Windowed Fourier Ridges Research Articles

Carrier fringe pattern analysis: Links between methods

Introducing a carrier into a fringe pattern makes the phase retrieval problem well-posed, and has become one of the pillar techniques in optical metrology. The carrier fringe pattern analysis for high-accuracy phase retrieval is hence important, for which, many well-known methods have been developed, such as the Fourier transform method, the windowed Fourier transform based methods (the windowed Fourier filtering and the windowed Fourier ridges), the spatial carrier phase-shifting method, and the sampling moiré method. These methods were developed independently and seemingly different. The purpose of this paper is to reveal the close relationships between them, and to connect these methods into a tight-knit methodology family.

Comparative study of sampling moiré and windowed Fourier transform techniques for demodulation of a single-fringe pattern.

Phase measurement techniques using a single-shot carrier fringe pattern play an important role in optical science and technology and have been widely used for various applications. In this paper, we focus on the comparative study of two major fringe analysis techniques, the sampling moiré (SM) and the windowed Fourier transform (WFT). While SM converts a single-fringe pattern to multiple phase-shifted moiré fringe patterns to extract the phase information in the spatial domain, WFT obtains the phase information in the windowed Fourier domain; thus, the two methods look entirely different. We evaluate the phase extraction errors of SM and windowed Fourier ridges (WFRs) as a typical WFT method for both linear and nonlinear phases with/without noise against the reference Fourier transform (FT) technique. For the simulated fringe patterns with linear or nonlinear phase and different random noise level, all the methods have high phase extraction accuracies. For a real experiment with more complicated phase and discontinuities, SM and WFR, both local methods, yield quite similar results and outperform FT.

Sampling moiré as a special windowed Fourier ridges algorithm in demodulation of carrier fringe patterns

In recent times, both the windowed Fourier ridges (WFR) and sampling moire (SM) algorithms have been extensively used due to their high effectiveness in the demodulation of carrier fringe patterns. As they are developed independently, they are mostly recognized as completely different techniques, but we theoretically prove that SM is a special WFR with a specific window shape and a preset local frequency. This unifies the two different algorithms and enhances the understanding of their theoretical aspects, which helps to simplify the selection of these algorithms in real applications.

Windowed Fourier ridges for demodulation of carrier fringe patterns with nonlinearity: a theoretical analysis.

Accurately extracting phase or phase derivative is the most important requirement in optical metrology. However, in practice, there are many error sources, among which nonlinear distortion in fringe patterns is often encountered. Several techniques have been proposed over time to remove the nonlinearity error. Among these techniques, the windowed Fourier ridges (WFR) algorithm has been shown to be an effective solution insensitive to nonlinearity, but it lacks a theoretical justification. In this paper, we theoretically analyze the local frequency estimation error and phase extraction error, which not only proves the mentioned insensitivity, but also supports the performance prediction and error control, and thus is very important and useful in optical measurement. The theoretical results have been verified by computer simulations. Other error sources such as model error and noise are also compared and discussed.

Boundary-aware single fringe pattern demodulation

Optical interferometric techniques offer non-contact, high accuracy and full field measurements, which are very attractive in various research and application fields. Single fringe pattern processing is often needed when measuring fast phenomenon. However, several difficulties are encountered in phase retrieval, among which the discontinuity problem of the fringe pattern is challenging and requires attention due to the increasing complexity of manufactured pieces. In this paper, we propose a complete flowchart for discontinuous single fringe pattern processing, which uses segmentation as universal pre-processing for all discontinuous fringe pattern problems. Within the flowchart, we also propose a systematic way to introduce boundary-awareness into demodulation methods by a masking function to improve demodulation accuracy and a quality-guided scanning strategy with a novel composite quality map to improve demodulation robustness. To the best of our knowledge, this is the first time a complete solution is proposed for single and discontinuous fringe pattern processing. Three typical demodulation methods, the frequency-guided regularized phase tracker with quadratic phase matching, the windowed Fourier ridges, and the spiral phase quadrature transform, are used to demonstrate how demodulation methods can be made boundary-aware. The proposed methods are verified by successful demodulation results from both simulated and experimental fringe patterns.

Complete dispersion characterization of microstructured optical fibers from a single interferogram using the windowed Fourier-ridges algorithm

If multiple pulses, either higher-order or polarization modes, simultaneously travel in a fiber, they might overlap in the time domain, hindering dispersion retrieval of the modes in question using conventional evaluation techniques. In this work, a high-resolution windowed Fourier-ridges (WFR) algorithm is developed for evaluation of spectrally resolved interferograms produced by light pulses that are overlapping in time. The sufficiency of one spectral interferogram to retrieve the differential group dispersion and the polarization dependent chromatic dispersion directly with high accuracy is demonstrated on a meter-long HC-800-02 photonic crystal fiber. Results are in accordance with previously published data.

Windowed Fourier ridges as a spatial carrier phase-shifting algorithm

The windowed Fourier ridges (WFR) algorithm is shown to be a phase-shifting algorithm for phase extraction from a carrier fringe pattern. However, the former only provides a phase estimation with a controllable phase error, whereas the latter pursues exact phase extraction. This link not only is interesting but also enhances the understanding of different phase extraction techniques. Advantages and disadvantages of the WFRs algorithm are discussed.

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.

Statistical analysis for windowed Fourier ridge algorithm in fringe pattern analysis

Based on the windowed Fourier transform, the windowed Fourier ridges (WFR) algorithm and the windowed Fourier filtering algorithm (WFF) have been developed and proven effective for fringe pattern analysis. The WFR algorithm is able to estimate local frequency and phase by assuming the phase distribution in a local area to be a quadratic polynomial. In this paper, a general and detailed statistical analysis is carried out for the WFR algorithm when an exponential phase field is disturbed by additive white Gaussian noise. Because of the bias introduced by the WFR algorithm for phase estimation, a phase compensation method is proposed for the WFR algorithm followed by statistical analysis. The mean squared errors are derived for both local frequency and phase estimates using a first-order perturbation technique. These mean square errors are compared with Cramer-Rao bounds, which shows that the WFR algorithm with phase compensation is a suboptimal estimator. The above theoretical analysis and comparison are verified by Monte Carlo simulations. Furthermore, the WFR algorithm is shown to be slightly better than the WFF algorithm for quadratic phase.

Dynamic three-dimensional sensing for specular surface with monoscopic fringe reflectometry

Dynamic full-field three-dimensional sensing of specular reflective surfaces can be conveniently implemented with fringe reflection technique. A monoscopic fringe reflectometric system can be adopted as a simple measuring setup. With the assistance of the windowed Fourier ridges method as an advanced fringe demodulation technique, only one cross grating is needed to reconstruct the three-dimensional surface shape changes. A suitable calibration enables determination of the actual three-dimensional surface profile. Experimental results of water wave variations are shown to demonstrate the feasibility of the proposed approach.

Fast full-field out-of-plane deformation measurement using fringe reflectometry

Full-field out-of-plane deformation measurement of specular surfaces can be implemented quickly and conveniently using fringe reflectometry. The system configuration is simple and processing is fast. With the assistance of an advanced fringe pattern processing technique, the windowed Fourier ridges method, only a single two-directional fringe pattern is necessary for determination of the deformed surface profile. Thus, the whole measurement only requires a single image with the potential for high speed or even real time measurement.

Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression

Extracting phase distribution from arbitrary phase-shifted fringe patterns, if possible, is very useful in phase-shifting interferometry. The advanced iterative algorithm (AIA) is introduced and the windowed Fourier ridges and least squares fitting (WFRLSF) is proposed. Both algorithms are sensitive to noise, which limits their applications to almost perfect fringe patterns. The windowed Fourier filtering (WFF) algorithm is proposed for both pre-filtering and post-filtering to suppress the noise. Simulation results show that with the effective noise suppression, the phase error is reduced to less than 0.1 rad. Experimental examples are also given for verification. The almost identical results produced by the AIA and the WFRLSF suggest that both algorithms can be used for phase extraction with cross-validation.

Frequency-guided windowed Fourier ridges technique for automatic demodulation of a single closed fringe pattern

A frequency-guided windowed Fourier ridges (FG-WFR) technique for phase extraction from a single closed fringe pattern is proposed. The algorithm begins with a selected seed point and is then guided by the local frequency of the computed pixels. For the seed point, the whole spectral range is examined with WFR to provide reliable local frequency and phase estimation. However, for the rest of the pixels, the WFR algorithm is performed in a more narrow spectral range centered at the determined frequency of its computed neighboring pixel, which not only enforces the continuity of the local frequency between adjacent pixels but also reduces computation consumption significantly. Besides, since phase unwrapping can be readily implemented after demodulation of each pixel, the algorithm directly offers a continuous phase map. The proposed FG-WFR algorithm is verified by successfully demodulating two computer simulated fringe patterns and one noisy experimental fringe pattern.

Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB ® environment

In the process of measurements such as optical interferometry and fringe projection, an important stage is fringe pattern analysis. Many advanced fringe analysis algorithms have been proposed including regularized phase tracking (RPT), partial differential equation based methods, wavelet transform, Wigner–Ville distribution, and windowed Fourier transform. However, most of those algorithms are computationally expensive. MATLAB ® is a general algorithm development environment with powerful image processing and other supporting toolboxes. It is also commonly used in photomechanical data analysis. With rapid development of multicore CPU technique, using multicore computer and MATLAB ® is an intuitive and simple way to speed up the algorithms for fringe pattern analysis. The paper introduces two acceleration approaches for fringe pattern processing. The first approach is task parallelism using multicore computer and MATLAB ® parallel computing toolbox. Since some algorithms are embarrassing problems, our first approach makes use of this characteristic to parallelize these algorithms. For this approach, parallelized windowed Fourier filtering (WFF) algorithm serves as an example to show how parallel computing toolbox accelerates the algorithm. Second, data parallelism using multicore computer and MATLAB ® parallel computing toolbox is proposed. A high level parallel wrapping structure is designed, which can be used for speeding up any local processing algorithms. WFF, windowed Fourier ridges (WFR), and median filter are used as examples to illustrate the speedup. At last, the results show that the parallel versions of former sequential algorithm with simple modifications achieve the speedup up to 6.6 times.

Windowed Fourier transform for fringe pattern analysis: theoretical analyses

A windowed Fourier ridges (WFR) algorithm and a windowed Fourier filtering (WFF) algorithm have been proposed for fringe pattern analysis and have been demonstrated to be versatile and effective. Theoretical analyses of their performances are of interest. Local frequency and phase extraction errors by the WFR and WFF algorithms are analyzed in this paper. Effectiveness of the WFR and WFF algorithms will thus be theoretically proven. Consider four phase-shifted fringe patterns with local quadric phase [c(20)=c(02)=0.005 rad/(pixel)(2)], and assume that the noise in these fringe patterns have mean values of zero and standard deviations the same as the fringe amplitude. If the phase is directly obtained using the four-step phase-shifting algorithm, the phase error has a mean of zero and a standard deviation of 0.7 rad. However, when using the WFR algorithm with a window size of sigma(x)=sigma(y)=10 pixels, the local frequency extraction error has a mean of zero and a standard deviation of less than 0.01 rad/pixel and the phase extraction error in the WFR algorithm has a mean of zero and a standard deviation of about 0.02 rad. When using the WFF algorithm with the same window size, the phase extraction error has a mean of zero and a standard deviation of less than 0.04 rad and the local frequency extraction error also has a mean of zero and a standard deviation of less than 0.01 rad/pixel. Thus, an unbiased estimation with very low standard deviation is achievable for local frequencies and phase distributions through windowed Fourier transforms. Algorithms applied to different fringe patterns, different noise models, and different dimensions are discussed. The theoretical analyses are verified by numerical simulations.

Windowed Fourier-filtered and quality-guided phase-unwrapping algorithm

We propose a windowed Fourier-filtered and quality-guided phase-unwrapping algorithm that is an extension and improvement of our previous phase-unwrapping algorithm based on windowed Fourier transform [Opt. Laser Technol.37, 458 (2005), Key Eng. Mater.326-328, 67 (2006). First, the filtered amplitude is used as a real-valued quality map, rather than a binary mask, which makes the phase-unwrapping algorithm more tolerant to low-quality regions in a wrapped-phase map, and the process is more automatic. Second, the window size selection is considered, which enables the algorithm to be adapted to tackle different phase-unwrapping problems. A large window size is useful for removing noise, building long barriers along phase discontinuities, and identifying invalid regions, while a small window size is useful for preserving local features, such as small regions and high-quality narrow channels. Eight typical examples in Ghiglia and Pritt's excellent book Two-Dimensional Phase Unwrapping: Theory, Algorithm and Software (Wiley, 1998) are used to evaluate the proposed algorithm. The proposed algorithm is able to unwrap all these examples successfully. The windowed Fourier ridges algorithm, another algorithm based on windowed Fourier transform, is also tested and found to be useful in building barriers along phase discontinuities.

Comparative analysis on some filters for wrapped phase maps

Some effective filtering methods for wrapped phase maps, a regularized phase-tracking method (RPT) without the regularization term, a multiple-parameter least-square method (MPLS), a windowed Fourier ridges method (WFR), an autocorrelation function method (ACF), and a sine/cosine average filter (SCAF), are analyzed in order to establish their transversal relationship. The analysis shows that principles of the RPT, MPLS, WFR, and ACF are equivalent and the SCAF also leads to the WFR by some extension, which elegantly unifies all these methods for filtering unwrapped phase maps.

On window size selection in the windowed Fourier ridges algorithm

The windowed Fourier ridges (WFR) algorithm can be used to extract the frequency and phase information from an exponential or carrier fringe pattern. The selection of the window size for this algorithm is investigated in this paper. For exponential phase fringe patterns, the window size does not need to be adjusted according to the signal frequency. Hence the scaling strategy of the wavelet transform is not necessary. Instead, the window size should be selected according to the balance between the linear phase approximation error and the noise level. For carrier fringe patterns, the influence of the conjugate component should also be considered. However it can be ignored when the lowest frequency of the fringe pattern is higher than 1.5/ σ ( σ is used to represent the window size).

On window size selection in the windowed Fourier ridges algorithm: Addendum

The windowed Fourier transform is a useful technique for fringe pattern analysis. It has been shown that the proper selection of the window size is a balance between the linear phase approximation error and the influence of noise [Q. Kemao, On widow size selection in windowed Fourier ridges algorithm. Opt Lasers Eng, 2007, accepted for publication]. Since the fringe intensity and noise level usually vary spatially, the window size should also be spatially adaptive in order to reach a good balance for each pixel of the fringe pattern. This addendum first shows that the windowed Fourier transform with a spatially fixed window size (SFWS) is still practically useful and then discusses the window size competition strategies for the windowed Fourier transform with a spatially adaptive window size (SAWS). The windowed Fourier transform with a SAWS is theoretically better than that with a SFWS but it is also more challenging in use. The windowed Fourier ridges algorithm is used for analysis throughout this paper. This analysis is also applicable to the windowed Fourier filtering algorithm.

Detection of Rotor Faults in Brushless DC Motors Operating Under Nonstationary Conditions

There are several applications where the motor is operating in continuous nonstationary operating conditions. Actuators and servo motors in the aerospace and transportation industries are examples of this kind of operation. Detection of faults in such applications is, however, challenging because of the need for complex signal processing techniques. Two novel methods using windowed Fourier ridges and Wigner-Ville-based distributions are proposed for the detection of rotor faults in brushless dc motors operating under continuous nonstationarity. Experimental results are presented to validate the concepts and illustrate the ability of the proposed algorithms to track and identify rotor faults. The proposed algorithms are also implemented on a digital signal processor to study their usefulness for commercial implementation