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Abstarct: S-Transform, as a time-frequency analysis approach, has been successfully introduced into fringe pattern demodulation in recent years. The paper presents a thorough discussion on the application of this technique in analyzing fringe pattern. A brief review of the S transform profilometry is followed by the comparison between the one-dimensional S-Transform method and the two-dimensional S-Transform method, including one-dimensional S-Transform ridge method, one-dimensional S-Transform filtering method, two-dimensional S-Transform ridge method and two-dimensional S-Transform filtering method in the application of the phase retrieval from the fringe patterns with different noise, contrast, nonlinearity and fringe breaking. Numerical computer simulations and experiments were carried out to show the 1DST method is sensitive to the noise but suitable to detect detail of the measured object, while 2DST method with capability of smooth processing for the surface of the measured object is adapt to demodulate the fringe pattern with serious noise. [Copyright &y& Elsevier]

Abstract: We discuss the wavelet transform profilometry based on the continuous wavelet transform technique as viewed from frequency analysis. We deduce the expression of one-dimensional (1-D) and two-dimensional (2-D) wavelet transform in frequency domain and analyze their characteristics in the application of demodulating the fringe patterns. We also compare 1-D CWT and 2-D CWT in demodulating the oblique fringe patterns with dual carrier frequency components. When oblique fringe patterns are processed, the direction normal to the grating line and x axis or y axis is not identical. By 1-D CWT, in which wavelet transform is carried out row by row, we cannot obtain the most similarity between local signal and the wavelet functions with different dilation values. While a fan 2-D continuous wavelet transformation can deal with the fringe pattern as a 2-D unit as well as has multi-directions, its advantage is that it can be used to exact the information in the spatial direction. However, its spatial localization ability is not very good, which leads that it is not suitable for demodulating the fringe patterns with high phase variation. Computer simulations and experiments have verified our analysis. [Copyright &y& Elsevier]

Abstract: An improved method is proposed to optimize the discrete wavelet decomposition, which can be used to decompose a deformed fringe by easily applying “ridge” information of continuous wavelet transform. With this method the zero component of a deformed fringe pattern can be extracted and eliminated automatically instead of using traditional repetitious tentative experiments. Computer simulations and experiments are carried out to verify our analysis by comparing the results from the correct level selection with that from the adjacent level selection. [Copyright &y& Elsevier]

Abstract: An off-axis Ronchi test based on the phase-shifting technique is presented to measure the aspheric mirror quantitatively. In this paper, slopes of deviations from the mirror''s ideal aspheric shape are derived when an off-axis point source is used. By integrating these slopes, the deviations of the shape are obtained and then the tested shape can be reconstructed. Compared with the coaxial Ronchi test, the measurement error caused by combination fringes can be avoided, and this system is more flexible. In the experiment, a marked point is used to guide the phase unwrapping procedure, which improves the accuracy of extracting corresponding phase points. In addition, the method has strong anti-noise ability. The validity of the method has been verified by both computer simulations and a preliminary experiment. [Copyright &y& Elsevier]

Abstract: A Hilbert assisted wavelet transform (HWT) method is presented for phase reconstruction of 3D profilometry and interferometry. A rigorous mathematical demonstration about this HWT method is given in this paper. An important conclusion that the phase of the optical fringe pattern is equal to the phase of its HWT coefficients on the ridge is theoretically clarified. The strict relation between the scale parameter and the phase gradient (also called the angular frequency) of the fringe pattern is also given. Computer simulations and experiments reveal the validity of the method and the correctness of the mathematical demonstration. Since the filtering process is avoided in this method, it can deal with the frequency overlapping problem to a certain extent. Applications of this method in both optical profilometry and interferometry are shown and discussed in the experiment section. [Copyright &y& Elsevier]

Abstract: A coordinate measuring method is presented, which is specially devised to perform the measurement of coordinates with projected fringe techniques of projectors in three dimensions. The system is composed of two parts: one is a target which can move freely in three dimensions, and the other is a stationary two-dimensional array of photodetectors. The mini-projector is tied to the target, and the projected fringe is monitored by the photodetectors. The phase of the photodetectors can be precisely measured with the phase-shifting algorithm, so that, the xyz location of the target can be determined with the geometric model of multilateration using the method of optimization. In this paper, the measuring principle, iterative method, computer simulation and preliminary results are given. The phase-shifting technique has the advantages of high accuracy and noise endurance. The method will provide the basis for follow-up iterative optimization calculation. The experimental results prove that the proposed coordinate measuring method is of high precision. [Copyright &y& Elsevier]

Abstract: The present study proposes a new method for measuring the surface shape of an object using elliptically pattern grating projection. Based on phase measuring profilometry (PMP), elliptically pattern grating is projected on the surface of the object to measure its shape in 3D. The computing formula for solving the phase by the phase-shifting fringe method is derived, and the 3D shape formula for the altitude is reestablished. The error-contrastive analysis of the object's surface shape is re-established based on the proposed method and traditional PMP. The proposed method was shown to have a strong anti-noise ability and able to measure objects under high noise. More accurate measurement results were obtained by computer simulation and experiment to verify the feasibility of the proposed method. [Copyright &y& Elsevier]