Fast phase recovery from a single closed-fringe pattern

DeepOrientation: convolutional neural network for fringe pattern orientation map estimation

Fringe pattern based measurement techniques are the state-of-the-art in full-field optical metrology. They are crucial both in macroscale, e.g., fringe projection profilometry, and microscale, e.g., label-free quantitative phase microscopy. Accurate estimation of the local fringe orientation map can significantly facilitate the measurement process in various ways, e.g., fringe filtering (denoising), fringe pattern boundary padding, fringe skeletoning (contouring/following/tracking), local fringe spatial frequency (fringe period) estimation, and fringe pattern phase demodulation. Considering all of that, the accurate, robust, and preferably automatic estimation of local fringe orientation map is of high importance. In this paper we propose a novel numerical solution for local fringe orientation map estimation based on convolutional neural network and deep learning called DeepOrientation. Numerical simulations and experimental results corroborate the effectiveness of the proposed DeepOrientation comparing it with a representative of the classical approach to orientation estimation called combined plane fitting/gradient method. The example proving the effectiveness of DeepOrientation in fringe pattern analysis, which we present in this paper, is the application of DeepOrientation for guiding the phase demodulation process in Hilbert spiral transform. In particular, living HeLa cells quantitative phase imaging outcomes verify the method as an important asset in label-free microscopy.

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 method based on local polynomial models and a maximum a posteriori model correction

Recently, a theory on local polynomial approximations for phase-unwrapping algorithms, considering a state space analysis, has been proposed in Appl. Opt.56, 29 (2017)APOPAI0003-693510.1364/AO.56.000029. Although this work is a suitable methodology to deal with relatively low signal to noise ratios observed in the wrapped phase, the methodology has been developed only for local-polynomial phase models of order 1. The resultant proposal is an interesting Kalman filter approach for estimating the coefficient or state vectors of these local plane models. Thus, motivated by this approach and simple Bayesian theory, and considering our previous research on local polynomial models up to the third order [Appl. Opt.58, 436 (2019)APOPAI0003-693510.1364/AO.58.000436], we propose an equivalent methodology based on a simple maximum a posteriori estimation, but considering a different state space: difference vectors of coefficients for the current high-order polynomial models. Specific estimations of the covariance matrices for difference vectors, as well as noise covariance matrices involved with the correct estimation of coefficient vectors, are proposed and reconstructions with synthetic and real data are provided.

Impact of background and modulation on parameter estimation using fractional Fourier transform and its solutions

Analysis of closed-fringe patterns with quadratic phase that are often encountered plays an important role in optical interferometry. But because the frequency spectra of the two exponential signals that compose closed-fringe patterns overlap in the Fourier domain while one is clearly distinct from the other in the fractional Fourier domain, fractional Fourier transform (FRFT) is a useful method for analyzing the images to provide parameter estimation. However, when the fringe pattern has varying background and/or modulation due to non-uniform illumination, parameter estimation accuracy based on FRFT is affected, which lacks theoretical justification. Thus, the impact of varying background and/or modulation on the FRFT is studied with theoretic analysis and presented in this paper. Key factors that contribute to the optimal results are discussed when employing three kinds of fringe normalization methods to eliminate the impact. Here, the fringe pattern is first processed by the normalization technique. Then the cosine-only term is used to estimate parameter by use of the FRFT-based method. Physical quantities are then obtained by parameter estimation. In comparison with our previous method based on FRFT, more accurate results are achieved. The feasibility and applicability of the proposed approach are demonstrated using simulation and experimental results.

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.

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.

Single shot fringe pattern phase demodulation using Hilbert-Huang transform aided by the principal component analysis

Hybrid single shot algorithm for accurate phase demodulation of complex fringe patterns is proposed. It employs empirical mode decomposition based adaptive fringe pattern enhancement (i.e., denoising, background removal and amplitude normalization) and subsequent boosted phase demodulation using 2D Hilbert spiral transform aided by the Principal Component Analysis method for novel, correct and accurate local fringe direction map calculation. Robustness to fringe pattern significant noise, uneven background and amplitude modulation as well as local fringe period and shape variations is corroborated by numerical simulations and experiments. Proposed automatic, adaptive, fast and comprehensive fringe analysis solution compares favorably with other previously reported techniques.

Analyzing closed-fringe images using two-dimensional Fan wavelets

In this paper, it will be shown how the use of two 2D Fan wavelets to analyze closed-fringe images can lead to a relatively fast and exceptionally noise-resistant algorithm capable of extracting not only local phase but also local frequency information. Our algorithm is up to 10 times faster than the current state-of-the-art in wavelet processing techniques and even up to 30 times faster than "windowed Fourier" transform programs, which achieve similar noise-resiliency figures. This improvement is mainly achieved by the use of Fan wavelets instead of Morlet wavelets, but a more efficient scale-space discretization strategy is also described, and three different alternatives are suggested capable of solving the phase sign-ambiguity problem in a quick and efficient manner. Finally, the application of the algorithm to real and numerically generated images shows that a precision of 1/30th of a fringe is achievable for noise levels going up to 1/5th of the input contrast.

Interferometric dynamic measurement: techniques based on high-speed imaging or a single photodetector

In recent years, optical interferometry-based techniques have been widely used to perform noncontact measurement of dynamic deformation in different industrial areas. In these applications, various physical quantities need to be measured in any instant and the Nyquist sampling theorem has to be satisfied along the time axis on each measurement point. Two types of techniques were developed for such measurements: one is based on high-speed cameras and the other uses a single photodetector. The limitation of the measurement range along the time axis in camera-based technology is mainly due to the low capturing rate, while the photodetector-based technology can only do the measurement on a single point. In this paper, several aspects of these two technologies are discussed. For the camera-based interferometry, the discussion includes the introduction of the carrier, the processing of the recorded images, the phase extraction algorithms in various domains, and how to increase the temporal measurement range by using multiwavelength techniques. For the detector-based interferometry, the discussion mainly focuses on the single-point and multipoint laser Doppler vibrometers and their applications for measurement under extreme conditions. The results show the effort done by researchers for the improvement of the measurement capabilities using interferometry-based techniques to cover the requirements needed for the industrial applications.

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.

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.

Quality-guided orientation unwrapping for fringe direction estimation

Fringe patterns produced by various optical interferometric techniques encode information such as shape, deformation, and refractive index. Denoising and demodulation are two important procedures to retrieve information from a single closed fringe pattern. Various existing denoising and demodulation techniques require fringe direction/orientation during the processing. Fringe orientation is often easier to obtain but fringe direction is needed in some demodulation techniques. A quality-guided orientation unwrapping scheme is proposed to estimate direction from orientation. Two techniques, one based on windowed Fourier ridges and the other based on fringe gradient, are proposed for the quality-guided orientation unwrapping scheme. The direction qualities are compared for both simulated and experimental fringe patterns. Their application to demodulation technique is also given.

Demodulation of a single-image interferogram using a Zernike-polynomial-based phase-fitting technique with a differential evolution algorithm

We propose a simple and robust polynomial-based phase-fitting (PPF) technique for single interferogram demodulation. Based on the smoothness assumption, the method employs a set of Zernike polynomials (ZPs) to fit the phase and estimates the expansion coefficients using a global optimization algorithm, i.e., differential evolution. The fitting order of the ZPs and the bounds of the coefficients can be intuitively determined according to the shape and number of fringes of the interferogram. Different from classical methods that need predefined scanning paths to guide the phase estimator, the PPF demodulates an interferogram globally and is insensitive to local defects, which allows it to deal with very noisy interferograms. Moreover, as the PPF gives the reconstructed phase by use of the ZPs, no further phase-unwrapping or wavefront-fitting procedures are needed. Experimental results have demonstrated the robustness and effectiveness of the method.

Robust detection scheme on noise and phase jump for phase maps of objects with height discontinuities--theory and experiment

This paper proposes a robust noise and phase jump detection scheme for noisy phase maps containing height discontinuities. The detection scheme has two primary functions, namely to detect the positions of noise and to locate the positions of the phase jumps. Generally speaking, the removal of noise from a wrapped phase map causes a smearing of the phase jumps and therefore leads to a loss of definition in the unwrapped phase map. However, in the proposed scheme, the boundaries of the phase jump regions are preserved during the noise detection process. The validity of the proposed approach is demonstrated using the simulated and experimental wrapped phase maps of a 3D object containing height discontinuities, respectively. It is shown that the noise and phase jump detection scheme enables the precise and efficient detection of three different types of noise, namely speckle noise, residual noise, and noise at the lateral surfaces of the height discontinuities. Therefore, the proposed scheme represents an ideal solution for the pre-processing of noisy wrapped phase maps prior to their treatment using a filtering algorithm and phase unwrapping algorithm.

Dynamic phase measurement in shearography by clustering method and Fourier filtering

Quantitative phase extraction is a key step in optical measurement. While phase shifting technique is widely employed for static or semi-static phase measurement, it requires several images with known phase shifts at each deformed stage, thus is not suitable for dynamic phase measurement. Fourier transform offer a solution to extract phase information from a single fringe pattern. However, a high frequency spatial carrier which is sometimes not easy to generate is required to solve the phase ambiguity problem. In this paper, we aim to propose an ideal solution for dynamic phase measurement. Four images with known phase shift are captured at the reference stage to analyze the initial phase information. After the object starts continuous deformation, only one image is captured at each deformed stage. A clustering phase extraction method is then applied for deformation phase extraction utilizing the phase clustering effect within a small region. This method works well for speckle image with low and medium fringe density. When the fringe density is high, especially in the case of shearographic fringe, information insufficiency inherent with merely one deformed speckle image often results in poor quality wrapped phase map with plenty of phase residues, which make phase unwrapping a difficult task. In the light of this limitation, a Fourier transform based phase filtering method is proposed for fringe frequency analysis and adaptive filtering, and effectively removes most of the phase residues to reconstruct a high quality wrapped phase map. Several real experiments based on shearography are presented. Comparison between the proposed solution and standard phase evaluation methods is also given. The results demonstrate the effectiveness of the proposed integrated dynamic phase extraction method.

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

The two-dimensional regularized phase-tracking (RPT) technique is one of the most powerful approaches to demodulate a single interferogram with either open or closed fringes. However, it often fails in the cases of complex interferograms and needs well-defined scanning strategies. An improved algorithm based on the RPT is presented in this paper. We use a paraboloid phase model to approximate the phase function and modify the cost functional to search the smoothest phase solutions in the function space C(2). With these modifications, the phase tracker preserves the robustness of the RPT while at the same time it is no more sensitive to stationary points and is capable of demodulating complex interferograms with arbitrary scanning schemes. Moreover, the phase reconstructed by the proposed algorithm is normally more accurate than that of the RPT both for noiseless and noisy interferograms under the same conditions. Computer simulations and experimental results are both presented.

Improved high-order ambiguity-function method for the estimation of phase from interferometric fringes

Interferometers often encode the information on the measurand in the phase of a fringe pattern, which is usually recorded by an imaging device. Accuracy of measurements carried out by interferometric techniques is thus strongly dependent on the accuracy with which the underlying phase distribution of these fringe patterns is estimated. Fringe analysis methods, which have been developed to accomplish this task, are in general characterized by their performance in terms of both accuracy of phase estimation and associated computational complexity. We propose an improved high-order ambiguity-function-based fringe-analysis method that is demonstrated to provide an accurate and direct estimation of the unwrapped phase distribution in a highly computationally efficient manner. Presented simulation and experimental results in digital holographic interferometry depict the potential utility of the proposed method.

Frequency guided methods for demodulation of a single fringe pattern

Phase demodulation from a single fringe pattern is a challenging task but of interest. A frequency-guided regularized phase tracker and a frequency-guided sequential demodulation method with Levenberg-Marquardt optimization are proposed to demodulate a single fringe pattern. Demodulation path guided by the local frequency from the highest to the lowest is applied in both methods. Since critical points have low local frequency values, they are processed last so that the spurious sign problem caused by these points is avoided. These two methods can be considered as alternatives to the effective fringe follower regularized phase tracker. Demodulation results from one computer-simulated and two experimental fringe patterns using the proposed methods will be demonstrated.