Path-independent phase unwrapping using phase gradient and total-variation (TV) denoising

Performance of a U2-net model for phase unwrapping

Phase unwrapping plays a pivotal role in optics and is a key step in obtaining phase information. Recently, owing to the rapid development of artificial intelligence, a series of deep-learning-based phase-unwrapping methods has garnered considerable attention. Among these, a representative deep-learning model called U 2-net has shown potential for various phase-unwrapping applications. This study proposes a U 2-net-based phase-unwrapping model to explore the performance differences between the U 2-net and U-net. To this end, first, the U-net, U 2-net, and U 2-net-lite models are trained simultaneously, then their prediction accuracy, noise resistance, generalization capability, and model weight size are compared. The results show that the U 2-net model outperformed the U-net model. In particular, the U 2-net-lite model achieved the same performance as that of the U 2-net model while reducing the model weight size to 6.8% of the original U 2-net model, thereby realizing a lightweight model.

Investigation of refractive index dynamics during in vitro embryo development using off-axis digital holographic microscopy

Embryo quality is a crucial factor affecting live birth outcomes. However, an accurate diagnostic for embryo quality remains elusive in the in vitro fertilization clinic. Determining physical parameters of the embryo may offer key information for this purpose. Here, we demonstrate that digital holographic microscopy (DHM) can rapidly and non-invasively assess the refractive index of mouse embryos. Murine embryos were cultured in either low- or high-lipid containing media and digital holograms recorded at various stages of development. The phase of the recorded hologram was numerically retrieved, from which the refractive index of the embryo was calculated. We showed that DHM can detect spatio-temporal changes in refractive index during embryo development that are reflective of its lipid content. As accumulation of intracellular lipid is known to compromise embryo health, DHM may prove beneficial in developing an accurate, non-invasive, multimodal diagnostic.

Phase unwrapping using deep learning in holographic tomography

Holographic tomography (HT) is a measurement technique that generates phase images, often containing high noise levels and irregularities. Due to the nature of phase retrieval algorithms within the HT data processing, the phase has to be unwrapped before tomographic reconstruction. Conventional algorithms lack noise robustness, reliability, speed, and possible automation. In order to address these problems, this work proposes a convolutional neural network based pipeline consisting of two steps: denoising and unwrapping. Both steps are carried out under the umbrella of a U-Net architecture; however, unwrapping is aided by introducing Attention Gates (AG) and Residual Blocks (RB) to the architecture. Through the experiments, the proposed pipeline makes possible the phase unwrapping of highly irregular, noisy, and complex experimental phase images captured in HT. This work proposes phase unwrapping carried out by segmentation with a U-Net network, that is aided by a pre-processing denoising step. It also discusses the implementation of the AGs and RBs in an ablation study. What is more, this is the first deep learning based solution that is trained solely on real images acquired with HT.

Evaluation of denoising techniques to remove speckle and Gaussian noise from dermoscopy images

Computerized methods provide analyses of skin lesions from dermoscopy images automatically. However, the images acquired from dermoscopy devices are noisy and cause low accuracy in automated methods. Therefore, various methods have been applied for denoising in the literature. There are some review-type papers about these methods. However, their authors have focused on either denoising with a specific approach or denoising from other images rather than dermoscopy images, which have a different characteristic. It is not possible to determine which method is the most suitable for denoising from dermoscopy images according to the results presented in them. Therefore, a review on the denoising approaches applied with dermoscopy images is required and, according to our knowledge, there is no such a review-type paper. To fill this gap in the literature, the required review has been performed in this work. Also, in this work, the methods in the literature have been implemented using the same data sets containing images with speckle or Gaussian types of noise. The results have been analyzed not only visually but also quantitatively to compare capabilities of the techniques. Our experiments indicated that each denoising technique has its own disadvantages and advantages. The main contributions of this paper are three-fold: (i) A comprehensive review on the denoising approaches applied with dermoscopy images has been presented. (ii) The denoising techniques have been implemented with the same images for meaningful comparisons. (iii) Both visual and quantitative analyses with different metrics have been performed and comparative performance evaluations have been presented.

Simple phase unwrapping method with continuous convex minimization

Phase unwrapping is a problem to reconstruct true phase values from modulo 2π phase values measured using various phase imaging techniques. This procedure is essentially formulated as a discrete optimization problem. However, most energy minimization methods using continuous optimization techniques have ignored the discrete nature and solved it as a continuous minimization problem directly, leading to losing exactness of the algorithms. We propose a new minimum norm method that can yield the optimal solution of the discrete problem by minimizing a continuous energy function. In contrast to the graph-cuts method, which is state of the art in this field, the proposed method requires much less memory space and a very simple implementation. Therefore, it can be simply extended to 3D or 4D phase unwrapping problems.

Robust particle-Kalman filtering phase unwrapping algorithm for wrapped fringe patterns

This paper presents a robust phase unwrapping algorithm based on a particle-Kalman filter for wrapped fringe patterns by combining a particle filter and an extended Kalman filter, which formulates the phase unwrapping problem of wrapped fringe patterns as an optimal state estimation problem under the frame of the particle-Kalman filter. First, a state space equation for state variables is extended to the second order of Taylor series, and a local phase gradient estimator based on a modified matrix pencil model is used to obtain the first-order and second-order phase gradient information required by the extended state space equation, which is conducive to enhancing the phase unwrapping accuracy of the proposed procedure. Second, the initial estimate of unwrapped phase is obtained through applying an efficient phase unwrapping program based on a particle filter to unwrap noisy wrapped pixels. Finally, the initial estimate of unwrapped phase obtained by the particle filter is taken as the predicted estimate of state variables and further processed by the extended Kalman filter to obtain the final estimate of unwrapped phase. In addition, an efficient quality-guided strategy that has been demonstrated well is used to guarantee that the particle-Kalman filter efficiently and accurately unwraps wrapped pixels along a suitable path. Results obtained with synthetic data and experimental data demonstrate the effectiveness of the proposed method and show that this new approach can obtain more acceptable solutions from noisy wrapped fringe patterns, with respect to some of the most commonly used methods.

Deep learning in optical metrology: a review

With the advances in scientific foundations and technological implementations, optical metrology has become versatile problem-solving backbones in manufacturing, fundamental research, and engineering applications, such as quality control, nondestructive testing, experimental mechanics, and biomedicine. In recent years, deep learning, a subfield of machine learning, is emerging as a powerful tool to address problems by learning from data, largely driven by the availability of massive datasets, enhanced computational power, fast data storage, and novel training algorithms for the deep neural network. It is currently promoting increased interests and gaining extensive attention for its utilization in the field of optical metrology. Unlike the traditional "physics-based" approach, deep-learning-enabled optical metrology is a kind of "data-driven" approach, which has already provided numerous alternative solutions to many challenging problems in this field with better performances. In this review, we present an overview of the current status and the latest progress of deep-learning technologies in the field of optical metrology. We first briefly introduce both traditional image-processing algorithms in optical metrology and the basic concepts of deep learning, followed by a comprehensive review of its applications in various optical metrology tasks, such as fringe denoising, phase retrieval, phase unwrapping, subset correlation, and error compensation. The open challenges faced by the current deep-learning approach in optical metrology are then discussed. Finally, the directions for future research are outlined.

Residue calibrated least-squares unwrapping algorithm for noisy and steep phase maps

This work proposes a robust unwrapping algorithm for noisy and steep phase maps based on the residue calibrated least-squares method. The proposed algorithm calculates and calibrates the residues in the derivative maps to get a noise-free Poisson equation. Moreover, it compensates for the residuals between the wrapped and unwrapped phase maps iteratively to eliminate approximation errors and the smoothing effect of the least-squares method. The robustness and efficiency of the proposed algorithm are validated by unwrapping simulated and experimentally wrapped phase maps. Compared with the other three typical algorithms, the proposed algorithm has the most effective performance in noisy and steep phase unwrapping, providing a reliable alternative for practical applications.

Tri-wavelength simultaneous ESPI for 3D micro-deformation field measurement

Electronic speckle pattern interferometry (ESPI), a well-established technique for micro-deformation measurement, can be used to determine both in-plane and out-of-plane displacement components. Although many works in ESPI have been reported for three-dimensional (3D) displacement measurement, few works have focused on the simultaneous measurement of 3D deformation fields. Here we present an ESPI system that consists of three sub-interferometers for simultaneous measurement of all three displacement components and in-plane strain fields. A 3CCD color camera, a specially designed shifting stage, and three lasers with optimal wavelengths are used in this system. The lasers and 3CCD camera provide independent interferograms with different color signals, while the shifting stage allows the sub-interferometers to achieve simultaneous phase shifting. The results of color separation and experimental measurement demonstrate the utility of the system.

Robust phase unwrapping algorithm for noisy and segmented phase measurements

This paper proposes a robust phase unwrapping algorithm (RPUA) for phase unwrapping in the presence of noise and segmented phase. The RPUA method presents a new model of phase derivatives combined with error-correction iterations to achieve an anti-noise effect. Moreover, it bridges the phase islands in the spatial domain using numerical carrier frequency and fringe extrapolation thus eliminating height faults to enable solving segmented phase unwrapping. Numerical simulation and comparison with three conventional methods were performed, proving the high robustness and efficiency of the RPUA. Further, three experiments demonstrated that the RPUA can obtain the unwrapped phase under different noise accurately and possesses the capability to process segmented phases, indicating reliable practicality.

Transform-based phase retrieval techniques from a single off-axis interferogram

Optical phase retrieval (OPR) methods are important because they are used to obtain the transverse phase profile information of a beam. Interference methods are extensively used to convert the phase information into an intensity pattern, which can then be processed further to retrieve the unknown phase. The most widely used interference method involves the interference of the unknown object beam and a known reference beam with an angle between them. There are several algorithms that retrieve the phase information from such a single off-axis interference pattern. For a particular application, the choice of an algorithm for OPR is very important. Therefore, it is necessary to choose between them, depending on the requirements. Three entirely different noniterative, transform-based algorithms, namely the Fourier transform (FT) method, the continuous wavelet transform (CWT) method, and the Hilbert transform (CWT) method, are explained in detail. A quantitative comparison is made using a combination of rms error and standard structural similarity measure. The advantages of using a standard unwrapping algorithm are also validated using the same combination of comparison metrics. We show that the HT method has a better response with object beam with higher spatial frequency content, but with the penalty of affected noise. The FT method and CWT method have better noise immunity, but have the limitation of the spatial frequency range of the object beam. The different constraints, advantages, and some practical limitations of the methods are discussed with the help of a quantitative phase imaging experiment of monodispersed polymethyl methacrylate beads.

Efficient phase unwrapping algorithm based on cubature information particle filter applied to unwrap noisy continuous phase maps

This paper presents a new phase unwrapping algorithm for wrapped phase fringes through combining a cubature information particle filter with an efficient local phase gradient estimator and an efficient quality-guided strategy based on heap-sort. The cubature information particle filter that not only is independent from noise statistics but also is not constrained by the nonlinearity of the model constructed is applied to retrieve unambiguous phase from modulus 2π wrapped fringe patterns through constructing a recursive cubature information particle filtering phase unwrapping procedure to perform simultaneously phase unwrapping and noise filtering for the first time to our knowledge, which can be expected to obtain more robust solutions from wrapped phase fringes. Phase gradient estimate is one of the key steps in almost all phase unwrapping algorithms and is directly related to the precision and the efficiency of phase unwrapping procedure. Accordingly, an efficient local phase gradient estimator that is more efficient than ones published previously is deduced to obtain phase gradient information required by the proposed algorithm, which can drastically decrease time consumption of unwrapping procedure and drastically improve the efficiency of the algorithm. The efficient quality-guided strategy based on heap-sort guarantees that the proposed algorithm efficiently unwraps wrapped pixels along the path from the high-reliance regions to the low-reliance regions of wrapped phase images. In addition, the accelerated version of the proposed algorithm is further developed through combing with reversible modulo wavelet operators to solve phase unwrapping problem of wrapped phase images in wavelet transform domain, which can reduce the amount of wrapped pixels that need to be unwrapped, and can further decrease time consumption of unwrapping procedure performing on wrapped phase images. This algorithm and its accelerated version under the frame of wavelet transform are demonstrated with various types of wrapped phase images, showing acceptable solutions.

Phase unwrapping in dual-wavelength digital holographic microscopy with total variation regularization

A two-step dual-wavelength phase imaging method with total variation (TV) regularization in digital holographic microscopy (DHM) is presented. The rough height map is first denoised with TV regularization, and then it is used as a guiding map for accurate phase unwrapping. By combining the principle of dual-wavelength interferometry and TV regularization, the problem of phase unwrapping is converted into a minimization problem that can be efficiently solved with the Split Bregman algorithm. The proposed algorithm allows a synthetic wavelength that can be arbitrarily large in principle to a single wavelength, while the accuracy of the phase map can achieve the same level in single-wavelength DHM. Moreover, the unwrapping method is robust to the change of weighting parameters. Comparative topographic measurements of a sample with abrupt steps are illustrated, and the results verified the effectiveness of the method.

Unscented information filtering phase unwrapping algorithm for interferometric fringe patterns

This paper proposes a new phase unwrapping (PU) algorithm based on an unscented information filter for interferometric fringes. The proposed algorithm is the result of combining an unscented information filter with a Levenberg-Marquardt method, a robust phase gradient estimator called the amended matrix pencil model, and an efficient quality-guided strategy based on heapsort. The unscented information filter, a new type of filter that has recently been well applied to traditional nonlinear object tracking fields, is introduced to estimate the unambiguous unwrapped phase of wrapped phase images for the first time, to the best of our knowledge. First, a recursive PU procedure based on an unscented information filter is established to perform PU and noise filtering at the same time by combining the unscented information filter and the amended matrix pencil model, where the amended matrix pencil model is applied to acquire phase gradient information needed for the recursive PU procedure. Second, the above recursive PU procedure is further optimized to improve the accuracy of the phase estimate by inserting the Levenberg-Marquardt method. This is also the first time that the Levenberg-Marquardt method is combined with the unscented information filter for the unwrapping of interferometric fringes, to the best of our knowledge. Finally, the efficient quality-guided strategy based on heapsort is used to efficiently route the path of the unwrapping procedure and to guide the proposed method to efficiently unwrap wrapped pixels along the path from the high-reliance region to the low-reliance region of the wrapped fringes. Results obtained with synthetic data and real data show more acceptable solutions with the proposed method, compared to some of the most used algorithms.

Dynamic wavefront measurement with a pinhole linear polarizer point-diffraction interferometer

We propose a pinhole linear polarizer point-diffraction interferometer (PLP-PDI) for dynamic wavefront measurements. The proposed interferometer uses a metallic wire grid linear polarizer that acts as a point-diffraction plate to generate an ideal spherical wave, from which we can obtain orthogonally polarized reference and test beams. The special polarization phase-shifting configuration allows four phase-shifted interferograms to be captured in a single shot with high precision and stability. The wavefront can then be reconstructed using a phase-unwrapping algorithm. In this paper, we describe the theory of the PLP-PDI and analyze the possible errors introduced by the device. The feasibility of the proposed PLP-PDI was verified by direct measurements of a wavefront. The experimental results show that the proposed PLP-PDI is an effective and efficient tool for the dynamic measurement of wavefronts.

Reconstruction of local frequencies for recovering the unwrapped phase in optical interferometry

In optics, when interferograms or digital holograms are recorded and their phase is recovered, it is common to obtain a wrapped phase with some errors, noise and artifacts such as singularities due to the non linearities of the demodulation process. This paper shows how to reconstruct the frequency field of the wrapped phase by using adaptive Gabor filters. Gabor filters are Gaussian quadrature filters tuned in at a certain frequency. We adapt these Gabor filters by tuning them locally and estimating the frequency using wrapped finite differences of the estimated phase. Doing this process iteratively, the frequency estimation is refined and smoothed. The unwrapped phase is easily recovered by integrating the recovered frequency field using, for example, a simple line raster integration. We don’t have problems with phase inconsistencies or residues while integrating the phase, because these are removed. The obtained unwrapped phase is clean, consistent and practically error-free. We show estimation errors with simulated data and the performance of the proposed method using real-world recorded wavefronts.

Phase-resolved heterodyne shearographic vibrometer for observation of transient surface motion: theory and model

A phase-resolved heterodyne shearing interferometer concept is under development for high-rate, whole field observations of transient surface motion. The sensor utilizes frequency and polarization multiplexing with two temporal carrier frequencies to separate each segment of a shearing Mach-Zehnder interferometer. Post-processing routines have been developed to recombine the segments by extracting the scattered object phase from Doppler shifted intermediate carrier frequencies. The processing routines provide quantitative relative phase changes and information required to generate phase resolved shearographic fringe patterns without temporal or spatial phase shifting. Separation of each segment allows for adjustment of shearing distance and direction as well as simultaneous whole field Doppler velocity (LDV) measurements. This paper presents background theory and numerical model results leading to a sensor concept.

Robust phase unwrapping for phase images in Fourier domain Doppler optical coherence tomography

To solve the 2 ? phase ambiguity for phase-resolved Doppler images in Doppler optical coherence tomography, we present a modified network programming technique for the first time to the best of our knowledge. The proposed method assumes that error of the discrete derivatives between unwrapped phase image and wrapped phase image can be arbitrary values instead of integer-multiple of 2 ? , which makes the real-phase restoration accurate and robust against noise. We compared our proposed method with the network programming method. Parameters including root-mean-square-error and noise amplification degree were adopted for comparison. The experimental study on simulated images, phantom, and real-vessel OCT images were performed. The proposed method consistently achieves optimal results.

Phase calibration unwrapping algorithm for phase data corrupted by strong decorrelation speckle noise

Robust phase unwrapping in the presence of high noise remains an open issue. Especially, when both noise and fringe densities are high, pre-filtering may lead to phase dislocations and smoothing that complicate even more unwrapping. In this paper an approach to deal with high noise and to unwrap successfully phase data is proposed. Taking into account influence of noise in wrapped data, a calibration method of the 1st order spatial phase derivative is proposed and an iterative approach is presented. We demonstrate that the proposed method is able to process holographic phase data corrupted by non-Gaussian speckle decorrelation noise. The algorithm is validated by realistic numerical simulations in which the fringe density and noise standard deviation is progressively increased. Comparison with other established algorithms shows that the proposed algorithm exhibits better accuracy and shorter computation time, whereas others may fail to unwrap. The proposed algorithm is applied to phase data from digital holographic metrology and the unwrapped results demonstrate its practical effectiveness. The realistic simulations and experiments demonstrate that the proposed unwrapping algorithm is robust and fast in the presence of strong speckle decorrelation noise.

Fast and robust estimation of ophthalmic wavefront aberrations

Rapidly rising levels of myopia, particularly in the developing world, have led to an increased need for inexpensive and automated approaches to optometry. A simple and robust technique is provided for estimating major ophthalmic aberrations using a gradient-based wavefront sensor. The approach is based on the use of numerical calculations to produce diverse combinations of phase components, followed by Fourier transforms to calculate the coefficients. The approach does not utilize phase unwrapping nor iterative solution of inverse problems. This makes the method very fast and tolerant to image artifacts, which do not need to be detected and masked or interpolated as is needed in other techniques. These features make it a promising algorithm on which to base low-cost devices for applications that may have limited access to expert maintenance and operation.

Two-dimensional phase unwrapping in Doppler Fourier domain optical coherence tomography

For phase-related imaging modalities using interferometric techniques, it is important to develop effective method to recover phase information that is mathematically wrapped. In this paper, we propose and demonstrate a two-dimensional (2D) method to achieve effective phase unwrapping in Doppler Fourier-domain (FD) optical coherence tomography (OCT), and recover the discontinuous phase distribution in retinal blood flow successfully for the first time in Doppler OCT studies. The proposed method is based on phase gradient approach in the axial dimension, with phase denoising performed through 2D window moving average in the sampled phase image using complex Doppler OCT data. The 2D unwrapping is carried out to correct phase discontinuities in the wrapped Doppler phase map, and the abrupt phase changes can be identified and corrected accurately. The proposed algorithm is computationally efficient and easy to be implemented.

Efficient multiscale phase unwrapping methodology with modulo wavelet transform

Many robust phase unwrapping algorithms are computationally very time-consuming, making them impractical for handling large datasets or real-time applications. In this paper, we propose a generic framework using a novel wavelet transform that can be combined with many types of phase unwrapping algorithms. By inserting reversible modulo operators in the wavelet transform, the number of coefficients that need to be unwrapped is significantly reduced, which results in large computational gains. The algorithm is tested on various types of wrapped phase imagery, reporting speedup factors of up to 500. The source code of the algorithm is publicly available.

Iterated unscented Kalman filter for phase unwrapping of interferometric fringes

A fresh phase unwrapping algorithm based on iterated unscented Kalman filter is proposed to estimate unambiguous unwrapped phase of interferometric fringes. This method is the result of combining an iterated unscented Kalman filter with a robust phase gradient estimator based on amended matrix pencil model, and an efficient quality-guided strategy based on heap sort. The iterated unscented Kalman filter that is one of the most robust methods under the Bayesian theorem frame in non-linear signal processing so far, is applied to perform simultaneously noise suppression and phase unwrapping of interferometric fringes for the first time, which can simplify the complexity and the difficulty of pre-filtering procedure followed by phase unwrapping procedure, and even can remove the pre-filtering procedure. The robust phase gradient estimator is used to efficiently and accurately obtain phase gradient information from interferometric fringes, which is needed for the iterated unscented Kalman filtering phase unwrapping model. The efficient quality-guided strategy is able to ensure that the proposed method fast unwraps wrapped pixels along the path from the high-quality area to the low-quality area of wrapped phase images, which can greatly improve the efficiency of phase unwrapping. Results obtained from synthetic data and real data show that the proposed method can obtain better solutions with an acceptable time consumption, with respect to some of the most used algorithms.

Grating-based x-ray differential phase contrast imaging with twin peaks in phase-stepping curves-phase retrieval and dewrapping

PURPOSE

X-ray differential phase contrast CT implemented with Talbot interferometry employs phase-stepping to extract information of x-ray attenuation, phase shift, and small-angle scattering. Since inaccuracy may exist in the absorption grating G2 due to an imperfect fabrication, the effective period of G2 can be as large as twice the nominal period, leading to a phenomenon of twin peaks that differ remarkably in their heights. In this work, the authors investigate how to retrieve and dewrap the phase signal from the phase-stepping curve (PSC) with the feature of twin peaks for x-ray phase contrast imaging.

METHODS

Based on the paraxial Fresnel-Kirchhoff theory, the analytical formulae to characterize the phenomenon of twin peaks in the PSC are derived. Then an approach to dewrap the retrieved phase signal by jointly using the phases of the first- and second-order Fourier components is proposed. Through an experimental investigation using a prototype x-ray phase contrast imaging system implemented with Talbot interferometry, the authors evaluate and verify the derived analytic formulae and the proposed approach for phase retrieval and dewrapping.

RESULTS

According to theoretical analysis, the twin-peak phenomenon in PSC is a consequence of combined effects, including the inaccuracy in absorption grating G2, mismatch between phase grating and x-ray source spectrum, and finite size of x-ray tube's focal spot. The proposed approach is experimentally evaluated by scanning a phantom consisting of organic materials and a lab mouse. The preliminary data show that compared to scanning G2 over only one single nominal period and correcting the measured phase signal with an intuitive phase dewrapping method that is being used in the field, stepping G2 over twice its nominal period and dewrapping the measured phase signal with the proposed approach can significantly improve the quality of x-ray differential phase contrast imaging in both radiograph and CT.

CONCLUSIONS

Using the phase retrieval and dewrapping methods proposed to deal with the phenomenon of twin peaks in PSCs and phase wrapping, the performance of grating-based x-ray differential phase contrast radiography and CT can be significantly improved.

Convex optimization-based windowed Fourier filtering with multiple windows for wrapped-phase denoising

The windowed Fourier filtering (WFF), defined as a thresholding operation in the windowed Fourier transform (WFT) domain, is a successful method for denoising a phase map and analyzing a fringe pattern. However, it has some shortcomings, such as extremely high redundancy, which results in high computational cost, and difficulty in selecting an appropriate window size. In this paper, an extension of WFF for denoising a wrapped-phase map is proposed. It is formulated as a convex optimization problem using Gabor frames instead of WFT. Two Gabor frames with differently sized windows are used simultaneously so that the above-mentioned issues are resolved. In addition, a differential operator is combined with a Gabor frame in order to preserve discontinuity of the underlying phase map better. Some numerical experiments demonstrate that the proposed method is able to reconstruct a wrapped-phase map, even for a severely contaminated situation.

Two-dimensional phase unwrapping using the transport of intensity equation

We report a method for two-dimensional phase unwrapping based on the transport of intensity equation (TIE). Given a wrapped phase profile, we generate an auxiliary complex field and propagate it to small distances to simulate two intensity images on closely spaced planes. Using the longitudinal intensity derivative of the auxiliary field as an input, the TIE is solved by employing the regularized Fourier-transform-based approach. The resultant phase profile is automatically in the unwrapped form, as it has been obtained as a solution of a partial differential equation rather than as an argument of a complex-valued function. Our simulations and experimental results suggest that this approach is fast and accurate and provides a simple and practical solution for routine phase unwrapping tasks in interferometry and digital holography.

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.

Efficient and robust phase unwrapping algorithm based on unscented Kalman filter, the strategy of quantizing paths-guided map, and pixel classification strategy

This paper presents an efficient and robust phase unwrapping algorithm which combines an unscented Kalman filter (UKF) with a strategy of quantizing a paths-guided map and a pixel classification strategy based on phase quality information. The advantages of the proposed method depend on the following contributions: (1) the strategy of quantizing the paths-guided map can accelerate the process of searching unwrapping paths and greatly reducing time consumption on the unwrapping procedure; (2) the pixel classification strategy proposed by this paper can reduce the error propagation effect by decreasing the amounts of pixels with equal quantized paths-guided value in the process of unwrapping; and (3) the unscented Kalman filter enables simultaneous filtering and unwrapping without the information loss caused by linearization of a nonlinear model. In addition, a new paths-guided map derived from a phase quality map is inserted into the strategy of quantizing the paths-guided map to provide a more robust path of unwrapping, and then ensures better unwrapping results. Results obtained from synthetic data and real data show that the proposed method can efficiently obtain better solutions with respect to some of the most used algorithms.

Quantitative phase microscopy: automated background leveling techniques and smart temporal phase unwrapping

In order for time-dynamic quantitative phase microscopy to yield meaningful data to scientists, raw phase measurements must be converted to sequential time series that are consistently phase unwrapped with minimal residual background shape. Beyond the initial phase unwrapping, additional steps must be taken to convert the phase to time-meaningful data sequences. This consists of two major operations both outlined in this paper and shown to operate robustly on biological datasets. An automated background leveling procedure is introduced that consistently removes background shape and minimizes mean background phase value fluctuations. By creating a background phase value that is stable over time, the phase values of features of interest can be examined as a function of time to draw biologically meaningful conclusions. Residual differences between sequential frames of data can be present due to inconsistent phase unwrapping, causing localized regions to have phase values at similar object locations inconsistently changed by large values between frames, not corresponding to physical changes in the sample being observed. This is overcome by introducing a new method, referred to as smart temporal unwrapping that temporally unwraps and filters the phase data such that small motion between frames is accounted for and phase data are unwrapped consistently between frames. The combination of these methods results in the creation of phase data that is stable over time by minimizing errors introduced within the processing of the raw data.

Isotropic inverse-problem approach for two-dimensional phase unwrapping

We propose a new technique for two-dimensional phase unwrapping. The unwrapped phase is found as the solution of an inverse problem that consists in the minimization of an energy functional. The latter includes a weighted data fidelity term that favors sparsity in the error between the true and wrapped phase differences, as well as a regularizer based on higher-order total variation. One desirable feature of our method is its rotation invariance, which allows it to unwrap a much larger class of images compared to the state of the art. We demonstrate the effectiveness of our method through several experiments on simulated and real data obtained through the tomographic phase microscope. The proposed method can enhance the applicability and outreach of techniques that rely on quantitative phase evaluation.

Large step-phase measurement by a reduced-phase triple-illumination interferometer

We present a reduced-phase triple-illumination interferometer (RPTII) as a novel single-shot technique to increase the precision of dual-illumination optical phase unwrapping techniques. The technique employs two measurement ranges to record both low-precision unwrapped and high-precision wrapped phases. To unwrap the high-precision phase, a hierarchical optical phase unwrapping algorithm is used with the low-precision unwrapped phase. The feasibility of this technique is demonstrated by measuring a stepped object with a height 2100 times greater than the wavelength of the source. The phase is reconstructed without applying any numerical unwrapping algorithms, and its noise level is decreased by a factor of ten.

Phase estimation from digital holograms without unwrapping

Digital holography is a convenient method for determining the phase induced by transparent objects. When the phase change is higher than 2π, an unwrapping algorithm is needed to provide a useful phase map. In the presence of noise, this process is not trivial and not fully resolved. In this paper a procedure is proposed to circumvent the need for unwrapping by estimating the phase from its gradient, which is directly computed from the reconstructed field. Application of the method to digital holograms of microscopic samples is demonstrated.

Enhanced phase unwrapping algorithm based on unscented Kalman filter, enhanced phase gradient estimator, and path-following strategy

This paper presents an enhanced phase unwrapping algorithm by combining an unscented Kalman filter, an enhanced local phase gradient estimator based on an amended matrix pencil model, and a path-following strategy. This technology is able to accurately unwrap seriously noisy wrapped phase images by applying the unscented Kalman filter to simultaneously perform noise suppression and phase unwrapping along the path from the high-quality region to the low-quality region of the wrapped phase images. Results obtained with synthetic data and real data validate the effectiveness of the proposed method and show improved performance of this new algorithm with respect to some of the most used algorithms.

Enhancement strategy based on three-layer filtering for a single fringe pattern

The quality enhancement of a single fringe pattern is a challenging task in speckle interferometry. Fringe patterns suffer greatly from three adverse variations (nonuniform background, speckle noise, and intensity modulation). In this Letter, we propose a three-layer filtering strategy for noisy fringe patterns. The first layer is aimed at high-frequency speckle noises and low-frequency background. The second layer is for remaining speckle noises distributed in the middle-frequency band. The third layer will further implement quality enhancement by a phase-recovery technique. The proposed strategy is quantitatively evaluated by different indexes and verified to be effective through numerous comparative experiments.

Quantitative phase maps denoising of long holographic sequences by using SPADEDH algorithm

We propose a denoising method for digital holography mod 2π wrapped phase map by using an adaptation of the SPArsity DEnoising of Digital Holograms (SPADEDH) algorithm. SPADEDH is a l(1) minimization algorithm able to suppress the noise components on digital holograms without any prior knowledge or estimation about the statistics of noise. We test our algorithm with either general numerical simulated wrapped phase, quantifying the performance with different efficiency parameters and comparing it with two popular denoising strategies, i.e., median and Gaussian filters, and specific experimental tests, by focusing our attention on long-sequence wrapped quantitative phase maps (QPMs) of in vitro cells, which aim to have uncorrupted QPMs. In addition, we prove that the proposed algorithm can be used as a helper for the typical local phase unwrapping algorithms.