Temporal Phase Unwrapping Algorithm Research Articles

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.

Unifying temporal phase unwrapping framework using deep learning.

Temporal phase unwrapping (TPU) is significant for recovering an unambiguous phase of discontinuous surfaces or spatially isolated objects in fringe projection profilometry. Generally, temporal phase unwrapping algorithms can be classified into three groups: the multi-frequency (hierarchical) approach, the multi-wavelength (heterodyne) approach, and the number-theoretic approach. For all of them, extra fringe patterns of different spatial frequencies are required for retrieving the absolute phase. Due to the influence of image noise, people have to use many auxiliary patterns for high-accuracy phase unwrapping. Consequently, image noise limits the efficiency and the measurement speed greatly. Further, these three groups of TPU algorithms have their own theories and are usually applied in different ways. In this work, for the first time to our knowledge, we show that a generalized framework using deep learning can be developed to perform the TPU task for different groups of TPU algorithms. Experimental results show that benefiting from the assistance of deep learning the proposed framework can mitigate the impact of noise effectively and enhance the phase unwrapping reliability significantly without increasing the number of auxiliary patterns for different TPU approaches. We believe that the proposed method demonstrates great potential for developing powerful and reliable phase retrieval techniques.

Real-time computer-generated frequency-carrier Moiré profilometry with three-frequency heterodyne temporal phase unwrapping

A real-time computer-generated frequency-carrier Moiré profilometry with three-frequency heterodyne temporal phase unwrapping is proposed. While three sinusoidal gratings shared the same direct current(DC) component and the same amplitude but with designed different frequencies are projected onto the measured object, three corresponding deformed patterns can be captured by camera. Then, the shared DC component is extracted from the highest-frequency pattern. And based on it, three AC components of the captured deformed patterns can also be effectively extracted by normalization. From this way, the number of the needed gratings can be reduced from six to three, and the disturbance of the object’s nonuniform reflectivity can be avoided efficiently. By using computer-generated frequency-carrier Moiré profilometry(CGFCMP), the carrier frequency designed as pure sinusoidal and cosinusoidal gratings is optimized in order to effectively extract the computer-generated frequency-carrier Moiré fringes, and finally-three wrapped phases at different frequencies can be obtained. Then the wrapped phase with the highest-frequency grating can be unwrapped by three-frequency heterodyne temporal phase unwrapping algorithm. The simulation and experimental results verify the feasibility and flexibility of the proposed method. It is suitable to measure the complex surface with abrupt slope and multiple isolated objects, and has prospective application in real-time three-dimensional measurement.

Time series phase unwrapping algorithm using LP-norm optimization compressive sensing

Time series phase unwrapping is an essential step in the time series interferometric synthetic aperture radar technique for deformation monitoring. However, traditional unwrapping algorithms are prone to unwrapping errors in the case of intense noise and steep deformation gradients, which directly affect the accuracy of the deformation information interpretation. To address this problem, a time series phase unwrapping algorithm using LP-norm optimization compressive sensing is proposed in this paper. This algorithm first transforms the minimizing L0-norm describing compressive sensing technique into a nonconvex problem with weaker constraints, that is, minimizing the LP-norm. Then, an optimized compressed sensing model was proposed by combining the constraint criterion of the phase triplet closure in the temporal domain and the solution strategy of the iterative reweighted least squares method. Improved temporal dimensional phase unwrapping and unwrapping error correction algorithms were developed based on this optimized model. Finally, the proposed algorithm was perfected by combining the above improved methods and the pseudo-three-dimensional phase unwrapping framework. Simulated and Sentinel-1A real datasets verify that the proposed algorithm has better noise robustness, unwrapping stability, and efficiency than traditional algorithms, particularly in the case of a steep deformation gradient and intense noise. The temporal coherence estimated by the proposed algorithm for the real dataset is generally greater than 0.99, which is far superior to traditional algorithms. Furthermore, the proposed algorithm facilitates high-precision interpretation of deformation information. The cross-validation results show that the deformation results obtained by traditional algorithms are underestimated by at least 73 mm/a and 68 mm at the maximum subsidence than the DSInSAR. However, the deformation result obtained by the proposed algorithm is highly consistent with that of the DSInSAR.

Phase-Shifting Temporal Phase Unwrapping Algorithm for High-Speed Fringe Projection Profilometry

Fringe projection profilometry based on temporal phase unwrapping (TPU) without error propagation has been widely used to measure the three-dimensional (3-D) shape of the complex scene. This paper develops a phase-shifting temporal unwrapping (PS-TPU) algorithm to ensure the measuring robustness and speed at the same time. In this method, a series of phase shifts are set as the cue of fringe order, and then embedded into the different periods of the single quasi-cosine pattern. High-quality wrapped phase and coded phase obtained from three conventional phase-shifting patterns and the single coded pattern can be utilized to generate fringe order pixel-by-pixel using the parallel and symmetrical relationship. Compared with the classical TPU algorithms, the number of projecting extra patterns is minimized to 1, while guaranteeing high noise immunity and less computational complexity. The static and dynamic experiments verify that the proposed PS-TPU is powerful for reconstructing 3-D shapes in our daily life.

Spatial-temporal phase unwrapping algorithm for fringe projection profilometry

In this paper, a generalized spatial-temporal phase unwrapping algorithm (STPUA) is proposed for extracting the absolute phase of the isolated objects with intricate surfaces. This proposed algorithm can eliminate thoroughly the order jumps of various temporal phase unwrapping algorithms (TPUAs), while inheriting the high measuring accuracy of quality-guided phase unwrapping algorithms (QGPUAs). Differing from the traditional phase unwrapping algorithms, wrapped phase is first divided into several regional wrapped phases, which can be extracted successively according to its areas and unwrapped individually by QGPUAs. Meanwhile, a series of reliable points from the fringe order map obtained from the code deformed patterns are selected to map the corresponding regional unwrapped phases into an absolute phase. The radii of selecting reliable points can provide the high measuring robustness compared with the classical point-to-point TPUAs for the complex surfaces and the motion blur, while keeping the same number of patterns. Therefore, the proposed STPUA combining SPUAs and TPUAs also can be employed in real-time three-dimensional (3D) reconstruction. Theoretical analysis and experimental results are performed to verify the effectiveness and capability of the proposed algorithm.

Modified three-wavelength phase unwrapping algorithm for dynamic three-dimensional shape measurement

Digital fringe projection (DFP) method has been widely used in three-dimensional (3D) shape measurement. Lots of studies have been conducted to enhance the measurement accuracy and speed by modulating the patterns. However, it is still hard to acquire enough number of 3D results to ensure the continuity of dynamic 3D shape measurement when phase shifting profilometry and temporal phase unwrapping algorithm is adopted. Besides, object motion can introduce phase error and thus measurement error for phase shifting profilometry. To this end, a dynamic 3D reconstruction framework based on modified three-wavelength phase unwrapping algorithm and phase error compensation method is proposed to solve this problem. By redefining the order of fringe patterns projected by the digital light processing (DLP) projector, the number of required patterns for 3D reconstruction can be reduced and the speed of dynamic 3D shape measurement can be improved. And using phase error compensation method based on Hilbert transform, the accuracy of 3D reconstruction can also be improved. Experiments were carried out to demonstrated the effectiveness of the proposed framework. And the modified algorithm successfully improved the speed of dynamic 3D shape measurement by 1∕3.

Three-Dimensional Measurement for Rigid Moving Objects Based on Multi-Fringe Projection

Multi-fringe projection profilometry, such as the widely used phase-shifting and temporal phase unwrapping algorithms, can achieve high accuracy and robustness in three-dimensional shape reconstruction of static objects. However, the motion of the objects will introduce errors. This paper proposes a new method to reconstruct 3D shape of rigid moving objects on an automated transmission line. First, a new pixel matching method for projected and captured images is proposed. Then, the pixel deviation caused by the motion of the objects in the three-dimensional direction is pre-detected, and the new projection fringes are obtained according to the above pixel matching. Third, the new projection fringes are binarized and the projector is defocused to realize high-speed projection. At last, the captured fringes are offset to reconstruct 3D shape. Experimental results show that the proposed method is effective in 3D shape reconstruction of rigid moving objects.

Pixelated polarizing system for dynamic interferometry events employing a temporal phase unwrapping approach

We describe a robust system to retrieve the phase using a temporal phase unwrapping approach and a pixelated polarizing camera. By considering the characteristics of the polarizing sensor and its capabilities to acquired simultaneously four phase shifted interferograms in a stable manner, we propose the use of a temporal phase unwrapping technique to simplify the phase retrieval procedure in dynamic events. We also show that by combining the polarizing sensor and temporal phase unwrapping algorithms, the phase unwrapping procedure is not only simplified, but also the computing time is reduced one order of magnitude compared with other algorithms. In the approach taken, each pixel is processed independently obtaining the phase difference for a specific time opening the possibility of real-time analysis. Validation experiments are presented.

Absolute Depth Measurement Using Multiphase Normalized Cross-Correlation for Precise Optical Profilometry.

In a multifrequency phase-shifting (MFPS) algorithm, the temporal phase unwrapping algorithm can extend the unambiguous phase range by transforming the measurement range from a short fringe pitch into an extended synthetic pitch of two different frequencies. However, this undesirably amplifies the uncertainty of measurement, with each single-frequency phase map retaining its measurement uncertainty, which is carried over to the final unwrapped phase maps in fringe-order calculations. This article analyzes possible causes and proposes a new absolute depth measurement algorithm to minimize the propagation of measurement uncertainty. Developed from normalized cross-correlation (NCC), the proposed algorithm can minimize wrong fringe-order calculations in the MFPS algorithm. The experimental results demonstrated that the proposed measurement method could effectively calibrate the wrong fringe order. Moreover, some extremely low signal-to-noise ratio (SNR) regions of a captured image could be correctly reconstructed (for surface profiles). The present findings confirmed measurement precision at one standard deviation below 5.4 µm, with an absolute distance measurement of 16 mm. The measurement accuracy of the absolute depth could be significantly improved from an unacceptable level of measured errors down to 0.5% of the overall measuring range. Additionally, the proposed algorithm was capable of extracting the absolute phase map in other optical measurement applications, such as distance measurements using interferometry.

Efficient phase retrieval of two-directional phase-shifting fringe patterns using geometric constraints of deflectometry.

In the phase measuring deflectometry, two groups of fringe patterns in orthogonal directions are usually applied to establish the correspondences between the pixel pairs on the screen and camera. Usually, 16 phase-shifting fringe patterns with different spatial frequencies are required in order to calculate the absolute phases in the conventional temporal phase unwrapping algorithms. This requirement makes the measurement inefficient and not robust against environmental noise. In this paper, an efficient phase retrieval strategy is developed, which requires only six fringe patterns. The modulating information in one-direction is obtained by first using four fringe patterns, and then it is applied to assist the phase calculation in the other direction, so that only two extra fringe patterns are needed. Subsequently the phases are unwrapped by using the geometric constraints of the software configurable optical test system without additional image acquisition. The measurement time is saved by 5/8, compared to the conventional methods. In this way, the influence of the low-frequency disturbances can be suppressed in the workshop condition. Experiments demonstrate that the proposed method can reliably retrieve the absolute phases, and it is of significance to improve the measuring efficiency and stability of in situ deflectometry.

Optimal wavelength selection strategy in temporal phase unwrapping with projection distance minimization.

Micro Fourier transform profilometry (μFTP) is a recently developed computational framework for high-speed dynamic 3D shape measurement of transient scenes based on fringe projection. It has been demonstrated that by using high-frame-rate fringe projection hardware, μFTP can achieve accurate, denser, unambiguous, and motion-artifact-free 3D reconstruction at a speed up to 10,000 Hz. μFTP utilizes a temporal phase unwrapping algorithm, so-called projection distance minimization (PDM), in which multiple wavelengths are used to solve the phase ambiguity optimally in the maximum-likelihood sense. However, it has been found that the choice of the wavelengths is essential to the unambiguous measurement range as well as the unwrapping reliability in the presence of noise. In this work, the relations between the wavelength combination and the noise resistance ability of PDM are analyzed and investigated in detail by analytical, emulational, and experimental means. This leads to a qualitative conclusion that the noise resistance ability of PDM is fundamentally determined by the value of each item in wavelength ratio: a smaller value of each item in wavelength ratio means better noise resistance ability in phase unwrapping. Our result provides a guideline for optimal wavelengths selection in order to improve the noise resistance ability of a practical fringe projection system. Simulations and experiments based on a microscopic fringe projection system are demonstrated to validate the correctness of our conclusion.

Accurate and fast 3D surface measurement with temporal-spatial binary encoding structured illumination.

Balancing the accuracy and speed for 3D surface measurement of object is crucial in many important applications. Binary encoding pattern utilizing the high-speed image switching rate of digital mirror device (DMD)-based projector could be used as the candidate for fast even high-speed 3D measurement, but current most schemes only enable the measurement speed, which limit their application scopes. In this paper, we present a binary encoding method and develop an experimental system aiming to solve such a situation. Our approach encodes one computer-generated standard 8 bit sinusoidal fringe pattern into multiple binary patterns (sequence) with designed temporal-spatial binary encoding tactics. The binary pattern sequence is then high-speed and in-focus projected onto the surface of tested object, and then captured by means of temporal-integration imaging to form one sinusoidal fringe image. Further the combination of phase-shifting technique and temporal phase unwrapping algorithm leads to fast and accurate 3D measurement. The systematic accuracy better than 0.08mm is achievable. The measurement results with mask and palm are given to confirm the feasibility.

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.

Iterative two-step temporal phase-unwrapping applied to high sensitivity three-dimensional profilometry

Although temporal phase unwrapping method can be applied to solve some problems to measure an object with steep shapes, isolated parts or fringe undersampling in three-dimensional (3D) shape measurement, it needs to acquire and process a sequence of fringe pattern images which would take much time. Servin et al. proposed a 2-step temporal phase unwrapping algorithm, which only needs the 2 extreme phase-maps to achieve exactly the same results as standard temporal unwrapping method. But this method is only validated by computer simulation, shortage of experimental demonstration, its sensitivity coefficient G is limited, and it cannot be used when the G value is larger. We proposed an iterative two-step temporal phase-unwrapping algorithm which is an extension of Servin׳s method. First, add a fringe pattern with an intermediate sensitivity, project the fringe patterns of different sensitivity onto the tested object’s surface, and collect deformed fringe patterns with a CCD camera. Then we obtain the unwrapped phase with larger sensitivity coefficient G by cascading the sensitivity coefficients. And we derive the initial phase conditions of the 2-step temporal phase unwrapping algorithm. Finally, the experimental evaluation is conducted to prove the validity of the proposed method. The results are analyzed and compared with Servin׳s method. The experimental results show that the proposed method can achieve higher sensitivity and more accurate measurement, and it can overcome the main disadvantages encountered by Servin׳s method.

Fast temporal phase unwrapping method for the fringe reflection technique based on the orthogonal grid fringes.

In traditional temporal phase unwrapping (TPU) algorithms, wrapped phases with different spatial frequencies are obtained from several groups of phase shift fringes to calculate the unwrapped phase. Therefore, the necessary quantity of captured fringes is very large, especially for the fringe reflection technique (FRT), since a pair of phases should be unwrapped to get the slopes of two perpendicular directions. In this paper, we propose a fast TPU algorithm based on the orthogonal grid fringes by which only one image is needed to extract the two integer phases for each frequency instead of two groups of phase shift fringes, and then they can be added into the wrapped phases separately to complete the unwrapping. There are ridge errors in the direct unwrapped phases, but they are significantly suppressed by our pseudo-phase-shift strategy without any extra captured fringes. The proposed method is robust and effective where the fringe amount used for unwrapping is only 1/4 of the previous similar algorithm and 1/6-1/8 of the traditional TPU methods. The detailed comparison of measurement time is also given, which demonstrate that the FRT measurement can be accelerated in most cases by our method. The algorithm is validated by the experiments, which still works well for the severely defocusing fringes or complex specimen.

Continuous casting slab surface crack depth measurement using sinusoidal phase grating method

With the development of hot charging and direct rolling technology in continuous slab casting, there are increasing demands for the surface quality of hot continuously cast slab evoked by modern metal material fabrication. In order to reduce noise, which is caused by scale, water film and oscillation marks, during on-line detection of surface defects, a sinusoidal phase grating projection method is presented to detect the depth and surface profile of surface cracks in continuous slab casting. This technology, based on a deformed projected grating pattern, which carries three-dimensional information of the measured surface of the slab, can provide an absolute measurement of the depth of the surface defect by using a temporal phase unwrapping algorithm. The main advantage is that objects containing depth discontinuities of the crack are profiled as easily as smooth ones. The experimental results show the method has a high signal to noise ratio, and good applicability for automatic crack depth testing in continuous casting.

An Error-Reduction Method for Temporal Phase Unwrapping Based on Double Three-Step Phase-Shifting and Least-Squares Fitting

Phase-shifting interferometry on structured light projection is widely used in 3-D surface measurement. An investigation shows that least-squares fitting can significantly decrease random error by incorporating data from the intermediate phase values, but it cannot completely eliminate nonlinear error. This paper proposes an error-reduction method based on double three-step phase-shifting algorithm and least-squares fitting, and applies it on the temporal phase unwrapping algorithm using three-frequency heterodyne principle. Theoretical analyses and experiment results show that this method can greatly save data acquisition time and improve the precision.

Flexible error-reduction method for shape measurement by temporal phase unwrapping: phase averaging method

Temporal phase unwrapping is an important method for shape measurement in structured light projection. Its measurement errors mainly come from both the camera noise and nonlinearity. Analysis found that least-squares fitting cannot completely eliminate nonlinear errors, though it can significantly reduce the random errors. To further reduce the measurement errors of current temporal phase unwrapping algorithms, in this paper, we proposed a phase averaging method (PAM) in which an additional fringe sequence at the highest fringe density is employed in the process of data processing and the phase offset of each set of the four frames is carefully chosen according to the period of the phase nonlinear errors, based on fast classical temporal phase unwrapping algorithms. This method can decrease both the random errors and the systematic errors with statistical averaging. In addition, the length of the additional fringe sequence can be changed flexibly according to the precision of the measurement. Theoretical analysis and simulation experiment results showed the validity of the proposed method.