Unwrapping Algorithm Research Articles

Dual-wavelength sub-Nyquist sampling scheme for clipping-avoidance photonic ADC

We propose a dual-wavelength scheme for a clipping-avoidance photonic analog-to-digital converter (PADC) operating at the sub-Nyquist sampling rate. The scheme utilizes two characteristics, the phase-wrapping feature of a PADC and the wavelength-sensitive feature of a phase modulator, equivalently performing a dual-modulus (DM) modulo operation to avoid clipping. Coupled with an unwrapping algorithm based on the Chinese remainder theorem (CRT), the proposed scheme enables signal reconstruction from the processed signals independent of the sampling rate. We demonstrated proof-of-concept experiments on a PADC chip fabricated on an LNOI platform. Experimental results show that 1G/2G/4G-baud 10-level pulse amplitude modulation (PAM-10) waveforms were successfully reconstructed at the sub-Nyquist sampling rate of 1/2/4Gs/s, respectively. Our work provides a potential solution to achieve on-chip clipping-avoidance PADCs operating at the sub-Nyquist sampling.

Optical property discrepancies found between healthy and unhealthy skin cells using digital holographic microscopy with three wavelengths

Cancer and other health disorders can be differentiated by changes in cell optical properties such as their refractive index, thickness, and topology (height and width). Here, we employ three wavelengths simultaneously in digital holographic microscopy (3λ-DHM) to visualize the whole cell topology as 3D images through a numerical reconstruction algorithm applied to a hologram. By identifying the cell state and the changes in its optical properties, it is possible to discern between healthy and unhealthy cells. The simultaneous use of three wavelengths provides a rapid and straightforward quantitative reconstruction of the whole cell without the need for an unwrapping algorithm. This is a benefit over traditional methods, which often require complicated procedures. The performance of the approach was first validated in a known sample, a silicon dioxide thin film, where we were able to corroborate its refractive index with the values reported in the literature. Then the method was applied to fixed skin cells finding a refractive index of 1.3443 for healthy cells and 1.3246 for cells found in tumor tissue. We discuss and highlight differences based on the refractive index to demonstrate that the employed process can provide reliable information to distinguish characteristics between healthy and unhealthy cells.

Simple and high-resolution method for displacement sensing using self-mixing interferometry

Self-mixing interferometry (SMI) is widely used for vibration, velocity, and displacement measurement applications. Many complex and computationally intensive phase unwrapping algorithms have already been proposed to obtain high-resolution measurements, but there remains a requirement for a high-resolution yet simple method for displacement measurement. In this work, a high-resolution yet simple improved unwrapping method (IUM) is proposed, which addresses the limitations of the previously proposed simple but low-resolution method, consecutive sample-based unwrapping (CSU). IUM shows a remarkable improvement of 49% over CSU. IUM was assessed for a broad range of SMI signals, and results revealed that IUM is capable of processing SMI signals from all three main feedback regimes with an average RMS and peak error of 37.7 nm and 113.3 nm, respectively. Such a high-resolution method with a straightforward processing methodology is a vital step towards real-time independent displacement sensors capable of performing high-resolution measurements.

Experimental Study on the Reconstruction of a Light Field through a Four-Step Phase-Shift Method and Multiple Improvement Iterations of the Least Squares Method for Phase Unwrapping

Phase unwrapping technology can reflect the true phase information of an image, but it is affected by adverse factors such as noise, shadows, and fractures when extracting the true phase information of an object. Therefore, corresponding unwrapping algorithms need to be studied for different interference images. This paper summarizes and analyzes various phase unwrapping algorithms and ultimately selects the required method based on their advantages and disadvantages. Using the four-step phase-shift method to reconstruct the phase of the optical field and then combining it with the least squares method to unwrap the phase through multiple improvement iterations, the simulated collected interference fringe images are simulated using the MATLAB program to complete the phase unwrapping of the interference information field. Based on the analysis of the final experimental results, the reliability of this research method was verified.

High-quality phase imaging by phase-shifting digital holography and deep learning

Quantitative phase imaging (QPI) technology is widely used in biomedical imaging and other fields because it can realize exact imaging for transparent phase-type samples, which is of great research significance. The complex amplitude distribution of the object wave obtained by phase-shifting digital holography (PSDH) reproduction can provide phase information for QPI, but its existence of phase wrapping and other problems limits its practical application. Although the traditional phase unwrapping algorithm provides a solution, it has problems such as low unwrapping accuracy or long time running. To solve these problems in QPI, a high-quality phase imaging (HQPI) method by PSDH and deep learning (DL) is proposed, where QPI is achieved by extracting the unknown phase shift using a generalized non-iterative phase shift extraction algorithm and unwrapping the wrapped phase by a DL network. Both numerical simulations and optical experiments verify the feasibility of the method. By comparing with the traditional phase unwrapping algorithm, it is demonstrated that the DL unwrapping method has higher unwrapping accuracy and more efficiency. The results show that the method of HQPI is capable of realizing comparatively fast and accurate QPI.

Spatial phase unwrapping approach for single-frame 3D shape measurement based on deep learning

To address the challenge of balancing accuracy and speed in traditional phase unwrapping algorithms, this paper proposes a deep-learning-based single-frame spatial phase unwrapping method. By leveraging extensive data learning, two neural networks are trained to directly acquire phase information and modulation from a single-frame fringe pattern. Then, through the integration of a modulation sorting phase unwrapping algorithm, we achieve high-precision 3D surface reconstruction from a single-frame fringe pattern, thereby enabling rapid object measurement. The experimental results demonstrate the remarkable accuracy of the proposed method in phase unwrapping, approaching the level achieved by the 12-step phase-shifting method. The integration of deep learning into phase unwrapping offers promising prospects for further developments in this area. This advancement holds significant implications for high-speed measurement in the manufacturing field.

Fast phase retrieval without unwrapping for 3D measurement based on digital phase difference

Phase unwrapping is still a tough task for fringe projection profilometry (FPP), due to the heavy computation burden and slow rate of the existing phase unwrapping algorithms. To speed up the retrieval of the absolute phase, a fast phase retrieval algorithm without unwrapping for 3D measurement based on digital phase difference is presented in this paper. Firstly, 4-step phase-shifting profilometry is utilized to retrieve a complex-valued map, who contains the complete phase information of the measurement object in its phase part. Instead of extracting the phase map from the complex-valued map directly, a novel complex-valued map containing the differential phase is obtained by the division between the complex-valued map and its pixel-shifted version. Then, the differential phase can be extracted from the novel complex-valued map by regular phase extraction operation easily. Finally, the absolute phase can be retrieved by integrating the differential phase along the inverse pixel-shifted direction. This proposed algorithm can retrieve the absolute phase easily and fast, while the measurement object has continuous surface with the phase variation in the range of (−π, π]. Simulated and experimental results were demonstrated to present that this algorithm has faster phase retrieval rate than existing algorithms and has no obvious jump error, which allows fast and high-quality phase retrieval.

Highly accurate phase unwrapping algorithm applied to comparative measurement in phase-shifting digital interferometry

Highly accurate phase unwrapping algorithm applied to comparative measurement in phase-shifting digital interferometry

Robust phase unwrapping algorithm for gear interferometry based on mutual information and adaptive spin-inpainting approaches

An enhanced robust phase unwrapping algorithm for gear tooth flank measurement by laser interferometry is proposed. The method integrates wrapped phase mutual information estimation with an adaptive spin-inpainting strategy, resulting in improved quality phase maps to guide the phase unwrapping process. Mutual information estimation is employed to investigate potential phase relationships among interference fringes in the wrapped phase map, forming the foundation for the suppression and recovery of abnormal fringes. The adaptive spin-inpainting method, which adjusts in response to fringe density, plays a crucial role in efficiently addressing the restoration of abnormal fringes and obtaining the optimized quality map. With the enhanced new quality map, the approach yields optimized results, enabling a concurrent assessment of unwrapping process efficiency and quality. Several simulations and experiments were conducted to validate the effectiveness of the proposed method. Comparative analyses against commonly employed methods consistently demonstrate the superior performance of the proposed algorithm.

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.

Dynamic phase-differencing profilometry with number-theoretical phase unwrapping and interleaved projection

High-speed 3D measurement is receiving increasing attention. However, simultaneously achieving high computational efficiency, algorithmic robustness, and reconstructing ratio is challenging. Therefore, a dynamic phase-differencing profilometry (DPDP) is proposed. By capturing the minimum three phase-shifting sinusoidal deformed patterns and establishing a brand-new model, the phase difference between the object on the reference plane and the reference plane is directly resolved to effectively improve computational efficiency. Although it is wrapped, by using only two auxiliary complementary gratings with a purposely designed lower frequency, a DPDP-based number-theoretical temporal phase unwrapping (NT-TPU) algorithm is also proposed to unwrap the wrapped phase difference rather than the phase itself with high robustness. Furthermore, compared to existing PSP-based NT-TPU, the proposed NT-TPU can normally work under more relaxed restrictions. In order to accomplish a high reconstructing ratio, a pentabasic interleaved projection (PIP) strategy based on time division multiplexing is proposed. It can improve the reconstructing ratio from one reconstruction per every five patterns to an equivalent of one reconstruction per every 1.67 patterns. Experimental results demonstrate that the proposed method achieves high computational efficiency, high algorithmic robustness, and high reconstructing ratio simultaneously and has prospective application in high-speed 3D measurement.

Snow water equivalent retrieval over Idaho – Part 1: Using Sentinel-1 repeat-pass interferometry

Abstract. Snow water equivalent (SWE) is identified as the key element of the snowpack that impacts rivers' streamflow and water cycle. Both active and passive microwave remote sensing methods have been used to retrieve SWE, but there does not currently exist a SWE product that provides useful estimates in mountainous terrain. Active sensors provide higher-resolution observations, but the suitable radar frequencies and temporal repeat intervals have not been available until recently. Interferometric synthetic aperture radar (InSAR) has been shown to have the potential to estimate SWE change. In this study, we apply this technique to a long time series of 6 d temporal repeat Sentinel-1 C-band data from the 2020–2021 winter. The retrievals show statistically significant correlations both temporally and spatially with independent in situ measurements of SWE. The SWE change measurements vary between −5.3 and 9.4 cm over the entire time series and all the in situ stations. The Pearson correlation and RMSE between retrieved SWE change observations and in situ stations measurements are 0.8 and 0.93 cm, respectively. The total retrieved SWE in the entire 2020–2021 time series shows an SWE error of less than 2 cm for the nine in situ stations in the scene. Additionally, the retrieved SWE using Sentinel-1 data is well correlated with lidar snow depth data, with correlation of more than 0.47. Low temporal coherence is identified as the main reason for degrading the performance of SWE retrieval using InSAR data. We also show that the performance of the phase unwrapping algorithm degrades in regions with low temporal coherence. A higher frequency such as L-band improves the temporal coherence and SWE ambiguity. SWE retrieval using C-band Sentinel-1 data is shown to be successful, but faster revisit is required to avoid low temporal coherence. Global SWE retrieval using radar interferometry will have a great opportunity with the upcoming L-band 12 d repeat-pass NASA-ISRO Synthetic Aperture Radar (NISAR) data and the future 6 d repeat-pass Radar Observing System for Europe in L-band (ROSE-L) data.

Phase unwrapping for MHz optical coherence elastography and application to brain tumor tissue.

During neuro-oncologic surgery, phase-sensitive optical coherence elastography (OCE) can be valuable for distinguishing between healthy and diseased tissue. However, the phase unwrapping process required to retrieve the original phase signal is a challenging and critical task. To address this issue, we demonstrate a one-dimensional unwrapping algorithm that recovers the phase signal from a 3.2 MHz OCE system. With a processing time of approximately 0.11 s per frame on the GPU, multiple 2π wraps are detected and corrected. By utilizing this approach, exact and reproducible information on tissue deformation can be obtained with pixel accuracy over the entire acquisition time. Measurements of brain tumor-mimicking phantoms and human ex vivo brain tumor samples verified the algorithm's reliability. The tissue samples were subjected to a 200 ms short air pulse. A correlation with histological findings confirmed the algorithm's dependability.

Method for Measuring the Three-Dimensional Morphology of Near-Wall Bubbles and Droplets Based on LED Digital Holography.

Digital holography, recognized for its noncontact nature and high precision in three-dimensional imaging, is effectively employed to measure the morphology of bubbles and droplets. However, in terms of near-wall bubbles and droplets, such as confined bubbles in microfluidic chips, the measurement of the interface morphology of bubbles near the glass surface has not yet been resolved due to the coherent noise resulting from glass surface reflections in microfluidic chips. Accordingly, an off-axis digital holography system was devised by using Linnik interferometry. Measuring the confined bubble interface near the wall within a microfluidic chip and droplet evaporation on solid surfaces was studied. Partially coherent LED sources and reference light modulation techniques were employed in the optical setup to mitigate the coherent noise. Dual exposure and weighted least-squares unwrapping algorithms were introduced to correct phase distortions, enhancing image quality. Imaging two confined CO2 bubbles was done near the wall in silicon oil within a porous microfluidic chip, and contact angles of 4.7 and 4.5° were measured. Additionally, the measurement of the three-dimensional morphology of vertically evaporating deionized water droplets on a glass surface was done, due to which calculation of contact angles at various orientations was possible. This work offers a feasible new method for measuring the 3D interface morphology of bubbles and droplets, particularly in microfluidic visualization, addressing current measurement gaps.

An Anti-Noise-Designed Residual Phase Unwrapping Neural Network for Digital Speckle Pattern Interferometry

DSPI (Digital Speckle Pattern Interferometry) is a non-destructive optical measurement technique that obtains phase information of an object through phase unwrapping. Traditional phase unwrapping algorithms depend on the quality of the images, which demands preprocessing such as filtering and denoising. Moreover, the unwrapping time is highly influenced by the size of the images. In this study, we proposed a new deep learning-based phase unwrapping algorithm combining the residual network and U-Net network. Additionally, we incorporated an improved SSIM function as the loss function based on camera characteristics. The experimental results demonstrated that the proposed method achieved higher quality in highly noisy phase unwrapping maps compared to traditional algorithms, with SSIM values consistently above 0.98. In addition, we applied image stitching to the network to process maps of various sizes and the unwrapping time remained around 1 s even for larger images. In conclusion, our proposed network is able to achieve efficient and accurate phase unwrapping.

Deep Learning in the Phase Extraction of Electronic Speckle Pattern Interferometry

Electronic speckle pattern interferometry (ESPI) is widely used in fields such as materials science, biomedical research, surface morphology analysis, and optical component inspection because of its high measurement accuracy, broad frequency range, and ease of measurement. Phase extraction is a critical stage in ESPI. However, conventional phase extraction methods exhibit problems such as low accuracy, slow processing speed, and poor generalization. With the continuous development of deep learning in image processing, the application of deep learning in phase extraction from electronic speckle interferometry images has become a critical topic of research. This paper reviews the principles and characteristics of ESPI and comprehensively analyzes the phase extraction processes for fringe patterns and wrapped phase maps. The application, advantages, and limitations of deep learning techniques in filtering, fringe skeleton line extraction, and phase unwrapping algorithms are discussed based on the representation of measurement results. Finally, this paper provides a perspective on future trends, such as the construction of physical models for electronic speckle interferometry, improvement and optimization of deep learning models, and quantitative evaluation of phase extraction quality, in this field.

Spatial phase unwrapping algorithm based on multi-anchors bidirectional location and suppression

Spatial phase unwrapping algorithm based on multi-anchors bidirectional location and suppression

A predictor-corrector phase unwrapping algorithm for temporally undersampled gradient-echo MRI.

To develop a method for unwrapping temporally undersampled and nonlinear gradient recalled echo (GRE) phase. Temporal unwrapping is performed as a sequential one step prediction of the echo phase, followed by a correction to the nearest integer wrap-count. A spatio-temporal extension of the 1D predictor corrector unwrapping (PCU) algorithm improves the prediction accuracy, and thereby maintains spatial continuity. The proposed method is evaluated using numerical phantom, physical phantom, and in vivo brain data at both 3 T and 9.4 T. The unwrapping performance is compared with the state-of-the-art temporal and spatial unwrapping algorithms, and the spatio-temporal iterative virtual-echo based Nyquist sampled (iVENyS) algorithm. Simulation results showed significant reduction in unwrapping errors at higher echoes compared with the state-of-the-art algorithms. Similar to the iVENyS algorithm, the PCU algorithm was able to generate spatially smooth phase images for in vivo data acquired at 3 T and 9.4 T, bypassing the use of additional spatial unwrapping step. A key advantage over iVENyS algorithm is the superior performance of PCU algorithm at higher echoes. PCU algorithm serves as a robust phase unwrapping method for temporally undersampled and nonlinear GRE phase, particularly in the presence of high field gradients.

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.

A phase unwrapping method with the sparse prior for diffraction phase microscopy

Phase unwrapping is widely used in interferometry. In the typical least square (LS) method, the noise in the wrapped phase cannot be removed. An improved algorithm based on total variation (TV) imposes L1-norm on the image gradient to improve the sparsity. However, it may lead to over-sparsity, and the unwrapped phase will be compressed if the penalty term is overlarge. In this study, we proposed a sparse prior phase unwrapping (SPUP) algorithm to solve the wrapped phase of interferograms of diffraction phase microscopy (DPM). Since DPM has a unique sample sparsity in the imaging field of high-power objectives, the L0-norm and L1-norm regularizations are introduced into the SPUP process. The results verify the strong denoising ability and high phase fidelity of the proposed SPUP method. Moreover, the uneven background noise and phase shifting of DPM interferograms could be corrected, demonstrating the great potential of the SPUP method in various interferometry measurements.