Eliminating the zero spectrum in Fourier transform ... - OSA Publishing

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Vol. 26, No. 5 / May 2009 / J. Opt. Soc. Am. A

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Eliminating the zero spectrum in Fourier transform profilometry using empirical mode decomposition Sikun Li,1,2 Xianyu Su,1,* Wenjing Chen,1 and Liqun Xiang1 1

Department of Opto-Electronics, Sichuan University, Chengdu 610064, China 2 email: [email protected] *Corresponding author: [email protected] Received November 12, 2008; accepted March 16, 2009; posted March 23, 2009 (Doc. ID 103729); published April 16, 2009

Empirical mode decomposition is introduced into Fourier transform profilometry to extract the zero spectrum included in the deformed fringe pattern without the need for capturing two fringe patterns with ␲ phase difference. The fringe pattern is subsequently demodulated using a standard Fourier transform profilometry algorithm. With this method, the deformed fringe pattern is adaptively decomposed into a finite number of intrinsic mode functions that vary from high frequency to low frequency by means of an algorithm referred to as a sifting process. Then the zero spectrum is separated from the high-frequency components effectively. Experiments validate the feasibility of this method. © 2009 Optical Society of America OCIS codes: 070.2025, 070.2615, 070.4790, 100.5070.

1. INTRODUCTION Optical 3-D sensing techniques based on structured illumination are becoming interesting tools for surface topography, industry inspection, quality control, biomedicine, and machine vision because of their qualities of noncontact, high measurement precision, ease of automation, etc. Among the existing techniques, Fourier transform profilometry [1,2] (FTP) is a popular one because of its high processing speed and need for only one fringe pattern. In FTP, a filtering procedure is performed to obtain the fundamental frequency spectrum in the frequency domain. This means frequency aliasing between the zero spectrum and the fundamental spectrum has a great influence on the measurement accuracy and measurable slope of height variation. To avoid this, sinusoidal grating projection and ␲ phase shifting techniques [3] have been used initially to eliminate the zero component, so the measurable slope of height variation has been extended to nearly three times. But a complex measurement system with a precision phase shifting device needs to be set up and two fringe images captured. At the second sampling, the grating is moved to half of the grating period, which reduces the instantaneous characteristic of FTP. To solve these problems the bi-color projecting method [4], windowed Fourier transform method [5], and wavelet transform method [6,7] have been proposed. Among the methods, the wavelet transform method of Xu et al. [6] is noteworthy. In this method, discrete wavelet transform [7–11] is adopted to eliminate the zero spectrum because of its strong multiresolution analysis capability. The captured fringe pattern is calculated initially by the wavelet transform, and then several low-frequency patterns are reconstructed from the low-frequency wavelet coefficients at different scales that represent different frequencies. Finally, the zero spectrum is subtracted. Al1084-7529/09/051195-7/$15.00

though it is efficient, this approach has the same shortcoming as the common wavelets signal processing method; that is, different signals need different mother wavelets. Once the mother wavelet is selected, analysis of the signal can be done by using it. However, since the wavelet decomposition can be considered a digital filter, it will lose some useful information during the data processing. To solve these problems, a new nonlinear technique, called empirical mode decomposition (EMD) [11] has been introduced. It has been regarded as a significant time series analysis tool in comparison with traditional methods such as Fourier methods, wavelet methods, and empirical orthogonal functions [12]. Since this method is based purely on the properties observed in the data without invoking the concept of stationarity, it can effectively decompose a linear signal into different components adaptively. It is also useful for analyzing nonstationary, nonlinear signals, and it has been widely used in many fields successfully [13–16]. In this paper, it is applied to eliminate the expansion of the zero spectrum in FTP. The deformed fringe pattern is decomposed into a finite number of intrinsic mode functions (IMF) that vary from high frequency to low frequency. Then the zero spectrum can be separated from the fundamental spectrum effectively. Experiments validate the feasibility of this method.

2. BASIC PRINCIPLE A sinusoidal grating pattern is projected onto an object surface to encode depth information following the optical geometry shown in Fig. 1. L0 and d are configuration parameters of the measurement system. In FTP, the extension of the background illumination is usually done by the projector, which radiates in one direc© 2009 Optical Society of America

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The procedure to extract the IMF is referred to as a sifting process, shown in Fig. 2. It mainly consists of three steps: (1) Identify all the local maxima of I共x兲 at the very beginning. Then connect all the local maxima by cubic spline line as the upper envelope Imax共x兲. Similarly, repeat the procedure for the local minima to produce the lower envelope Imin共x兲. All the points of the data are between the two envelopes. (2) Subtract the mean of the two envelopes from the signal; a new signal is obtained as Fig. 1.

Experimental geometry.

N共x兲 = I共x兲 −

tion only, so the expansion of the zero spectrum effect can be treated as a one-dimensional vector. Thus, here the fringe pattern is expressed as I共x兲 = A共x兲 + B共x兲cos关2␲f0x + ␸共x兲兴,

共1兲

where ␸共x兲 is the depth-related phase of the measured object and h共x兲 is the height. When L0 Ⰷ h共x兲 it can be calculated as

␸共x兲 = −

2 ␲ f 0d L0

共2兲

f0 is the fundamental frequency of the carrier, and A共x兲 and B共x兲 are the background intensity and the fringe contrast, respectively, with A共x兲 representing the zero spectrum in the frequency domain. Usually A共x兲 changes slowly. Here, we assume that it is constant in each half period of the fringe pattern [17]. Then the ith half-period background intensity Ai can be decomposed approximately from the corresponding 共Ii兲max and 共Ii兲min as Ai = 关共Ii兲max + 共Ii兲min兴 / 2. For higher accuracy, a particular decomposition is performed on the fringe pattern by using EMD. The essence of EMD is to identify the intrinsic oscillatory modes by their characteristic time scales in the data empirically, and then decompose the data accordingly [11]. The number of modes and frequencies of each mode are inherently determined by these time scales. A time signal is decomposed into a finite number of IMFs adaptively, which vary from high-frequency to low-frequency. An IMF is a function, which satisfies the following two conditions:

2

m

兺

x=0

兩hk−1共x兲 − hk共x兲兩2 h2k−1共x兲

.

共3兲

共4兲

When the SD is smaller than a threshold, we get the first IMF expressed as c1共x兲, which contains the highestfrequency component of the signal. (3) Separate c1共x兲 from the rest of the data by I共x兲 − c1共x兲 = r1共x兲. Note that the residue r1共x兲 still contains information of longer-period components. We can therefore treat the

(1) in the whole data set, the number of extrema and the number of zero crossings must either equal or differ at most by one; (2) at any point, the mean value of the envelope defined by the maxima and the envelope defined by the local minima is zero. The first condition is similar to the traditional narrowband requirements for a stationary Gaussian process. The second condition modifies the classical global requirement to a local one. It is necessary so that the instantaneous frequency will not have the unwanted fluctuations induced by asymmetric wave forms. Here, it will ensure that the back ground illumination can be eliminated effectively.

.

Ideally, N共x兲 should be an IMF while in reality it is not. So repeat the sifting procedure k times, until N共x兲 is an IMF. To guarantee that the IMF components retain enough physical sense of both amplitude and frequency modulations, some stopping criteria are used to terminate the sifting process, such as the sum of differences (SD) [11]

SD = h共x兲.

Imax共x兲 + Imin共x兲

Fig. 2.

Flow chart of EMD.

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residue as a new signal and apply the same sifting process as described above to decompose the rest of the signal as r1共x兲 − c2共x兲 = r2共x兲, r2共x兲 − c3共x兲 = r3共x兲, ... rn−1共x兲 − cn共x兲 = rn共x兲.

共5兲

Thus, locally, each IMF contains lower-frequency oscillations than the one that was extracted before. The sifting process is stopped when there are no IMFs in the residue. Thus, the time signal is decomposed into a finite number of IMFs 共c1共x兲 , c2共x兲 . . . cn共x兲兲 that vary from highfrequency to low-frequency and a residue rn共x兲 that represents the lowest trend component of the signal. Although this sifting process can work effectively, as a practical matter, it faces several issues that require further attention [18]. Among them, the end effect is the one that the decomposition is very sensitive to. EMD uses the cubic spline interpolation to create the top and bottom envelopes that are implemented in the first step of the sifting process. It is difficult to interpolate data near the beginnings or end, where the cubic spline fitting can have large swings. Estimation of top and bottom envelopes is

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difficult as there are not enough data. If the ends are left unattended, the end swings will eventually propagate inward and progressively corrupt the whole dataset, especially the low-frequency IMFs. To solve this issue, several useful methods have been proposed [19–22]. In FTP, the background illumination, which represents the zero spectrum in the frequency domain, is always changing slowly compared to the fringe pattern. By using the EMD method, the deformed fringe pattern is decomposed into a series of IMFs that vary from high-frequency to low-frequency. Because the background illumination is low-frequency, we can directly extract the high-frequency after decomposition, or subtract the decomposed lowfrequency from the original fringe pattern. By doing this, we can eliminate the zero spectrum. Figure 3 shows the process of the decomposition applied to one line of the captured fringe pattern. Figure 3(a) is the first time decomposition, where the solid curve is the signal, the dotted–dashed curve depicts the upper envelope obtained by using cubic spline interpolation, and the dashed curve represents the lower envelope. Figure 3(b) shows the new signal after the first time decomposition, where because it is not an IMF, the same process has operated on it again. The signal is finally decomposed as shown in Fig. 3(c) into six IMFs 共c1 , c2 . . . c6兲 that vary from high-frequency to low-frequency, and a residue R7. 4 cn共x兲, Filter out the high-frequency components g共x兲 = 兺n=1

Fig. 3. (a) Fringe pattern and its upper and lower envelopes during the first step. (b) Signal with the low frequency component subtracted by the first step. (c) Final result. (d) Fringe with zero component eliminated.

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Fig. 4.

Cross section of the object.

and the desired fringe pattern without the zero component is obtained shown in Fig. 3(d).

3. EXPERIMENT An experiment is carried out to test the effectiveness of the EMD method in eliminating the zero spectrum. Since the illumination effect is mainly caused by the light source, we use a one-dimensional algorithm [21] instead of a two-dimensional algorithm [23].

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In FTP, a common reason for the aliasing between the zero spectrum and the fundamental spectrum is the variation of height slope of the measured object [7]. Here, the measured triangle object’s maximum slope is tan ␪ = 1.8, which is high enough to cause frequency aliasing in our measuring system. Figure 4 shows the cross section of the object. A sinusoidal grating pattern is projected onto the object surface to encode depth information. The deformed fringe pattern is captured by a CCD camera and is shown in Fig. 5(a). The size of the figure is 512 ⫻ 512 pixels. Figure 5(b) shows the intensity distribution of the 256th row. Figure 5(c) depicts its frequency spectrum. Because the measured triangle is steep, the zero spectrum can be widely displayed, and there exists a serious frequency aliasing between the zero spectrum and the fundamental spectrum. Using traditional FTP, we cannot get the correct depth-related phase distribution, which is shown in Fig. 5(d). The frequency aliasing has apparently influenced the filtering procedure to filter out the useful information. To eliminate the zero spectrum, we use both Xu’s wavelet transform method and our method here, and select the same filter as used in Fig. 5. In the wavelet method, we choose the Daubechies wavelet [6,24] and the Symlets wavelet [24] as the mother wavelet separately. Figure 6(a) is the frequency distribution of that row after the zero spectrum is eliminated by using Daubechies wavelet de-

Fig. 5. (a) Deformed fringe pattern. (b) Intensity distribution of the 256th row. (c) Frequency distribution of that row. (d) Retrieved depth-related phase.

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Fig. 6. (a), (b) Spectrum treated by the Daubechies wavelet and the retrieved phase, respectively. (c), (d) Spectrum treated by Symlets wavelet and the retrieved phase, respectively. (e), (f) Spectrum treated by EMD and the restored depth-related phase, respectively.

composition method. Figure 6(b) shows the depth-related phase corresponding to that method. Figures 6(c) and 6(d) are the frequency distribution and the depth-related phase, respectively, by using the Symlets wavelet. In comparing Fig. 6(b) and Fig. 6(d), it can be seen that some useful information is lost when we use the Daubechies wavelet as the mother wavelet. By using the EMD method, the deformed fringe pattern is decomposed into a series of IMFs that vary from highfrequency to low-frequency adaptively. We directly extract the high-frequency after decomposition, or subtract the decomposed low-frequency from the original fringe pat-

tern. Thus the zero component is eliminated. Figure 6(e) shows the spectrum treated by the EMD method on the 256th row, from which we can see that the zero spectrum is well eliminated. Figure 6(f) shows retrieved depthrelated phase after the zero component is eliminated. By using the EMD method, the zero spectrum is eliminated and we get the correct phase. Comparing Fig. 6(b), Fig. 6(d), and Fig. 6(f), we can see that both wavelet methods and the EMD method can well eliminate zero spectrum effectively and enlarge the measurable range of FTP, while the EMD method is more adaptive. To further exemplify the method reported in this paper,

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Fig. 7. (a) Cat’s face model. (b) Spectrum of the fringe pattern. (c) Spectrum of the fringe pattern with zero spectrum eliminated. (d) Restored depth-related phase distribution.

we measure a cat’s face model. Figures 7(a) and 7(b) show the deformed fringe pattern captured by CCD and its spectrum, respectively. We perform our operation on the fringe pattern line by line. Figure 7(c) shows the spectrum of the fringe pattern after the zero component is eliminated by using the EMD method. Comparing Fig. 7(b) and Fig. 7(c), we can see that the zero spectrum is largely eliminated. The correct depth-related phase shown in Fig. 7(d) can be retrieved from the spectrum shown in Fig. 7(c), which proves that the useful information is not lost in the process of removing the zero spectrum. Therefore, it is an effective method.

ACKNOWLEDGMENTS The authors acknowledge the support by the National Natural Science Foundation of China (NSFC) (10876021, 60677028).

REFERENCES 1. 2. 3.

4. CONCLUSION EMD has been introduced to eliminate the zero spectrum in FTP. EMD decomposes the data according to the intrinsic oscillatory modes by their characteristic time scales in the data. There is no need to choose analysis tools such as wavelets, so it has a high adaptive capacity. It is also fast, without the need for capturing two fringe patterns with ␲ phase difference; only a sifting process is needed. The fringe pattern with the zero component eliminated is subsequently demodulated by using a standard FTP algorithm. The depth-related phase can be retrieved correctly. Experiments validate the feasibility of the proposed method, even when there is a serious frequency aliasing between the zero spectrum and the foundational spectrum.

4. 5. 6. 7.

8.

M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983). X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35, 263–284 (2001). J. Li, X. Su, and L. Guo, “Improved Fourier transform profilometry of the automatic measurement of threedimensional object shapes,” Opt. Eng. (Bellingham) 29, 1439–1444 (1990). W. Chen, P. Bu, and S. Zheng, “Study on Fourier transform profilometry based on bi-color projecting,” Opt. Laser Technol. 39, 821–827 (2007). W. Chen, X. Su, Y. Cao, Q. Zhang, and L. Xiang, “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267–1276 (2005). Xu Qinghong, Zhong Yuexian, and You Zhifu, “Study on phase demodulation technique based on wavelet transform,” Acta Opt. Sin. 20, 1617–1622 (2000). M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using a two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482–489 (2006). S. Mallat and W. L. Hwang, “Singularity detection and processing with wavelets,” IEEE Trans. Inf. Theory 38, 617–643 (1992).

Li et al. 9. 10. 11.

12.

13.

14. 15.

16. 17.

S. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989). W. Chen, J. Sun, and X. Su, “Discuss the structure condition and sampling condition of wavelet transform profilometry,” J. Mod. Opt. 54, 2747–2762 (2007). N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. R. Soc. London, Ser. A 454, 903–995 (1998). F. Wu and L. Qu, “An improved method for restraining the end effect in empirical mode decomposition and its applications to the fault diagnosis of large rotating machinery,” J. Sound Vib. 314, 586–602 (2008). M. Blanco-Velasco, B. Weng, and K. E. Barner, “ECG signal denoising and baseline wander correction based on the empirical mode decomposition,” Comput. Biol. Med. 38, 1–13 (2008). Q. Gao, C. Duan, H. Fan, and Q. Meng, “Rotating machine fault diagnosis using empirical mode decomposition,” Mech. Syst. Signal Process. 22, 1072–1081 (2008). P. D. Spanos, A. Giaralis, and N. P. Politis, “Time-frequency representation of earthquake accelerograms and inelastic structural response records using the adaptive chirplet decomposition and empirical mode decomposition,” Soil Dyn. Earthquake Eng. 27, 675–689 (2002). A. Boudraa and J. Cexus, “EMD-based signal filtering,” IEEE Trans. Instrum. Meas. 56, 2196–2202 (2007). Tang Yan, Chen Wen-jing, and Su Xian-yu, “Neural


18. 19.

20.

21.

22.

23. 24.

1201

network applied to reconstruction of complex objects based on fringe projection,” Opt. Commun. 278, 274–278 (2007). N. E. Huang, “Introduction to the Hilbert–Huang transform and its related mathematical problems,” Interdisciplinary Mathematics 5, 1–26 (2005). Zhao Zhidong and Wang Yang, “A new method for processing end effect in empirical mode decomposition,” in Proceedings of 2007 IEEE Conference on Communications, Circuits and Systems (IEEE, 2007), pp. 841–845. K. Zeng and M. He, “A simple boundary process technique for empirical mode decomposition,” in Proceedings of 2004 IEEE Conference on Geoscience and Remote Sensing Symposium (IEEE, 2004), pp. 4258–4261. G. Rilling, P. Flandrin, and P. Goncalves, “On empirical mode decomposition and its algorithms,” in IEEEEURASIP Workshop on Nonlinear Signal and Image Processing, NSIP-03, GRADO(I) (IEEE, 2003). R. Deering and J. F. Kaiser, “The use of a mask signal to improve empirical mode decomposition,” in Proceedings of IEEE 2005 Conference on Acoustics, Speech, and Signal Processing (IEEE, 2005), pp. 485–488. C. Damerval and S. Meignen, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett. 12, 701–704 (2005). G. Hu, Z. Ren, and Y. Tou “Symmetrical wavelet construction and its application for electrical machine fault signal reconstruction,” in Proceedings of the Fifth IEEE Conference on Machine Learning and Cybernetics (IEEE, 2006), pp. 3734–3737.