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Abstract
By utilizing 1-bit binary fringe patterns instead of conventional 8-bit sinusoidal patterns, binary defocusing techniques have been successfully applied for high-speed 3D shape measurement. However, simultaneously achieving high accuracy and high speed remains challenging. To overcome this limitation, we propose a high-efficiency and robust binary fringe optimization method for superfast 3D shape measurement, which consists of 1D optimization and 2D modulation. Specifically, for 1D optimization, the three-level OPWM technique is introduced for high-order harmonics elimination, and an optimization framework is presented for generating the ‘best’ three-level OPWM pattern especially for large fringe periods. For 2D modulation, a single-pattern three-level OPWM strategy is proposed by utilizing all the dimensions for intensity modulation to decrease the required projection patterns. Thus, the proposed method essentially belongs to the 2D modulation technique, yet iterative optimization is carried out along one dimension, which drastically improves the computational efficiency while ensuring high accuracy. With only one set of optimized patterns, both simulations and experiments demonstrate that high-quality phase maps can be consistently generated for a wide range of fringe periods (e.g., from 18 to 1140 pixels) and different amounts of defocusing, and it can achieve superfast and high-accuracy 3D shape measurement.
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1. Introduction
High-speed and high-accuracy 3D shape measurement is of great significant in numerous fields such as manufacturing, entertainment, and healthcare industry [1–3]. Due to the characters of non-contact, high resolution and low cost, the fringe projection profilometry (FPP) stands out and becomes one of the most popular and promising 3D imaging method [4–6].
Conventional FPP generally illuminates 8-bit sinusoidal fringes on the target object, thus its measurement speed is restricted by the maximum frame rate of the projector (typically 120 Hz), which makes it difficult to achieve superfast measurement. In addition, both the nonlinear gamma effect of the projector and the lack of precise synchronization between the projector and camera may deteriorate the phase quality [7]. To overcome these limitations, Heist et al. proposed the GOBO projector [8,9], which used a rotating slide structure to project aperiodic quasi-sinusoidal fringe patterns at high frame rate, but an extra high-speed camera was required for 3D reconstruction. More generally, the binary defocusing technique has been proposed by utilizing 1-bit binary patterns instead of conventional 8-bit sinusoidal patterns [10]. By properly defocusing the projector lens for specific binary patterns [11], pseudo-sinusoidal patterns can be generated, and the speed bottleneck has been broken on the digital-light-processing (DLP) projection platform, of which the maximum projection speed can be boosted up to over 20 kHz. Moreover, there is no need for gamma calibration and strict synchronization. Though with success of aforementioned advantages, the binary defocusing technique only works well within a small out-of-focus range, thus limiting its depth measurement capability.
To improve the phase quality and enhance the measurable depth range, various modulation methods have been explored, which can be divided into two major categories as single-pattern [12–14] and multi-pattern methods [15,16], by whether requiring additional patterns or not. For the single-pattern modulation category, several different pulse width modulation (PWM) methods have been proposed such as the sinusoidal pulse width modulation (SPWM) technique [17,18] and the optimal pulse width modulation (OPWM) technique [19]. Both techniques of them could improve phase quality when the fringe period is relatively small, however, they fail to produce high-quality phase if the fringe period becomes too large. PWM techniques only modulate the patterns in one dimension with two grayscale levels, thus their ultimate enhancements are rather limited. To improve the phase quality under these extreme cases, two-dimensional (2D) optimization methods have been developed such as area modulation technique [20] and dithering/halftoning technique [21], which demonstrate their advantages in improvement on measurement quality for a certain range of fringe periods. However, their 2D optimization methods [22,23] to further minimizing phase and intensity errors are very time consuming especially for large periods of fringes, and sometimes the algorithm does not converge if the initial pattern is not good.
Unlike the single-pattern methods, lots of research have also been conducted by taking advantage of the temporally acquired information to further improve the phase quality with more patterns. Zhang [24] proposed to use two or four sets of phase-shifted binary patterns with an initial phase offset of π/6 or π/12 for phase error compensation, and finally high-quality phase maps were achieved even when the projector is slightly defocused. Zhu et al. [25] developed an interesting temporal-spatial binary encoding method to improve the phase quality, but it still requires four optimized binary dithered patterns for representing one sinusoidal pattern. Zuo [26], Wang [27] and Silva [28] et al. respectively proposed to employ two or three SPWM patterns to shift the carrier signal or eliminate undesired high-order harmonics, for better quality phase generation. Wang et al. [29] proposed a multilevel symmetric pattern design and optimization method to provide more flexibility for eliminating undesired high-frequency harmonics, but l-1 binary patterns are required for l-level grayscale. Though with the success of improving the phase quality, these multi-pattern optimization methods greatly slow down the measurement speed by at least 50%, which is not acceptable for lots of high-speed applications.
In this paper, to overcome the problems of low computational efficiency of 2D optimization methods and slow speed of multi-pattern optimization methods, we propose a high-efficiency and robust binary fringe optimization method for superfast 3D shape measurement, which consists of 1D optimization and 2D modulation. First, by introducing one more grayscale level, the three-level OPWM technique can better suppress the high-order harmonics and improve the phase quality. Second, the proposed 1D optimization framework assists to find the ‘best’ patterns to avoid the challenge of solving nonlinear transcend equations especially for large fringe period. In addition, through 2D intensity modulation in the spatial domain, a single-pattern three-level OPWM strategy can achieve three-level grayscale projection, thereby ensuring the speed and accuracy at the same time. It should be noted that the proposed method needs to be calculated and iteratively optimized only along one dimension, which drastically improves the computational efficiency while ensuring high quality. With only three optimized binary patterns, we verify that very high-quality phase maps can be consistently generated for a wide range of fringe periods and different amounts of defocusing. And the experimental results demonstrate the proposed technique can achieve high-speed and high-accuracy 3D shape measurement.
Section 2 explains the principle of the proposed method. Section 3 shows simulation and experimental results to verify the performance of the proposed method; and Section 4 summarizes the paper.
2. Principle
In order to achieve high-speed and high-accuracy 3D shape measurement, high-efficiency and robust binary fringe optimization is of great significance. To this end, we propose a new binary fringe optimization method, and Fig. 1 demonstrates its overall flowchart. The proposed method mainly includes:
- 1) 1D optimization. Firstly, we introduce the three-level OPWM technique for high-order harmonics elimination and high-quality phase retrieval. Comparing with conventional 1D optimization methods, one more additional grayscale level is used, providing more flexibility for harmonics elimination. Secondly, the increase of fringe period would lead to high computational cost or even impossible calculation, thus an optimization framework is presented for generating the ‘best’ three-level OPWM pattern especially for large fringe periods.
- 2) 2D modulation. Since the three-level OPWM pattern needs to be decomposed into two binary patterns for implementation, we propose the single-pattern three-level OPWM strategy that modulates the intensity of the previously optimized mode along another dimension.
It should be noted that the proposed method essentially modulates and optimizes along three dimensions, which, to the best of our knowledge, has never been seen in previous studies. But the iterative optimization is only carried out along one dimension, which drastically improves the computational efficiency while ensuring high quality. In addition, we can successfully achieve or even exceed the superior accuracy of multi-pattern optimization methods with only one set of patterns.
2.1. Three-step phase-shifting algorithm
Phase-shifting algorithms are widely employed in the optical metrology due to their high accuracy capability [30]. In this paper, a three-step phase-shifting algorithm with a phase shift of 2π/3 is adopted, and the fringe patterns can be described as,
where I'(x, y) denotes the average intensity, I''(x, y) is the intensity modulation, and Φ(x, y) represents the phase to be solved for. The phase can be calculated from Eqs. (1)-(3):The result of this equation is the wrapped phase that limited in the range of [-π, +π) with 2π discontinuities owing to the nature of arctangent function. Phase unwrapping algorithms such as multi-frequency algorithms can be applied to remove the 2π discontinuities and obtain the absolute phase map. Finally, a simple calibration method based on reference plane is applied to converting the absolute phase into height information for 3D reconstruction.
2.2. Three-level optimal pulse width modulation (OPWM) technique
Instead of conventional 8-bit sinusoidal patterns, binary defocusing techniques has enabled speed breakthroughs for 3D shape measurement, but it only works well within a limited out-of-focus range. The conventional sinusoidal waveform and squared binary waveform are shown in Figs. 2(a)-(b). To overcome this limitation, Conventional OPWM method modulates the squared binary pattern by inserting notches with different switching angles [19], as shown in Fig. 2(c). Since it does not change the grayscale values, the conventional OPWM waveform just has two grayscale levels. For the proposed three-level OPWM method, one more additional grayscale level is added while the quarter-wave symmetric character is still persevered, as shown in Fig. 2(d). This introduced grayscale level brings more flexibility for selectively eliminating undesired high frequency components, thus poses the superiority for producing high phase quality.
Fig. 2. Principle of three-level OPWM. (a) Conventional sinusoidal waveform; (b) Conventional squared binary method; (c) Conventional OPWM method; (d) Three-level OPWM method.
A three-level OPWM wave can be rewritten with the following harmonics form
Because the half-cycle symmetry of the OPWM waveform, only odd-order harmonics exist. And the amplitude of the harmonics can be simplified as:
The n chops in the waveform afford n degrees of freedom. Thus n-1 number of selected harmonics can be eliminated while keeping the fundamental frequency component with a certain magnitude. Specifically, one can set the corresponding coefficients in the above equation to be desired values (0 for the n-1 undesired harmonics and the desired magnitude m for the fundamental frequency). Considering for the three-step phase-shifting algorithm, the 3nth high-order harmonics have no effects on the final phase quality, therefore these components are not necessarily eliminated. And the problem becomes to solve the following set of nonlinear transcendental equations.
Numerous research has been conducted on how to solve for this type of nonlinear equations. For example, some research has been conducted to solve for transcendental equations [31]. Due to the ability to eliminate undesired high-order harmonics, three-level OPWM waveform could generate high-quality phase with a small degree of defocusing.
2.3. Iterative optimization of OPWM pattern for large fringe periods
The three-level OPWM pattern can be quickly obtained when the fringe period is relatively small. However, with the increase of the fringe period, more notches with more switching angles are required to better depress the high-order harmonics, which makes the solution of the aforementioned nonlinear transcendental equations much harder or even impossible. Therefore, in order to find the OPWM pattern especially for large fringe periods, we propose an iterative optimization method as follows.
- Step 1: Pattern initialization. Since the designed three-level OPWM pattern has quarter-wave symmetric character, for a given fringe period T, the initial quarter-wave pattern with T/4 points can be obtained by randomly choosing the 0 or 1 value for each point. By using a quarter portion of the initial pattern, a three-level OPWM pattern can be finally formed through symmetry and periodicity.
- Step 2: Iterative optimization. For the initial OPWM patterns, alternate some bit of the OPWM pattern and reevaluate the phase error. If the phase quality is improved, the alternation is accepted till the best candidate is obtained. And this process is iteratively executed until the algorithm converges when the phase root-mean-square (RMS) error difference for a new round of iteration is less than 0.001%.
- Step 3: Best candidate selection. Comparing the optimized OPWM patterns with different initials, the best OPWM pattern is selected if the resultant phase errors are consistently small under different defocusing levels.
Figures 3(a)–3(c) respectively show the representative optimization OPWM patterns for fringe period T = 120 pixels with three different initials. And Fig. 3(d)–3(f) show the corresponding phase errors plots under different defocusing levels. Comparing these results, it can be found that when the filter size is 5 and 7 pixels, initializations 1 and 2 provide the minimum phase errors, respectively. However, the phase errors increase with the increase of the filter size. And for initialization 3, the phase errors are consistently small under all defocusing levels with a decreasing trend, thus this pattern is chosen as the ‘best’ candidate.
Fig. 3. Optimized three-level OPWM patterns for fringe period T = 120 pixels with different initials. Optimized pattern of (a) initialization 1; (b) initialization 2; (c) initialization 3; (d)-(f) Corresponding phase error plots with different defocusing levels.
2.4. Single-pattern three-level OPWM strategy
The well designed three-level OPWM method has the advantages of better eliminating specific high-order harmonics due to more flexibility. However, it requires at least two sets of binary phase-shifting patterns for retrieving high-quality phase maps, similar with the double-pattern TPWM method, which is not desirable for high-speed measurements. In order to solve this problem, a single-pattern three-level OPWM strategy is proposed by utilizing all the dimensions for 2D modulation, so as to guarantee the measurement speed and accuracy at the same time.
Figure 4 illustrates the modulation principle. For the conventional OPWM methods, the optimization is mainly applied in the U-X dimension. And the distribution of all the rows along Y dimension is actually identical, thus this dimension is not well exploited. This paper presents an interesting way to use this dimension for intensity modulation and OPWM pattern optimization. As shown in Fig. 4, the specific modulation rules are as follows:
- 1) Rule 1: 1 → ‘white’. If the intensity values of the three-level OPWM waveform are 1, the pixels of the single-pattern OPWM pattern along Y dimension are all set as ‘white’.
- 2) Rule 2: -1 → ‘black’. If the intensity values of the three-level OPWM waveform are -1, the pixels of the single-pattern OPWM pattern along Y dimension are all set as ‘black’.
- 3) Rule 3: 0 → ‘white/black’ or ‘black/white’. When the intensity values are 0, the modulation should follow three sub-rules, which are represented by a, b, and c respectively in Fig. 4.
- a) Rule 3-1. Along Y dimension, the pixels should be alternatively set as ‘white’ and ‘black’, or ‘black’ and ‘white’.
- b) Rule 3-2. Considering that several adjacent pixels along X dimension could be all with the intensity value of 0, such as two and three adjacent pixels in Fig. 4, the adjacent distribution then should be changed to form a chessboard pattern in order to ensure the uniformness of the intermediate intensity.
- c) Rule 3-3. For each isolated region with intensity 0 separated by 1 or - 1, such as the isolated regions c1, c2, c3 and c4 indicated by the red arrows in Fig. 4, they can all start with two different Y dimensional distributions, such as ‘white’ and ‘black’, or ‘black’ and ‘white’. Therefore, different realization ways should be compared to find the best single-pattern three-level OPWM map, of which the resultant phase errors are consistently small under different defocusing levels.
Figure 5 shows two different single-pattern OPWM maps for fringe period T = 12 pixels. Figures 5(a) and 5(b) are the fringe patterns, while Figs. 5(c) and 5(d) respectively shows the phase errors under different defocusing levels. The difference between these two OPWM patterns is the Y dimensional distribution in Rule 3-3. The first pattern utilizes the same Y dimensional distributions for the two adjacent isolated intermediate regions, while the second one use different Y dimensional distribution. From the phase error results, it can be found that for the second OPWM pattern, the improvement is significant especially when the pattern is nearly focused.
Fig. 5. Single-pattern three-level OPWM maps for fringe period T = 12 pixels. (a)-(b) Two different single-pattern OPWM maps. (c)-(d) The corresponding phase error plots with different defocusing levels.
After modulating the three-level OPWM pattern using the abovementioned rules in the X-Y dimension, the single-pattern three-level OPWM pattern can be obtained, which can not only provide almost the same superior measurement accuracy as the three-level OPWM method, but also decrease the required projection patterns and then improves the measurement speed.
3. Experiments
Experiments have been conducted to verify the performance of our proposed single-pattern three-level OPWM method. As shown in Fig. 6, the experimental system incorporates a DLP projection unit (model: DLP VisionFly 6500) with 1920 × 1080 pixels native resolution and a high-speed camera (model: Photron FASTCAM Mini AX200) that is attached with a lens (model: AF-S NIKKOR 28-300 mm f/3.5-5.6G ED VR). The camera resolution was set at 1024 × 672 pixels and the camera was synchronized by a trigger signal from a projector.
3.1. Comparative analysis
We first applied the optimization method described in Section 2 to determine the best three-level patterns for different fringe periods, and then performed intensity modulation to obtain single-pattern three-level OPWM structure (Single OPWM). To evaluate the performance of the proposed strategy, we conducted simulations to compare their phase errors with the conventional squared binary method (SBM) [11], the previously proposed 2D dithering technique (Dithering) [21] and double three-level OPWM method (Double OPWM). Figure 7 show the phase RMS errors of different fringe pitches (T = 18, 36, 90, 120, 540 and 1140 pixels) under different defocusing levels for these four methods. Different defocusing levels are emulated by different sizes of Gaussian filters. These simulation results demonstrate that the Single OPWM strategy successfully achieves almost the same effect as the Double OPWM method and generates superior and much smoother phase quality nearly for all fringe periods. Especially, the phase RMS errors of three-level patterns are smaller than 0.05 rad for fringe periods ranging from T = 18 to T = 1140 pixels, even when fringe pattern is nearly focused (i.e., with a Gaussian filter size of 3 × 3 pixels and standard deviation of 1/3 pixels). This means that the proposed method can work well for a wide range of fringe patterns even under small amount of defocusing. Furthermore, it can be noted that as the filter size increases, the phase error generally decreases. However, the phase error of the narrow fringe patterns (e.g., T = 18 pixels) will actually increase when the amount of defocusing is too large, due to the reduction in fringe contrast.
Fig. 7. Phase errors under different levels of defocusing in simulations on different fringe periods of (a) T = 18 pixels; (b) T = 36 pixels; (c) T = 90 pixels; (d) T = 120 pixels; (e) T = 540 pixels and (f) T = 1140 pixels.
To verify the simulation results, a flat white board was measured, and the amount of defocusing from nearly focused to significantly defocused was realized by manually adjusting the focal length of the projector. For each fringe period, we carried out phase error evaluation by comparing the phase Φ with a reference phase Φr. The reference map was generated by employing a set of 18-step phase-shifted fringe patterns with a fringe period of Tr = 18 pixels. To compute the phase error, we took the difference between the measured phase map Φ (fringe period T) and the reference phase map Φr (fringe period Tr) as
Figure 8 shows the phase RMS errors obtained by abovementioned four methods through experiments under different defocusing levels and fringe pitches. It can be seen that when the fringe pitch is relatively small (e.g., T = 18 and 36 pixels), the proposed three-level methods apparently outperform SBM and Dithering methods with different defocusing levels. The differences are more obvious especially for small amount of defocusing. Clearly, these experimental results are consistent with the simulation, and thus confirm that the proposed single-pattern three-level OPWM strategy not only approximates the double three-level OPWM method in accuracy, but also realizes the speed breakthrough. In addition, phase errors of the proposed method have a more stable trend for any fringe period. Therefore, it is suitable for high-speed 3D shape measurement in large depth range.
Fig. 8. Phase errors under different levels of defocusing in experiments on different fringe periods of (a) T = 18 pixels; (b) T = 36 pixels; (c) T = 90 pixels; (d) T = 120 pixels; (e) T = 540 pixels and (f) T = 1140 pixels.
3.2. Accuracy evaluation under slightly defocusing
To further evaluate the performance of the proposed method under slightly defocusing with relatively narrow fringe periods, we measured a sphere and took the reconstructed results obtained through 18-step phase-shifted binary patterns with T = 18 pixels as the “ground truth” for quantitative analysis. Here we employed a three-wavelength temporal phase unwrapping algorithm to obtain the absolute phase map [32].
Firstly, we conducted comparative evaluations of the SBM, Dithering and proposed method with the fringe period of 18 pixels under nearly focused, and related fringe periods for three-wavelength phase unwrapping were set as T = 120 and 1140 pixels. Figures 9(a)-(c) respectively show the captured fringes of different periods for the three methods. From these figures, it can be seen that the fringes are with nearly focused effect. Specifically, the granular spots of dithering fringes and the ribbon thin lines of the proposed optimized fringes are clearly visible. The results of 3D shape reconstruction are shown in Figs. 10(a)-(c) and cross sections of these 3D results obtained from different methods are plotted in Fig. 10(d) to better visualize the difference. Visually we can see that the reconstructed surface is much smoother for the proposed method. It should be noted that the phase accuracy of the relatively large-period fringes of the traditional SBM is not high enough, and then the three-wavelength algorithm cannot be used for correct phase unwrapping. Therefore, the absolute phase is obtained by using the medium and low frequency fringes of either of the other two methods. To quantitatively evaluate the measurement accuracy, we compared the measurement results with the “ground truth”, and then calculated RMS errors for both methods. We obtain the estimated errors for different methods as shown in Figs. 10(e)-(g) and plot the corresponding cross sections as shown in Fig. 10(h). For the three methods, the RMS errors are 0.1489 rad, 0.0706 rad and 0.0428 rad respectively. This experiment demonstrates that the proposed method provides higher accuracy under nearly focused when the fringe period is small comparing with the SBM and Dithering method.
Fig. 9. Captured fringes with (a) SBM; (b) dithering method and (c) proposed method when measuring a sphere (fringe periods are respectively T = 18, 120, and 1140 pixels from top to bottom).
Fig. 10. Accuracy evaluation under nearly focused. (a)-(c) 3D reconstruction results for the SBM, Dithering and proposed method respectively with fringe period T = 18 pixels; (d) Cross sections of absolute phase maps; (e)-(g) Corresponding phase errors of (a)-(c); (h) Cross sections of absolute phase error.
Then, the proposed method is evaluated for different fringe periods under slight defocusing. It is well known that when fringe stripes are wide, the random noise is very large, and with the fringe period decreasing, the quality of the 3D results becomes better. Thus, the fringe periods T = 18, 36 and 60 pixels are selected to verify the effectiveness of the proposed method, and Figs. 11(a)–11(c) respectively show their representative captured fringe patterns. Three phase-shifting optimized patterns are captured and used to recover the 3D shape, utilizing respective wide fringes to assist in three-wavelength phase unwrapping. Figures 11(d)–11(f) show the 3D result for each case, and the RMS errors are 0.0397 rad, 0.0573 rad and 0.0888 rad respectively. It is worth noting that when the fringe with period T = 18 pixels is under nearly focused and slightly defocusing, the RMS errors are 0.0428 rad and 0.0397 rad, respectively. Both of them have excellent results, but with the increase of defocusing level, the reconstruction quality will become higher. The experimental results shows that the 3D shape measurement for each period have superior quality, especially when the fringe period is T = 18, which demonstrates that the proposed method can stably obtain high-quality results for different fringe periods.
Fig. 11. Experimental results of the sphere with different fringe periods using the proposed method under slightly defocusing. (a)-(c) Representative optimized patterns. (d)-(f) 3D results using the optimized patterns.
3.3. Comparative measurement on dynamic scenes
To demonstrate that the proposed method can also achieve extremely high measurement speed, we then carried out 3D shape measurements on the translated sculpture and nodding doll with the abovementioned three methods at a reconstructed rate of 9524/4 = 2381 fps. We chose the high-frequency fringe period as 36 pixels for phase calculation and employed a robust gray-code-based algorithm for phase unwrapping [33]. Finally, a simple calibration method based on reference plane was applied to converting the absolute phase into height information for 3D reconstruction [34]. And the complete reconstruction results are provided in Visualization 1. Figures 12(a)-(c) respectively show the high-frequency fringe patterns with fringe period T = 36 pixels captured by the camera using three methods. One can clearly see that the projector was nearly focused because the binary structures are clearly visible. And the corresponding reconstructed 3D shape is shown in Figs. 12(d)-(f). It is clear that our proposed method provides better and smoother quality. In contrast, the reconstructed surface dithering method has lots of noise, and the SBM shows inferior results with obvious stripes.
Fig. 12. Measurement results under nearly focused on the dynamic experiment. (a)-(c) Fringe pattern at T = 36 pixels for the SBM, Dithering and proposed method; (d)-(f) Corresponding 3D reconstructed result of (a)-(c) (Visualization 1).
To better view the differences, Fig. 13 shows the close-up view of 3D results for the different methods. Figures 13(a)-(b) respectively present the zoom-in reconstructed results for different methods when the projector is more significantly defocused. Figures 13(c)-(d) display the corresponding results when the projector is slightly defocused, while Figs. 13(e)-(f) show the results when the projector is nearly focused. When the patterns are more defocused, the 3D results of the latter two methods have higher quality. In contrast, the proposed method has superior reconstruction results in the whole depth range. These experimental results verified that the proposed method could perform superfast and high-quality dynamic scene measurements.
Fig. 13. Comparisons on zoom-in reconstructed results for the SBM, Dithering and proposed method with fringe pitch T = 36 under different defocusing levels when the projector is (a)-(b) significantly defocused; (c)-(d) slightly defocused and (e)-(f) nearly focused.
4. Discussion and conclusion
In conclusion, this paper has presented a high-efficiency and robust binary fringe optimization method for superfast 3D shape measurement. This method is mainly divided into 1D optimization and 2D modulation. Both simulations and experiments demonstrated that the proposed method can consistently generate high-quality phase for a wide range of fringe periods and different defocusing levels. In order to evaluate the accuracy, we measured a sphere, and it provides superior accuracy even in the case of nearly focused. Furthermore, it could perform superfast and high-accuracy dynamic scene measurements in a large depth range.
Compared with conventional high-speed and high-accuracy 3D shape measurement techniques, our proposed method stands out in following aspects,
- 1) High efficiency. Compared with the conventional 2D optimization methods such as dithering-based methods, the proposed method is much more efficient especially for large fringe period, because it only needs iterative optimization along one dimension. For example, it will be very time-consuming when T > 120 pixels for 2D optimization methods since the pixel variation is along two different dimensions.
- 2) Both high speed and high accuracy. On the one hand, compared with the multi-pattern optimization methods, it requires only one set of patterns to conduct 3D measurement, improving the measurement speed. On the other hand, the proposed single-pattern method can provide almost the same superior measurement accuracy as the three-level OPWM method. Therefore, it can simultaneously realize both high-speed and high-accuracy 3D shape measurement.
- 3) Robustness. The proposed method not only consistently performs well for a wide range of fringe periods (e.g., from 18 to 1140 pixels), but also can consistently generate high-quality phase maps under different amounts of defocusing. Thus, it can achieve high-quality 3D measurement in a large depth range.
It should be noted that this paper only presents and demonstrates the single-pattern three-level OPWM strategy, but the grayscale level can be flexible, such as five-level pattern. More grayscale levels could provide more flexibility for eliminating high-order harmonics. Therefore, the proposed method can also be extended to single-pattern multilevel OPWM strategy, especially for large fringe periods.
Funding
National Natural Science Foundation of China (62075143); Open Fund of Key Laboratory of Icing and Anti/De-icing (Grant No. IADL20200308).
Disclosures
The authors declare 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.
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