Profilometry measures surface roughness and surface variation. Measured fringe patterns contain phase information, which is extracted using different algorithms to deduce the corresponding surface variation. Fringe-profile profilometry uses both incoherent (white light) and coherent (laser) interference-fringe patterns, while actual projection or recording conditions include noise and potentially nonsinusoidal waveforms. For dynamic profile characterization, where only a single fringe-pattern measurement is available, phase retrieval is especially crucial.

One-frame fringe-pattern analysis techniques include Fourier-transform (FT),^{1} wavelet-transform (WT),^{2} composite structured light, coding-based, and color phase-shifting methods. A subset of FT methods, windowed FTs (WFTs), include windowed Fourier filtering (WFF) and windowed Fourier ridges (WFR).^{3} We compare the performance of FT, WFT, and WT methods in the presence of random and speckle noise as well as nonsinusoidal waveforms. Transform-based fringe-pattern processing methods^{4} either convert a real into an analytic signal through filtering (FT ^{1} and WFF^{3}) or find the best match (‘ridges’) between a real signal and an analytic basis (WFR^{3} and WT^{2}).

The principle of Fourier transformation is to extract the fundamental component in the frequency domain using an FT. Its phase is then retrieved by taking an inverse FT of the selected component. The principle of the WFF method is similar to that of the FT technique, but with additional computational steps of windowing and summation before and after the forward and inverse FTs. Both methods retrieve the phase values from the filtered exponential fringe pattern.

Unlike the WFF approach, the WFR method uses only a single forward FT. Similarly, the WT method uses a single WT only once for each window. The phase-retrieval strategy in both the WFR and WT methods is to determine the phase and frequencies of the target pixel (the central pixel in the window) using ridge frequencies that best match the fringe in the window of interest. WFR and WT differ in their window size. WFR uses a constant window, while the WT window varies with fringe frequency. Additionally, the phases extracted by the WFR and WT ridge-based algorithms require application of a phase-compensation process to reduce the theoretical phase errors.

In our simulation, the true phase is set as 3.5 times a known function. The simulated function, *peaks*(*x*, *y*), is a MATLAB^{®} built-in function. The size of the fringe patterns is 256×256pixels^{2}. Figure 1 presents several simulated fringe patterns under different conditions, including an ideal pattern and its counterparts with random noise, speckle noise, and a nonsinusoidal waveform.

**Figure 1. **Fringe pattern under different conditions. (a) Ideal fringe pattern and its counterparts with (b) random noise, (c) speckle noise, and (d) a nonsinusoidal waveform.

Real measurements contain noise. To compare the methods' sensitivity to noise, we simulated different levels of random and speckle noise and examined the resulting errors (see Figure 2). The captured fringe patterns are commonly not standard sinusoidal waveforms because of the nonlinear response of the entire measurement system. Therefore, we also simulated the effect of a nonsinusoidal waveform to determine its effect on the different phase-retrieval methods. We found that ridge-based methods (WFR and WT) are superior to filtering-based techniques (FT and WFF: see Figure 2), 2D methods perform better than 1D approaches, and local methods are superior to their global counterparts. Overall, in the presence of noise and nonlinear waveforms, the 2D WFR and WT methods perform best.

**Figure 2. **Phase error under noise conditions. (a) Normally distributed random noise, (b) speckle noise, and (c) nonsinusoidal waveform for the Fourier-transform (FT), windowed FT (WFT), windowed Fourier ridges (WFR), and wavelet-transform (WT) methods. The WFR and WT methods perform best. a_{1}, a_{2}: Amplitudes.

We generated an observable fringe pattern using a laser diode and diffractive optical element with both a nonsinusoidal waveform and speckle noise. Figure 3 shows the unwrapped phase difference from the experimental fringe patterns using these 2D phase-retrieval methods. Figure 3(a) shows the performance of the 2D FT method, with obvious errors, because global spectral filtering is difficult under noisy and nonsinusoidal-waveform conditions. Figure 3(b) shows the result of the 2D WFF method, with significant errors, since spectrum overlap is more severe because of spectral expansion when using the spatially windowed strategy.

**Figure 3. **Retrieved, unwrapped phase difference between deformed and reference phase. (a) 2D FT method with obvious errors. (b) 2D WFF with significant errors. (c) 2D WFR. (d) 2D WT. The 2D WFR and WT results are superior.

Figure 3(c) and (d) show phase results from the 2D WFR and WT methods, which continue to perform best, consistent with our simulation results. However, the 2D WFR and WT methods are more computationally expensive. We will next investigate different applications of these fringe-pattern processing methods in fringe-based optical metrology.

Anand K. Asundi, Lei Huang, Kemao Qian

Nanyang Technological University

Singapore, Singapore

References:

1. X. Su, W. Chen, Fourier transform profilometry: a review, *Opt. Lasers Eng*. 35, pp. 263-284, 2001.

2. Z. Wang, H. Ma, Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing, *Opt. Eng. *45, pp. 045601, 2006.

3. Q. Kemao, Windowed Fourier transform for fringe pattern analysis, *Appl. Opt*. 43, pp. 2695-2702, 2004.

4. L. Huang, Q. Kemao, B. Pan, A. K. Asundi, Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry, *Opt. Lasers Eng*. 48, pp. 141-148, 2010.