Fourier transform profilometry:: a review

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Abstract

Fourier transform profilometry is one of the popular non-contact 3-D measurement methods, where a Ronchi grating or sinusoidal grating is projected onto a diffuse three-dimensional surface, and the resulting deformed grating image is detected by a CCD camera and processed by a computer. This method requires only one frame (or two frames) of the deformed fringe pattern in some algorithms to retrieve the surface of measured object, so it has obvious advantage for real time data acquisition and 3-D measurement of dynamic process. In this paper, we review some algorithms in FTP, discuss some important problems, including frequency spectra overlapping, phase unwrapping, sampling, and 3-D measurement of dynamic process. With the development of computer hardware and software and availability of high-resolution image grabber, FTP method will be a promising one for acquiring 3-D data of object, and more and more researchers pay attention to it.

Introduction

Optical 3-D non-contract profilometry has been widely used for 3-D sensing, mechanical engineering, machine vision, intelligent robots control, industry monitoring, biomedicine, dressmaking, etc. Several 3-D object profilometry methods that use structured light pattern, including Moiré technique (MT) [1], [2], phase-measuring profilometry (PMP) [3], [4], [5], [6], [7], [8], [9], [10], [11], Fourier transformation profilometry (FTP) [12], [13], modulation measurement profilometry (MMP) [14], [15], spatial phase detection (SPD) [16], [17], [18], laser triangulation measurement [19], [20], [21], [22], color-coded fringe projection [23], gray-coded binary fringe sequences [24] have been exhaustively studied. Among them, Fourier transform profilometry, by Takeda et al. is a popular one, because of following merits, only one (or two) fringe(s) needed, full field analysis, and high precision, etc. In FTP, a Ronchi grating or a sinusoidal grating is projected onto the object surface, the depth information of the object is encoded into a deformed fringe pattern recorded by an image acquisition sensor. The surface shape can be decoded by calculating Fourier transformation, filtering in spatial frequency domain, and calculating inverse Fourier transformation. Compared with the Moiré topography (MT), FTP can accomplish fully automatic distinction between a depression and an evaluation of the object shape. It requires no fringe order assignments or fringe center determination, and it requires no interpolation between fringes because it gives height distribution at every pixel over the entire fields. Compared with the phase-measuring profilometry (PMP) and modulation measurement profilometry (MMP), FTP requires only one or two images of the deformed fringe pattern, which makes real-time data processing and dynamic data processing possible. Whereas, PMP and MMP require many images with fixed phase variation between neighboring two images to retrieve the height distribution, and they take much time to finish phase-shifting procedure using mechanical device. So it is impossible to use them to measure dynamic object.

After Takeda et al. the FTP method has been extensively studied [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41]. A grating π phase shifting technique [28], [29] can extend the measurable slope of height variation to nearly three times that of the unimproved FTP. Two-dimensional Fourier transform and 2-D hanning filtering are applied to provide a better separation of the height information from noise when speckle-like structures and discontinuities exist in the fringe pattern [31]. FTP based on TDI camera can be used to measure 360° shape [33]. The frequency-multiplex technology [35], [36] permits the 3-D shape measurement of objects that have discontinues height step and/or spatially isolated surfaces. The phase error caused by sampling in FTP is discussed in detail [40]. This paper reviews some of the developments in Fourier transform profilometry over the past years, including frequency spectra overlapping, phase unwrapping, sampling, and 3-D measurement of dynamic process, complex object phase unwrapping, 3-D phase unwrapping, etc. With the availability of high resolving CCD and high frame rate grabber, FTP has become an effective method for 3-D shape measurement.

Section snippets

Principal

FTP was introduced by Takeda et al. The general geometry is shown in Fig. 1, in which the optical axes P1P2 of a projector lens crosses that of camera lens I1I2 at point O on a reference plane R, which is a fictitious plane normal to I1I2 and serves as a reference, from which object height h(x,y) is measured. D expresses a tested point, A, C express points on R, d is the distance between P2 and I2, and L0 is the distance between I2 and O. Ronchi grating G has its lines normal to the plane of

Quasi-sine projection and π phase shifting technique [28,29]

In FTP, we apply quasi-sine projection technique and π phase shifting technique to merely make the fundamental component exist in spatial frequency domain. In this case, the lower frequency part of the fundamental component can extend to zero, and the higher part can extend to 2f0 without overlaps. FTP can measure the object with higher variation.

The intensity distribution on the object surface can be expressed asg(x,y)=a(x,y)+b(x,y)cos(2πf0x+φ(x,y)).To eliminate the influence of background

The influence of sampling in FTP [40]

Employing continuous Fourier transform analytical method, Takeda et al. analyzed the problems that existed in FTP, such as frequency aliasing, measurable range, etc. [13]. But in practice, the deformed fringe pattern captured by CCD camera is discrete, and the discrete Fourier transform (DFT) is applied. The obvious shortcoming in theory analysis results in the inconsistency between theory and experience, especially if high-order spectra existed. To obtain the correct reconstruction of the

Dynamic 3-D shape measurement method based on FTP

Now we review the method based on Fourier transform profilometry for dynamic 3-D shape measurement. First, we record a fringe pattern for h(x,y)=0:g0(x,y)=n=−∞Anexp(i(2πnf0x+nϕ0(x,y))),then put a dynamic 3-D object in the optical field, and record and store a sequence of deformed fringe patterns rapidly. The intensity distributions of these fringe patterns can be expressed asg(x,y,t)=n=−∞Anr(x,y,t)exp(i(2πnf0x+nϕ(x,y,t)))(t=1,2,…,s)in , , r(x,y,t) and ϕ(x,y,t) represent a non-uniform

Phase unwrapping in FTP method

Phase unwrapping is a critical but challenging step in any grating projection profilometry, including Fourier transform profilometry. In recent years, phase unwrapping problem has been widely studied [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56]. In this paper, we will concentrate on the phase unwrapping based on the modulation ordering and extend 2-D phase unwrapping algorithm into 3-D phase space for dynamic 3-D measurement.

Conclusions

As a non-contract measurement method, FTP method will play an important role in 3-D sensing field in the future because of only one or two images of the deformed fringe needed. Especially, with the availability of high-resolution CCD, for example, up to 5k∗5k pixels CCD sensor are commercially available. Combined with macroscanning technique, the resolution of CCD can be as high as 20k∗20k. So we can project high-density-grating image onto the measured object to separate the fundamental

Acknowledgements

This project was supported by the National Natural Science Foundation of China.

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