it suffers from increased noise, which reduces phase sensitivity, especially when a larger beat wavelength is synthesized. We present a simple, robust, and fast dual-wavelength phase unwrapping method, without increased phase noise, by segmenting a single wavelength phase map into multiple regions along its phase discontinuities and calculating the fringe order of each region. The method works properly with a large beat wavelength, while reducing phase noise to the level of a single wavelength. Simulations were performed for imaging a smooth profile sample and a multi-heights step. We also demonstrated its practical implementation in quantitative imaging of microstructures where a beat wavelength was up to 20.475μm.

Deep Learning-Enabled Spatial Phase Unwrapping for 3d Measurement

In terms of 3D imaging speed and system cost, the single-camera system projecting single-frequency patterns is the ideal option among all proposed Fringe Projection Profilometry (FPP) systems. This system necessitates a robust spatial phase unwrapping (SPU) algorithm. However, robust SPU remains a challenge in complex scenes. Quality-guided SPU algorithms need more efficient ways to identify the unreliable points in phase maps before unwrapping. End-to-end deep learning SPU methods face generality and interpretability problems. This paper proposes a hybrid method combining deep learning and traditional path-following for robust SPU in FPP. This hybrid SPU scheme demonstrates better robustness than traditional quality-guided SPU methods, better interpretability than end-to-end deep learning scheme, and generality on unseen data. Experiments on the real dataset of multiple illumination conditions and multiple FPP systems differing in image resolution, the number of fringes, fringe direction, and optics wavelength verify the effectiveness of the proposed method.

Efficient intensity-based fringe projection profilometry method resistant to global illumination.

Intensity-based fringe projection profilometry (IBFPP) is used widely because of its simple structure, high robustness, and noise resilience. Most IBFPP methods assume that any scene point is illuminated by direct illumination only, but global illumination effects introduce strong biases in the reconstruction result for many real-world scenes. To solve this problem, this paper describes an efficient IBFPP method for reconstructing three-dimensional geometry in the presence of global illumination. First, the average intensity of two sinusoidal patterns is used as a pixel-wise threshold to binarize the codeword patterns. The binarized template pattern is then used to convert other binarized fringe patterns into traditional Gray-code patterns. A proprietary compensation algorithm is then applied to eliminate fringe errors caused by environmental noise and lens defocusing. Finally, simple, efficient, and robust phase unwrapping can be achieved despite the effects of subsurface scattering and interreflection. Experimental results obtained in different environments show that the proposed method can obtain three-dimensional information reliably when influenced by global illumination.

Single-shot absolute 3D shape measurement with deep-learning-based color fringe projection profilometry.

Recovering the high-resolution three-dimensional (3D) surface of an object from a single frame image has been the ultimate goal long pursued in fringe projection profilometry (FPP). The color fringe projection method is one of the technologies with the most potential towards such a goal due to its three-channel multiplexing properties. However, the associated color imbalance, crosstalk problems, and compromised coding strategy remain major obstacles to overcome. Inspired by recent successes of deep learning for FPP, we propose a single-shot absolute 3D shape measurement with deep-learning-based color FPP. Through "learning" on extensive data sets, the properly trained neural network can "predict" the high-resolution, motion-artifact-free, crosstalk-free absolute phase directly from one single color fringe image. Compared with the traditional approach, our method allows for more accurate phase retrieval and more robust phase unwrapping. Experimental results demonstrate that the proposed approach can provide high-accuracy single-frame absolute 3D shape measurement for complicated objects.

PhUn-Net: ready-to-use neural network for unwrapping quantitative phase images of biological cells.

We present a deep-learning approach for solving the problem of 2π phase ambiguities in two-dimensional quantitative phase maps of biological cells, using a multi-layer encoder-decoder residual convolutional neural network. We test the trained network, PhUn-Net, on various types of biological cells, captured with various interferometric setups, as well as on simulated phantoms. These tests demonstrate the robustness and generality of the network, even for cells of different morphologies or different illumination conditions than PhUn-Net has been trained on. In this paper, for the first time, we make the trained network publicly available in a global format, such that it can be easily deployed on every platform, to yield fast and robust phase unwrapping, not requiring prior knowledge or complex implementation. By this, we expect our phase unwrapping approach to be widely used, substituting conventional and more time-consuming phase unwrapping algorithms.

Rapid and robust two-dimensional phase unwrapping via deep learning.

Two-dimensional phase unwrapping algorithms are widely used in optical metrology and measurements. The high noise from interference measurements, however, often leads to the failure of conventional phase unwrapping algorithms. In this paper, we propose a deep convolutional neural network (DCNN) based method to perform rapid and robust two-dimensional phase unwrapping. In our approach, we employ a DCNN architecture, DeepLabV3+, with noise suppression and strong feature representation capabilities. The employed DCNN is first used to perform semantic segmentation to obtain the segmentation result of the wrapped phase map. We then combine the wrapped phase map with the segmentation result to generate the unwrapped phase. We benchmarked our results by comparing them with well-established methods. The reported approach out-performed the conventional path-dependent and path-independent algorithms. We also tested the robustness of the reported approach using interference measurements from optical metrology setups. Our results, again, clearly out-performed the conventional phase unwrap algorithms. The reported approach may find applications in optical metrology and microscopy imaging.

High-speed three-dimensional shape measurement based on shifting Gray-code light.

The measuring technique combining a phase-shifting algorithm and Gray-code light has been widely used in three-dimensional (3D) shape measurement for static scenes because of its high robustness and anti-noise ability. However, in the high-speed measurement, phase unwrapping errors occur easily on the boundaries of adjacent Gray-code words because of the defocus of the projector, the motion of the objects and the non-uniform reflectivity of the surface. To overcome this challenge, a high-speed 3D shape measurement method based on shifting Gray-code light has been proposed in this paper. Firstly, the average intensity of three captured phase-shifting fringe images are used as a pixel-wise threshold to binarize the Gray codes and to eliminate most phase unwrapping errors caused by the non-uniform reflectivity, ambient light variations, and the defocus of projector. Then, the shifting Gray-code (SGC) coding strategy is proposed to avoid the remaining errors of phase unwrapping on the edge of the code words. In this strategy, no additional patterns are projected, and two sets of decoding words with staggered boundaries are built in the temporal sequences for one wrapped phase. Finally, the simple, efficient, and robust phase unwrapping can be achieved in the high-speed dynamic measurement. This proposed method has been applied to reconstruct 3D shape of randomly collapsing objects in a large depth range, and the experimental results demonstrate that it can reliably obtain high-quality shape and texture information at 310 frames per second.

Robust phase unwrapping by probabilistic consensus

Structured light scanning works by projecting over the scene a supplement of controlled information: the captured signal is processed to provide a unique label (namely a code) for each observed point, and then proceed to geometrical triangulation. In phase shift profilometry sinusoidal patterns are projected and each point is labelled according to the observed phase. Then, due to the periodic nature of the signal, a disambiguation method (known as phase unwrapping) is needed. Several unwrapping techniques have been proposed in the literature, since noisy signals lead to inaccuracies in phase estimation.This paper presents a novel phase unwrapping approach based on a probabilistic framework. The method involves the projection of multiple sinusoidal patterns with distinct period lengths, encoding different phase values at each point location. Phase values are then modelled as samples from a Wrapped Gaussian distribution with an unknown mean, determined by the projector code that generated the values. This formulation allows us to robustly perform phase unwrapping via Maximum Likelihood Estimation, recovering code values from the observed phases. Furthermore, the same likelihood function can be exploited to identify and correct faulty unwrappings by gauging mutual support in a spatial neighbourhood. An extensive experimental assessment validates the Gaussian distribution hypothesis and verifies the improvements in coding accuracy when compared to other classical unwrapping techniques.

Light-field-based absolute phase unwrapping.

Ambiguity caused by a wrapped phase is an intrinsic problem in fringe projection-based 3D shape measurement. Among traditional methods for avoiding phase ambiguity, spatial phase unwrapping is sensitive to sensor noise and depth discontinuity, and temporal phase unwrapping requires additional encoding information that leads to an increase of image sequence acquisition time or a reduction of fringe contrast. Here, to the best of our knowledge, we report a novel method of absolute phase unwrapping based on light field imaging. In a recorded light field under structured illumination, i.e., a structured light field, a wrapped phase-encoded field can be retrieved and resampled in diverse image planes associated with several possible fringe orders in a measurement volume. Then, by leveraging phase consistency constraint in the resampled wrapped phase-encoded field, correct fringe orders can be determined to unwrap the wrapped phase without any additional encoding information. Experimental results demonstrated that the proposed method was suitable for accurate and robust absolute phase unwrapping.

Robust and efficient multi-frequency temporal phase unwrapping: optimal fringe frequency and pattern sequence selection.

Temporal phase unwrapping (TPU) is an essential algorithm in fringe projection profilometry (FPP), especially when measuring complex objects with discontinuities and isolated surfaces. Among others, the multi-frequency TPU has been proven to be the most reliable algorithm in the presence of noise. For a practical FPP system, in order to achieve an accurate, efficient, and reliable measurement, one needs to make wise choices about three key experimental parameters: the highest fringe frequency, the phase-shifting steps, and the fringe pattern sequence. However, there was very little research on how to optimize these parameters quantitatively, especially considering all three aspects from a theoretical and analytical perspective simultaneously. In this work, we propose a new scheme to determine simultaneously the optimal fringe frequency, phase-shifting steps and pattern sequence under multi-frequency TPU, robustly achieving high accuracy measurement by a minimum number of fringe frames. Firstly, noise models regarding phase-shifting algorithms as well as 3-D coordinates are established under a projector defocusing condition, which leads to the optimal highest fringe frequency for a FPP system. Then, a new concept termed frequency-to-frame ratio (FFR) that evaluates the magnitude of the contribution of each frame for TPU is defined, on which an optimal phase-shifting combination scheme is proposed. Finally, a judgment criterion is established, which can be used to judge whether the ratio between adjacent fringe frequencies is conducive to stably and efficiently unwrapping the phase. The proposed method provides a simple and effective theoretical framework to improve the accuracy, efficiency, and robustness of a practical FPP system in actual measurement conditions. The correctness of the derived models as well as the validity of the proposed schemes have been verified through extensive simulations and experiments. Based on a normal monocular 3-D FPP hardware system, our method enables high-precision unambiguous 3-D shape measurement with the highest fringe frequency up to 180 by using only 7 fringe patterns achieving a depth precision ∼ 38μm across a field of view of 400 × 300 × 400 mm.

Realization of Absolute‐Phase Unwrapping and Speckle Suppression in Laser Digital Holography

In this paper, an absolute‐phase unwrapping and speckle suppression approach to reconstruct a three‐dimensional (3‐D) image of an object with laser digital holography is described. This method offers three advantages to enhance the performance of the phase reconstruction technique. First, both speckle suppression and phase unwrapping are processed in the complex amplitude domain rather than in the single phase or amplitude domain. With this approach, the phase details of the object are better preserved upon phase reconstruction. Second, the proposed algorithm requires no threshold determination and thus achieves self‐adaptive speckle suppression and robust phase unwrapping, in contrast to other methods. Finally, an improved dual‐domain image denoising method is applied to further remove speckle‐remnant‐induced phase distortion. Ideal 3‐D phase reconstruction results are obtained both theoretically and experimentally for the first time.

Parisar: Patch-Based Estimation and Regularized Inversion for Multibaseline SAR Interferometry

Reconstruction of elevation maps from a collection of synthetic aperture radar (SAR) images obtained in interferometric configuration is a challenging task. Reconstruction methods must overcome two difficulties: the strong interferometric noise that contaminates the data and the $2\pi $ phase ambiguities. Interferometric noise requires some form of smoothing among pixels of identical height. Phase ambiguities can be solved, up to a point, by combining linkage to the neighbors and a global optimization strategy to prevent from being trapped in local minima. This paper introduces a reconstruction method, Parisar, that achieves both a resolution-preserving denoising and a robust phase unwrapping (PhU) by combining nonlocal denoising methods based on patch similarities and total-variation regularization. The optimization algorithm, based on graph cuts, identifies the global optimum. Combining patch-based speckle reduction methods and regularization-based PhU requires solving several issues: 1) computational complexity, the inclusion of nonlocal neighborhoods strongly increasing the number of terms involved during the regularization, and 2) adaptation to varying neighborhoods, patch comparison leading to large neighborhoods in homogeneous regions and much sparser neighborhoods in some geometrical structures. Parisar solves both issues. We compare Parisar with other reconstruction methods both on numerical simulations and satellite images and show a qualitative and quantitative improvement over state-of-the-art reconstruction methods for multibaseline SAR interferometry.

Weighted least-squares phase unwrapping algorithm based on a non-interfering image of an object.

Conventional quality maps for weighted least-squares phase unwrapping are not appropriate in regions of abnormal fringes. In this paper, the use of a non-interfering image of an object for a reliability map is proposed for robust phase unwrapping. First, the conventional weighted least-squares algorithm is applied, and the unreliable region is detected. Then, the unreliable region is unwrapped iteratively using the non-interfering image of the object. Finally, both phase maps are combined and smoothed by a continuity operation. Experimental results show that the proposed algorithm is appropriate for unwrapping phase maps of abnormal fringes.

Closed-form, robust and accurate multi-frequency phase unwrapping: Frequency design and algorithm

A closed-form algorithm called “concerto” is proposed for phase-based distance estimation in multi-frequency phase unwrapping (MFPU) systems. The concerto method consists of three coherent estimation stages, i.e., initial modified beat wavelength (BW) estimation, residual error estimation and least squares (LS) estimation, each of which has a closed-form expression and interacts closely with the others like a concerto. Due to a specially designed frequency pattern, concerto is reliable, accurate and computationally simple, and simplifies measurement frequency selection. Performance comparisons show that our method outperforms the BW, excess fractions (EF) and Chinese remainder theorem (CRT) methods in terms of both accuracy and reliability, and can asymptotically achieve the Cramér-Rao bound (CRB).

Pixel-by-pixel absolute three-dimensional shape measurement with modified Fourier transform profilometry

The single-pattern Fourier transform profilometry (FTP) and double-pattern modified FTP methods have great value in high-speed three-dimensional shape measurement, yet it is difficult to retrieve absolute phase pixel by pixel. This paper presents a method that can recover absolute phase pixel by pixel for the modified FTP method. The proposed method uses two images with different frequencies, and the recovered low-frequency phase is used to temporally unwrap the high-frequency phase pixel by pixel. This paper also presents the computational framework to reduce noise impact for robust phase unwrapping. Experiments demonstrate the success of the proposed absolute phase recovery method using only two fringe patterns.