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Two Dimensional Phase Unwrapping Theory Algorithms And Software

Phase unwrapping is an ill-posed classical problem in many practical applications of significance such as 3D profiling through fringe projection, synthetic aperture radar and magnetic resonance imaging. Conventional phase unwrapping techniques estimate the phase either by integrating through the confined path (referred to as path-following methods) or by minimizing the energy function between the wrapped phase and the approximated true phase (referred to as minimum-norm approaches). However, these conventional methods have some critical challenges like error accumulation and high computational time and often fail under low SNR conditions. To address these problems, this paper proposes a novel deep learning framework for unwrapping the phase and is referred to as “PhaseNet 2.0”. The phase unwrapping problem is formulated as a dense classification problem and a fully convolutional DenseNet based neural network is trained to predict the wrap-count at each pixel from the wrapped phase maps. To train this network, we simulate arbitrary shapes and propose new loss function that integrates the residues by minimizing the difference of gradients and also uses $L_{1}$ loss to overcome class imbalance problem. The proposed method, unlike our previous approach PhaseNet, does not require post-processing, highly robust to noise, accurately unwraps the phase even at the severe noise level of −5 dB, and can unwrap the phase maps even at relatively high dynamic ranges. Simulation results from the proposed framework are compared with different classes of existing phase unwrapping methods for varying SNR values and discontinuity, and these evaluations demonstrate the advantages of the proposed framework. We also demonstrate the generality of the proposed method on 3D reconstruction of synthetic CAD models that have diverse structures and finer geometric variations. Finally, the proposed method is applied to real-data for 3D profiling of objects using fringe projection technique and digital holographic interferometry. The proposed framework achieves significant improvements over existing methods while being highly efficient with interactive frame-rates on modern GPUs.

PhaseNet 2.0: Phase Unwrapping of Noisy Data Based on Deep Learning Approach

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.

Iterated unscented Kalman filter for phase unwrapping of interferometric fringes.

Phase unwrapping is a crucial process to obtain the absolute phase profile in many optical phase measurement techniques such as interferometry, holography, profilometry, etc. In this paper, we have studied several phase unwrapping algorithms based on solving the discrete Poisson equation. The differences among those algorithms lie in two aspects: one is the way to calculate the input for the Poisson equation using the wrapped phase data and the other is the way to compute the output (unwrapped phase data) using the corresponding input. Firstly, the method to compute the input for the Poisson equation was investigated using the finite difference (FD) and fast Fourier transform(FFT) method. Then different methods, based on FFT or discrete cosine transform (DCT), were employed to calculate the unwrapped phase, and their performances were compared in terms of accuracy and efficiency. To enhance the precision of those algorithms, an iteration strategy was introduced and its performance was investigated under different noise conditions. Finally, several real phase data were tested by using the direct and iterative methods. The detailed software package can be found in https://www.mathworks.com/matlabcentral/fileexchange/71810-phase- unwrapping-algorithms-by-solving-the-poisson-equation.

Comparative study of phase unwrapping algorithms based on solving the Poisson equation

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.

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

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.

Particle-filter-based phase estimation in digital holographic interferometry

Phase information recovered through interferometric techniques is mathematically wrapped in the interval (−π, π). Obtaining the original unwrapped phase is very important in numerous number of applications. This paper discusses a Fourier transform based phase unwrapping method. Kalman filter is proposed for denoising in post processing step to restore the unwrapped phase without any noise. The proposed method is highly robust to noise and performs better even at lower SNR values (5–10dB) with a very less value of RMS error. Also, the time taken for execution is very less compared to the many available methods in the literature.

P. Ram Sukumar

Papers

Particle-filter-based phase estimation in digital holographic interferometry

Phase unwrapping with Kalman filter based denoising in digital holographic interferometry