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Threedimensional shape measurement of complex surfaces based on optimized dithering fringe patterns
Journal of the European Optical SocietyRapid Publicationsvolume 14, Article number: 26 (2018)
Abstract
Background
Optimized dithering fringe pattern is a promising method for highspeed, highaccuracy threedimensional shape measurement. The recently proposed dithering optimization technology optimizes the fringe quality in either phase domain or intensity domain according to their objective functions. Phasebased optimization is direct and effective, but it is sensitive to projector defocusing levels. Intensitybased optimization is robust to projector defocusing levels, but it does not fully improve the phase quality. In practice, it is difficult to control defocusing levels so it is still a challenge to get high quality fringe patterns which affects the measurement quality under different defocusing levels.
Methods
In order to get highquality binary dithered patterns which are robust to defocusing levels, this paper proposes a weight object function. This function combines two parts: a global intensity part and a local structure part. The global intensity measurement is based on the normalized mean squared error. The local structure measurement is based on residual error of intensity. To generate high quality fringe patterns, the weight object function is applied to the best patch framework.
Results and discussion
Both simulation and experimental results demonstrate that the phasebased optimization method and the proposed method perform better than the intensitybased optimization method under nearly focused. However the quality of measurement results from phasebased optimization will decrease with the defoucsing levels increasing. The proposed method is robust to the defocusing levels and it can still reduce the phase error when the projector is strongly defocused.
Conclusions
The proposed method can get highquality binary dithered patterns under different defocusing levels by combining global similarity and local residual error of intensity. It inherits the merits of binary fringe pattern so that gamma calibration of projector is not required. In practice, the proposed method can be used to generate high quality fringe patterns. The experiment results verify that the proposed method can get better measurement results without considering the projector defocusing levels.
Introduction
In the past decades, digital fringe projection (DFP) methods have been proposed as one of the most reliable techniques to reconstruct the surface of objects because of their accuracy and efficiency [1,2,3]. Because it is easier and more convenient to generate fringe patterns and project patterns with a high speed, 3D shape measurement of dynamic objects has been rapidly expanding [4, 5]. However, conventional DFP technique has major limitations: the projection nonlinearity and speed bottleneck [6,7,8,9]. These shortcomings make it difficult to be applied to highquality and highspeed 3D shape measurement.
To overcome the bottleneck, binary defocusing technique is recently developed which has successfully made speed breakthrough [10,11,12]. However, binary defocusing technique cannot achieve the same measurement capability compared with conventional DFP methods: (1) the measurement accuracy decreases due to highfrequency harmonics influence and (2) the measurement range is smaller since the projector must be properly focused for highquality measurement [13,14,15]. Lately, pulse width modulation (PWM) [16] and optimal pulse width modulation (OPWM) [17] techniques are proposed to improve the fringe quality by eliminating highfrequency harmonics. However improvement is limited to wide fringe stripes, because the modulation is only onedimensional without making full use of twodimensional information.
Dithering technology is developed to take advantage of twodimensional binary images so it could improve fringe quality of wide fringe stripes [18,19,20,21]. These techniques maintain lowfrequency component so that the overall image is similar to the original pattern when a lowpass filter is applied. However, if fringe stripes are narrow, the improvement is rather small because highfrequency sinusoidal fringe patterns are usually desirable since they provide better measurement quality when the period of fringe pattern is small.
Taking advantage of twodimensional (2D) nature of the structured patterns, dithering optimization methods are developed for DFP systems [22,23,24]. According to their objective functions, these optimization methods can be classified into two categories: intensitybased optimization [25, 26] and phasebased optimization [27, 28]. The former makes the dithered pattern approximation for an ideal sinusoidal fringe after defocusing. They are robust to projector defocusing levels but they can not reduce phase errors efficiently. Since the ultimate goal of optimization is to generate highquality phase, it is natural to optimize the pattern in the phase domain [29]. The latter makes the phase close to the desired linear phase when properly focused. However, the phasebased optimization method is sensitive to different defocusing levels.
In order to address the aforementioned challenge, this paper proposes a weight object function combining normalized mean squared error and intensity residual error. Instead of optimizing the whole pattern, our framework is optimizing pixels group by group to find the best binary patches, and then tiling the best patch to generate the fullsize pattern utilizing symmetry and periodicity structure of the sinusoidal pattern. The proposed fringe patterns belong to dithering binary stripes so they inherit the property of dithering binary pattern which means that they can increase the measurement speed without using 8bit information. It can also be used to overcome the gamma of projector without projector nonlinearity calibration. Moreover, these patterns can overcome the shortcomings from phasebased optimization and intensitybased optimization.
Both simulation and experiments demonstrate that the proposed method can achieve substantial phase quality improvements when the projector is at different defocusing levels.
The rest of the paper is organized as follows. Section Methods explains the principles concerned in this technique and the process of fringe pattern generation. Section Results and Discussion presents simulation results, experimental results and the merits of the proposed technique. Section Conclusions summarizes this paper.
Methods
Threestep phaseshifting algorithm
Sinusoidal phase shifting algorithms are very widely used in conventional optical metrology due to their fast speed and high accuracy [30]. Typically, the more numbers of fringe patterns are used, the better measurement quality can be achieved. For highspeed 3D shape measurement, a threestep phaseshifting algorithm is usually adopted since it requires the minimum number of patterns to get the phase uniquely point by point. Since our research focuses on highspeed 3D shape measurement, a simple threestep phaseshifting algorithm with a phase shift of 2π/3is used to evaluate the proposed optimization algorithm. Three fringe images can be described as:
Where A(x, y) is average intensity. B(x, y) is intensity modulation. ϕ(x, y) is the phase to be solved for:
This equation provides the phase ranging [−π, π) with 2π discontinuities. A continuous phase map can be obtained by adopting a spatial or temporal phase unwrapping algorithm. In this research, we use the temporal phase unwrapping method with three frequency phaseshifting algorithms.
Bayerdithering technique
Dithering techniques have been developed to convert a higher bit depth into a lower bit depth, and this is analogous to halftone technique used in printing. Various dithering algorithms can be used to approximate an ideal sinusoidal fringe pattern, such as simple thresholding, random dithering, and ordered dithering [31].
Among these dithering methods, Bayerordered dithering has been extensively utilized in many applications due to its simplicity and efficiency, which is adopted in this study. The Bayer dithering technique compares the original image with a 2D grid of thresholds called Bayer kernel, and then the original image is quantized according to corresponding pixels in the Bayer kernel: if the grayscale value is larger than the kernel, the pixel is turned to 1 (or 255 grayscale value), otherwise to 0. Neighboring pixels do not affect each other. Different kernels can generate completely different dithering effects. Among the kernels used, Bayer has shown that if the sizes of the matrices are 2^{N} (N is an integer), there is an optimal dither kernel that results in the pattern noise being as highfrequency as possible. A lowpass filter can suppress the highfrequency noises. The Bayer kernel can be obtained by:
Where M_{1} is the smallest base, and other larger patterns can be obtained by:
Where U_{n} is ndimensional unit matrix. In this research, we find that the Bayer kernel 8 × 8 produces the best results for all tested fringe stripes, and then it is utilized.
Weight objection function
Dithering optimization technique has been researched in reducing its overall phase error. The objective function of all these optimization techniques is to get the best fit of the binary patterns [32]. The optimization process can be described as a norm function such as Frobenius norm function:
Where ‖•‖ represents the Frobenius norm. I(x, y) is the ideal sinusoidal intensity pattern. G(x, y) represents a 2D Gaussian function. B(x, y) is a 2D binary pattern and ∗ represents convolution. The optimization function evaluates the global intensity similarity between the dithering pattern and its corresponding ideal pattern, but it just considers the global similarity without focusing on local structure similarity. In addition, NP problem makes it difficult to calculate the object function [15].
The objective of all these optimization techniques is to obtain the best fit of the binary patterns which are close to the ideal sinusoidal pattern. The optimized binary patterns should be as close as possible to the ideal sinusoidal patterns after applying Gaussian filter. The local detailed structure is one of the important resources in an image. It contains particular highfrequency components. In order to contain local similarity in the objective function, we propose a weighted object function (WOF). The objective function combines two parts: a global intensity part and a local structure part. The global intensity measurement is based on the normalized mean squared error (NMSE). The local structure measurement is based on the residual error of intensity (REI). If the intensity of ideal pattern are I and the intensity of dithering pattern is I_{d}, WOF can be expressed as:
Where NMSE(I, I_{d}) measures the global similarity and REI(I, I_{d}) measures the local similarity, respectively. WOF gets the synthetic error between the ideal pattern I and the dithering pattern I_{d}. ω_{1} and ω_{2} are weighted factors limited by ω_{1} + ω_{2} = 1. Figure 1 illustrates phase root mean square error values RMS (rad) when weighted factor ω_{1} = 0~1. In order to simplify the description, we call it phase RMS error in the following parts. The lowest point of the broken line in Fig. 1 means the root mean square error values RMS value is the smallest when ω_{1} is 0.7. Thus, in this research, we set ω_{1} = 0.7.
First a global intensity similarity is introduced into the WOF expression. NMSE(I, I_{d}) can evaluate the global intensity similarity between the dithering pattern and the corresponding ideal pattern. NMSE(I, I_{d}) can be shown as:
Where H and W respectively represent height and width of the pattern. NMSE(I, I_{d}) method reflects statistical errors from the whole fringe pattern intensity. Thus, the smaller the NMSE(I, I_{d}) is, the better the whole pattern quality becomes. However NMSE(I, I_{d}) only focuses on the global similarity, so REI(I, I_{d}) is introduced as an optimization term to evaluate the local structure similarity between the dithering pattern and the corresponding ideal pattern.
Because the existence of the third harmonics does not induce any phase errors for threestep phaseshifting algorithm [33], it can be proved that if the intensity error varies periodically at the frequency which is multiple of 3f_{0} (f_{0}is the fundamental frequency of the desired sinusoidal pattern). The obtained phase map could be as same as the ideal phase map. The intensity error could not vary periodically but it is possible to separate the 3f_{0} harmonics from the total intensity error. Therefore, the intensity residual error function REI(I, I_{d}) is designed as:
Where E(x, y) is the total intensity error between the defocused binary pattern and the ideal sinusoidal pattern. \( {E}_{3{f}_0}\left(x,y\right) \) is the periodically harmonics. G(x, y) is a Gaussian function. B(x, y) is the binary dithered pattern. * is convolution. I(x, y) is the ideal sinusoidal pattern intensity.
It can be assumed that a periodical signal s(x) varies along x direction for one period T(T = 3 N) and s_{3N}(x) is one part of s(x). In order to extract the harmonics s_{N}(x) from the whole signal s_{3N}(x), discrete Fourier transform is employed to obtain its spectrum S_{3N}(k). After that, all those harmonics at the frequency (3l)th should be retained and other harmonics should be removed. Therefore, the new sampled spectrum S_{N}(k) can be described as:
Where H(k) is the Dirac comb function and every δ function is separated from each other at 3. According to discrete Fourier transform theory and convolution principle, the filtering process in the spatial domain can be described as:
Where h(x) is the spatial filtering function. ∗ represents convolution. It turns out that h(x) is a Dirac comb function, but it is composed of three δ functions and the extent decreases to onethird. Therefore, the (3l)th harmonics of s_{N}(x) can be solved for:
Therefore, the (3l)th intensity error \( {E}_{3{f}_0}\left(x,y\right) \) can be extracted from the intensity error E(x, y), which can be described as the following function:
This function demonstrates a simple way to extract the (3l)th harmonics s_{N}(x) out of signal without discrete Fourier transform. Firstly, s_{3N}(x) is divided into three segments at the same length N. Secondly, three segments are added together to calculate an average segment. And finally, the (3l)th harmonics s_{N}(x) are generated by periodically extending the average segment along x direction three times. Then, the intensity residual error REI(I, I_{d}) can be calculated.
Implementation in the best patch optimization framework
Instead of optimizing the whole fringe pattern, we use an idea which called binary patch, and then the patch are tiled to generate fullsize patterns using symmetry and periodicity. Assuming that the desired sinusoidal fringe patterns vary along x direction: the bestfit binary pattern should be symmetric along x direction for one fringe period (T); and it should be periodic along the y direction. The period in every row S_{y} is defined as the period along y direction. In this paper, we take S_{x} = T/2 as the patch column period and T is the fringe period. The row period S_{y} for a best patch is set to different widths which are optimized respectively and from which the best one is chosen. This framework also includes two parts, the global intensity optimization and the further combined optimization. The process of optimization can be divided into the following major steps:

Step 1: Initial patch generation. Bayer dithered sinusoidal fringes are generated firstly. The patch size of the row period S_{y} ranges from 2 to 10. Then, according to the symmetry of the sinusoidal pattern and the optimization efficiency, the patch column period S_{x} is set to T/2 which is one half of fringe period.

Step 2: Patch optimization based on weighted error function. Each pixel (i, j) are mutated from 0 to 1 or from 1 to 0. Because three step phaseshifting algorithm is used, there are eight groups: (0,0,0), (0,01), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), (1,1,1). They are filtered by a Gaussian filter with size G = 5. Weighted error function is further optimized for each group and the group error ζ is kept from the one with biggest absolute phase error to the smallest one. The further optimization is performed iteratively until phase RMS error is less than 0.01% or the number of iteration reaches maximum times. Figure 2 shows the phase RMS error after each round of optimization.

Step 3: Patch shifting. The global optimization binary patch is tiled and shifted to produce phase shifted fringes.

Step 4: Patch selection. After a number of patch mutations, a set of optimized patches are generated. From these patches, the best patch is selected based on the following two rules: (1) phase error does not change drastically if different sizes of Gaussian filter are applied; and (2) the resultant phase error is consistently small. These two rules imply that the best patch under one defocusing level may not be chosen.

Step 5: Fringe pattern generation. Utilizing symmetry and periodicity properties of fringe patterns, the desired size fringe pattern is generated by tiling the best patch together.
Figure 3 shows binary patterns before and after applying the optimization algorithms. Figure 3a shows the desired sinusoidal pattern. The period of every fringe pattern is T = 20pixels. Figure 3b, c and d show three binary patterns comparison, which are respectively generated by utilizing intensitybased optimization, phasebased optimization and the proposed method.
Phasebased optimization (popt) and intensitybased optimization (iopt) are also two main categories to improve the quality of fringe patterns. In order to show the superiority of the proposed optimization method, we compare it with the norm Frobenius (NF), the popt [17], iopt [14] and ideal sinusoidal fringe patterns under the same condition. Figure 3 shows the comparison among different optimization. Figure 4a shows the comparison between the ideal sinusoidal curve and crosssections from different fringe patterns. Figure 4b and c give the intensity difference and phase difference with iopt, oopt, NF and the proposed method. From the results we can find that the optimization framework can generate better quality and less phase error. Because NF just compares the global similarity without considering the local structure similarity, in the following section we just compare the intensitybased optimization, phasebased optimization and the proposed optimization.
Results and discussion
We evaluate the proposed algorithm through simulation where different periods are used. In this simulation, we use fringe periods T = 30,50,70 and 90 pixels. Different defocusing levels are expressed by applying different sizes of Gaussian filters. The size of Gaussian filterGranges from 5 to 17 and standard deviation is σ = G/3.
When the size of Gaussian filter is 5 × 5 (σ = 5/3), it represents the case that the projector is nearly focused. When the size of Gaussian filter is 17 × 17 (σ = 17/3), it represents the case that the projector is defocused to a much more degree. In this simulation, the wrapped phase is obtained based on threestep phaseshifting algorithm and the phase RMS error is obtained by calculating the difference between the phase obtained from the binary patterns and the phase obtained from the ideal sinusoidal fringe pattern. Figure 5 shows the phase RMS error comparison among phasebased optimization (Phaseopt) [17], intensitybased optimization (Intensityopt) [14] and the proposed bestpatch optimization (Proposedopt).
By analyzing the simulation, we can find that when the projector is nearly focused (Gaussian filter size is 5 × 5 and standard deviation is 5/3), the phasebased optimization has the lowest phase RMS error. When the period changes from 20 pixels to 90 pixels, the phasebased has the best phase quality. While with the Gaussian filter size increasing, the quality from phasebased optimization decreases because it is sensitive to filter sizes. When the projector is slightly defocused (Gaussian filter size is 11 × 11 and standard deviation is 11/3 pixels). The phase error from the intensitybased optimization and the phase error from the proposed method decrease greatly as the filter size increases and they produce better results. They have the same tendency when the filter sizes increases. From simulation results, we can also find when the filter size is small, the proposed optimization almost have smaller phase RMS errors than the intensitybased optimization. It means that when projector is nearly focused(the filter size is small), the proposed optimization can perform better than intensitybased optimization. Besides it is robust to defocusing levels. When the projector is slightly focused and strongly defocused (the filter size is large), the proposed optimization can still get high quality fringe pattern with less phase RMS error. Simulation has verified the superiority of the proposed optimization technique which can produce fringes with high phase quality under different defocusing levels.
We also develop a 3D shape measurement system to verify the proposed method which consists of a DLP projector (Samsung SPP310MEMX) and a digital CCD camera (Daheng MER50014U3M/CL). The camera is attached with a 16 mm focal length lens (Computar M1614MP) and the lens resolution is 1024 × 768. The resolution of projector is 800 × 600 and it has 0.49–2.80 m projection distance.
A uniform flat board is measured based on different optimization under three defocusing levels. Because phaseshifting algorithm with (n + 2) steps is insensitive to highorder harmonics up to the nthorder [20]. So in this research, the ideal phase is obtained by a tenstep phaseshifting algorithm and the period of fringe patterns is set to 20 pixels. Figure 6ac show dithering fringe patterns which are optimized based on the proposed method from nearly focused, slightly defocused and strongly defocused respectively. Figure 6df show the corresponding phase RMS errors and the phase RMS errors comparison from intensitybased optimization and phasebased optimization. This experimental results show that when the projector is nearly focused, the proposed method and phasebased method have better phase quality and the intensitybased method has the largest rms error. With the defocusing levels increasing, the intensitybased method performs better than the phasebased method while the proposed method has small fluctuate. However phase errors are still smaller. It indicates that the proposed method performs consistently well to get the better phase quality under three defousing levels.
Furthermore, a more complex 3D statue is measured to compare these optimization methods under the same projector defocusing levels respectively. Figure 7 shows the deformed fringe pattern under nearly focused, slightly defocused and strongly defocused respectively. The measurement results are shown in Fig. 8. In order to compare the quality of results, Figs. 9, 10 and 11 show the results based on phase optimization and the results based on intensity optimization at the same defocusing levels respectively. From the measurement results, it can be found that when the projector is nearly focused, the result from the phasebased optimization appears slightly better than the intensitybased optimization method. Under slightly defocused and strongly defoucsed, the results from the intensitybased optimization method have less errors than the results from phasebased optimization method. Because the proposed method decreases the phase error and it optimizes the intensity errors, the proposed method can reconstruct smoother surface than both optimization methods.
In order to give a more clear comparison, the phase errors of three dithering methods are calculated and they are shown in Fig. 12 according to Figs. 8, 9, 10 and 11. When the projector is nearly focused, phase RMS errors of intensitybased optimization, phasebased optimization and the proposed optimization are 0.057 rad, 0.048 rad, and 0.045 rad. When the projector is slightly defocused, the corresponding phase RMS errors are 0.049 rad, 0.055 rad, and 0.045 rad for slightly defocused and when the projector is strongly defocused, the phase RMS errors are 0.045 rad, 0.058 rad and 0.042 rad. Table 1 lists the corresponding measurement time based on Fig. 12. By comparison between the measurement time, it can be found that the proposed method can increase measurement speed.
These results also clearly show that phasebased optimization technology can improve phase quality under a certain condition but it is sensitive to the amount of defocusing. The intensitybased optimization technology is able to improve the fringe quality with the defocusing levels increasing, but it can not reduce phase errors efficiently. The proposed technique has the same tendency with intensitybased optimization but it can increase the phase quality under different defocusing levels. The results are consistent with the simulation analysis.
Conclusions
This paper has presented an optimization method based on the weight objective function to generate highquality sinusoidal fringe patterns. By using normalized mean squared error and residual error of intensity, the global similarity and local similarity can be optimized. Utilizing symmetry and periodicity properties of the fringe patterns, the desired fringe pattern is generated by tiling the best patch together. Simulation shows that the proposed fringes maintain high phase quality under different filter sizes. Experimental results also verify the feasibility and advantage of the proposed technique that the proposed method can generate highquality binary fringes under different defocusing levels.
Abbreviations
 2D:

Twodimensional
 3D:

Threedimensional
 DFP:

Digital fringe projection
 Iopt:

Intensitybased optimization
 NMSE:

Normalized mean squared error
 OPWM:

Optimal pulse width modulation
 Popt:

Phasebased optimization
 PWM:

Pulse width modulation
 REI:

Residual error of intensity
 RMS:

Phase root mean square
 WOF:

Weighted object function
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Acknowledgments
This work was supported by National Key R&D Program of China (2017YBF1300900), the the National Natural Science Foundation of China (U1713216) and the Fund of Shenyang (1787000).
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Faculty of Robot and Engineering of Northeastern University, Shenyang, China provides the funding for this research.
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Correspondence to Feng Lu.
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Keywords
 Fringe analysis
 Binary defocusing
 Dithering
 Optimization