Unwrapping Algorithm Research Articles

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.

Multi-echo gradient-recalled-echo phase unwrapping using a Nyquist sampled virtual echo train in the presence of high-field gradients.

To develop a spatio-temporal approach to accurately unwrap multi-echo gradient-recalled echo phase in the presence of high-field gradients. Using the virtual echo-based Nyquist sampled (VENyS) algorithm, the temporal unwrapping procedure is modified by introduction of one or more virtual echoes between the first lower and the immediate higher echo, so as to reinstate the Nyquist condition at locations with high-field gradients. An iterative extension of the VENyS algorithm maintains spatial continuity by adjusting the phase rotations to make the neighborhood phase differences less than π. The algorithm is evaluated using simulated data, Gadolinium contrast-doped phantom, and in vivo brain, abdomen, and chest data sets acquired at 3 T and 9.4 T. The unwrapping performance is compared with the standard temporal unwrapping algorithm used in the morphology-enabled dipole inversion-QSM pipeline as a benchmark for validation. Quantitative evaluation using numerical phantom showed significant reduction in unwrapping errors in regions of large field gradients, and the unwrapped phase revealed an exact match with the linear concentration profile of vials in a gadolinium contrast-doped phantom data acquired at 9.4 T. Without the need for additional spatial unwrapping, the iterative VENyS algorithm was able to generate spatially continuous phase images. Application to in vivo data resulted in better unwrapping performance, especially in regions with large susceptibility changes such as the air/tissue interface. The iterative VENyS algorithm serves as a robust unwrapping method for multi-echo gradient-recalled echo phase in the presence of high-field gradients.

Accurate and practical feature extraction from noisy holograms.

Quantitative phase imaging using holographic microscopy is a powerful and non-invasive imaging method, ideal for studying cells and quantifying their features such as size, thickness, and dry mass. However, biological materials scatter little light, and the resulting low signal-to-noise ratio in holograms complicates any downstream feature extraction and hence applications. More specifically, unwrapping phase maps from noisy holograms often fails or requires extensive computational resources. We present a strategy for overcoming the noise limitation: rather than a traditional phase-unwrapping method, we extract the continuous phase values from holograms by using a phase-generation technique based on conditional generative adversarial networks employing a Pix2Pix architecture. We demonstrate that a network trained on random surfaces can accurately generate phase maps for test objects such as dumbbells, spheres, and biconcave discoids. Furthermore, we show that even a rapidly trained network can generate faithful phase maps when trained on related objects. We are able to accurately extract both morphological and quantitative features from the noisy phase maps of human leukemia (HL-60) cells, where traditional phase unwrapping algorithms fail. We conclude that deep learning can decouple noise from signal, expanding potential applications to real-world systems that may be noisy.

Toward an Estimation of the Optical Feedback Factor C on the Fly for Displacement Sensing.

In this paper, a method based on the inherent event-based sampling capability of laser optical feedback interferometry (OFI) is proposed to assess the optical feedback factor C when the laser operates in the moderate and strong feedback regimes. Most of the phase unwrapping open-loop OFI algorithms rely on the estimation of C to retrieve the displacement with nanometric precision. Here, the proposed method operates in open-loop configuration and relies only on OFI’s fringe detection, thereby improving its robustness and ease of use. The proposed method is able to estimate C with a precision of <5%. The obtained performances are compared to three different approaches previously published and the impacts of phase noise and sampling frequency are reported. We also show that this method can assess C on the fly even when C is varying due to speckle. To the best of the authors’ knowledge, these are the first reported results of time-varying C estimation. In addition, through C estimation over time, it could pave the way not only to higher performance phase unwrapping algorithms but also to a better control of the optical feedback level via the use of an adaptive lens and thus to better displacement retrieval performances.

Enlarging Dynamic Strain Range in UWFBG Array-Based Φ-OTDR Assisted With Polarization Signal

Phase-sensitive optical time-domain reflectometry ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Phi $ </tex-math></inline-formula> -OTDR) measures the dynamic strain by extracting the phase of the scattering signal. However, the dynamic strain measurement range in conventional <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Phi $ </tex-math></inline-formula> -OTDR is restricted by the phase unwrapping algorithm. In this letter, a new phase unwrapping algorithm, combining the phase and polarization information, is proposed and demonstrated to enlarge the dynamic strain measurement range in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Phi $ </tex-math></inline-formula> -OTDR based on ultra-weak fiber Bragg grating (UWFBG) array. By correcting the phase signal with the assistance of polarization signal, the large dynamic strain can be recovered. In experiments, the dynamic strain measurement range is demonstrated to increase by about three times, which cannot be achieved by the conventional unwrapping algorithm. Further, the proposed method has the potential to achieve a large amplitude range determined by polarization signal and a good sensitivity determined by phase signal at the same time.

Photometric constraint for absolute phase unwrapping from single-frequency fringe patterns.

As a fundamental step in fringe projection profilometry, absolute phase unwrapping via single-frequency fringe patterns is still a challenging ill-posed problem, which attracts lots of interest in the research area. To solve the problem above, additional constraints were constructed, such as spatial smoothness constraint (SSC) in spatial phase unwrapping algorithm and viewpoint consistency constraint (VCC) in multi-view systems (e.g., stereo and light-field cameras). However, there still exists phase ambiguity in the unwrapping result based on SSC. Moreover, VCC-based methods rely on additional cameras or light-field cameras, which makes the system complicated and expensive. In this paper, we propose to construct a novel constraint directly from photometric information in captured image intensity, which has never been fully exploited in phase unwrapping. The proposed constraint, named photometric constraint (PC), provides a prospective constraint for absolute phase unwrapping from single-frequency fringe patterns without any additional cameras. Extensive experiments have been conducted for the validation of the proposed method, which achieved comparable performance with the state-of-the-art method, given a traditional camera-projector setup and single high-frequency fringe patterns.

The spatial phase-shifting measuring profilometry based on dual-frequency grating

A real-time three-dimensional(3D) shape measuring method based on dual-frequency spatial phase-shifting measuring profilometry (SPSMP) is proposed. To combine the low-frequency and high-frequency gratings into one single composite pattern, each individual grating is modulated along orthogonal direction with a distinct carrier frequency. Since the spectrum components are staggered in the spectrum of the composite deformed pattern, the low-frequency and high-frequency deformed patterns with background light can be demodulated easily. Then two groups of spatial phase-shifting deformed sub-patterns are extracted from the corresponding deformed patterns to calculate the low-frequency wrapped phase and the high-frequency wrapped phase. According to the dual-frequency phase unwrapping algorithm, the high-frequency wrapped phase is guided by the low-frequency unwrapped phase. Compared to the existing multi-frequency single-shot fringe projection, the filtering process of the proposed method is more concise, it does not involve multiple filtering and the extraction of the single spectrum component. The static and real-time experimental results show that the proposed method inherits the high accuracy of PMP, while guaranteeing single-shot feature.

Zernike integrated partial phase error reduction algorithm

A modification to the error reduction algorithm is reported in this paper for determining the prescription of an imaging system in terms of Zernike polynomials. The technique estimates the Zernike coefficients of the optical prescription as part of a modified Gerchberg-Saxton iteration combined with a new gradient-based phase unwrapping algorithm. Zernike coefficients are updated gradually as the error reduction algorithm converges by recovering the partial pupil phase that differed from the last known pupil phase estimate. In this way the wrapped phase emerging during each iteration of the error reduction algorithm does not represent the entire wrapped phase of the pupil electric field and can be unwrapped with greater ease.The algorithm is tested in conjunction with a blind deconvolution algorithm using measured laboratory data with a known optical prescription and is compared to a baseline approach utilizing a combination of the error reduction algorithm and a least-squares phase unwrapper previously reported in the literature. The combination of the modified error reduction algorithm and the new least-squares Zernike phase unwrapper is shown to produce superior performance for an application where it is desirable that Zernike coefficients be estimated during each iteration of the blind deconvolution procedure.

Geometric dimension assisted absolute phase recovery in 3D shape measurement with digital fringe projection

A novel method of geometric dimension assisted absolute phase recovery in 3D shape measurement is presented. This method mainly includes two steps: (a) using the window Fourier filter-quality guided phase unwrapping algorithm to obtain the relative phase distribution of the tested object; (b) using the geometric dimension of the object as a clue to convert the relative phase distribution to the absolute phase distribution. This method is convenient, and only three fringe image acquisitions are needed to recover the absolute phase. In addition, there is no limit to the depth range of the tested object. The correctness of this method is verified by several experiments.

Novel phase unwrapping technique based on extended information filter

A general framework of a phase unwrapping procedure based on extended information filter is built. Two phase unwrapping algorithms with different observed models, are proposed by combining an extended information filter with an efficient phase gradient estimator, and an efficient quality-guided strategy based on heap-sort. The extended information filter that is a new type of filter with high efficiency and high robustness, is applied to acquire the unambiguous unwrapped phase of wrapped phase images for the first time to our knowledge. Through the application of the extended information filter, simultaneous phase unwrapping and noise suppression have been performed. The efficient phase gradient estimator that has been demonstrated well, is used to efficiently acquire phase gradient information required for the proposed algorithms. The efficient quality-guided strategy based on heap-sort is used to define the path of the proposed algorithms and to guarantee that they efficiently unwrap wrapped pixels along the path, from high-reliance to low-reliance regions. Synthetic and experimental data is used to demonstrate that our proposed algorithms can obtain robust solutions with reasonable time consumption, compared with some of the more traditional algorithms.

Hybrid Technique of the Branch-Cut and the Quality-Guided for Insar Phase Unwrapping

Phase unwrapping is a key step for interferometric synthetic aperture radar imaging. It is widely used for earth mapping and surface change detection. Several residue-immune phase unwrapping algorithms have been proposed; among them, we find branch-cut and quality-guided in the path-following category. Branch-cut methods are usually faster than the quality-guided techniques; however, the accuracy of their unwrapped phase images is lower. In this paper, a hybrid model which combines both algorithms is proposed in order to establish a satisfactory compromise between processing time and accuracy. In order to verify the usefulness of the proposed hybridization, it is tested on simulated and real inSAR data. The obtained results are compared with the two methods under several relevant metrics.

Implementation of a high-accuracy phase unwrapping algorithm using parallel-hybrid programming approach for displacement sensing using self-mixing interferometry

Phase unwrapping is an integral part of multiple algorithms with diverse applications. Detailed phase unwrapping is also necessary for achieving high-accuracy metric sensing using laser feedback-based self-mixing interferometry (SMI). Among SMI specific phase unwrapping approaches, a technique called Improved Phase Unwrapping Method (IPUM) provides the highest accuracy. However, due to its complex, sequential, and compute-intensive nature, this method requires a high-performance computing architecture, capable of scalable parallel processing so that such a high-accuracy algorithm can be used for high-bandwidth sensing applications. In this work, the existing sequential IPUM C program is parallelized by using hybrid OpenMP/MPI (Open Multi-Processing/Message Passing Interface) parallel programming models and tested on Barcelona Supercomputing Center Nord-III Supercomputer. The computational performance of the proposed parallel-hybrid IPUM algorithm is compared with existing IPUM sequential code by executing multi-core and uni-core processor architecture, respectively. While comparing the performance of sequential IPUM with the parallel-hybrid IPUM algorithm on 16 nodes of Nord-III supercomputer, the results show that the parallel-hybrid algorithm gets 345.9x times performance improvement as compared to IPUM’s standard, sequential implementation on a single node system. The results show that the parallel-hybrid version of IPUM gives a scalable performance for different target velocities and a different number of processing cores.

Dynamic Three-Dimensional Surface Reconstruction Approach for Continuously Deformed Objects

For dynamic three-dimensional (3-D) surface reconstruction of continuously deformed objects, an efficient method employing one fringe pattern is proposed. First, a two-dimensional wavelet transform profilometry (2-D WTP) employing an improved cost function is adopted to obtain the wrapped phase, which has the better noise suppression ability. Then, a new 3-D phase unwrapping algorithm is introduced to obtain the unwrapped phase, which includes four steps. a) A new 3-D residues recognition method is proposed. b) A 3-D branch cut construction algorithm based on an improved GA is proposed. c) A 3-D phase derivative variance quality map is established. d) A 3-D flood filling phase unwrapping algorithm guided by 3-D quality map avoiding 3-D branch cut (3-D FGQAB) is introduced to obtain the unwrapped phase. Simulation and experimental comparisons are carried out, the results show that the proposed method can be applied to dynamic 3-D reconstruction of the continuously deformed objects.

Robust phase unwrapping algorithm based on enhanced denoising and fringe quality improvement routines

Phase unwrapping algorithms have widely been studied and implemented with efforts aimed at unwrapping wrapped phase signals. However, the presence of noise and unreliable fringe quality poses a major obstacle for the retrieval of reliable phase signals. While many techniques have been implemented to deal with the aforementioned issues, most algorithms are application dependent or difficult to implement. Here we present a simple yet effective global phase unwrapping algorithm, that does not resort to Least-Squares Minimization, making use of Fast-Fourier Transform (FFT) based spectral differentiation, Signal Dependent Rank Ordered Mean (SD-ROM) filtering, and Fuzzy Logic Edge Detection (FLED). The proposed algorithm was tested using simulated, noisy, wrapped phaseograms and has shown to improve image and fringe quality, as well as overall retrieved phase reliability.

Light-Field-Assisted Phase Unwrapping of Fringe Projection Profilometry

A phase unwrapping algorithm assisted by light field without additional time coding is proposed, which can be used in various aspects of optical imaging and metrology. The phase consistency between the light field and the industrial camera is the sticking point to convert the angular resolution of the light field into the order of phase. By virtue of the angle information recorded by the light field, the correct order can be uniquely determined from several candidate orders without confusion in the measurement volume. The mandatory requirements for system fixation and pre-calibration and the connection between the light field and industrial camera are prerequisites for order uniqueness. The mapping relationship established by the phase implies coordinate conversion, lens distortion and other factors, thus providing a high degree of stability for the transmission of phase order information. The achieved results demonstrate that our method is robust and can be taken as an effective tool for means of phase unwrapping.

Fast phase unwrapping algorithm based on region segmenting with mathematical morphology

Fast phase unwrapping algorithm based on region segmenting with mathematical morphology

Low-noise hierarchical phase unwrapping method for dual-wavelength digital holography using two synthetical wavelengths

Dual-wavelength digital holography can expand the unambiguous measurement depth in phase unwrapping by using a differential synthetic wavelength which is longer than the single illumination wavelength. However, the phase noise is significantly amplified due to the magnification of the differential synthetic wavelength, resulting in a lower measurement accuracy. On the other hand, a lower noise level can be achieved by using additive synthetic-wavelength which is shorter than the single illumination wavelength. However, the corresponding unambiguous measurement depth is greatly reduced due to the phase ambiguity. In this case, combining the merits of the differential synthetic-wavelength and the additive synthetic-wavelength, different low noise phase unwrapping algorithms have been developed in recent years. However, these algorithms are complex and time consuming because they need to calculate multiple intermediate variables or search for the constrained boundary conditions in two-dimensional space. Therefore, in this paper, we develop a hierarchical phase unwrapping algorithm by using the two synthetic wavelengths for dual-wavelength digital holography to realize low noise and fast unambiguous measurement with large depth. In this algorithm, the unwrapped phase difference obtained by the differential synthetic wavelength is used to guide the wrapped phase of one single wavelength to realize phase unwrapping, and then the optical path difference obtained by the single-wavelength unwrapped phase is employed to guide the wrapped phase sum, and thus realizing phase unwrapping. As a result, the phase noise is attenuated and the depth sensitivity is preserved for dual-wavelength phase unwrapping. After theoretical analysis, a series of simulation experiments is carried out on the reconstructed quality, anti-noise characteristics and speed through comparing with state-of-the-art dual-wavelength phase unwrapping algorithms, including the conventional algorithm, the linear programming algorithm and the direct linear programming algorithm. In this case, a flipping dual-wavelength common-path digital holography with orthogonal carrier is built to acquire multiplexed off-axis hologram in one shot and illustrate the operation of the algorithm with circular step target, and stability test of the setup. Both the simulation and experimental results show that the proposed method can be simplified and deterministic, resulting in a lower noise phase unwrapping in a time of 20.5 ms for a phase map of one megapixel. We expect that the proposed method can have practical applications in measurement that requires high accuracy, fast speed, and large depth.

Improved fast phase unwrapping algorithm based on parallel acceleration

Aiming at the shortcoming of low serial operational efficiency in the quality-map-guided phase-unwrapping algorithm proposed by Miguel, an improved algorithm for parallel merging of multiple low-reliability blocks is pro-posed. Under the condition that the original algorithm design idea is satisfied, the unwrapping path is redefined as the largest reliable edge of the block. In addition, based on the non-continuous characteristic of the unwrapping path of the original algorithm, a low-reliability block out-of-order merging strategy is proposed to make multiple merging tasks can be performed simultaneously. The improved algorithm uses a multi-threaded software architecture. The main thread is responsible for looping through the unprocessed blocks to check whether they meet the requirements of merging, and the child threads receive and perform the merge tasks. The experimental results show that the improved method is completely consistent with the processing results of the original algorithm, and the parallel improvement strategy can effectively use the computer's multi-core resources, so that the operational efficiency of the phase unwrapping algorithm is improved by more than 50%.

L 1-Norm Sparse 2-D Phase Unwrapping Algorithm Based on Reliable Redundant Network

Multitemporal interferometric synthetic aperture radar (InSAR) technology has become an important tool for digital elevation model (DEM) generation and surface deformation monitoring. The prerequisite is to obtain the correct unwrapped phase of the interferograms. However, it is still challenging to obtain reliable results by traditional 2-D phase unwrapping (PU) algorithms and even 3-D PU algorithms, when the phase continuity assumption fails due to large deformation gradients or abrupt topographic changes. This letter proposes an L¹-norm sparse 2-D PU algorithm based on reliable redundant network in the phase discontinuous region, which is obtained by implementing ambiguity detector on free network within a given distance. Experiment on real synthetic aperture radar (SAR) interferograms covering urban tall buildings verified that the algorithm can obtain more reliable results in phase discontinuous regions than traditional PU algorithms.

Phase unwrapping in ICF target interferometric measurement via deep learning.

This paper proposes an unwrapping algorithm based on deep learning for inertial confinement fusion (ICF) target interferograms. With a deep convolutional neural network (CNN), the task of phase unwrapping is transferred into a problem of semantic segmentation. A method for producing the data set for the ICF target measurement system is demonstrated. The noisy wrapped phase is preprocessed using a guided filter. Postprocessing is introduced to refine the final result, ensuring the proposed method can still accurately unwrap the phase even when the segmentation result of the CNN is not perfect. Simulations and actual interferograms show that our method has better accuracy and antinoise ability than some classical unwrapping approaches. In addition, the generalization capability of our method is verified by successfully applying it to an aspheric nonnull test system. By adjusting the data set, the proposed method may be transferred to other systems.