Fringe pattern analysis based on the two-dimensional synchrosqueezing transform.

The windowed Fourier transform (WFT) has been recognized as an effective tool for extracting phase map from a single fringe pattern. This paper presents a new WFT-based algorithm, which is guided by the principal component analysis (PCA) of the fringe pattern. With it, the principal direction of frequency at each pixel is determined first by using the PCA of the gradients of the fringe pattern, and then the windowed Fourier ridge for each pixel is detected by searching a one-dimensional space of frequency, so that the computational burden for phase extraction is alleviated significantly. In addition, this technique enables automatically identifying and excluding the regions of singular points from the fringe pattern by using the eigenvalues resulting from the PCA just mentioned. Numerical simulation and experiment are carried out to demonstrate the validity of this method.

Series of thermodiffusion experiments using optical digital interferometry (ODI) have been conducted onboard the International Space Station. Conventionally, the two-dimensional (2D) fast Fourier transform (FFT) fringe analysis technique has been applied as a fast and reliable technique to extract data. In this study, for the first time, the windowed Fourier transform (WFT) method is used to analyse the same experiments. In this method, a Fourier transformation is applied on the fringes at two different stages: initially, during the filtration of the non-zero peaks and then on the wrapped phase image. We provide a detailed comparison between FFT and WFT results of binary and ternary mixtures for ODI thermodiffusion experiments. The substantial enhancements of this method are presented and discussed for different experiments conducted for both binary and ternary mixtures. We show that while disturbances in the phase fringe pattern can cause significant error in FFT techniques, if the windowed Fourier filtration (WFF) parameters are properly chosen this type of noise can be eliminated during WFF analysis. The importance of replacing the FFT method becomes more pronounced for the ternary system, as this method fails to reconcile reliable concentration profiles. The results of this chapter can show that the application of the windowed Fourier transform in optical digital interferometry investigations show improved results over the same experiments analysed using FFT methods, especially for experiments involving very small heat and mass fluxes such as the Soret effect in multicomponent mixtures.

3D surface profilometry based on 2D S-Transform method with optimized window

Optical full-field measurement techniques have been widely studied in academia and applied to many actual fields of automated inspection, reverse engineering, cosmetic surgery, and so on. With the advent of color CCD cameras and DMD (Digital Micromirror Device) based color DLP (Digital Light Processing) projectors, their major red, green, and blue channels have been used as a carrier to code fringe patterns. Since three fringe patterns can be simultaneously projected and captured at one shot, the acquisition time reduces to 1/3 of the value by the gray fringe pattern projection. This chapter will introduce two kinds of applications of red, green, and blue as a carrier: 1) modulation and demodulation method of coding sinusoidal fringe patterns into RGB channels of a composite color image; and 2) modulation and demodulation method of coding sinusoidal and binary fringe patterns into RGB channels of multiple composite color images. Experiments on testing the two kinds of applications were carried out by measuring the shape of objects’ surface. The results confirm that red, green, and blue channels can be used as a carrier to reduce the acquisition time.

Conditions of the digital recording of the fringe pattern determine the phase reconstruction procedure, which in turn directly shapes the final accuracy and throughput of the full-field (non-scanning) optical measurement technique and defines the system capabilities. In this way, the fringe pattern analysis plays a crucial role in the ubiquitous optical measurements and thus is under constant development focused on high temporal/spatial resolution. It is especially valuable in the quantitative phase imaging technology, which emerged in the high-contrast label-free biomedical microscopy. In this paper, I apply recently blossomed two-frame phase-shifting techniques to the QPI and merge them with advanced adaptive interferogram pre-filtering algorithms. Next, I comprehensively test such frameworks against classical and adaptive single-shot methods applied for phase reconstruction in dynamic QPI enabling highest phase time-space-bandwidth product. The presented study systematically tackles important question: what is the gain, if any, in QPI realized by recording two phase-shifted interferograms? Counterintuitively, the results show that single-shot demodulation exhibited higher phase reconstruction accuracy than two-frame phase-shifting methods in low and medium interferogram signal-to-noise ratio regimes. Thus, the single-shot approach is promoted due to not only high temporal resolution but also larger phase-information throughput. Additionally, in the majority of scenarios, the best option is to shift the paradigm and employ two-frame pre-filtering rather than two-frame phase retrieval. Experimental fringe analysis in QPI of LSEC/RWPE cell lines successfully corroborated all novel numerical findings. Hence, the presented numerical-experimental research advances the important field of fringe analysis solutions for optical full-field measurement methods with widespread bio-engineering applications.

Optical interferometry is a popular technique for high precision measurement. It produces fringe patterns for analysis. The output fringe patterns usually have noises, and thus a filtering process for fringe patterns is necessary. A few filtering algorithms, such as windowed Fourier filtering (WFF), have been proposed specially for fringe pattern denoising. Meanwhile, filtering techniques have been intensively studied for a long time in the general image processing area. It is curious that how the filtering techniques in general image processing area perform on fringe pattern denoising. In this paper, a state-of-the-art filtering algorithm, block-matching 3D filtering (BM3D), is selected and compared with the WFF.

A principled approach, based on Bayesian estimation theory and complex-valued Markov random-field prior models, is introduced for the design of a new class of adaptive quadrature filters. These filters are capable of adapting their tuning frequency to the local dominant spatial frequency of the input image while maintaining an arbitrarily narrow local frequency response; therefore they may be effectively used for the accurate recovery of the phase of broadband spatial-carrier fringe patterns, even when they are corrupted by a significant amount of noise. Also, by constraining the spatial variation of the adaptive frequency to be smooth, they permit the completely automatic recovery of local phase from single closed fringe pattern images, since the spurious discontinuities and sign reversals that one obtains from the classical Fourier-based methods are avoided in this case. Although the applications discussed here come from fringe pattern analysis in optics, these filters may also be useful in the solution of other problems, such as texture characterization and segmentation and the recovery of depth from stereoscopic pairs of images.

The Window Fourier Ridge (WFR) is one of the useful methods in local frequency extraction from a fringe pattern. But the background of the pattern and the inherent noises existed in the pattern will disturb the extracted result, which need to be removed before operation. To solve this problem, we modify the WFR method by spectrum normalization of the fringe pattern in its frequency domain. Then, the accuracy of the extracted local fringe frequency is enhanced greatly due to the inhibition of the disturbance from its background. The principle of the normalized spectrum method is detailed by spectrum analysis of the fringe pattern. The conducted simulations and experiment results show that the local fringe-frequency estimated from the fringe pattern with background by the proposed method is as good as that from the fringe pattern without background by traditional WFR method.

Nonlinear dynamics of toroidally confined, initially cold, collisionless pure electron plasma has been numerically simulated in a tight aspect ratio, axisymmetric device, confined using a toroidal magnetic field, using a 3D3V particle-in-cell code PEC3PIC. A set of three numerical experiments are conducted by loading the toroidal electron cloud at varying radial distances from the central axis at the vertical midplane, and a comparative analysis of the progression of cloud dynamics and particle transport in the three experiments is carried out. In each experiment, the cloud is seen to initiate toroidal Diocotron oscillations with the following interesting features: (i) initial nonlinear reshaping and density peaking, (ii) elliptical orbital path in the poloidal cross section along with chirp or rotational frequency dynamics and the increase and decrease in the peak density of the filled electron cloud, (iii) cross-field transport and particle loss, and (iv) the measured wall probe signals showing close similarity to experimental signals. It is demonstrated that relatively better confinement of electrons in the toroidal configuration is achieved by loading the initial plasma at the vertical midplane, close to the inner wall of the chamber, supporting the mean-field theoretical predictions. For all cases, the density distribution profiles in the (r−θ) and (r − z) planes of the cylindrical coordinate system (r,θ,z) have consistent peaked density central profiles. The time dependency of the dominant frequencies of the dynamics, obtained from wall probe data using Hilbert–Huang transformation and windowed Fourier transformation, suggests toroidicity induced low poloidal number m (∼1−12) coupling and dynamical chirping.

Application of fourier analysis to the laser based interferometric strain/displacement gage

With the development of science and technology and industrial production, the application of three-dimensional shape measurement technology in the fields of machine vision, industrial monitoring, mechanical engineering and medical testing has become increasingly widespread. The optical three-dimensional shape measurement technology is a method that uses a series of optical methods to obtain the three-dimensional information of the measured object, and the structured light measurement method is a widely used measurement method. In the structured light measurement method, the generation of fringe patterns is a very important link. Here, we use holographic method to generate a fringe pattern whose period and phase can be easily modulated. First, a black and white fringe pattern with a certain spatial frequency is generated according to the cosine structured light period and phase. The spatial frequency of the black and white fringe is determined by the spatial frequency of the structured light to be generated and the magnification of the projection system. A prism phase that can cause lateral movement is applied to the black part of the black and white stripes, so that the light incident on the black area deviates from the optical axis to the first order of diffraction, while the light incident on other parts (without being modulated) continues to travel along the optical axis. In this way, the deviated beam is bright and dark, and finally structured illumination is obtained. Theories and experiments verify the effectiveness of the method. The method to generate fringe patterns is simple, fast, accurate and can be conveniently controlled, and can be well applied to three-dimensional shape measurement technology.

Time–frequency (TF) analysis is a powerful tool to describe the frequency response of subsurface rocks and hydrocarbon reservoirs. In this letter, the matching demodulation synchrosqueezing S transform (MDSST) that achieves a highly concentrated TF representation was introduced to seismic TF analysis. The MDSST includes two steps: the matching demodulation technique is preprocessing that provides a rough estimation of instantaneous frequency (IF) of the analyzed signal, and the synchrosqueezing transform (SST) is postprocessing that produces an accurate IF. Numerical experiments on synthetic and real signals demonstrate the superiority of the MDSST in seismic TF analysis.

Single-frame fringe pattern processing and analysis is an important task of optical interferometry, structural illumination and moire techniques. In this contribution we present several algorithmic solutions based on the notion of Hilbert-Huang transform consisting of empirical mode decomposition algorithm and Hilbert spectral analysis. EMD adaptively dissects a meaningful number of intrinsic mode functions from the analyzed pattern. Appropriately managing this set of functions results in a powerful fringe processing tool. We describe in detail especially tailored manners proposed to extend the EMD algorithm to 2D and perform Hilbert-transform-aided efficient fringe pattern denoising, detrending and amplitude/phase demodulation.

A novel Hilbert transform algorithm for detecting phase from fringe pattern is proposed.Only one interference pattern is required for the phase evaluation.In practice,the Hilbert transform based on the Fourier transformation eliminates all negative frequency components,and any phase variation in the fringe pattern having negative local spatial frequency can not be correctly reconstructed.A procedure for detection of regions having zero spatial frequency in the fringe pattern is proposed by an estimation function for complicated phase distribution.A binary matrix is used for modifying wrapped phase of complicated phase distribution fringe pattern, and a method for removing the bias component by twice Hilbert transform is proposed.The correct values of phase is obtained by an unwrapping process.The performance of the proposed method is evaluated by computer simulation.

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.