Windowed Fourier Ridges Algorithm Research Articles

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.

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.

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.

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.

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.