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Envelope peak detection algorithm based on the CEEMDAN in white light interferometry

Yinrui Li, Yingzhe Yang, Jiale Zhang, Qun Yuan, and Yiyong Liang

Open Access Open Access

Abstract

White light scanning interferometry (WLSI) is a key non-contact three-dimensional (3D) morphometry method. The traditional envelope peak detection (EPD) algorithm is a typical WLSI surface recovery algorithm without phase calculation, but its accuracy is susceptible to noise and fitting errors. In this paper, we propose an improved EPD algorithm based on the complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) to solve the above problem. The proposed algorithm can accurately locate the position of the envelope peak. The simulation and experiment results demonstrate that the proposed method is effective in improving the accuracy of the measurements.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The development of precision optics components demands more advanced detection technology [1]. White light scanning interferometry (WLSI) is a mature non-contact optical measurement method with a broad spectrum and short optical coherence length and is a key and widely researched technology [2–4]. The mainstream surface topography solving methods for white light interferometry (WLI) can be divided into two categories: time-domain modulation algorithms based on the envelope amplitude information of the interference signal [5–8] and frequency-domain algorithms based on the signal spectrum phase analysis [9–11].

The time-domain modulation algorithm is mainly used to find the peak of the interferometric signal’s envelope, which is generally obtained by the Fourier transform or Hilbert transform [12,13]. The frequency-domain algorithm gets the phase information of the interference signal by phase shift or Fourier transform algorithm, then the height information is solved by the phase information [10]. The frequency-domain algorithm is generally more accurate than the time-domain modulation algorithm [14], especially the frequency domain algorithm (FDA) proposed by P. de Groot [3]. While the phase calculated by phase shift is susceptible to phase ambiguity, thus additional means are required to correct the phase [15]. The envelope peak detection method (EPD) can avoid this problem because it does not calculate the phase. Paul J. Caber first proposed to determine the envelope peak by fitting simple curves to discrete points near the peak [16]. Min-Cheol Park built on this to determine the envelope peak simply with a single-step matrix multiplication [14]. Larkin proposed to find the peak position of the envelope by fitting a Gaussian function [8]. However, the EPD algorithms select a small number of discrete signals near the peak to fit the envelope curve, which inevitably brings about fitting errors. Furthermore, mechanical vibrations and optical noise during the scanning process, and the overlap of zero-order and second-order fringes caused by low reflections on the surface, can lead to the wrong detection of envelope peaks [17].

In this paper, a new envelope peak detection algorithm is proposed to reduce system noise by the complete ensemble empirical mode decomposition with adaptive noise algorithm (CEEMDAN) algorithm. The CEEMDAN algorithm, proposed by Torres et al. [18], is an improvement of the empirical mode decomposition (EMD) method [19], which decomposes the signal to be processed into a series of single-component signals by adding a moderate amount of Gaussian white noise. Each single-component signal contains only one mode of oscillation, and these decomposed signals are called intrinsic mode functions (IMFs). The noise can be effectively decreased by selecting the IMF closest to the original signal to reconstruct the signal. The proposed algorithm is divided into three steps: firstly, the CEEMDAN algorithm is used to reconstruct the interference signal. Then the FFT algorithm is used to extract the envelope of the reconstructed signal. Finally, the centroid method [10] is used to locate the peak of the envelope. The proposed algorithm can effectively improve the accuracy of envelope peak detection by simulation and experimental verification.

2. Principles

2.1 Interference signals

WLSI is an optical non-contact 3D morphometry method. The interference fringes appear near the position of zero optical path difference (ZOPD) due to the broad spectrum and short coherence length of WLI. During the vertical scan, a series of interferograms are acquired by the CCD. The height information of each pixel point can be obtained by solving the ZOPD position of each pixel point on the interferogram, and then the 3D profile of the sample can be recovered. The intensity of each pixel point on the interferogram can be expressed as [9]:

$$I(z) = {I_\beta } + \alpha {I_\beta }\exp \left[ { - {{\left( {\frac{{z - {z_0}}}{{{l_c}}}} \right)}^2}} \right]\cos \left[ {\frac{{4\pi }}{{{\lambda_0}}}({z - {z_0}} )+ {\mathrm{\varphi }_0}} \right], $$
where Iβ is the background intensity; α is the fringe contrast; lc is the coherence length of the light source; z is the scanning position along the optical axis; z0 is the position of the ZOPD; φ0 is the phase difference between the reference beam and the object beam. It is worth noting that Eq. (1) only applies to very small NA values and broadband light sources with Gaussian spectrum, and is not a generalized equation [20]. For convenience, Eq. (1) can be rewritten as:
$$I(\textrm{z}) = {I_\beta }\{ 1 + \mathrm{\alpha }g(z)\cos [\phi (z)]\}, $$
where $g(z )= \exp \left[ { - {{\left( {\frac{{z - {z_0}}}{{{l_c}}}} \right)}^2}} \right]$ is the envelope function of the signal and $\phi (z )= \frac{{4\pi }}{{{\mathrm{\lambda }_0}}}({z - {z_0}} )+ {\mathrm{\varphi }_0}$ is the phase function of the signal. The height information can be obtained by detecting the peak of the envelope. The phase ambiguity can be avoided without calculating the phase, yet the peak of the envelope may be incorrectly located. As shown in Fig. 1, there is a deviation of Δl from the peak of the envelope to the actual ZOPD position.

 figure: Fig. 1.

Fig. 1. The illustration of the WLSI signal and its envelope [9].

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2.2 Envelope extraction method

The envelope of the WLI signal is generally obtained by the Fast Fourier transform (FFT).

According to Eq. (1), the FFT of the WLI signal can be written as [9]

$$\begin{aligned} \textrm{FFT}[\textrm{I}(\textrm{z})] &= 2\pi {I_\beta }\delta (k) + \alpha G(k){e^{ - hkj}}\\& \textrm{ }\ast \frac{1}{2}\left[ {{e^{\varphi j - \frac{{4\pi h}}{{{\lambda_0}}}j}}\delta \left( {k - \frac{{4\pi }}{{{\lambda_0}}}} \right) + {e^{\frac{{4\pi h}}{{{\lambda_0}}}j - \varphi j}}\delta \left( {k + \frac{{4\pi }}{{{\lambda_0}}}} \right)} \right], \end{aligned}$$
where δ(k) is the impact function; k = 2π / z is the spatial angular frequency; G(k) is the Fourier transform of g(z - h); λ0 is the central wavelength; h is the position of ZOPD; φ is the phase information and the asterisk means convolution. After the FFT, the spectrum contains equal positive and negative frequency components and a zero-frequency component. The zero frequency and negative frequency components are removed, and the positive frequency components are moved to the center of the spectrum. After the inverse Fourier transform, the following expression can be obtained [9]:
$$\begin{aligned} IFFT \left[ {\frac{\alpha }{2}{e^{\varphi j - hkj}}\textrm{G}(k)} \right] &= \frac{\alpha }{2}{e^{\varphi j}}g(z)\ast \delta (z - h)\\& \textrm{ } = \frac{\alpha }{2}{e^{\varphi j}}g(z - h). \end{aligned}$$

It can be seen that the envelope curve g(z - h) is directly proportional to the amplitude in Eq. (4). The process of Fourier transform is shown in Fig. 2.

 figure: Fig. 2.

Fig. 2. The interference signal’s envelope is obtained by Fourier transform. (a) Original WLI signal. (b) The FFT spectrum with equal amounts of positive and negative frequencies and zero-frequency. (c) The envelope obtained by FFT.

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After obtaining the interference signal’s envelope by Fourier transform, the traditional EPD method determines the envelope peak location by selecting a few data points around the maximum value of the envelope function and fitting them by a parabolic or Gaussian function, as shown in Fig. 3. However, the envelope peak position is often incorrectly located by noise and fitting errors, etc. similar to Fig. 1.

 figure: Fig. 3.

Fig. 3. Fitting the envelope using the EPD method.

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2.3 New EPD algorithm with CEEMDAN

CEEMDAN is an improvement algorithm for EMD, which is used to analyze non-linear and non-stationary signals. The EMD has the problem of modal mixing, which can be reduced by adding white noise, but this will introduce extra noise that affects the subsequent analysis and processing of the signal. The CEEMDAN algorithm solves the problem by the following process to mitigate the effect of noise [21]. The algorithm is described as follows [18]:

  • (1) Adding N sets of Gaussian white noise signals to the interference signal to get $B(i) = I + \varepsilon \omega (i)$(i = 1, …, N), the first IMF is obtained after EMD decomposition.
    $$E(B(i)) = {C_1}(i) + r(i), $$
    where I is the interference signal, ω(i) is the Gaussian white noise signal with zero mean and unit variance, ε is the amplitude of the added noise, E is the operator of EMD decomposition, C1 is the first IMF of EMD and r is the residual after EMD decomposition
  • (2) The first IMF of CEEMDAN is obtained by taking its average value:
    $${\tilde{C}_1} = \frac{1}{N}\sum\limits_{i = 1}^N {{C_1}(i)}. $$
  • (3) Calculate the residual after removing the first IMF of CEEMDAN:
    $${r_1} = I - {\tilde{C}_1}. $$
  • (4) The second IMF and the second residual of CEEMDAN are obtained in the same way:
    $$E({r_1} + \varepsilon \omega (i)) = {C_2}(i) + r(i), $$
    $${\tilde{C}_2} = \frac{1}{N}\sum\limits_{i = 1}^N {{C_2}(i)}, $$
    $${r_2} = {r_1} - {\tilde{C}_2}. $$
  • (5) For k = 3, …, K, the kth IMF and the residual of CEEMDAN are calculated as:
    $${\tilde{C}_k} = \frac{1}{N}\sum\limits_{i = 1}^N {{C_k}(i)}, $$
    $${r_k} = {r_{k - 1}} - {\tilde{C}_k}. $$
  • (6) The above steps are repeated until the obtained residual signal is a monotonic function, and the EMD decomposition cannot be continued, at which point the algorithm ends.

With the CEEMDAN algorithm, the interference signal is decomposed into a series of IMFs, and we need to select one of them as the reconstructed signal. Calculating the correlation coefficient of each IMF and the original signal, and selecting the IMF with the largest correlation coefficient as the reconstructed signal to reconstruct the 3D shape of the sample. The expression for the correlation coefficient is shown as [22]:

$$\rho = \frac{{\sum {({F_k}(i) - {{\bar{F}}_k})(F(i) - \bar{F})} }}{{\sqrt {\sum {{{({F_k}(i) - {{\bar{F}}_k})}^2} \cdot \sum {{{(F(i) - \bar{F})}^2}} } } }}, $$
where ${F_k}$ is the kth IMF and ${\bar{F}_k}$ denotes its average value; F is the original signal and $\bar{F}$ denotes its average value. After obtaining the reconstructed signal with the CEEMDAN algorithm, the envelope curve is obtained using FFT, and then the envelope is processed by the centroid method [10]. The final height information is:
$$H = \frac{{\sum {{{(M(i))}^2} \times i} }}{{\sum {{{(M(i))}^2}} }} \times \Delta z, $$
where H is the calculated result of the height information, Δz is the step length per scan, and M denotes the envelope curve function. The flow chart of the method is shown in Fig. 4.

 figure: Fig. 4.

Fig. 4. Flow chart of the proposed method.

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3. Simulation results

In order to verify the effectiveness of the proposed method, a set of simulated white light interference signal data are generated to compare the proposed method with EPD algorithm, the FDA algorithm and the white light phase-shifting interferometry (WLPSI) algorithm. The discrete WLI signal is set to:

$$I(i) = 200 + 200\exp [ - \frac{{{{(i - 72)}^2}{\Delta ^2}}}{{{l_c}^2}}]\cos [\frac{{4\pi }}{{{\lambda _0}}}(i - 72)\Delta ], $$
where lc is the coherence length of the light source, set to 600 nm; Δ is the scanning step length 72 nm; λ0 is the central wavelength of the light source, and is set to 576 nm. The position of ZOPD is located at 72, and the corresponding height is 5184 nm. To demonstrate that CEEMDAN can suppress the impact of noise, we add a random noise with an intensity of 5% of the average light intensity to the ideal interference signal as shown in Fig. 5(a). The IMFs decomposed by the CEEMDAN algorithm are shown in Fig. 6, where IMF1 is used as the reconstructed signal as shown in Fig. 5(b). It can be seen that IMF1 can separate the background light and has a certain suppression effect on noise.

 figure: Fig. 5.

Fig. 5. (a) Interference signal with 5% random noise. (b) IMF1.

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 figure: Fig. 6.

Fig. 6. The IMFs obtained by decomposing the signal of Fig. 5. (a) using the CEEMDAN algorithm.

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We add a random noise with an intensity of 10% of the average light intensity to the simulated ideal interference signal. The FFT transform is used to extract the signal envelope and compare the results of different algorithms. To reduce the uncertainty of the results caused by randomness, each algorithm is simulated 10,000 times and the average of the processing results is calculated. As shown in Table 1, the EPD (P) denotes fitting the envelope by a polynomial and EPD (G) denotes fitting the envelope by a Gaussian function. The simulation results demonstrate that the accuracy of both envelope peak detection methods is improved to some extent by combining the CEEMDAN algorithm. The centroid method combined with CEEMDAN has the smallest error. The FDA has the worst accuracy due to noise, phase unwrapping and then fitting a straight line will get the wrong slope. Due to the additional process of the CEEMDAN algorithm, the time lost increases correspondingly. However, the time consumed is within the acceptable range with MATLAB’s multi-core parallel computing capability.

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Table 1. The processing results of different algorithms

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To further investigate the effect of system noise on the measurement results, six sets of random noise are added to the original interference signal separately. The noise intensities are 5%, 10%, 15%, 20%, 25%, and 30% of the average interference fringe intensity, respectively. As shown in Table 2, the absolute error of the surface height is calculated 10000 times with the different algorithms. The absolute errors of these algorithms become larger as the noise intensity increases. The three envelope peak detection algorithms combined with CEEMDAN feature stronger noise resistance with smaller absolute error at different noise levels The RMS errors of these algorithms at different noise levels are calculated as shown in Table 3. It can be seen that the centroid method with CEEMDAN is the most stable with the smallest RMS error. In general, the above results demonstrate that the centroid algorithm with CEEMDAN can improve the accuracy of envelope peak detection. It should be added additionally that the FDA algorithm has a disproportionately large RMS Error.

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Table 2. The absolute error of each algorithm with different noise levels

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Table 3. The RMS error of each algorithm with different noise levels

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4. Experimental results

To further verify the feasibility and effectiveness of the proposed method, the experiments were conducted. The experimental setup included a 10x Mirau interference objective with a numerical aperture of 0.3, a nano-precision piezoelectric transducer (PZT), and a CCD camera with an 8-bit gray level and a resolution of 1024 pixels × 1024 pixels. The light source was an LED with a broad continuous spectrum. The test sample was a standard step block with a height of 7.805 µm. The PZT moved 72 nm per scan, and then the CCD camera recorded the interference fringes and stored them on the computer. The system setup and interferogram are shown in Fig. 7.

 figure: Fig. 7.

Fig. 7. (a) Experimental setup. (b) Interferogram of the 7.805 µm standard step block.

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The height of the standard steps can be obtained after calculating the height data of the upper and lower surfaces. Ten times tests were conducted with the centroid algorithm combined with CEEMDAN and the results are shown in Table 4. It can be observed that the average step height is 7.829 µm, which is close to the standard step height of 7.805 µm. Furthermore, the standard deviation is tiny, only 0.008 µm.

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Table 4. The height results calculated by the proposed method

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To compare these algorithms, the sample was tested with the above system. The 3D morphology and 2D cross-sectional profile of the sample are shown in Fig. 8–15, respectively.

 figure: Fig. 8.

Fig. 8. Measurement of 7.805 µm standard step sample with EPD (P) algorithm without CEEMDAN: (a) 3D profile and (b) 2D cross-sectional profile.

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 figure: Fig. 9.

Fig. 9. Measurement of 7.805 µm standard step sample with EPD (P) algorithm with CEEMDAN: (a) 3D profile and (b) 2D cross-sectional profile.

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 figure: Fig. 10.

Fig. 10. Measurement of 7.805 µm standard step sample with EPD (G) algorithm without CEEMDAN: (a) 3D profile and (b) 2D cross-sectional profile.

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 figure: Fig. 11.

Fig. 11. Measurement of 7.805 µm standard step sample with EPD (G) algorithm with CEEMDAN: (a) 3D profile and (b) 2D cross-sectional profile.

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 figure: Fig. 12.

Fig. 12. Measurement of 7.805 µm standard step sample with Centroid algorithm without CEEMDAN: (a) 3D profile and (b) 2D cross-sectional profile.

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 figure: Fig. 13.

Fig. 13. Measurement of 7.805 µm standard step sample with Centroid algorithm with CEEMDAN: (a) 3D profile and (b) 2D cross-sectional profile.

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 figure: Fig. 14.

Fig. 14. Measurement of 7.805 µm standard step sample with WLPSI algorithm: (a) 3D profile and (b) 2D cross-sectional profile.

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 figure: Fig. 15.

Fig. 15. Measurement of 7.805 µm standard step sample with FDA algorithm: (a) 3D profile and (b) 2D cross-sectional profile.

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It can be seen that using the centroid algorithm to extract the envelope peak can effectively reduce the burr in Fig. 8–13. Due to other factors such as noise, the envelope obtained from the actual tested WLI signal sometimes does not match the single-peak characteristic shown in Fig. 16(a). When the envelope has a bimodal character as shown in Fig. 16(b), the polynomial fit or Gaussian fit will have a large deviation. The centroid method is not overly affected since it is solved by the entire envelope.

 figure: Fig. 16.

Fig. 16. Envelope of WLI signal obtained from actual test: (a) Single-peak (b) Bimodal.

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Meanwhile, to compare the accuracy of different algorithms, each algorithm was tested with 10 sets of data, the results are shown in Table 5. The experimental results demonstrate that the centroid algorithm with CEEMDAN is more precise and stable compared to the WLPSI and EPD (with/without CEEMDAN) algorithms, which is close to the accuracy of the FDA algorithm. Previous simulations focused on comparing the noise levels that different algorithms can withstand. However, the added noise level is much greater than in actual measurements, so the FDA algorithm does not behave the same way in actual tests as it does in simulations. In fact, the FDA algorithm is extremely accurate in real tests.

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Table 5. The height results calculated by different algorithms

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5. Conclusion

In this paper, we propose a novel 3D morphology recovery algorithm for WLI based on the CEEMDAN algorithm and centroid method to reduce the error of envelope peak detection. The simulation results demonstrate that the proposed method provides higher accuracy and stronger noise immunity compared to the conventional EPD and WLPSI algorithms. The experimental verification was completed using a sample with a height of 7.805 µm, and the results show that the accuracy of our proposed method is close to that of the FDA algorithm and higher than that of the traditional EPD algorithm and WLPSI algorithm. The simulation and experimental results validate the effectiveness of the method.

Funding

National Natural Science Foundation of China (62175107).

Disclosures

The authors declare no conflicts of interests.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author upon reasonable request.

References

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16. P. Caber, “Interferometric profiler for rough surfaces,” Appl. Opt. 32(19), 3438–3441 (1993). [CrossRef]  

17. Z. Lei, X. Liu, L. Chen, W. Lu, and S. J. M. Chang, “A novel surface recovery algorithm in white light interferometry,” Measurement 80, 1–11 (2016). [CrossRef]  

18. M. E. Torres, M. A. Colominas, G. Schlotthauer, and P. Flandrin, “A complete ensemble empirical mode decomposition with adaptive noise,” 2011 IEEE international conference on acoustics, speech and signal processing (ICASSP), IEEE (2011), 4144–4147.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author upon reasonable request.

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Figures (16)
Fig. 1.
Fig. 1. The illustration of the WLSI signal and its envelope [9].

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Fig. 2.
Fig. 2. The interference signal’s envelope is obtained by Fourier transform. (a) Original WLI signal. (b) The FFT spectrum with equal amounts of positive and negative frequencies and zero-frequency. (c) The envelope obtained by FFT.

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Fig. 3.
Fig. 3. Fitting the envelope using the EPD method.

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Fig. 4.
Fig. 4. Flow chart of the proposed method.

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Fig. 5.
Fig. 5. (a) Interference signal with 5% random noise. (b) IMF1.

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Fig. 6.
Fig. 6. The IMFs obtained by decomposing the signal of Fig. 5. (a) using the CEEMDAN algorithm.

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Fig. 7.
Fig. 7. (a) Experimental setup. (b) Interferogram of the 7.805 µm standard step block.

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Fig. 8.
Fig. 8. Measurement of 7.805 µm standard step sample with EPD (P) algorithm without CEEMDAN: (a) 3D profile and (b) 2D cross-sectional profile.

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Fig. 9.
Fig. 9. Measurement of 7.805 µm standard step sample with EPD (P) algorithm with CEEMDAN: (a) 3D profile and (b) 2D cross-sectional profile.

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Fig. 10.
Fig. 10. Measurement of 7.805 µm standard step sample with EPD (G) algorithm without CEEMDAN: (a) 3D profile and (b) 2D cross-sectional profile.

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Fig. 11.
Fig. 11. Measurement of 7.805 µm standard step sample with EPD (G) algorithm with CEEMDAN: (a) 3D profile and (b) 2D cross-sectional profile.

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Fig. 12.
Fig. 12. Measurement of 7.805 µm standard step sample with Centroid algorithm without CEEMDAN: (a) 3D profile and (b) 2D cross-sectional profile.

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Fig. 13.
Fig. 13. Measurement of 7.805 µm standard step sample with Centroid algorithm with CEEMDAN: (a) 3D profile and (b) 2D cross-sectional profile.

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Fig. 14.
Fig. 14. Measurement of 7.805 µm standard step sample with WLPSI algorithm: (a) 3D profile and (b) 2D cross-sectional profile.

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Fig. 15.
Fig. 15. Measurement of 7.805 µm standard step sample with FDA algorithm: (a) 3D profile and (b) 2D cross-sectional profile.

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Fig. 16.
Fig. 16. Envelope of WLI signal obtained from actual test: (a) Single-peak (b) Bimodal.

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Tables (5)

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Table 1. The processing results of different algorithms

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Table 2. The absolute error of each algorithm with different noise levels

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Table 3. The RMS error of each algorithm with different noise levels

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Table 4. The height results calculated by the proposed method

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Table 5. The height results calculated by different algorithms

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Equations (15)

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I ( z ) = I β + α I β exp ⁡ [ − ( z − z 0 l c ) 2 ] cos ⁡ [ 4 π λ 0 ( z − z 0 ) + φ 0 ] ,
I ( z ) = I β { 1 + α g ( z ) cos ⁡ [ ϕ ( z ) ] } ,
FFT [ I ( z ) ] = 2 π I β δ ( k ) + α G ( k ) e − h k j   ∗ 1 2 [ e φ j − 4 π h λ 0 j δ ( k − 4 π λ 0 ) + e 4 π h λ 0 j − φ j δ ( k + 4 π λ 0 ) ] ,
I F F T [ α 2 e φ j − h k j G ( k ) ] = α 2 e φ j g ( z ) ∗ δ ( z − h )   = α 2 e φ j g ( z − h ) .
E ( B ( i ) ) = C 1 ( i ) + r ( i ) ,
C ~ 1 = 1 N ∑ i = 1 N C 1 ( i ) .
r 1 = I − C ~ 1 .
E ( r 1 + ε ω ( i ) ) = C 2 ( i ) + r ( i ) ,
C ~ 2 = 1 N ∑ i = 1 N C 2 ( i ) ,
r 2 = r 1 − C ~ 2 .
C ~ k = 1 N ∑ i = 1 N C k ( i ) ,
r k = r k − 1 − C ~ k .
ρ = ∑ ( F k ( i ) − F ¯ k ) ( F ( i ) − F ¯ ) ∑ ( F k ( i ) − F ¯ k ) 2 ⋅ ∑ ( F ( i ) − F ¯ ) 2 ,
H = ∑ ( M ( i ) ) 2 × i ∑ ( M ( i ) ) 2 × Δ z ,
I ( i ) = 200 + 200 exp ⁡ [ − ( i − 72 ) 2 Δ 2 l c 2 ] cos ⁡ [ 4 π λ 0 ( i − 72 ) Δ ] ,
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