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Abstract
Single-shot high-accuracy reconstruction of three-dimensional (3D) objects plays an essential role in many fields. Wavelet-based fringe projection profilometry (WFPP) is a popular single-shot imaging method due to its multiple-resolution analysis and non-contact property. However, the construction of mother wavelets for high-accuracy WFPP is still challenging. In this paper, a general framework of two-dimensional (2D) compact support complex wavelet employing the dual-tree concept combined with the one-dimensional (1D) asymmetric wavelet function is proposed. The constructed wavelet has an asymmetric frequency envelope in the radial direction and concentrated energy in a single quadrant of the frequency domain. A method that the constructed wavelet applied in dual-angle rotation-based 2D WFPP is presented for higher accuracy. The theoretical analysis of the proposed method is presented, and a comparison of frequency characteristics between the constructed wavelet and other popular wavelets is conducted. Computer simulations and experiments have demonstrated that the proposed method provides a better reconstruction of 3D shapes in single-shot WFPP.
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1. Introduction
Fringe projection profilometry (FPP) based on structured illumination is a type of non-contact optical profilometry technique to reconstruct the three-dimensional (3D) surface [1,2]. The application of a digital fringe projection system makes the structured light technique more diverse [3,4]. It has become an exciting tool for surface topography, industry inspection, intelligent manufacturing, biomedicine, and human-computer interaction because of its high measurement precision, whole-field analysis, ease of automation, etc.
In the past few decades, the FPP has developed tremendously. Demodulating the underlying phase distribution from the deformed fringe pattern is the main process of FPP. Implementations of the FPP can be mainly divided into two major groups. The first is multiple-shot fringe pattern methods such as phase shift profilometry (PSP) [5,6], which provides high measurement accuracy. Multiple-shot methods in real-time 3D shape measurement have been researched deeply [7]. With the improvement of hardware and software of the measurement system, some critical issues in dynamic measurement have been solved, which pushes dynamic object measurement forward [8,9]. The second is the methods based on transformation analysis, such as Fourier transform profilometry (FTP) [10,11], Windowed Fourier Transform profilometry (WFTP) [12,13], S-transform profilometry (STP) [14], and Wavelet Transform profilometry (WTP) [15,16]. They theoretically require only one pattern to be captured, namely called single-shot methods, and are suitable for dynamic object measurement. Huang [17] compared these methods, and Qian [18] compared the characteristics of the different carrier fringe pattern methods. Recently, deep learning was discussed for single-shot fringe absolute 3D shape measurement [19].
The most popularly used single-shot method is FTP. However, the FTP lacks local analytic ability as a global transform method. Moreover, the inherent issue of spectrum leakage and spectrum aliasing limits the measurement accuracy and the range of FTP, which makes it impossible to measure complex objects with discontinuous surface areas and large gradient variations. Other single-shot methods mentioned above can be regarded as alternative algorithms [12–16]. WFTP uses a sliding window function to make up for the shortcomings of FTP to analyze the fringes locally. However, the fixed window is difficult to satisfy the optimal spatial and frequency resolution requirement of local fringes. The STP employed a scalable window function according to the frequency of local fringes in two orthogonal directions to analyze the fringe. WFTP and STP can obtain the Fourier spectrum by adding all local spectra along the window sliding direction. Therefore, they provide two stratagems to calculate the phase information from the fringe, the “ridge” method and the “filter” method. However, in WFTP and STP, the window function cannot be rotated to optimally extract the local fundamental spectrum. WTP is suited to obtain phase information from fringes with bigger density fluctuation in practice measurement because of the characteristics of multi-resolution analysis and rotation ability, offering good localization in the time-frequency domains.
As a research topic, many achievements using one-dimensional (1D) wavelets or two-dimensional (2D) wavelets to obtain phase map information from fringe patterns have been presented. M. Afif et al. [20] employed a 1D Paul wavelet to extract the phase from fringe patterns, and Gdeisat M. et al. [21] proposed using the 2D continuous Paul wavelet transform to extract the phase of spatial carrier fringe patterns. Zhong [22] proposed the Morlet wavelet to analyze the phase distributions of the spatial carrier-fringe pattern and discussed the robust applicability condition based on the evaluation of the local linearity. Abid et al. [23] proposed a cost function to detect the wavelet ridge, which provides a more reliable result in ridge extraction. Jiang et al. [24] presented a multi-frequency fringe projection profilometry based on wavelet transform, and the wrapped phase maps are calculated using the 1D complex Morlet continuous wavelet transform (CWT). Li et al. [25] proposed a 2D CWT algorithm employing a real 2D Mexican hat (Mexihat) mother wavelet and presented a comparison with the Morlet and Fan wavelet. Hani A. et al. [26] proposed a shadow detection method in FPP, where the Haar wavelet is required for differentiating shadow and non-shadow areas. However, these methods naturally focus on applying existing wavelets and traditional single-angle rotation wavelet transform. Considering the asymmetric spectrum envelope of some wavelets, the Dual-angle rotation 2D WTP [27], which introduces an additional rotation operation to improve the directivity of wavelets, was proposed and applied to calculate the phase map from a single fringe accurately.
In this paper, a general framework of 2D compact support complex wavelet in FPP employing the dual-tree concept combined with the basic wavelet function is proposed. The framework provides a perceptual intuition flow to form a 2D compact support complex wavelets suitable for carrier fringe demodulation. As long as the selected 1D wavelet function has an asymmetric spectrum, the constructed 2D wavelet has an asymmetric frequency envelope in the radial direction, resulting in better frequency match ability to the local fringe. Moreover, the constructed wavelets can be used not only in traditional 2D Wavelet-based fringe projection profilometry (2D WFPP) but also in Dual-angle rotation-based 2D WFPP for higher accuracy. A comparative analysis between our method and some popular 2D wavelets used in 3D measurement, including Fan wavelet and 2D complex Mexican hat selectivity (Cmexhs) wavelet, is conducted.
The rest of this paper is arranged as follows. In Section 2, we described the system of fringe projection profilometry and the principle of 2D WTP based on the Dual-Angle rotation algorithm. Section 3 is devoted to the detail of the general framework and the comparative analysis of some popular wavelets. In Section 4 and Section 5, computer simulations and experiments are carried out to show the performance and comparison between the proposed method and other methods. Lastly, the paper is concluded in Section 6.
2. System and principle
2.1 System of fringe projection profilometry
The schematic diagram of the optical geometry of a typical FPP system based on the trigonometric principle is shown in Fig. 1. A sinusoidal fringe pattern is projected to the surface of the measured object. Then a deformed fringe pattern is captured by a CCD camera at another position. The deformed fringe pattern captured by the CCD camera can be expressed as
where $A(x,y)$ and $B(x,y)$ are the background intensity and the fringe contrast, respectively. ${f_0}$ is the spatial carrier frequency, $\varphi (x,y)$ is phase modulation resulting from the object height distribution, and $n(x,y)$ is the noise.Different phase demodulation algorithms are used to extract the phase map according to the numbers of the captured fringes. FTP, WFTP, WTP, and STP are popular algorithms for the single-shot fringe pattern. The phase directly obtained from the fringe(s) is wrapped within [-π, π]. It needs to be unwrapped by a suitable phase unwrapping method to obtain the continuous unwrapped phase. The object's height is calculated by employing system parameters or an existing phase-to-height look-up table in advance.
The phase-to-height look-up table is established by system calibration based on a moving plane according to the polynomial model. For example, employing the quadratic polynomial model, the relationship is defined as
2.2 Principle of 2D wavelet transform profilometry
Without loss of generality, the 2D wavelet transform of a function $I(x,y)$ based on mother-wavelet $\psi (x,y)$ can be expressed as a general Dual-Angle rotation model [27]. The wavelet coefficients of the fringe pattern can be calculated by Eq. (3) in the space domain.
The rotation characteristic of the Fourier transform indicates that rotating the wavelet with an angle $\theta$ in the spatial domain is equivalent to its spectrum being rotated by the same angle in the frequency domain and vice versa. According to the Fourier transform property, the wavelet coefficients calculated by Eq. (3) can be carried out in the frequency domain, which is expressed as
The phase of the local fringe at the position $({b_x},{b_y})$ is calculated from the 3D wavelet coefficient matrix, which can be expressed as
The “ridge” phase corresponding to the maximum amplitude of the wavelet coefficient calculated the wrapped phase
After all the points are treated, the wrapped phase map ${\varphi _{wrapped}}(x,y)$ can be obtained. A suitable phase unwrapping algorithm works on the wrapped phase map, and the continuous phase distribution can be obtained
The detailed flow chart of complex wavelet-based 2D WTP is shown in Fig. 2. The phase map can be obtained by either Traditional 2D WTP or Dual-Angle rotation 2D WTP.
3. General framework of 2D compact support complex wavelet
3.1 Construction for 2D compact support complex wavelet
In the FPP based on wavelet transform, some wavelets have been used to retrieve the 3D surface of the object. If real wavelets are selected, the Hilbert pro-processing operation is required to work on fringes to form the analytic signal for extracting the phase information. In contrast, complex wavelets with a single-side-band (SSB) positive spectrum can be directly used in FPP. The dual-tree concept [30], which explains the process of constructing directional complex wavelets using two 1D wavelets, is an enhancement to the wavelet transform. Although it has been used in Discrete Wavelet Transform (DWT), the concept of dual-tree provides a strategy to construct 2D continuous complex wavelets.
Without loss of generality, six ways are presented to form the 2D complex wavelet according to the dual-tree idea [30]. For example, the 2D complex wavelet $\alpha (x,y)$ can be formed by two 1D complex wavelets in the orthogonal directions
where $\alpha (x)$ and $\beta (y)$ are 1D complex wavelets associated with row and column, respectively. For satisfying the perfect reconstruction (PR) conditions, the wavelet is also to be analytic or approximately analytic, $\alpha (x)$ is defined as $\alpha (x) = {\alpha _{real}}(x) + j{\alpha _{imag}}(x)$. The real part ${\alpha _{real}}(x)$ is a real wavelet, and the imaginary part is ${\alpha _{imag}}(x) = Hilbert[{\alpha _{real}}(x)]$. The definition of $\beta (y)$ is similar to $\alpha (x)$. The 2D wavelet function $\alpha (x,y)$ is expressedThe other five ways are $\alpha (x)\alpha (y)$, $\alpha (x)\overline {\alpha (y)}$, $\alpha (x)\overline {\beta (y)}$, $\beta (x)\alpha (y)$ and $\beta (x)\overline {\alpha (y)}$, where $\overline {\alpha ({\cdot} )}$ stands for the complex conjugation of $\alpha ({\cdot} )$, and $\overline {\beta ({\cdot} )}$ stands for the complex conjugation of $\beta ({\cdot} )$. After the above six different combinations are performed, the real part formulas of six 2D complex wavelets are expressed as
Two examples are given to display the constructed 2D complex wavelets by our method using 1D real wavelet functions according to Eq. (13) and (15). In the first case ${\alpha _{real}}({\cdot} ) = {\beta _{real}}({\cdot} )\textrm{ = }Mexhat({\cdot} ) = (1 - {({\cdot} )^2})\exp ( - {({\cdot} )^2}/2)$, where $({\cdot} )$ means $(x)or(y)$, is selected to form 2D complex wavelets according to Eq. (16). The 2D complex wavelets have two different styles, and the corresponding spectrum distribution is shown in Fig. 3.
In the second case, different 1D real wavelets are used to form 2D complex wavelets. The 1D real Mexican hat wavelet and 1D real Morlet wavelet $Morlet({\cdot} ) = \cos (5({\cdot} ))\exp ( - {({\cdot} )^2}/2)$ are selected to form 2D complex wavelets according to Eq. (17). The 2D compact support complex wavelets have six different styles, and the corresponding spectrum distribution is shown in Fig. 4.
Fig. 3. The spectrum distribution of wavelets using 1D real $Mexhat$ function (top view). (a) ${\sigma _i} + j{\tau _i}(i = 1,2,3)$; (b) ${\sigma _i} + j{\tau _i}(i = 4,5,6)$.
Fig. 4. The spectrum distribution of wavelets using 1D $Mexhat$ and 1D $Morlet$ functions (top view). (a)-(f) ${\sigma _i} + j{\tau _i}(i = 1,2,...,5,6)$, respectively.
In Figs. 3 and 4, the frequency origin is set at the center of each sub-image. The spectrum distribution of each sub-figures is an “island” in a single quadrant, and the “island” spectrum envelope has an asymmetrical characteristic because one of the selected 1D real wavelets is asymmetric. They can be used in FPP to extract the fringe's phase information. In the above two examples, Fig. 3(a) is the same as Fig. 4(c), and Fig. 3(b) is the same as Fig. 4(f). The spectrum of the 2D complex shown in Fig. 3(a) is similar to that of the 2D Cmexhs wavelet reported in Ref. [31], but it is smoother and has no cutoff frequency. So for comparison, in this paper, the constructed wavelet is applied in simulations and experiments for phase calculation. In addition, the constructed wavelet also satisfies the admissible condition of the wavelet: a)$\Psi (\overrightarrow v ){|_{\overrightarrow v \to 0}} = 0,\Psi (\overrightarrow v ){|_{\overrightarrow v \to \infty }} = 0$; b)$|{{C_h}} |= \int\limits_{ - \infty }^{ + \infty } {\frac{{{{|{\Psi (\overrightarrow v )} |}^2}}}{{|{\overrightarrow v } |}}} d\overrightarrow v < \infty$, where $\overrightarrow v = {[u,v]^T}$. Making Fig. 3(a) as an example, when it is rotated ${\theta _\textrm{1}}\textrm{ = }\pi \textrm{/4,}{\theta _2}\textrm{ = }0$, the spatial domain distribution and the frequency distribution of the constructed wavelet are shown in Fig. 5.
Fig. 5. The spatial and frequency distribution of the constructed 2D compact support complex wavelets. (a) the real part; (b) the imaginary part; (c) 3D view in the frequency domain; (d) top view in the frequency domain.
According to the proposed method, other 1D real wavelets can also form 2D complex wavelets. In fringe projection profilometry, generally speaking, 2D compact support complex wavelet with smooth and asymmetrical frequency characteristics is more suitable because of the property of the fringes, which will result in better phase extraction in Dual-Angle rotation 2D WTP. Therefore, the proposed framework of 2D compact support complex wavelet is general. The schematic diagram of the framework is shown in Fig. 6. For comparison, in the frequency domain, an analysis of other popular wavelets used in 2D WTP is given in Section 3.2.
3.2 Comparison analysis of common 2D complex wavelets
Morlet wavelet and its extension and the Mexihat wavelet are the most popular wavelets in WTP. Traditional Mexihat wavelet, as the second derivative of the Gaussian function, is a real wavelet. When it is used in WTP for fringe analysis, the Hilbert transform operation is required to work on the fringe captured by CCD to form an analytic single for phase calculation. The 2D complex Mexihat wavelet can be formed by 1D Hilbert transform processing line to line. Its spectral distribution is a half-circular ring, as shown in Figs. 7(a) and 7(d), with the 3D view and top view, respectively. It has an asymmetrical envelope in radian but lacks direction selectivity. The 2D Cmexhs wavelet [31] obtained by 2D single-quadrant Hilbert transforms processing has direction selectivity expressed as Eq. (18), and its spectral distribution is shown in Figs. 7(b) and 7(e).
The Fan wavelet [32], which is formed by superposing a set of 2D complex Morlet wavelets, is another common wavelet in WTP for its direction selectivity. The Fourier spectrum of the Fan wavelet is defined as Eq. (19), and the spectrum distribution is shown in Figs. 7(c) and 7(f).
Fig. 7. The frequency distribution of different wavelets. (a)-(c) 3D view of the 2D complex Mexihat wavelet, the 2D Cmexhs wavelet, and the Fan wavelet, respectively; (d)-(f) top view of the above wavelets.
The 2D Cmexhs wavelet can be carried out rotation operations because of its asymmetrical property. However, it has a sharp cutoff in the spectrum distribution, which causes slight oscillation in the reconstructed surface shape. The Fan wavelet has several symmetrical sub-spectral distribution regions. It can form a complex wavelet filter when its scale is adjusted. Fan wavelet has strong anti-noise ability but lower spatial resolution. Its spectral distribution is long-narrow size with multi-peaks. Dual-angle rotation 2D WTP based on Fan wavelet does not have the advantage in FPP because the local rotation operation of Fan wavelet may extract useless information, which results in low reconstruction accuracy. Compared with the above-mentioned 2D complex wavelet, the proposed method can flexibly construct a 2D complex wavelet with the property of SSB and asymmetrical for FPP. Moreover, it has a compact support characteristic and a smoother envelope without sharp cutoff in the frequency domain, which provides good frequency localization ability.
4. Simulations
Computer simulation verifies the feasibility of our method. The simulation object is described by a “peaks” function, as expressed in Eq. (20). Due to existing regions with both slow and rapid variations, it has become a popular benchmark for testing the performance of different fringe analysis algorithms.
In the simulation, the size of the images is $512 \times 512$ pixels. The surface of the object is shown in Fig. 8(a), and the deformed fringe pattern is expressed by Eq. (21). The deformed fringe is shown in Fig. 8(b).
where ${f_0}$ is the carrier frequency and the value is set to 0.0625/pixel. The parameter $\beta $, from 1 to 3 with 0.5 as an interval, is set to adjust the phase caused by the height variation of the object. The parameter $R(x,y)$ is the uneven reflectivity of the object, where $R(x,y) = 0.1\ast ({peaks(512) + 6 + abs(\min (\min (peaks(512))))} )$. Operators “$abs$” and “$\min $” denote calculating absolute value and minimum value, $Normalization$ stands for the amplitude normalization operation.Fig. 8. Computer-generated fringe pattern and the simulated object with $\beta = 2$. (a) Simulated object; (b) Deformed fringe.
Different wavelet-based 2D WTP is used to calculate the phase map, which is unwrapped by using the flood phase unwrapping algorithm. And the accuracy evaluation is calculated using the root mean square error (RMSE).
The simulated object and the deformed fringe were as shown in Fig. 8. The reconstructed phases using the traditional 2D WTP based on the 2D Cmexhs wavelet method (Tra-2D CWM), the Fan wavelet method, the 2D compact support wavelet constructed by our method (Tra-2D CSWM), and Dual-angle rotation 2D WTP based on the 2D compact support wavelet method (abbreviated as the proposed method) are shown in Figs. 9(a)–9(d), respectively. The corresponding phase error distributions by different methods are shown in Figs. 9(e)–9(h), respectively. Because the spectrum of the constructed wavelet is more narrow than that of the 2D Cmexhs wavelet in traditional 2D WTP, a bigger error in the local region with bigger phase variation is existed, as shown in Figs. 9(c) and 9(g). The enlarged detailed result of the 150th row is shown in Figs. 10(a) and 10(b). Compared with other methods, the proposed method provides high accuracy and smoother result. The RMSE ($\beta = 2$) of the four methods is 0.0559 rad, 0.1574 rad, 0.1516 rad, and 0.0171 rad, respectively. Different depth of the simulation object was tested, and the RMSE of the reconstructed error was calculated. The Tra-2D CWM, Fan wavelet method, and the Tra-CSWM introduce bigger reconstruction errors with height variation increasing, while the proposed method provides good reconstruction, as shown in Fig. 10(c). The details comparative results are shown in Table 1.
Fig. 9. Reconstructed results and errors. (a)-(d) Reconstructed result using Tra-2D CWM, Fan wavelet, Tra-2D CSWM, and the proposed method, respectively; (e),(f) Error distribution of Tra-2D CWM, Fan wavelet, Tra-CSWM, and the proposed method, respectively.
Fig. 10. The detailed comparison of four methods. (a) The result of the 150th row; (b) details of the marked red area; (c) RMSE of the object with different depth ranges.
Table 1. The RMSE comparison of four methods
5. Experiments
Experiments have been conducted to test our method. The experimental system includes a computer, reference plane, translation stages, a DLP projector (LightCrafter 4500), and a CCD camera (Baumer TXG13). The resolutions of the projector and camera are $912 \times 1140$ and $1392 \times 1040$, respectively. In computer simulation, the Tra-2D CSWM and Fan wavelet method cannot provide the correct reconstruction for complex objects with a bigger height variation. Thus, Tra-2D CWM, eight-step PSP, and the proposed method were compared in the experiment.
In the first experiment, to quantitatively evaluate the accuracy of our method, a standard step-shaped workpiece with a height difference of 30 mm for each step and a standard ceramic sphere with a diameter (Dia.) of 50.7991 mm were measured. The experimental setup and the objects are shown in Figs. 11(a)–11(c). The corresponding deformed fringe pattern is shown in Figs. 11(d) and 11(e), respectively. First, the height differences between the three areas (Areas 1, 2, and 3) were measured. The 3D results reconstructed using the Tra-2D CWM, eight-step PSP, and the proposed method are shown in Figs. 11(f)–11(h), respectively. The mean height of the three areas and error rate (the percentage of the mean error of area to the standard value of 30 mm) using three methods are shown in Table 2. Then, the flatness of the measured three areas in the workpiece was evaluated. The fitting values of the results in the three areas are treated as the truth value, respectively, and the error distribution of each area is shown in Figs. 11(i)–11(k), respectively. The RMSE of the measurement results using three methods is shown in Table 3.
Fig. 11. The 3D results of standard objects. (a) Experiment setup; (b) standard step-shaped workpiece; (c) standard ceramic sphere; (d) deformed fringe of the workpiece; (e) deformed fringe of the sphere; (f)-(h), and (l)-(n) 3D results of the workpiece and the sphere using Tra-2D CWM, eight-step PSP, and the proposed method, respectively; (i)-(k), and (o)-(q) flatness and RMSE using Tra-2D CWM, eight-step PSP, and the proposed method, respectively.
Table 2. The mean height comparison of the three methods
Table 3. The RMSE comparison of the three methods
The 3D results of the sphere dome using Tra-2D CWM, eight-step PSP, and the proposed method for the standard ceramic sphere were correctly restored, as shown in Figs. 11(l)–11(n). Employing the least square sphere-fitting via commercial software Geomagic, the diameters of the reconstructed standard sphere from the 3D point clouds obtained by the three methods are 50.9284 mm, 50.7365 mm, and 50.7549 mm, respectively. The three methods’ diameter deviations are 0.1293 mm, 0.0626 mm, and 0.0442 mm, respectively. The point clouds error distribution of the three methods is shown in Figs. 11(o)–11(q). Because of the proposed method's compact support characteristic and directional selectivity, the method has higher accuracy in surface reconstruction. The detailed comparison is shown in Table 4.
Table 4. The sphere dome comparison of the three methods
In the next experiment, two complex portraits (David and Pirate) were measured. The deformed fringes and the images of the objects are shown in Figs. 12(a)–11(d), respectively. The modulation masks were used to select the valuable phase region in our experiment. The 3D surface results of the David using the Tra-2D CWM, the eight-step PSP, and the proposed method are shown in Figs. 12(e)–12(g), respectively. In the white dotted box, as shown in Figs. 12(h)–12(j), the smooth surface reconstruction was obtained using either the proposed method or eight-step PSP, standing for higher accuracy. While the Tra-2D CWM provides an oscillated surface reconstruction. Although the filter operation of the wavelet method and space phase unwrapping affect the 3D result of the proposed method, the reconstructed result of the proposed method provides a better 3D surface shape. The 3D surface results of the Pirate using the Tra-2D CWM, the eight-step PSP, and the proposed method are shown in Figs. 12(k)–12(m), respectively. In the white dotted box, as shown in Figs. 12(n)–12(p), the proposed method still provides smooth surface reconstruction. The proposed method can provide more depth information because of the high compact support ability, as shown in the black dotted area of Fig. 12(p). All the results of the two experiments and the comparative analysis show that the proposed method has high accuracy and better performance in the single-shot FPP system.
Fig. 12. The 3D results of complex objects. (a)-(d) The deformed fringes and the images of two portraits; (e)-(g) 3D reconstructed result of David using the Tra-2D CWM, the eight-step PSP, and the proposed method, respectively; (h)-(j) details of the white dotted area of David using the Tra-2D CWM, the eight-step PSP, and the proposed method, respectively; (k)-(m) 3D reconstructed result of Pirate using the Tra-2D CWM, the eight-step PSP, and the proposed method, respectively; (n)-(p) details of the white dotted area of Pirate using the Tra-2D CWM, the eight-step PSP, and the proposed method, respectively.
6. Conclusion
A general framework for constructing 2D compact support complex wavelets has been proposed and applied in the single-shot WFPP. The framework employed the dual-tree concept combined with the 1D asymmetric wavelet function to form a 2D complex wavelet. The proposed 2D wavelet framework for FPP is flexible and general. The constructed wavelets have an asymmetric spectrum without sharp cutoff frequency. A comparative analysis of the popular and the constructed wavelet was also conducted. The computer simulations verified the effectiveness of our method, and the different large-depth-range objects were measured to test the performance of our method. In experiments, we compared the 3D reconstructed results of standard objects and complex portraits using the Tra-2D CWM, eight-step PSP, and the proposed method, respectively. The reconstructed 3D results of the standard workpiece by our method are better than that of Tra-2D CWM, which is close to that of the eight-step PSP, and the reconstructed accuracy of the standard ceramic sphere achieves the same accuracy level as the eight-step PSP. Besides, the proposed method correctly reconstructed 3D shapes of complex portraits and provided more depth information. All the results demonstrated the proposed method has advantages in single-shot WFPP.
Funding
National Key Scientific Instrument and Equipment Development Projects of China (2013YQ490879); National Natural Science Foundation of China (62075143, U20A20215).
Disclosures
The authors have declared no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. S. S. Gorthi and P. Rastogi, “Fringe projection techniques: Whither we are?” Opt. Lasers Eng. 48(2), 133–140 (2010). [CrossRef]
2. F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39(1), 10–22 (2000). [CrossRef]
3. S. Zhang, “Absolute phase retrieval methods for digital fringe projection profilometry: a review,” Opt.Lasers Eng. 107, 28–37 (2018). [CrossRef]
4. Z. Wang, “Review of real-time three-dimensional shape measurement techniques,” Measurement 156, 107624 (2020). [CrossRef]
5. X. Su and Q. Zhang, “Dynamic 3D shape measurement: a review,” Opt. Lasers Eng. 48(2), 191–204 (2010). [CrossRef]
6. C. Zuo, S. Feng, L. Huang, T. Tao, W. Yin, and Q. Chen, “Phase shifting algorithms for fringe projection profilometry: A review,” Opt. Lasers Eng. 109, 23–59 (2018). [CrossRef]
7. S. Van der Jeught and J. J. J. Dirckx, “Real-time structured light profilometry: a review,” Opt.Lasers Eng. 87, 18–31 (2016). [CrossRef]
8. Z. Wu, C. Zuo, W. Guo, T. Tao, and Q. Zhang, “High-speed three-dimensional shape measurement based on cyclic complementary gray-code light,” Opt. Express 27(2), 1283–1297 (2019). [CrossRef]
9. W. Guo, Z. Wu, Q. Zhang, and Y. Wang, “Real-time motion-induced error compensation for 4-step phase-shifting profilometry,” Opt. Express 29(15), 23822–23834 (2021). [CrossRef]
10. X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Lasers Eng. 35(5), 263–284 (2001). [CrossRef]
11. E. Zappa and G. Busc, “Static and dynamic features of Fourier transform profilometry: A review,” Opt. Lasers Eng. 50(8), 1140–1151 (2012). [CrossRef]
12. K. Qian, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Laser Eng. 45(2), 304–317 (2007). [CrossRef]
13. H. Lin and C. Yang, “Two-dimensional multiscale windowed Fourier transform based on two-dimensional wavelet transform for fringe pattern demodulation,” Opt. Laser Technol. 43(1), 72–81 (2011). [CrossRef]
14. M. Zhong, W. Chen, and X. Su, “A comparison of one and two-dimensional s-transform in fringe pattern demodulation,” Opt. Lasers Eng. 55, 212–220 (2014). [CrossRef]
15. Z. Wang, J. Ma, and M. Vo, “Recent progress in two-dimensional continuous wavelet transform technique for fringe pattern analysis,” Opt. Lasers Eng. 50(8), 1052–1058 (2012). [CrossRef]
16. J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. 30(19), 2560–2562 (2005). [CrossRef]
17. L. Huang, K. Qian, B. Pan, et al., “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(2), 141–148 (2010). [CrossRef]
18. K. Qian, “Carrier fringe pattern analysis: Links between methods,” Opt. Laser Eng. 150, 106874 (2022). [CrossRef]
19. Y. Li, J. Qian, S. Feng, Q. Chen, and C. Zuo, “Deep-learning-enabled dual-frequency composite fringe projection profilometry for single-shot absolute 3D shape measurement,” Opto-Electron. Adv. 5(5), 210021 (2022). [CrossRef]
20. M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Comm. 211(1-6), 47–51 (2002). [CrossRef]
21. M. Gdeisat, D. Burton, F. Lilley, M. Lalor, and C. Moore, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous Paul wavelet transform,” AIP Conf. Proc. 1236, 112–117 (2010). [CrossRef]
22. Z. Zhang and J. Zhong, “Applicability analysis of wavelet-transform profilometry,” Opt. Express 21(16), 18777–18796 (2013). [CrossRef]
23. A. Z. Abid, M. A. Gdeisat, D. R. Burton, M. J. Lalor, and F. Lilley, “Spatial fringe pattern analysis using the two-dimensional continuous wavelet transform employing a cost function,” Appl. Opt. 46(24), 6120–6126 (2007). [CrossRef]
24. C. Jiang, S. Jia, J. Dong, Q. Lian, and D. Li, “Multi-frequency fringe projection profilometry based on wavelet transform,” Opt. Express 24(11), 11323 (2016). [CrossRef]
25. S. Li, X. Su, and W. Chen, “Wavelet ridge techniques in optical fringe pattern analysis,” J. Opt. Soc. Am. A 27(6), 1245–1254 (2010). [CrossRef]
26. A. F. M. Hani, A. A. Khoiruddin, N. Walter, and I. Faye, “Wavelet analysis for shadow detection in fringe projection profilometry,” IEEE ISIEA, 336–340 (2012).
27. M. Han and W. Chen, “Dual-angle rotation two-dimensional wavelet transform profilometry,” Opt. Lett. 47(6), 1395–1398 (2022). [CrossRef]
28. W. Zhao, X. Su, and W. Chen, “Whole-field high precision point to point calibration method,” Opt. Lasers Eng. 111, 71–79 (2018). [CrossRef]
29. Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. 22(11), 1330–1334 (2000). [CrossRef]
30. I. W. Selesnick, R. G. Baraniuk, N. C. Kingsbury, et al., “The dual-tree complex wavelet transform,” IEEE Signal Process. Mag. 22(6), 123–151 (2005). [CrossRef]
31. M. Han and W. Chen, “Two-dimensional complex wavelet with directional selectivity used in fringe projection profilometry,” Opt. Lett. 46(15), 3653–3656 (2021). [CrossRef]
32. S. Li, X. Su, and W. Chen, “Spatial carrier fringe pattern phase demodulation by use of a two-dimensional real wavelet,” Appl. Opt. 48(36), 6893–6906 (2009). [CrossRef]
