A Robust and Simple Measure for Quality-Guided 2D Phase Unwrapping Algorithms

Phase unwrapping algorithm based on phase diversity wavefront reconstruction and virtual Hartmann-Shack technology

Phase unwrapping (PU) algorithms play a crucial role in various phase measurement techniques. Traditional algorithms cannot work well in strong noise environments, which makes it very difficult to obtain the accurate absolute phase from the noisy wrapped phase. In this Letter, we introduce a novel, to the best of our knowledge, phase unwrapping algorithm named PD-VHS. This algorithm innovatively employs point spread function (PSF) filtering to eliminate noise from the wrapped phase. Furthermore, it combines a phase diversity (PD) wavefront reconstruction technology with a virtual Hartmann-Shack (VHS) technology for phase reconstruction and phase unwrapping of the filtered PSFs. In simulations, hundreds of random noise wrapped phases, containing the first 45 Zernike polynomials (excluding piston and the two tilt terms) and the wavefront RMS = 0.5λ and 1λ, are used to compare the classical quality-map guided algorithm, the VHS algorithm with decent noise immunity, with our PD-VHS algorithm. When signal-to-noise ratio (SNR) drops to just 2 dB, the mean root mean square errors (RMSEs) of the residual wavefront between the unwrapped result and the absolute phase of the quality-map guided algorithm and the VHS algorithm are up to 3.99λ, 0.44λ, 4.29λ, and 0.85λ, respectively; however, our algorithm RMSEs are low: 0.11λ and 0.17λ. Simulation results demonstrated that the PD-VHS algorithm significantly outperforms the quality-map guided algorithm and the VHS algorithm under large-scale noise conditions.

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.

An On-Site InSAR Terrain Imaging Method with Unmanned Aerial Vehicles

An on-site InSAR imaging method carried out with unmanned aerial vehicles (UAVs) is proposed to monitor terrain changes with high spatial resolution, short revisit time, and high flexibility. To survey and explore a specific area of interest in real time, a combination of a least-square phase unwrapping technique and a mean filter for removing speckles is effective in reconstructing the terrain profile. The proposed method is validated by simulations on three scenarios scaled down from the high-resolution digital elevation models of the US geological survey (USGS) 3D elevation program (3DEP) datasets. The efficacy of the proposed method and the efficiency in CPU time are validated by comparing with several state-of-the-art techniques.

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

Background

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.

Methods

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.

Results

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.

Conclusion

The proposed method demonstrates robust and accurate phase unwrapping capabilities, positioning it as a promising option for abdominal QSM applications.

Two-dimensional phase unwrapping based on U2-Net in complex noise environment

This paper proposes applying the nested U2-Net to a two-dimensional phase unwrapping (PU). PU has been a classic well-posed problem since conventional PU methods are always limited by the Itoh condition. Numerous studies conducted in recent years have discovered that data-driven deep learning techniques can overcome the Itoh constraint and significantly enhance PU performance. However, most deep learning methods have been tested only on Gaussian white noise in a single environment, ignoring the more widespread scattered noise in real phases. The difference in the unwrapping performance of deep network models with different strategies under the interference of different kinds of noise or drastic phase changes is still unknown. This study compares and tests the unwrapping performance of U-Net, DLPU-Net, VUR-Net, PU-GAN, U2-Net, and U2-Netp under the interference of additive Gaussian white noise and multiplicative speckle noise by simulating the complex noise environment in the real samples. It is discovered that the U2-Net composed of U-like residual blocks performs stronger anti-noise performance and structural stability. Meanwhile, the wrapped phase of different heights in a high-level noise environment was trained and tested, and the network model was qualitatively evaluated from three perspectives: the number of model parameters, the amount of floating-point operations, and the speed of PU. Finally, 421 real-phase images were also tested for comparison, including dynamic candle flames, different arrangements of pits, different shapes of grooves, and different shapes of tables. The PU results of all models are quantitatively evaluated by three evaluation metrics (MSE, PSNR, and SSIM). The experimental results demonstrate that U2-Net and the lightweight U2-Netp proposed in this work have higher accuracy, stronger anti-noise performance, and better generalization ability.

Phase unwrapping using deep learning in holographic tomography

Holographic tomography (HT) is a measurement technique that generates phase images, often containing high noise levels and irregularities. Due to the nature of phase retrieval algorithms within the HT data processing, the phase has to be unwrapped before tomographic reconstruction. Conventional algorithms lack noise robustness, reliability, speed, and possible automation. In order to address these problems, this work proposes a convolutional neural network based pipeline consisting of two steps: denoising and unwrapping. Both steps are carried out under the umbrella of a U-Net architecture; however, unwrapping is aided by introducing Attention Gates (AG) and Residual Blocks (RB) to the architecture. Through the experiments, the proposed pipeline makes possible the phase unwrapping of highly irregular, noisy, and complex experimental phase images captured in HT. This work proposes phase unwrapping carried out by segmentation with a U-Net network, that is aided by a pre-processing denoising step. It also discusses the implementation of the AGs and RBs in an ablation study. What is more, this is the first deep learning based solution that is trained solely on real images acquired with HT.

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).

Temporal phase unwrapping based on unequal phase-shifting code

In fringe projection profilometry (FPP) based on temporal phase unwrapping (TPU), reducing the number of projecting patterns has become one of the most important works in recent years. To remove the 2π ambiguity independently, this paper proposes a TPU method based on unequal phase-shifting code. Wrapped phase is still calculated from N-step conventional phase-shifting patterns with equal phase-shifting amount to guarantee the measuring accuracy. Particularly, a series of different phase-shifting amounts relative to the first phase-shifting pattern are set as codewords, and encoded to different periods to generate one coded pattern. When decoding, Fringe order with a large number can be determined from the conventional and coded wrapped phases. In addition, we develop a self-correction method to eliminate the deviation between the edge of fringe order and the 2π discontinuity. Thus, the proposed method can achieve TPU but need to only project one additional coded pattern (e. g. 3+1), which can significantly benefit dynamic 3D shape reconstruction. The theoretical and experimental analysis verify that the proposed method performs high robustness on the reflectivity of the isolated object while ensuring the measuring speed.

Using neural networks to create a reliable phase quality map for phase unwrapping

Two-dimensional phase unwrapping is a crucial step in interferometric signal processing. A phase quality map can help the unwrapping algorithm deal with low-quality and fast-changing regions. However, because existing algorithms cannot calculate a quality map representing the gradient quality directly, it is usually necessary to approximate the gradient quality with phase quality to assist the network-based phase unwrapping algorithm. Furthermore, they cannot withstand intense noise in low-quality regions, resulting in many errors in path-based algorithms. To address the aforementioned issues, this paper analyzes the essence of a quality map and proposes a quality map generation method based on a convolutional neural network. The generated quality maps are a pair, each indicating the quality of horizontal and vertical gradients. Experiments show that the quality map generated by this method can help path-based and network-based algorithms perform better.

Automated phase unwrapping in digital holography with deep learning

Digital holography can provide quantitative phase images related to the morphology and content of biological samples. After the numerical image reconstruction, the phase values are limited between -π and π; thus, discontinuity may occur due to the modulo 2π operation. We propose a new deep learning model that can automatically reconstruct unwrapped focused-phase images by combining digital holography and a Pix2Pix generative adversarial network (GAN) for image-to-image translation. Compared with numerical phase unwrapping methods, the proposed GAN model overcomes the difficulty of accurate phase unwrapping due to abrupt phase changes and can perform phase unwrapping at a twice faster rate. We show that the proposed model can generalize well to different types of cell images and has high performance compared to recent U-net models. The proposed method can be useful in observing the morphology and movement of biological cells in real-time applications.

An Improved Phase Unwrapping Method Based on Hierarchical Networking and Constrained Adjustment

Accurate phase unwrapping (PU) is a precondition and key for using synthetic aperture radar interferometry (InSAR) technology to successfully invert topography and monitor surface deformations. However, most interferograms are seriously polluted by noise in the low-quality regions, which poses difficulties for PU. Therefore, using the strategy of leveling network adjustment, this paper proposes an improved PU method based on hierarchical networking and constrained adjustment. This method not only limits the phase error transfer of low-quality points, but also takes the PU results of high-quality points as control points and uses the network adjustment method with constraints to unwrap low-quality points, which effectively inhibits the influence of noise and improves the accuracy of unwrapping. Regardless of the unwrapping method used for high-quality points, the unwrapping accuracy of low-quality points can always be improved. Compared with other traditional two-dimensional phase unwrapping workflows, this method can more accurately recover the phase of low-coherence regions only through the interferogram. A simulation experiment showed that the local noise of the interferogram was effectively inhibited, and the PU accuracy of the low-quality regions was improved by 16–46% compared with different traditional methods. For a real-data experiment of mining area with low coherence, the PU result of our proposed method had fewer residues and lower phase standard deviation than traditional methods, further indicating the practicability and robustness of the proposed method. The work in this paper has considerable practical significance for recovering the decoherence phase with serious local noise such as mining centers and groundwater subsidence centers.

Robust phase unwrapping algorithm for noisy and segmented phase measurements

This paper proposes a robust phase unwrapping algorithm (RPUA) for phase unwrapping in the presence of noise and segmented phase. The RPUA method presents a new model of phase derivatives combined with error-correction iterations to achieve an anti-noise effect. Moreover, it bridges the phase islands in the spatial domain using numerical carrier frequency and fringe extrapolation thus eliminating height faults to enable solving segmented phase unwrapping. Numerical simulation and comparison with three conventional methods were performed, proving the high robustness and efficiency of the RPUA. Further, three experiments demonstrated that the RPUA can obtain the unwrapped phase under different noise accurately and possesses the capability to process segmented phases, indicating reliable practicality.

Phase unwrapping based on a residual en-decoder network for phase images in Fourier domain Doppler optical coherence tomography

To solve the phase unwrapping problem for phase images in Fourier domain Doppler optical coherence tomography (DOCT), we propose a deep learning-based residual en-decoder network (REDN) method. In our approach, we reformulate the definition for obtaining the true phase as obtaining an integer multiple of 2π at each pixel by semantic segmentation. The proposed REDN architecture can provide recognition performance with pixel-level accuracy. To address the lack of phase images that are noise and wrapping free from DOCT systems for training, we used simulated images synthesized with DOCT phase image background noise features. An evaluation study on simulated images, DOCT phase images of phantom milk flowing in a plastic tube and a mouse artery, was performed. Meanwhile, a comparison study with recently proposed deep learning-based DeepLabV3+ and PhaseNet methods for signal phase unwrapping and traditional modified networking programming (MNP) method was also performed. Both visual inspection and quantitative metrical evaluation based on accuracy, specificity, sensitivity, root-mean-square-error, total-variation, and processing time demonstrate the robustness, effectiveness and superiority of our method. The proposed REDN method will benefit accurate and fast DOCT phase image-based diagnosis and evaluation when the detected phase is wrapped and will enrich the deep learning-based image processing platform for DOCT images.

Optimal processing scheme for restoration of phase data corrupted by strong decorrelation noise and dislocations

The presence of speckle noise and dislocations makes phase restoration potentially difficult in quantitative phase imaging and metrology. Unfortunately, there is no appropriate approach to deal with phase data corrupted by high speckle noise and phase dislocations. Usually, processing schemes may deal with low-pass phase filtering, phase unwrapping, or phase inpainting. This paper discusses the efficient processing to deal with noisy phase maps corrupted with phase dislocations. Six processing schemes, combining four operations, are evaluated. The investigation is carried out by realistic numerical simulations in which strong decorrelation phase noise and phase dislocations are generated. As a result, most robust and faster processing is established. The applicability of the optimal scheme is demonstrated through deformation measurement in dental materials.

Hierarchical quality-guided phase unwrapping algorithm

A hierarchical quality-guided phase unwrapping algorithm in a shared memory environment is proposed. First, the wrapped phase is divided into regular blocks, and local wrap counts of every block are obtained by a quality-guided strategy. Then, the gradient of the block wrap counts of adjacent blocks and the quality of every block are defined by the boundary local wrap counts of every block. Each block's data can be regarded as an abstract phase point, and the quality-guided strategy can be used again to solve the block wrap counts of each block. Finally, the absolute wrap counts of each phase point are obtained by adding local wrap counts and corresponding block wrap counts, and then the final unwrapped phase is obtained. The performance of the proposed algorithm is verified through an unwrapping experiment performed on simulated data and the real interferometric synthetic aperture sonar wrapped phase in a shared memory environment. The results show that the proposed method greatly improves phase unwrapping efficiency while maintaining the correctness of unwrapped results.

Weighted least-squares phase-unwrapping algorithm based on the orientation coherence for discontinuous optical phase patterns

Phase unwrapping is one of the key steps of optical interferogram analysis, among which phase discontinuity is still a challenge. In this paper, we propose a new weighted least-squares phase-unwrapping algorithm for discontinuous optical phase patterns. In the proposed algorithm, the orientation coherence is introduced to define the new weighting coefficient, which can accurately show the wrapped phase quality. According to our proposed algorithm, the new weighting coefficient has a good performance on distinguishing the continuous regions and the discontinuous regions in wrapped phase patterns. This advantage of our algorithm can ensure a more reliable unwrapped result for discontinuous optical phase patterns. We test the proposed algorithm on the computer-simulated speckle phase images and two experimentally obtained phase images, respectively, and compare them with the other five widely used methods. The experimental results demonstrate the performance of our new weighted least-squares phase-unwrapping algorithm.

An efficient phase error self-compensation algorithm for nonsinusoidal gating fringes in phase-shifting profilometry

The phase-shifting method is widely used in fringe projection profilometry. Since both the digital light projector and camera used in a grating projection measurement system are nonlinear pieces of equipment, the grating fringes captured by using the camera do not have a good sinusoidal property, which leads to a three-dimensional measurement error. Although the double-step phase-shifting method has proved that the phase error can be reduced to a large extent, the number of grating fringes is doubled, which affects the measurement efficiency. In this paper, we present an efficient phase error self-compensation algorithm. It transforms the initial wrapped phase into a second wrapped phase and integrates the initial and second wrapped phases to reduce the phase error. The advantage is that the measurement accuracy is close to that of the double-step phase-shifting method without increasing the number of projection fringes; at the same time, the measurement time is shortened, and the measurement efficiency is improved. We have elaborated the proposed algorithm in detail and compared it with the traditional single-step phase-shifting method and the double-step phase-shifting method. Finally, we utilize the proposed algorithm to measure different objects. The results prove its effectiveness.

A Robust Wrap Reduction Algorithm for Fringe Projection Profilometry and Applications in Magnetic Resonance Imaging

In this paper, we present an effective algorithm to reduce the number of wraps in a 2D phase signal provided as input. The technique is based on an accurate estimate of the fundamental frequency of a 2D complex signal with the phase given by the input, and the removal of a dependent additive term from the phase map. Unlike existing methods based on the discrete Fourier transform (DFT), the frequency is computed by using noise-robust estimates that are not restricted to integer values. Then, to deal with the problem of a non-integer shift in the frequency domain, an equivalent operation is carried out on the original phase signal. This consists of the subtraction of a tilted plane whose slope is computed from the frequency, followed by a re-wrapping operation. The technique has been exhaustively tested on fringe projection profilometry (FPP) and magnetic resonance imaging (MRI) signals. In addition, the performance of several frequency estimation methods has been compared. The proposed methodology is particularly effective on FPP signals, showing a higher performance than the state-of-the-art wrap reduction approaches. In this context, it contributes to canceling the carrier effect at the same time as it eliminates any potential slope that affects the entire signal. Its effectiveness on other carrier-free phase signals, e.g., MRI, is limited to the case that inherent slopes are present in the phase data.