Demodulation of a single complex fringe interferogram with a path-independent regularized phase-tracking technique

One step accurate phase demodulation from a closed fringe pattern with the convolutional neural network HRUnet

Retrieving a phase map from a single closed fringe pattern is a challenging task in optical interferometry. In this paper, a convolutional neural network (CNN), HRUnet, is proposed to demodulate phase from a closed fringe pattern. The HRUnet, derived from the Unet model, adopts a high resolution network (HRnet) module to extract high resolution feature maps of the data and employs residual blocks to erase the gradient vanishing in the network. With the trained network, the unwrapped phase map can be directly obtained by feeding a scaled fringe pattern. The high accuracy of the phase map obtained from HRUnet is demonstrated by demodulation of both simulated data and actual fringe patterns. Compared results between HRUnet and two other CNNS are also provided, and the results proved that the performance of HRUnet in accuracy is superior to the two other counterparts.

Fringe pattern demodulation using Zernike polynomials and a l1-norm regularized extended Kalman filter

A novel algorithm for closed fringe demodulation for an absolute phase estimation, to the best of our knowledge, is proposed. The two-dimensional phase is represented as a weighted linear combination of a certain number of Zernike polynomials (ZPs). Essentially, the problem of phase estimation is converted into the estimation of ZP coefficients. The task of ZP coefficient estimation is performed based on a state space model. Due to the nonlinear dependence of the fringe intensity measurement model on the ZP coefficients, the extended Kalman filter (EKF) is used for the state estimation. A pseudo-measurement model is considered based on the state vector sparsity constraint to improve the convergence performance of the EKF. Simulation and experimental results are provided to demonstrate the noise robustness and the practical applicability of the proposed method.

Variable mesh optimization applied to fringe pattern demodulation using a Bézier surface

We present a parametric method to carry out a demodulation process in complex fringe pattern images with either open or closed fringes; this method is based on the parallel demodulation algorithm and introduces a novel way, to the best of our knowledge, to approximate the phase map using the Bezier surface control points. For this study, a recently developed population meta-heuristic called Variable Mesh Optimization is introduced to implement the optimization process. The results of the proposed method improve the phase error and the run time scores in comparison with other global optimization algorithms, which address this type of problem.

Phase unwrapping based on adaptive image in-painting of fringe patterns in measuring gear tooth flanks by laser interferometry

Phase unwrapping in regions of abnormal fringes remains an unresolved issue. In this paper, we present an approach that combines an image-inpainting strategy based on an adaptive window which is obtained according to the density and orientation of fringe patterns and a quality-guided algorithm for phase unwrapping. First, a threshold is set to a quality map to detect the target region. Second, the target region is filled with new phase values by the adaptive image-inpainting method. Then, a quality-guided phase unwrapping algorithm is applied to this newly generated wrapped phase map. Finally, postprocessing of the unwrapped result is performed. The method is validated through several simulation and experiments. The results demonstrate that the proposed algorithm is effective in the presence of abnormal fringes.

Automatic fringe pattern enhancement using truly adaptive period-guided bidimensional empirical mode decomposition

Fringe patterns encode the information about the result of a measurement performed via widely used optical full-field testing methods, e.g., interferometry, digital holographic microscopy, moiré techniques, structured illumination etc. Affected by the optical setup, changing environment and the sample itself fringe patterns are often corrupted with substantial noise, strong and uneven background illumination and exhibit low contrast. Fringe pattern enhancement, i.e., noise minimization and background term removal, at the pre-processing stage prior to the phase map calculation (for the measurement result decoding) is therefore essential to minimize the jeopardizing effect the mentioned error sources have on the optical measurement outcome. In this contribution we propose an automatic, robust and highly effective fringe pattern enhancement method based on the novel period-guided bidimensional empirical mode decomposition algorithm (PG-BEMD). The spatial distribution of the fringe period is estimated using the novel windowed approach and then serves as an indicator for the truly adaptive decomposition with the filter size locally adjusted to the fringe pattern density. In this way the fringe term is successfully extracted in a single (first) decomposition component alleviating the cumbersome mode mixing phenomenon and greatly simplifying the automatic signal reconstruction. Hence, the fringe term is dissected without the need for modes selection nor summation. The noise removal robustness is ensured employing the block matching 3D filtering of the fringe pattern prior to its decomposition. Performance validation against previously reported modified empirical mode decomposition techniques is provided using numerical simulations and experimental data verifying the versatility and effectiveness of the proposed approach.

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.

Direct phase unwrapping method based on a local third-order polynomial fit

When recovering smooth phases by phase unwrapping algorithms, many noniterative algorithms are available. However, normally those algorithms offer approximations of the current phase that cannot be accurate enough. This is because the majority of them are based on global approaches instead of local ones. Although smooth estimations are not often expected in phase reconstructions for real applications, a smooth initial guess could be useful for robust iterative techniques. Therefore, based on the most recent local polynomial approaches, we propose a simple least-squares fitting of the partial derivatives of the phase, normally estimated from the wrapped operator, by considering local polynomial models of the phase up to the third order. Synthetic and real data of wrapped phases are considered in our work.

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.

Phase retrieval in two-shot phase-shifting interferometry based on phase shift estimation in a local mask

Fringe analysis in two-shot phase-shifting interferometry is important but meets challenges due to a limited number of images, corrupting noise, and background modulation. Here we propose an effective algorithm for phase retrieval from two interferograms with unknown phase shifts. The algorithm first evaluates the phase shift in a local mask through phase fitting and global optimization and then computes a full-field phase map using an arctangent function. Since the phase shift evaluation is performed within a local mask, the algorithm is fast compared with conventional optimization-based algorithms and typically needs tens of seconds to complete the processing. Computer simulation and experimental results show that the proposed algorithm has excellent performance compared with state-of-the-art algorithms. A complete software package of the algorithm in MATLAB is available at http://two-shot.sourceforge.io/.

Simplified paraboloid phase model-based phase tracker for demodulation of a single complex fringe

A regularized phase tracker (RPT) is an effective method for demodulation of single closed-fringe patterns. However, lengthy calculation time, specially designed scanning strategy, and sign-ambiguity problems caused by noise and saddle points reduce its effectiveness, especially for demodulating large and complex fringe patterns. In this paper, a simplified paraboloid phase model-based regularized phase tracker (SPRPT) is proposed. In SPRPT, first and second phase derivatives are pre-determined by the density-direction-combined method and discrete higher-order demodulation algorithm, respectively. Hence, cost function is effectively simplified to reduce the computation time significantly. Moreover, pre-determined phase derivatives improve the robustness of the demodulation of closed, complex fringe patterns. Thus, no specifically designed scanning strategy is needed; nevertheless, it is robust against the sign-ambiguity problem. The paraboloid phase model also assures better accuracy and robustness against noise. Both the simulated and experimental fringe patterns (obtained using electronic speckle pattern interferometry) are used to validate the proposed method, and a comparison of the proposed method with existing RPT methods is carried out. The simulation results show that the proposed method has achieved the highest accuracy with less computational time. The experimental result proves the robustness and the accuracy of the proposed method for demodulation of noisy fringe patterns and its feasibility for static and dynamic applications.

Closed fringe demodulation using phase decomposition by Fourier basis functions

We report a new technique for the demodulation of a closed fringe pattern by representing the phase as a weighted linear combination of a certain number of linearly independent Fourier basis functions in a given row/column at a time. A state space model is developed with the weights of the basis functions as the elements of the state vector. The iterative extended Kalman filter is effectively utilized for the robust estimation of the weights. A coarse estimate of the fringe density based on the fringe frequency map is used to determine the initial row/column to start with and subsequently the optimal number of basis functions. The performance of the proposed method is evaluated with several noisy fringe patterns. Experimental results are also reported to support the practical applicability of the proposed method.

Demodulation of two-shot fringe patterns with random phase shifts by use of orthogonal polynomials and global optimization

We propose a simple and robust phase demodulation algorithm for two-shot fringe patterns with random phase shifts. Based on a smoothness assumption, the phase to be recovered is decomposed into a linear combination of finite terms of orthogonal polynomials, and the expansion coefficients and the phase shift are exhaustively searched through global optimization. The technique is insensitive to noise or defects, and is capable of retrieving phase from low fringe-number (less than one) or low-frequency interferograms. It can also cope with interferograms with very small phase shifts. The retrieved phase is continuous and no further phase unwrapping process is required. The method is expected to be promising to process interferograms with regular fringes, which are common in optical shop testing. Computer simulation and experimental results are presented to demonstrate the performance of the algorithm.

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.

Continuous phase estimation from noisy fringe patterns based on the implicit smoothing splines

We introduce the algorithm for the direct phase estimation from the single noisy interferometric pattern. The method, named implicit smoothing spline (ISS), can be regarded as a formal generalization of the smoothing spline interpolation for the case when the interpolated data is given implicitly. We derive the necessary equations, discuss the properties of the method and address its application for the direct estimation of the continuous phase in both classical interferometry and digital speckle pattern interferometry (DSPI). The numerical illustrations of the algorithm performance are provided to corroborate the high quality of the results.

Improved generalized regularized phase tracker for demodulation of a single fringe pattern

A generalized regularized phase tracker (GRPT) for demodulation of a single fringe pattern was recently proposed. It is very successful for many fringe patterns. However, the GRPT has poor performance in the area where the fringe pattern is sparse. An improved GRPT (iGRPT) with two novel improvements is proposed to overcome the problem. First, the fixed window used in the GRPT is replaced by a spatially adaptive window. Second, a background regularization term and a modulation regularization term are incorporated in the cost function. With these two improvements, the proposed iGRPT can successfully demodulate sparse fringes and thus improves the demodulation capability of the GRPT. Simulation and experimental results are presented to verify the performance of the iGRPT.

Adaptive enhancement of optical fringe patterns by selective reconstruction using FABEMD algorithm and Hilbert spiral transform

Presented method for fringe pattern enhancement has been designed for processing and analyzing low quality fringe patterns. It uses a modified fast and adaptive bidimensional empirical mode decomposition (FABEMD) for the extraction of bidimensional intrinsic mode functions (BIMFs) from an interferogram. Fringe pattern is then selectively reconstructed (SR) taking the regions of selected BIMFs with high modulation values only. Amplitude demodulation and normalization of the reconstructed image is conducted using the spiral phase Hilbert transform (HS). It has been tested using computer generated interferograms and real data. The performance of the presented SR-FABEMD-HS method is compared with other normalization techniques. Its superiority, potential and robustness to high fringe density variations and the presence of noise, modulation and background illumination defects in analyzed fringe patterns has been corroborated.

A generalized regularized phase tracker for demodulation of a single fringe pattern

The regularized phase tracker (RPT) is one of the most powerful approaches for demodulation of a single fringe pattern. However, two disadvantages limit the applications of the RPT in practice. One is the necessity of a normalized fringe pattern as input and the other is the sensitivity to critical points. To overcome these two disadvantages, a generalized regularized phase tracker (GRPT) is presented. The GRPT is characterized by two novel improvements. First, a general local fringe model that includes a linear background, a linear modulation and a quadratic phase is adopted in the proposed enhanced cost function. Second, the number of iterations in the optimization process is proposed as a comprehensive measure of fringe quality and used to guide the demodulation path. With these two improvements, the GRPT can directly demodulate a single fringe pattern without any pre-processing and post-processing and successfully get rid of the problem of the sensitivity to critical points. Simulation and experimental results are presented to demonstrate the effectiveness and robustness of the GRPT.

Tomographic reconstruction of three-dimensional refractive index fields by use of a regularized phase-tracking technique and a polynomial approximation method

We present a complete data-processing procedure for quantitative reconstruction of three-dimensional (3D) refractive index fields from limited multidirectional interferometric data. The proposed procedure includes two parts: (1) extraction of the projection data from limited multidirectional interferograms by a regularized phase-tracking technique and wavefront fitting and (2) reconstruction of the 3D refractive index fields by a modified polynomial approximation method. It has been shown that the procedure gives a satisfactory solution to the reconstruction issue in interferometric tomography, from the initial projection data extraction to the final image reconstruction. Computer simulation and experimental results are both presented.

Nonnull interferometer simulation for aspheric testing based on ray tracing

The nonnull interferometric method that employs a partial compensation system to compensate for the longitude aberration of the aspheric under test and a reverse optimization procedure to correct retrace errors is a useful technique for general aspheric testing. However, accurate system modeling and simulation are required to correct retrace errors and reconstruct fabrication error of the aspheric. Here, we propose a ray-tracing-based method to simulate the nonnull interferometer, which calculates the optical path difference by tracing rays through the reference path and the test path. To model a nonrotationally symmetric fabrication error, we mathematically represent it with a set of Zernike polynomials (i.e., Zernike deformation) and derive ray-tracing formulas for the deformed surface, which can also deal with misalignment situations (i.e., a surface with tilts and/or decenters) and thus facilitates system modeling extremely. Simulation results of systems with (relatively) large and small Zernike deformations and their comparisons with the lens design program Zemax have demonstrated the correctness and effectiveness of the method.

Regularized frequency-stabilizing method for single closed-fringe interferogram demodulation

We present a simple and fast regularized frequency-stabilizing method for single open- or closed-fringe interferogram demodulation. The proposed method recovers the phase maps of interferograms by establishing a cost function, according to prior knowledge. Because only the phase field to be estimated is employed in the cost function, the optimization process is fast. Moreover, the recovered phase is continuous, and no further phase unwrapping is necessary. Computer simulation and experimental results have demonstrated both the rapidity and the efficiency of the method.