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Circular fringe projection profilometry (CFPP), as a branch of carrier fringe projection profilometry, has attracted research interest in recent years. Circular fringe Fourier transform profilometry (CFFTP) has been used to measure out-of-plane objects quickly because the absolute phase can be obtained by employing fewer fringes. However, the existing CFFTP method needs to solve a quadratic equation to calculate the pixel displacement amount related to the height of the object, in which the root-seeking process may get into trouble due to the phase error and the non-uniform period of reference fringe. In this paper, an improved CFFTP method based on a non-telecentric model is presented. The calculation of displacement amount is performed by solving a linear equation instead of a quadratic equation after introducing an extra projection of circular fringe with circular center translation. In addition, Gerchberg iteration is employed to eliminate phase error of the region close to the circular center, and the plane calibration technique is used to eliminate system error by establishing a displacement-to-height look-up table. The mathematical model and theoretical analysis are presented. Simulations and experiments have demonstrated the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: Both two-dimensional (2D) Windowed Fourier Transform and 2D S Transform are popular time–frequency analysis tools for processing signals. But fewer articles reported the application of 2D S Transform in Fringe pattern analysis. This paper studies the 2D S Transform and compares it with 2D Windowed Fourier Transform. The phase calculation formula based on the 2D Windowed Fourier Transform “ridge” coefficients and 2D S Transform “ridge” coefficients are deduced respectively. A comparison between them is carried out to show their own characteristic in fringe patterns analysis based on the structured-light illumination. It verifies that the 3D reconstruction result of 2D ST is better than that of 2D WFT because of the employment of the adjusted windows by frequency parameters. [Copyright &y& Elsevier]

Abstract: S transform combines the advantages of the windowed Fourier transform and the wavelet transform. In optical three-dimensional (3D) surface measurement based on the structured-light illumination, a two-dimensional S-Transform filtering (2DSTF) method is proposed to demodulate phase maps from fringe patterns. Unlike the two-dimensional S transform ridge (2DSTR) technique, in which the first-order approximation of Taylor expansion is introduced to express the phase information modulated in the fringe patterns, 2DSTF method reconstructs one of the fundamental components of the fringe pattern by adding the extracted local fundamental spectra of S-Transform, and the filtering operation and the inverse Fourier transform are used to calculate the phase maps. In this paper, a detailed theoretical analysis of 2DSTF method is given and its applicability is demonstrated on both simulations and experiments by comparing it with 2DSTR and the previous 1D ST (1DSTF and 1DSTR), which verify that the proposed method has a better performance than other methods in demodulating the phase of fringe patterns. [Copyright &y& Elsevier]

Abstract: S-transform proposed in 1996 by Stockwell R.G is a simple and popular technique for the time–frequency analysis. It has been introduced in optical three-dimensional shape measurement, recently. In this paper, a study about applications of S-transform in the demodulation of deformed fringe patterns is performed. We focus on discussing not only the S-transform spectrum filtering technique, the S-transform ridge technique and the phase gradient calculation method based on S-transform used in fringe pattern demodulation, but also the phase unwrapping technique. In addition, a generalized S-transform was introduced to analyze fringe patterns, which is helpful to improve the measurement accuracy and flexibility of the method based on S-transform. The reconstruction results based on S-transform were compared with that on wavelet transform and windowed Fourier transform in fringe analysis. [Copyright &y& Elsevier]

Abstract: A modified Fourier transform profilometry (FTP) based on a fringe pattern with two frequency components is proposed, which provides a larger measuring range than that of the traditional FTP. We analyze the maximum measuring range and give an expression to describe the measurable slope of the height variation limitation of this method. The modified FTP provides us another approach to eliminate frequency overlapping. When the spectra distribution of a measured object is not spherical symmetry, we can avoid the frequency aliasing through projecting a fringe pattern with two frequency components, instead of increasing the density of the projected fringe and the resolution of a CCD camera. The theoretical analysis and primary experiments verified the method. [Copyright &y& Elsevier]

Abstract: Projecting a bicolor sinusoidal fringe pattern consisting of two interlaced RGB format base color fringe patterns with π phase difference onto an object thought digital light projector, we can capture a deformed color pattern by color digital camera, then decode two individual sinusoidal fringe patterns with π phase difference by color-separating technique. Accessing these two fringe patterns, not only are zero-order spectra eliminated, but mask function is also built to mark valid unwrapping area in FTP, automatically. Moreover, because the wrapped phase just inside the valid areas is needed unwrapping, we can mark these areas with mask function, which avoids the error transferring resulting from unwrapping the invalid areas and shortens the unwrapping time. Furthermore, in Fourier transform processing, the full-field deformed fringe pattern generally needed to guarantee measurement precision can be formed by expanding non-full-field fringe pattern captured using the mask function. [Copyright &y& Elsevier]

Abstract: Windowed Fourier transform (FT) is introduced into Fourier transform profilometry (FTP) to extract zero component included in deformed fringe pattern without capturing two fringe patterns with phase difference. First, we extract zero component of deformed fringe image using windowed FT and normalize it, then calculate the normalized Fourier spectra of deformed fringe image, finally subtract the zero component from Fourier spectra of deformed fringe image. We give the theoretical analysis of the proposed method. The experiment is carried out to demonstrate that the method can eliminate zero component and extend the measurable slope of height variation to nearly three times of the conventional FTP. [Copyright &y& Elsevier]

The high-speed three-dimensional (3-D) shape measurement technique has become more and more popular recently, because of the strong demand for dynamic scene measurement. The single-shot nature of Fourier Transform Profilometry (FTP) makes it highly suitable for the 3-D shape measurement of dynamic scenes. However, due to the band-pass filter, FTP method has limitations for measuring objects with sharp edges, abrupt change or non-uniform reflectivity. In this paper, an improved Temporal Fourier Transform Profilometry (TFTP) algorithm combined with the 3-D phase unwrapping algorithm based on a reference plane is presented, and the measurement of one deformed fringe pattern producing a new 3-D shape of an isolated abrupt objects has been achieved. Improved TFTP method avoids band-pass filter in spatial domain and unwraps 3-D phase distribution along the temporal axis based on the reference plane. The high-frequency information of the measured object can be well preserved, and each pixel is processed separately. Experiments verify that our method can be well applied to a dynamic 3-D shape measurement with isolated, sharp edges or abrupt change. A high-speed and low-cost structured light pattern sequence projection has also been presented, it is capable of projection frequencies in the kHz level. Using the proposed 3-D shape measurement algorithm with the self-made mechanical projector, we demonstrated dynamic 3-D reconstruction with a rate of 297 Hz, which is mainly limited by the speed of the camera. [ABSTRACT FROM AUTHOR]

Abstract: Here a flexible optical geometry of Fourier Transform Profilometry (FTP) is discussed, and a strict theoretical analysis about an improved phase-height mapping formula based on a new description of a reference fringe and a deformed fringe proposed based on a general case, in which the exit pupil of the projecting lens and the entrance pupil of the imaging lens are not horizontal and do not have the equal distance to the reference plane; moreover, the optical axes of the projecting device and of the imaging device are not coplanar. Compared with the traditional Fourier Transform Profilometry, it is easier for us to obtain the full-field fringe through adjusting the location of either the projector or the imaging device. Compared with the Improved Fourier Transform Profilometry (IFTP), only three length system parameters need to be measured, so the operation is simple and the result is more reliable. [ABSTRACT FROM AUTHOR]