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

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.

Deep learning phase-unwrapping method based on adaptive noise evaluation

To address the problem of phase unwrapping for interferograms, a deep learning (DL) phase-unwrapping method based on adaptive noise evaluation is proposed to retrieve the unwrapped phase from the wrapped phase. First, this method uses a UNet3+ as the skeleton and combines with a residual neural network to build a network model suitable for unwrapping wrapped fringe patterns. Second, an adaptive noise level evaluation system for interferograms is designed to estimate the noise level of the interferograms by integrating phase quality maps and phase residues of the interferograms. Then, multiple training datasets with different noise levels are used to train the DL network to achieve the trained networks suitable for unwrapping interferograms with different noise levels. Finally, the interferograms are unwrapped by the trained networks with the same noise levels as the interferograms to be unwrapped. The results with simulated and experimental interferograms demonstrate that the proposed networks can obtain the popular unwrapped phase from the wrapped phase with different noise levels and show good robustness in the experiments of phase unwrapping for different types of fringe patterns.

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.

Adaptive Square-Root Unscented Kalman Filter Phase Unwrapping with Modified Phase Gradient Estimation

Phase unwrapping (PU) is a key program in data processing in the interferometric synthetic aperture radar (InSAR) technique, and its accuracy directly affects the quality of final SAR data products. However, PU in regions with large gradient changes and high noise has always been a difficult problem. To overcome the limitation, this article proposes an adaptive square-root unscented Kalman filter PU method. Specifically, a modified phase gradient estimation (PGE) algorithm is proposed, in which a Butterworth low-pass filter is embedded, and the PGE window can be adaptively adjusted according to phase root-mean-square errors of pixels. Furthermore, the outliers of the PGE results are detected and revised to obtain high-precision vertical and horizontal phase gradients. Finally, the unwrapped phase is calculated by the adaptive square-root unscented Kalman filter method. To the best of our knowledge, this article is the first to combine the modified PGE with an adaptive square-root unscented Kalman filter for PU. Two sets of simulated data and a set of TerraSAR-X/TanDEM-X real data were used for experimental verification. The experimental results demonstrated that the various improvement measures proposed in this article were effective. Additionally, compared with the minimum-cost flow algorithm (MCF), statistical-cost network-flow algorithm (SNAPHU) and unscented Kalman filter PU (UKFPU), the proposed method had better accuracy and model robustness.

Phase unwrapping algorithm based on a rank information filter

A robust phase unwrapping algorithm based on a rank information filter is proposed to retrieve the unambiguous unwrapped phase from noisy wrapped phase images. First, a recursive phase unwrapping program, based on a rank information filter, is proposed to transform the problem of phase unwrapping for wrapped phase into the problem of the state estimation for state variables under the framework of a rank information filter, where a local phase gradient estimator based on the amended matrix pencil model (AMPM) is used to obtain phase gradient information required by the recursive phase unwrapping program. Second, an efficient path-following strategy based on heap-sort is used to guide the phase unwrapping path, which ensures that the recursive phase unwrapping program based on a rank information filter unwraps wrapped phase images along the path from high-quality pixels to low-quality pixels. Finally, the results obtained from synthetic data and experimental measured data demonstrate the effectiveness of the proposed method and show this method can obtain robust solutions from noisy wrapped phase images.

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.

Adaptive Unscented Kalman Filter Phase Unwrapping Method and Its Application on Gaofen-3 Interferometric SAR Data

Phase unwrapping (PU) is a key step in the reconstruction of digital elevation models (DEMs) and the monitoring of surface deformation from interferometric synthetic aperture radar (SAR, InSAR) data. In this paper, an improved PU method that combines an amended matrix pencil model, an adaptive unscented kalman filter (AUKF), an efficient quality-guided strategy based on heapsort, and a circular median filter is proposed. PU theory and the existing UKFPU method are covered. Then, the improved method is presented with emphasis on the AUKF and the circular median filter. AUKF has been well used in other fields, but it is for the first time applied to interferometric images PU, to the best of our knowledge. First, the amended matrix pencil model is used to estimate the phase gradient. Then, an AUKF model is used to unwrap the interferometric phase based on an efficient quality-guided strategy based on heapsort. Finally, the key results are obtained by filtering the results using a circular median. The proposed method is compared with the minimum cost network flow (MCF), statistical cost network flow (SNAPHU), regularized phase tracking technique (RPTPU), and UKFPU methods using two sets of simulated data and two sets of experimental GF-3 SAR data. The improved method is shown to yield the greatest accuracy in the interferometric phase maps compared to the methods considered in this paper. Furthermore, the improved method is shown to be the most robust to noise and is thus most suitable for PU of GF-3 SAR data in high-noise and low-coherence regions.

Phase unwrapping algorithm using polynomial phase approximation and linear Kalman filter

A noise-robust phase unwrapping algorithm is proposed based on state space analysis and polynomial phase approximation using wrapped phase measurement. The true phase is approximated as a two-dimensional first order polynomial function within a small sized window around each pixel. The estimates of polynomial coefficients provide the measurement of phase and local fringe frequencies. A state space representation of spatial phase evolution and the wrapped phase measurement is considered with the state vector consisting of polynomial coefficients as its elements. Instead of using the traditional nonlinear Kalman filter for the purpose of state estimation, we propose to use the linear Kalman filter operating directly with the wrapped phase measurement. The adaptive window width is selected at each pixel based on the local fringe density to strike a balance between the computation time and the noise robustness. In order to retrieve the unwrapped phase, either a line-scanning approach or a quality guided strategy of pixel selection is used depending on the underlying continuous or discontinuous phase distribution, respectively. Simulation and experimental results are provided to demonstrate the applicability of the proposed 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.

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.

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.

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.

Particle-filter-based phase estimation in digital holographic interferometry

In this paper, we propose a particle-filter-based technique for the analysis of a reconstructed interference field. The particle filter and its variants are well proven as tracking filters in non-Gaussian and nonlinear situations. We propose to apply the particle filter for direct estimation of phase and its derivatives from digital holographic interferometric fringes via a signal-tracking approach on a Taylor series expanded state model and a polar-to-Cartesian-conversion-based measurement model. Computation of sample weights through non-Gaussian likelihood forms the major contribution of the proposed particle-filter-based approach compared to the existing unscented-Kalman-filter-based approach. It is observed that the proposed approach is highly robust to noise and outperforms the state-of-the-art especially at very low signal-to-noise ratios (i.e., especially in the range of -5 to 20 dB). The proposed approach, to the best of our knowledge, is the only method available for phase estimation from severely noisy fringe patterns even when the underlying phase pattern is rapidly varying and has a larger dynamic range. Simulation results and experimental data demonstrate the fact that the proposed approach is a better choice for direct phase estimation.

Practical phase unwrapping of interferometric fringes based on unscented Kalman filter technique

A phase unwrapping algorithm for interferometric fringes based on the unscented Kalman filter (UKF) technique is proposed. The algorithm can bring about accurate phase unwrapping and good noise suppression simultaneously by incorporating the true phase and its derivative in the state vector estimation through the UKF process. Simulations indicate that the proposed algorithm has better accuracy than some widely employed phase unwrapping approaches in the same noise condition. Also, the time consumption of the algorithm is reasonably acceptable. Applications of the algorithm in our different optical interferometer systems are provided to demonstrate its practicability with good performance. We hope this algorithm can be a practical approach that can help to reduce the systematic errors significantly induced by phase unwrapping process for interferometric measurements such as wavefront distortion testing, surface figure testing of optics, etc.

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.