Phase Unwrapping Research Articles

Three-dimensional reconstruction of a light field based on phase restoration for highly reflective surfaces

This paper proposes, a novel, to our knowledge, phase-restoration-based light field method to achieve 3D reconstruction of highly reflective surfaces. First, a focused light field camera whose angular and spatial resolutions can be adjusted according to the needs has been designed and fabricated to capture 4D light field information. Then, according to the pixel offsets between different sub-aperture images, a phase restoration method based on multi-view complementary information is proposed to restore the missing absolute phase information caused by highlights. Finally, a cubic B-spline curve method is used to directly fit the relationship between absolute phase and coordinates to achieve 3D reconstruction of highly reflective surfaces. The experimental results demonstrate that the proposed method effectively utilizes the multi-view information from the light field to restore missing absolute phase data in the phase unwrapping, ensuring accurate 3D reconstruction of highly reflective surfaces. What is more, our method requires no additional hardware, camera angle calibration, or point cloud fusion, which significantly reduces both hardware complexity and computational demands.

Fringe projection profilometry based on deep learning phase demodulation combined with temporal phase unwrapping

Fringe projection profilometry based on deep learning phase demodulation combined with temporal phase unwrapping

Research on hydroacoustic signal processing algorithm based on B-spline and Hilbert transform

Purpose This paper aims to solve the problems of baseline drift, susceptibility to abnormal data interference during baseline drift processing, and phase inconsistency in underwater acoustic target detection and signal processing of single microelectromechanical systems (MEMS) vector hydrophone. To this end, this paper proposes a baseline drift removal algorithm based on Huber regression model with B-spline interpolation (H-BS) and a phase compensation algorithm based on the Hilbert transform. Design/methodology/approach First, the Huber regression model is innovatively introduced into the conventional B-spline interpolation (B-spline) to solve the control point vectors more accurately and to improve the anti-interference ability of the abnormal data when the B-spline interpolation fitting removes baseline drift and the baseline drift components in the signals are fitted accurately and removed by the above method. Next, the Hilbert transform is applied to the three-channel output signals of the MEMS vector hydrophone to calculate the instantaneous phase and the phase compensation is performed on the vector X/Y signals with the scalar P signal as the reference. Findings Through simulation experiments, it is found that H-BS proposed in this paper has smaller processing error and better robustness than variational modal decomposition and B-spline, but the H-BS algorithm takes slightly longer than the B-spline. In the actual lake test experiments, the H-BS algorithm can effectively remove the baseline drift component in the original signal and restore the signal waveform, and after the Hilbert transform phase compensation, the direction of arrival estimation accuracy of the signal is improved by 1°∼2°, which realizes high-precision orientation to the acoustic source target. Originality/value In this paper, the Huber regression model is introduced into B-spline interpolation fitting for the first time and applied in the specialized field of hydroacoustic signal baseline drift removal. Meanwhile, the Hilbert transform is applied to phase compensation of hydroacoustic signals. After simulation and practical experiments, these two methods are verified to be effective in processing hydroacoustic signals and perform better than similar methods. This study provides a new research direction for the signal processing of MEMS vector hydrophone, which has important practical engineering application value.

Sinusoidal Fitting Decomposition for Instantaneous Characteristic Representation of Multi-Componential Signal.

The research on how to effectively extract the instantaneous characteristic components of non-stationary signals continues to be both a research hotspot and a very challenging topic. In this paper, a new method of multi-component decomposition is proposed to decompose a signal into finite mono-component signals and extract their Instantaneous Amplitude (IA), Instantaneous Phase (IP), and Instantaneous Frequency (IF), which is called Sinusoidal Fitting Decomposition (SFD). The proposed method can ensure that the IA extracted from the given signal must be positive, the IP is monotonically increasing, and the signal synthesized by both IA and IP must be mono-componential and smooth. It transforms the decomposition process into a synthesis iterative process and does not rely on any dictionary or basis function space or carry out the sifting operation. In addition, the proposed method can describe the instantaneous-frequency-amplitude characteristics of the signal very well on the time-frequency plane. The results of numerical simulation and the qualitative analysis of the amount of calculation show that the proposed method is effective.

Novel CMR-assessed right ventricular-pulmonary artery coupling index independently associates with exercise capacity and clinical risk in pulmonary arterial hypertension

Abstract Background We defined a novel noninvasive right ventricle (RV)–pulmonary artery (PA) coupling index as the ratio of cardiovascular magnetic resonance (CMR)-assessed RV direct flow (RVDF) and PA pulse wave velocity (PWV), and investigated its associations with exercise capacity and clinical risk in pulmonary arterial hypertension (PAH). Method 43 PAH patients (mean age 46±12 years, M:F 7:36) and 60 healthy volunteers (mean age 46±13 years, M:F 11:48) underwent CMR and cardiopulmonary exercise test (CPET) within one week. From cine CMR, RV ejection fraction (EF) and RV end-diastolic volume (RVEDV) were calculated using volumetric analysis; and tricuspid annular plane systolic excursion (TAPSE), feature tracking. PAPWV was the ratio of early systolic DPA flow and DPA area measured from instantaneous phase and magnitude through plane 2D flow CMR images across the pulmonary trunk, respectively. RVDF was the 4D flow CMR-assessed blood volume entering and exiting the RV in one cardiac cycle, indexed to RVEDV. RV-PA coupling was RVDF/PAPWV. Based on CPET-assessed peak oxygen consumption (PVO2), all 103 participants were stratified into two groups: PVO2>15 ml/kg/min; and PVO2≤15 ml/kg/min. Based on the REVEAL (Registry to Evaluate Early and Long-term PAH Disease Management) 2.0 score, the 43 PAH patients were stratified into low risk (score ≤6) and intermediate to high risk (score >6) groups. Results RVDF/PAPWV was significantly lower in PAH vs. controls (7.9 vs. 20.4 %/(m/s), P<0.001); and in PAH patients with intermediate to high risk vs. low risk (4.6 vs. 9.4 %/(m/s), P=0.001). On multivariable analyses, RVDF/PAPWV was an independent predictor of PVO2≤15 ml/kg/min (=0.302, P=0.001) among all participants, and of REVEAL 2.0 score >6 (=0.621, P=0.009) among PAH patients. Compared to RVEF and TAPSE, RVDF/PAPWV had numerically higher discrimination (p values non-significant based on Delong test) for PVO2≤15 ml/kg/min (AUC=0.914 vs. 0.872, 0.843); and among PAH patients, REVEAL 2.0 score >6 (AUC=0.851 vs. 0.723 and 0.728) (Figure). Conclusion CMR-derived noninvasive RVDF/PAPWV independently associates with exercise capacity and PAH clinical risk, supporting its utility as an imaging marker of disease progression and therapeutic response.

Coherent linking between confocal amplitude image and confocal phase image in dual-comb microscopy

We propose a method for integrating confocal amplitude and phase images obtained through dual-comb microscopy (DCM). DCM combines the benefits of confocal laser microscopy and quantitative phase microscopy, offering high axial resolution and scan-less imaging. By leveraging the coherence between confocal amplitude and phase images within the same DCM system, we accurately determine the number of phase wrapping iterations, thereby eliminating ambiguity in phase wrapping. We demonstrate this approach using samples with micrometer-range optical thickness and nanometer-scale surface roughness. The results demonstrate an expanded axial range, spanning from micrometers to millimeters, while maintaining nanometer-level axial resolution. This integrated DCM imaging technique enables the simultaneous acquisition of confocal amplitude image and absolute phase image, thus enhancing its potential for wide-axial-dynamic-range imaging across various applications.

Fringe photometric stereo

In fringe projection profilometry (FPP), a high spatial frequency of fringe typically indicates powerful error suppression capability; however, it complicates phase unwrapping, necessitating a greater number of patterns and thus compromising the real-time performance of scanning. In this letter, we report a fringe photometric stereo (FPS) to completely bypass phase unwrapping for recovering the continuous 3D surface. We reveal the linear mapping relationship between the phase gradient and depth gradient, thereby facilitating the computation of depth gradients from phase gradients. Thus, we can employ depth gradients to reconstruct high-quality 3D profiles. Experimental results demonstrate that our FPS surpasses traditional phase-shifting profilometry in mitigating Gaussian noise. Our approach enables high-quality and unambiguous 3D reconstruction using single-frequency fringe patterns, relying solely on a camera and a projector without the need for dual, multiple, or light field cameras, thereby paving a path to high-speed and precision 3D imaging.

Dual frequency composite pattern temporal phase unwrapping for 3D surface measurement

Phase shifting profilometry (PSP) is widely used in three-dimensional (3D) optical metrology applications such as mechanical engineering, industrial monitoring, computer vision, and biomedicine because of its high measurement accuracy. In PSP, multi-frequency temporal phase unwrapping (TPU) plays a dominant role due to its high-accuracy measurement of surfaces with discontinuities and isolated objects. However, it requires a large number of fringe patterns. To reduce the number of required patterns, a new dual-frequency composite-pattern TPU method is developed, which needs only three patterns. The new method combines two fringe patterns of different frequencies into three composite fringe patterns, reconstructs rough 3D points by demodulating the low-frequency fringe pattern using geometric constraints, and reprojects it to obtain a rough low-frequency absolute phase, which is then refined to correct edge errors. Finally, the high-frequency wrapped phase is unwrapped under the guidance of the refined low-frequency phase, thereby the accurate 3D points are reconstructed based on stereo-vision technology. Experimental results demonstrated that the proposed method can achieve 3D surface measurement with only three images, which greatly improves the measurement efficiency. The proposed method has great application potential for the rapid 3D measurement of discontinuous surfaces and isolated objects.

Identifying Potential Landslides in Low-Coherence Areas Using SBAS-InSAR: A Case Study of Ninghai County, China

The southeastern coastal regions of China are characterized by typical hilly terrain with abundant rainfall throughout the year, leading to frequent geological hazards. To investigate the measurement accuracy of surface deformation and the effectiveness of error correction methods using the small baselines subset–interferometry synthetic aperture radar (SBAS-InSAR) method in identifying potential geological hazards in such areas, this study processes and analyzes 129 SAR images covering Ninghai County, China. By processing coherence coefficients using the Stacking technique, errors introduced by low-coherence images during phase unwrapping are mitigated. Subsequently, interferograms with high coherence are selected for time-series deformation analysis based on the statistical parameters of coherence coefficients. The results indicate that, after mitigating errors from low-coherence images, applying the SBAS-InSAR method to only high-coherence SAR datasets provides reliable surface deformation results. Additionally, when combined with field geological survey data, this method successfully identified landslide boundaries and potential landslides not accurately detected in previous geological surveys. This study demonstrates that using the SBAS-InSAR method and selecting high-coherence SAR images based on interferogram coherence statistical parameters significantly improves measurement accuracy and effectively identifies potential geological hazards.

Multi task deep learning phase unwrapping method based on semantic segmentation

Abstract Phase unwrapping is a key step to obtain continuous phase distribution in optical phase measurement. When the wrapped phase obtained from the interference pattern is unclear and noisy, estimating the unwrapped phase becomes more challenging. As deep learning advances in optical image processing, it will enhance processing efficiency and accuracy, bringing broader possibilities for various applications. This paper introduces an innovative phase unwrapping method based on multi-task learning, aiming to simultaneously enhancing denoised images and predicting wrap count. The proposed network, named ICER-Net, comprises an encoder and two decoders, transforming the input low-luminance, noisy wrapped phase into two intermediate outputs: enhanced wrapped phase and wrap count. Finally, these two intermediate results are fused to obtain the unwrapped phase. Experimental results demonstrate that ICER-Net not only enhances the accuracy of phase unwrapping, particularly when facing challenges of various noise levels and luminance sizes but also exhibits outstanding performance in actual collected speckle phase images. This indicates that ICER-Net holds significant superiority in addressing complex issues in optical image processing.

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.

Emergence of the Molecular Geometric Phase from Exact Electron-Nuclear Dynamics.

Geometric phases play a crucial role in diverse fields. In molecules, they appear when a reaction path encircles an intersection between adiabatic potential energy surfaces and the molecular wave function experiences quantum-mechanical interference effects. This intriguing effect, closely resembling the magnetic Aharonov-Bohm effect, crucially relies on the adiabatic description of the dynamics, and it is an open issue whether and how it persists in an exact quantum dynamical framework. Recent works suggest that the molecular geometric phase is an artifact of the adiabatic approximation, thereby challenging the entire concept. Here, building upon a recent investigation (Martinazzo, R.; Burghardt, I. Phys. Rev. Lett. 2024, 132, 243002), we address this issue using the exact factorization of the total wave function. We introduce instantaneous gauge-invariant phases separately for the electrons and nuclei and use them to monitor the phase difference between the trailing edges of a wavepacket encircling a conical intersection between adiabatic surfaces. The transition from the time-dependent open-path phase differences to the closed-path limit is examined, revealing how the phase differences in the electronic and nuclear subspaces compensate for each other upon path closure. In this way, we unambiguously demonstrate the role of the geometric phase in the interference process and shed light on its persistence beyond the adiabatic approximation.

High-accuracy fringe projection profilometry without phase unwrapping based on multi-view geometry constraints

The number of fringes and phase unwrapping in fringe projection profilometry result in two key factors. The first is to avoid the problems of excessive fringe patterns, and the second is phase ambiguity. This paper presents a three-dimensional (3D) measurement method without phase unwrapping. This method benefits from the geometric constraints and does not require additional images. Meanwhile, epipolar rectification is performed to calibrate the rotation matrix relationship between the new plane of the dual camera and the plane of the projector. Subsequently, using depth constraints, the point pairs with incorrect 3D positions are effectively eliminated, and the initial parallax map is obtained by establishing epipolar lines of the left and right matching points in the projector domain, obtaining the intersection points, and setting up the threshold for filtering. Finally, a function combining the modulation intensity and phase is proposed to refine the parallax map such that the 3D result is insensitive to phase error. The standard step block and standard ball were used to verify the validity of the proposed method, and the experimental results showed that the root mean square error of the method was 0.052 mm.

Fast concrete crack depth detection using low frequency ultrasound array SH waves data

Fast detection of the depths of surface-open cracks plays an important role in evaluating the damage conditions of concrete elements. The presence of surface-open cracks and other anomalies inside concrete complicates the ultrasonic wave field and thus severely undermines the precision of traditional nondestructive testing methods. This study introduces independently developed low-frequency ultrasonic array detection equipment. The detector adopts a doubled-ray coverage strategy to enhance the imaging stability under noisy conditions. Moreover, we propose an imaging method called the crack focusing-synthetic aperture focusing technique (CF-SAFT), through which both reflected and transmitted surface waves are removed so that only diffracted SH waves converge to their origins. An extra instantaneous phase analysis is supplemented to highlight the diffraction points. We test the effectiveness of our method through a multitude of numerical examples and a model experiment. Successful depth identification was obtained regardless of different geometries of the cracks or interference from the steel reinforcements. The superiority of our method is further verified through noisy ultrasonic data and complex scenarios.

Phase unwrapping via fully exploiting global and local spatial dependencies

Phase unwrapping (PU) is the process of extracting the authentic phase image from its noisy wrapped measurements, playing a crucial role in scientific imaging techniques. PU requires solving a challenging non-linear ill-posed problem, particularly in the presence of noticeable noise. In recent years, deep learning has emerged as a promising approach for PU. Inspired by the success of convolutional neural networks (CNNs) in image restoration, many existing works trained CNNs for PU. However, due to the locality of convolutional kernels, CNNs are not efficient in capturing global spatial dependencies, a critical cue for PU. As an alternate, recent studies employed recurrent neural networks (RNNs) defined on handcrafted pixel paths. Nonetheless, a limited number of pre-defined pixel paths cannot fully exploit global spatial dependencies existing in complex phase structures. In this paper, we introduce a vision transformer (ViT) model that effectively captures both global and local spatial dependencies using a hierarchical structure with a multi-scale process. The proposed ViT model employs a series of global transformer blocks to capture global spatial dependencies at the roughest scale. The resulting global features are used to guide a set of local transformer blocks to analyze local spatial dependencies in a coarse-to-fine progressive manner for unwrapping. Extensive experiments show that, our proposed ViT model produces higher-quality unwrapped phases over existing CNN/RNN-based methods, while maintaining a lightweight nature.

Processing group delay spectrograms for study of formant and harmonic contours in speech signals.

This paper deals with study of formant and harmonic contours by processing the group delay (GD) spectrograms of speech signals. The GD spectrum is the negative derivative of the phase spectrum with respect to frequency. Recent study shows that the GD spectrogram can be obtained without phase wrapping. Formant frequency contours can be observed in the display of the peaks of the instantaneous wideband equivalent GD spectrogram, derived using the modified single frequency filtering (SFF) analysis of speech signals. Harmonic frequency contours can be observed in the display of the peaks of the instantaneous narrowband equivalent GD spectrogram, derived using the modified SFF analysis of speech signals. For synthetic speech signals, the observed formant contours match the ground truth formant contours from which the signal is derived. For natural speech signals, the observed formant contours match approximately with the given ground truth formant contours mostly in the voiced regions. The results are illustrated for several randomly selected utterances from the TIMIT database. While this study helps to observe the contours of formants in the display, automatic extraction of the formant frequencies needs further processing, requiring logic for eliminating the spurious points, without forcing the number of formants.

Optical 3D Surface Vertical Measurement Based on 2D Generalized S-Transform

Abstract The optical 3D surface measurement technique aligns the optical axis of the projector with that of the camera, enabling the measurement of objects with significant height variations, such as deep holes and grooves. This technique encodes the height of the object into the modulation of the fringe pattern, eliminating the requirement for phase unwrapping and avoiding issues such as shadow occlusion. To further enhance the noise reduction capability of fringe analysis and reconstruct objects with high-frequency detail, this paper introduces the two-dimensional generalized S-transform (2D-GST) method for modulation extraction. By incorporating two additional parameters px and py to adjust the resolution in the time domain/frequency domain, 2D-GST can provide higher reconstruction precision. The root mean square (RMS) for tested plane with 2D-GST is 4.35 μm, whereas for the traditional 1D S-transform (1D-ST), the RMS is 4.97 μm.

TMS-induced phase resets depend on TMS intensity and EEG phase

Objective. The phase of the electroencephalographic (EEG) signal predicts performance in motor, somatosensory, and cognitive functions. Studies suggest that brain phase resets align neural oscillations with external stimuli, or couple oscillations across frequency bands and brain regions. Transcranial Magnetic Stimulation (TMS) can cause phase resets noninvasively in the cortex, thus providing the potential to control phase-sensitive cognitive functions. However, the relationship between TMS parameters and phase resetting is not fully understood. This is especially true of TMS intensity, which may be crucial to enabling precise control over the amount of phase resetting that is induced. Additionally, TMS phase resetting may interact with the instantaneous phase of the brain. Understanding these relationships is crucial to the development of more powerful and controllable stimulation protocols.Approach.To test these relationships, we conducted a TMS-EEG study. We applied single-pulse TMS at varying degrees of stimulation intensity to the motor area in an open loop. Offline, we used an autoregressive algorithm to estimate the phase of the intrinsicµ-Alpha rhythm of the motor cortex at the moment each TMS pulse was delivered.Main results. We identified post-stimulation epochs whereµ-Alpha phase resetting and N100 amplitude depend parametrically on TMS intensity and are significantversusperipheral auditory sham stimulation. We observedµ-Alpha phase inversion after stimulations near peaks but not troughs in the endogenousµ-Alpha rhythm.Significance. These data suggest that low-intensity TMS primarily resets existing oscillations, while at higher intensities TMS may activate previously silent neurons, but only when endogenous oscillations are near the peak phase. These data can guide future studies that seek to induce phase resetting, and point to a way to manipulate the phase resetting effect of TMS by varying only the timing of the pulse with respect to ongoing brain activity.

Two-plus-two fringe projection profilometry based on phase-shifted coding

In fringe projection profilometry based on temporal phase unwrapping, determining a fringe order map commonly requires a large number of fringes. To reduce the fringe number, this paper proposes a concise absolute phase retrieval algorithm just by projecting four fringes. The first two orthogonal fringes with relatively large frequency can collect reliable height information. The second two fringes are designed the same as the first two, but the only difference is that each 2π-phase of them is shifted by a unique amount, which can robustly label a large number of fringe orders. For decoding the fringes, we develop an average intensity one-time extraction algorithm, which allows for the rapid acquisition of the two pairs of alternating current components. From this, the wrapped phase containing height information and the stair-coded phase providing fringe orders can be directly extracted by arctangent operation in a point-to-point manner. Furthermore, we also develop a universal fringe order correction algorithm that can simultaneously correct the common errors and the misalignment between the wrapped phase and fringe orders. Experiment results demonstrate that this method achieves comparable accuracy and adaptability to the phase-coding method, while utilizing two fewer fringes.

4D flow MRI velocity uncertainty quantification.

An automatic method is presented for estimating 4D flow MRI velocity measurement uncertainty in each voxel. The velocity distance (VD) metric, a statistical distance between the measured velocity and local error distribution, is introduced as a novel measure of 4D flow MRI velocity measurement quality. The method uses mass conservation to assess the local velocity error variance and the standardized difference of means (SDM) velocity to estimate the velocity error correlations. VD is evaluated as the Mahalanobis distance between the local velocity measurement and the local error distribution. The uncertainty model is validated synthetically and tested in vitro under different flow resolutions and noise levels. The VD's application is demonstrated on two in vivo thoracic vasculature 4D flow datasets. Synthetic results show the proposed uncertainty quantification method is sensitive to aliased regions across various velocity-to-noise ratios and assesses velocity error correlations in four- and six-point acquisitions with correlation errors at or under 3.2%. In vitro results demonstrate the method's sensitivity to spatial resolution, venc settings, partial volume effects, and phase wrapping error sources. Applying VD to assess in vivo 4D flow MRI in the aorta demonstrates the expected increase in measured velocity quality with contrast administration and systolic flow. The proposed 4D flow MRI uncertainty quantification method assesses velocity measurement error owing to sources including noise, intravoxel phase dispersion, and velocity aliasing. This method enables rigorous comparison of 4D flow MRI datasets obtained in longitudinal studies, across patient populations, and with different MRI systems.