Temporal Phase Unwrapping Research Articles

One-Step Three-Dimensional Phase Unwrapping Approach Based on Small Baseline Subset Interferograms

One of the most critical steps in a multitemporal D-InSAR analysis is the resolution of the phase ambiguities in the context of phase unwrapping. The Extended Minimum Cost Flow approach is one of the potential phase unwrapping algorithms used in the Small Baseline Subset analysis. In a first step, each phase gradient is unwrapped in time using a linear motion model and, in a second step, the spatial phase unwrapping is individually performed for each interferogram. Exploiting the temporal and spatial information is a proven method, but the two-step procedure is not optimal. In this paper, a method is presented which solves both the temporal and spatial phase unwrapping in one single step. This requires some modifications regarding the estimation of the motion model and the choice of the weights. Furthermore, the problem of temporal inconsistency of the data, which occurs with spatially filtered interferograms, must be considered. For this purpose, so called slack variables are inserted. To verify the method, both simulated and real data are used. The test region is the Lower-Rhine-Embayment in the southwest of North Rhine-Westphalia, a very rural region with noisy data. The studies show that the new approach leads to more consistent results, so that the deformation time series of the analyzed pixels can be improved.

Precision Position Measurement of PMSLM Based on ApFFT and Temporal Sinusoidal Fringe Pattern Phase Retrieval

To improve the accuracy of the mover position detection with longer measurement range, a linear motor displacement measurement method is proposed based on digital image measurement and temporal sinusoidal fringe pattern phase retrieval. A simple computer-generated sinusoidal fringe pattern is designed as the target image, and sequences of fringe signals are recorded by a line scan camera that moves with the linear motor. Fourier phase analysis is employed to extract the initial phase of each signal curve, and then temporal phase unwrapping is applied to reconstruct the mover displacement evolution according to the system magnification parameter and fringe width. This method removes the phase ambiguity problem of traditional temporal phase analysis, and can be used in complex motion of permanent magnet synchronous linear motors. To suppress the spectrum leakage of fast Fourier transform (FFT), all-phase FFT with double Hanning windows is used to improve the detection accuracy of initial phase and fringe width. Simulation and experimental results demonstrate that the proposed method can estimate the mover position with longer measurement range precisely and robustly under different working conditions.

A Triangle-Oriented Spatial–Temporal Phase Unwrapping Algorithm Based on Irrotational Constraints for Time-Series InSAR

The sparse 3-D phase unwrapping (PU) is a significant problem in the time-series interferometric synthetic aperture radar (TSInSAR), which is a popular technique on the extraction of the ground deformation. The popular 1-D + 2-D strategies in 3-D PU provide an initial temporal estimation of the phase gradients followed by a spatial integration procedure. However, the temporal PU (TPU) solution is prone to be spatially inconsistent, which increases the difficulty of the subsequent spatial PU (SPU). To solve the problem, in this article, an improved global TPU formulation targeting at a totally irrotational phase-gradient field is first proposed, which, however, has a heavy computation complexity, because numerous variables have to be optimized simultaneously. Consequently, a more feasible and practical triangle-oriented TPU (TOTPU) strategy is subsequently suggested, in which a triangle rather than an arc is the elementary unwrapping unit with the spatial irrotational constraints imposed. Furthermore, a residue-minimization algorithm is proposed to settle the problem of the “conflict edges” introduced by TOTPU, which occurs in just a small portion of arcs. Finally, the improved TPU phase gradients are integrated using an $L^{1}$ -norm SPU method with a novel weight function to obtain more reliable unwrapped phases. The proposed triangle-oriented spatial–temporal (TOST) algorithm is validated to be effective and reliable with both the simulated data and the real TerraSAR-X images. Moreover, it can be transplanted into a parallel platform for large-scale applications.

Temporal phase unwrapping using deep learning

The multi-frequency temporal phase unwrapping (MF-TPU) method, as a classical phase unwrapping algorithm for fringe projection techniques, has the ability to eliminate the phase ambiguities even while measuring spatially isolated scenes or the objects with discontinuous surfaces. For the simplest and most efficient case in MF-TPU, two groups of phase-shifting fringe patterns with different frequencies are used: the high-frequency one is applied for 3D reconstruction of the tested object and the unit-frequency one is used to assist phase unwrapping for the wrapped phase with high frequency. The final measurement precision or sensitivity is determined by the number of fringes used within the high-frequency pattern, under the precondition that its absolute phase can be successfully recovered without any fringe order errors. However, due to the non-negligible noises and other error sources in actual measurement, the frequency of the high-frequency fringes is generally restricted to about 16, resulting in limited measurement accuracy. On the other hand, using additional intermediate sets of fringe patterns can unwrap the phase with higher frequency, but at the expense of a prolonged pattern sequence. With recent developments and advancements of machine learning for computer vision and computational imaging, it can be demonstrated in this work that deep learning techniques can automatically realize TPU through supervised learning, as called deep learning-based temporal phase unwrapping (DL-TPU), which can substantially improve the unwrapping reliability compared with MF-TPU even under different types of error sources, e.g., intensity noise, low fringe modulation, projector nonlinearity, and motion artifacts. Furthermore, as far as we know, our method was demonstrated experimentally that the high-frequency phase with 64 periods can be directly and reliably unwrapped from one unit-frequency phase using DL-TPU. These results highlight that challenging issues in optical metrology can be potentially overcome through machine learning, opening new avenues to design powerful and extremely accurate high-speed 3D imaging systems ubiquitous in nowadays science, industry, and multimedia.

Multi-Wavelength Digital-Phase-Shifting Moiré Based on Moiré Wavelength

Multi-wavelength digital-phase-shifting moiré was demonstrated using multiple moiré wavelengths determined by system calibration over the full working depth. The method uses the extended noisy phase map as a reference to unwrap the phase map with a shorter wavelength, and thus achieve a less noisy and more accurate continuous phase map. The moiré wavelength calibration determines a moiré-wavelength to height relationship that permits pixelwise refinement of the moiré wavelength and height during 3D reconstruction. Only a single pattern has to be projected and, thus, a single image captured to compute each phase map with a different wavelength to perform digital-phase-shifting moiré temporal phase unwrapping. Only two captured images are required for two-wavelength phase unwrapping and three captured images for three-wavelength phase unwrapping. The method has been demonstrated in the 3D surface-shape measurement of an object with surface discontinuities and spatially isolated objects.

Automatic look-up table based real-time phase unwrapping for phase measuring profilometry and optimal reference frequency selection

For temporal phase unwrapping in phase measuring profilometry, it has recently been reported that two phases with co-prime frequencies can be absolutely unwrapped using a look-up table; however, frequency selection and table construction has been performed manually without optimization. In this paper, a universal phase unwrapping method is proposed to unwrap phase flexibly and automatically by using geometric analysis, and thus we can programmatically build a one-dimensional or two-dimensional look-up table for arbitrary two co-prime frequencies to correctly unwrap phases in real time. Moreover, a phase error model related to the defocus effect is derived to figure out an optimal reference frequency co-prime to the principal frequency. Experimental results verify the correctness and computational efficiency of the proposed method.

High-speed three-dimensional shape measurement using geometry-constraint-based number-theoretical phase unwrapping

In this paper, we propose a high-speed three-dimensional (3-D) shape measurement technique for dynamic scenes using geometry-constraint-based number-theoretical phase unwrapping. As a classical algorithm for temporal phase unwrapping (TPU), the number-theoretical approach is suitable for the binary defocusing fringe projection system since it can retrieve an absolute phase without using low-frequency fringe patterns. However, the conventional number-theoretical TPU approach cannot provide sufficient stability to unwrap a high-frequency phase since it requires the two fringe frequencies to be coprime within the global range of the projector coordinate. In contrast, using low-frequency fringe patterns tends to make phase unwrapping more reliable, but at the expense of the measurement precision. By introducing depth constraint into the traditional number-theoretical TPU, we only need to eliminate the phase ambiguity of each pixel within a small period range defined by the depth range, which means that our method just requires the two fringe frequencies to be coprime within the local period range instead of the conventional global range. Due to the reduction of fringe order candidates and the unambiguous phase range, the reliability of phase unwrapping can be significantly improved compared with the traditional number-theoretical TPU approach even when high-frequency fringe patterns are used. The proposed method has been successfully implemented on a high-frame-rate fringe projection system, achieving high-precision, robust, and absolute 3-D shape measurement at 3333 frames per second.

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.

An arc-clustering-based phase ambiguity estimation method for persistent scatterer interferometry

ABSTRACTThe reliability of persistent scatterer interferometry (PSI) is directly related to the accurate phase unwrapping (PU) of the 3-D (2-D space and time) sparse data stack. Phase ambiguities estimated using temporal phase unwrapping (TPU) are crucially exploited in most 3-D PU algorithms. However, most TPU methods are not noise-robust owing to the independent processing of each arc in the persistent scatterer network. To solve this problem, the cluster-analysis-based PU method for digital elevation model (DEM) construction is adapted to the phase ambiguity estimation problem for PSI in this letter. First, for each arc, the interior relation among phase ambiguities in multiple interferograms is derived to be linear and can be represented by an intercept vector. Subsequently, arcs with the same ambiguities are clustered according to their intercept vectors because arcs with identical phase ambiguities are verified to have the same intercept vectors. Moreover, a more reliable intercept vector can be calculated for each arc cluster by combining the intercept vectors within the cluster. Finally, more precise phase differences are estimated for each cluster using the corresponding intercept vector so that all the arcs are better unwrapped. Both the simulated and real experiments indicate the effectiveness and superiority of the proposed algorithm.

Extended depth-range dual-wavelength interferometry based on iterative two-step temporal phase-unwrapping

For dual-wavelength interferometry (DWI), we propose an extended depth-range phase-shifting interferometry technique of extracting phase distributions from randomly phase-shifted interferograms. First, the generalized phase-shifting algorithm is utilized to retrieve the single-wavelength phase from interferograms with random phase shifts. Then, the phase difference and phase-sum are calculated from the wrapped single-wavelength phases, and iterative two-step temporal phase-unwrapping is introduced to unwrap the phase-sum. Finally, the height of objects is determined. Simulations and experiments are conducted to demonstrate the excellent precision and overall performance of the proposed method. As the limitation of being exclusively applicable to interferograms with special phase shifts is broken, the proposed method will greatly facilitate the practical application of DWI. Furthermore, because the phase-sum and iterative two-step temporal phase-unwrapping is introduced to extend the depth-range, the sensitivity and precision of the retrieved phase are greatly improved.

Dual-sensitivity profilometry with defocused projection of binary fringes.

A dual-sensitivity profilometry technique based on defocused projection of binary fringes is presented. Here, two sets of fringe patterns with a sinusoidal profile are produced by applying the same analog low-pass filter (projector defocusing) to binary fringes with a high- and low-frequency spatial carrier. The high-frequency fringes have a binary square-wave profile, while the low-frequency binary fringes are produced with error-diffusion dithering. The binary nature of the binary fringes removes the need for calibration of the projector's nonlinear gamma. Working with high-frequency carrier fringes, we obtain a high-quality wrapped phase. On the other hand, working with low-frequency carrier fringes we found a lower-quality, nonwrapped phase map. The nonwrapped estimation is used as stepping stone for dual-sensitivity temporal phase unwrapping, extending the applicability of the technique to discontinuous (piecewise continuous) surfaces. We are proposing a single defocusing level for faster high- and low-frequency fringe data acquisition. The proposed technique is validated with experimental results.

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.

A generalized multi-sensitivity temporal phase unwrapping method for absolute phase retrieval

In this paper, a noble phase unwrapping method based on 2-sensitivity temporal phase unwrapping (TPU) is proposed. The algorithm considers the maximum ratio between two wavelengths in a measurement system and deals with wrapped phase error in detail. In addition, a generalized multi-sensitivity TPU is also proposed to improve the capability and accuracy in the phase unwrapping process. Simulations and experiments are carried out to verify the validity of the proposed method. The results show that the measurement accuracy is significantly improved using the proposed algorithm.

Monitoring of ground movement in open pit iron mines of Carajás Province (Amazon region) based on A-DInSAR techniques using TerraSAR-X data

Persistent scatterer interferometry (PSI) analysis of a large area is always a challenging task regarding the removal of the atmospheric phase component. This work presents an investigation of ground movement measurements based on a combination of differential SAR interferometry time-series (DTS) and PSI techniques, applied on a large area of extent with open pit iron mines located in Carajas (Brazilian Amazon Region), aiming at detecting linear and nonlinear ground movement. These mines have presented a history of instability, and surface monitoring measurements over sectors of the mines (pit walls) have been carried out based on ground-based radar and total station (prisms). Using a priori information regarding the topographic phase error and a phase displacement model derived from DTS, temporal phase unwrapping in the PSI processing and the removal of the atmospheric phases can be performed more efficiently. A set of 33 TerraSAR-X (TSX-1) images, acquired during the period from March 2012 to April 2013, was used to perform this investigation. The DTS analysis was carried out on a stack of multilook unwrapped interferograms using an extension of SVD to obtain the least-square solution. The height errors and deformation rates provided by the DTS approach were subtracted from the stack of interferograms to perform the PSI analysis. This procedure improved the capability of the PSI analysis for detecting high rates of deformation, as well as increased the numbers of point density of the final results. The proposed methodology showed good results for monitoring surface displacement in a large mining area, which is located in a rain forest environment, providing very useful information about the ground movement for planning and risk control.

Absolute three-dimensional shape measurement with a known object.

This paper presents a novel method for absolute three-dimensional (3D) shape measurement that does not require conventional temporal phase unwrapping. Our proposed method uses a known object (i.e., a ping-pong ball) to provide cues for absolute phase unwrapping. During the measurement, the ping-pong ball is positioned to be close to the nearest point from the scene to the camera. We first segment ping-pong ball and spatially unwrap its phase, and then determine the integer multiple of 2π to be added such that the recovered shape matches its actual geometry. The nearest point of the ball provides zmin to generate the minimum phase Φmin that is then used to unwrap phase of the entire scene pixel by pixel. Experiments demonstrated that only three phase-shifted fringe patterns are required to measure absolute shapes of objects moving along depth z direction.

Temporal phase unwrapping for fringe projection profilometry aided by recursion of Chebyshev polynomials

This paper presents a temporal phase-unwrapping method for fringe projection profilometry. With it, a sequence of phase-shifting fringe patterns is projected onto the measured object for getting the wrapped phase map and achieving a high measurement resolution, and an additional sequence corresponding to Chebyshev polynomials is used for determining their fringe orders. For effectuating this method, we deduce an algorithm by use of the recursive property of Chebyshev polynomials. This algorithm, combined with a correction operation in the least-squares sense, allows us to accurately estimate the fringe orders in the presence of noise. Experimental results demonstrate the proposed method to be effective in restoring the absolute phase maps of fringe patterns.

Digital holographic interferometry in the long-wave infrared and temporal phase unwrapping for measuring large deformations and rigid body motions of segmented space detector in cryogenic test

We present digital holographic interferometry (DHI) in the long-wave infrared for monitoring the deformation under cryogenic conditions of a segmented focal plane array to be used in a space mission. The long wavelength was chosen for its ability to allow measurement of displacements 20 times larger than DHI in the visible, which were foreseen with the test object under large temperature variations. The latter is a mosaic of 4×4 detectors assembled on a frame. DHI was required to assess the global deformation of the assembly, the deformation of each detector, and out-of-plane movements of each of them with respect to their neighbors. For that reason, we incorporated the temporal phase unwrapping by capturing a sufficiently high number of holograms between which the phase does not undergo large variations. At last, since the specimen exhibits specular reflectivity at that wavelength, it is illuminated by means of a reflective diffuser.

Enhanced two-frequency phase-shifting method.

One of the major challenges of employing a two-frequency (or two-wavelength) phase-shifting algorithm for absolute three-dimensional shape measurement is its sensitivity to noise. Therefore, three- or more-frequency phase-shifting algorithms are often used in lieu of a two-frequency phase-shifting algorithm for applications where the noise is severe. This paper proposes a method to use geometric constraints of digital fringe projection system to substantially reduce the noise impact by allowing the use of more than one period of equivalent phase map for temporal phase unwrapping. Experiments successfully verified the enhanced performance of the proposed method without increasing the number of patterns.

Temporal phase unwrapping algorithms for fringe projection profilometry: A comparative review

In fringe projection profilometry (FPP), temporal phase unwrapping is an essential procedure to recover an unambiguous absolute phase even in the presence of large discontinuities or spatially isolated surfaces. So far, there are typically three groups of temporal phase unwrapping algorithms proposed in the literature: multi-frequency (hierarchical) approach, multi-wavelength (heterodyne) approach, and number-theoretical approach. In this paper, the three methods are investigated and compared in detail by analytical, numerical, and experimental means. The basic principles and recent developments of the three kind of algorithms are firstly reviewed. Then, the reliability of different phase unwrapping algorithms is compared based on a rigorous stochastic noise model. Furthermore, this noise model is used to predict the optimum fringe period for each unwrapping approach, which is a key factor governing the phase measurement accuracy in FPP. Simulations and experimental results verified the correctness and validity of the proposed noise model as well as the prediction scheme. The results show that the multi-frequency temporal phase unwrapping provides the best unwrapping reliability, while the multi-wavelength approach is the most susceptible to noise-induced unwrapping errors.

Spatiotemporal three-dimensional phase unwrapping in digital speckle pattern interferometry.

We propose a hybrid spatiotemporal three-dimensional phase unwrapping algorithm for use in digital speckle pattern interferometry (DSPI). The feature of the proposed algorithm is the integration of one-dimensional temporal and two-dimensional spatial phase unwrapping algorithms. By demodulating the phase on a single reference point or multiple reference points using temporal phase unwrapping and on each separated phase map region using spatial phase unwrapping, the DSPI with the spatiotemporal three-dimensional phase unwrapping algorithm can realize the measurement of dynamic absolute displacements and the determination of abrupt phase changes which are usually caused by object discontinuities. We demonstrate that the presented algorithm can overcome the drawbacks of the traditional spatial and temporal phase unwrapping algorithms.