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On the basis of the objective functions, dithering optimization techniques can be divided into the intensity-based optimization technique and the phase-based optimization technique. However, both types of techniques are spatial-domain optimization techniques, while their measurement performances are essentially determined by the harmonic components in the frequency domain. In this paper, a novel genetic optimization technique in the frequency domain is proposed for high-quality fringe generation. In addition, to handle the time-consuming difficulty of genetic algorithm (GA), we first optimize a binary patch, then join the optimal binary patches together according to periodicity and symmetry so as to generate a full-size pattern. It is verified that the proposed technique can significantly enhance the measured performance and ensure the robustness to various amounts of defocusing.
Three-dimensional (3D) depth data acquisition techniques are becoming increasingly important for video game design, animation, movies, virtual reality, telesurgery, medicine, homeland security and manufacturing.[1] Among these techniques, the digital fringe projection (DFP) techniques have the great potential to be an integrated part of intelligent systems due to their simplicity, reliability and flexibility.[2] To reconstruct the 3D depth data, the DFP technique utilizes a projector to project a sequence of sinusoidal phase-shifted fringe patterns onto the surface of an object, and a camera to capture the deformed fringe patterns.[3] However, the conventional DFP technique needs 8 bits to present sinusoidal fringes, which is limited by a projector’s frame rate (typically
To break the speed limitation, the binary defocusing techniques[6,7] were introduced, with which a properly defocused binary pattern is considered as a sinusoidal pattern in a good approximation. Based on the structure of digital micro-mirror device (DMD), we are able to present the binary patterns by simply toggling DMD, which can improve the frame rate.[8] Moreover, the 1-bit depth of a binary pattern can eliminate the projector nonlinearity error easily. However, the measurement quality based on the binary defocusing technique is not comparable to that from the conventional DFP technique due to the harmonic errors.[9,10]
The squared binary defocusing method (SBM) was proposed at the very beginning.[11] Later, sinusoidal pulse width modulation (SPWM),[12] optimal pulse width modulation (OPWM),[13] and tripolar OPWM[14] were proposed and developed to reduce the harmonics errors. These techniques can achieve good performance with narrow fringe stripes, but all fail to produce high-precision defocused sinusoidal fringes with long fringe periods. These modulation techniques are one-dimensional techniques, which cannot take advantage of two-dimensional characteristics of a binary pattern.
The dithering technique,[15] which is also called halftoning, takes advantage of two-dimensional characteristics and was introduced to enhance the phase quality for wide fringe stripes. However, the improvements have still been limited by narrow fringe stripes.[16] Since dithering technique is only a simple matrix transformation, the measurement quality has great room to be improved. Dithering optimization techniques are introduced for a DFP system with defocused projectors. On the basis of the objective function, the previous techniques can be divided into the intensity-based optimization technique[17] and the phase-based optimization technique.[18] These two techniques are referred to as i-opt and p-opt hereafter, respectively. The p-opt technique can obtain better performance, while the phase quality directly determines the measurement performance. However, the different sensitivities to different amounts of defocusing make the p-opt technique troublesome to use. Although the i-opt technique cannot improve the measurement quality directly nor efficiently, it ensures the robustness to different amounts of defocusing. The two techniques can achieve pretty good measurement results, and they are both spatial-domain optimized. Since the most troublesome noise is induced by the high-frequency harmonics, it is necessary to optimize a binary pattern based on the frequency to improve the measurement quality.
The genetic algorithm (GA)[19] is an evolutionary algorithm on the basis of the theory of natural selection. The GA is a non-deterministic global optimization well suitable for binary fringe generation, which iterates through selection, crossover and mutation stages. A genetic method[20] was proposed to optimize the dithering technique. However, it is a time-consuming process to achieve good measurement quality.
In this paper, a genetic optimization framework in the frequency domain is proposed, which essentially improves the measurement quality by eliminating the harmonic errors. In view of the fact that defocusing amount is continuous in reality, our framework optimizes the binary pattern not only at a certain amount of defocusing. On the basis of the periodicity and symmetry property of sinusoidal pattern, we search a best binary patch instead of the whole pattern to handle the time-consuming difficulty of GA. The optimal patch is then tied together to generate the whole pattern. It is verified that the proposed technique is able to consistently obtain higher measurement performance under various amounts of defocusing.
Some related principles of the proposed technique are introduced in Section
Phase-shifting algorithms have been increasingly utilized in a DFP system due to their high measurement accuracy and flexibility.[21] Typically, these algorithms obtain better 3D measurement result with more fringe images. A minimal number of fringe patterns is required in a three-step phase-shifting algorithm with a phase shift of 2π / 3. The intensities of these sinusoidal fringe patterns can be described as
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Dithering techniques used in computer graphics are introduced to represent a color depth pattern with a limited bit depth.[23] To approximate an ideal sinusoidal pattern with a binary pattern, a variety of dithering algorithms have been proposed, such as error-diffusion dithering, pattering dithering, simple thresholding, ordered dithering and random dithering.[16] The error-diffusion dithering algorithm has been widely utilized due to its high accuracy. For an error-diffusion algorithm, it adopts an error diffusion kernel d(i,j) to quantize pixels in a specified order, and it then compensates for the quantization error through the use of feedback. The process is shown as
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![](cpb_28_8_084202/cpb_28_8_084202_ieqn2.gif)
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Two-dimensional discrete Fourier transform (DFT2) is often used in mathematical imaging and vision. The DFT2 of an M × N image f(x,y), can be described as
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Here, F(u,v) is the coefficient matrix in frequency domain,
To reduce the computation time of DFT2, two-dimensional fast Fourier transform algorithm (FFT2) containing a variety of tricks is proposed.[24] The FFT2 can push the algorithm complexity from N
2 into
The key issue of the binary defocusing technique is how to produce binary patterns which can imitate the original sinusoidal patterns well after being defocused by a projector. The previously proposed techniques all optimized a binary pattern based on the spatial domain. Actually, the harmonic components in the frequency domain essentially determine the measurement quality. A sinusoidal fringe pattern is shown in Fig.
Our proposed method translates the process of optimization from the spatial domain to the frequency domain. Its aim is to search an optimal binary fringe to approximate the sinusoidal fringe in the frequency domain, so the problem is formulated as
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![](cpb_28_8_084202/cpb_28_8_084202_ieqn6.gif)
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In view of the fact that defocusing amount is continuous in reality, our framework optimizes the binary patch not only at one certain amount of defocusing. Let S be the sinusoidal patch, and P be the patch to be optimized. The approximation error in the frequency domain with a certain sized Gaussian filter can be defined as
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![](cpb_28_8_084202/cpb_28_8_084202_ieqn8.gif)
Gaussian filters of 5 × 5, 9 × 9, and 13 × 13 are, respectively, modeled as nearly focused, slightly defocused, and severely defocused projectors. To make the measurement quality robust to various amounts of defocusing, the frequency error should be minimized at various amounts of defocusing. Mathematically, the objective function can be selected as
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The GA was first introduced by John Holland in the 1960s.[19] According to the theory of natural selection, the iteration of GA has different stages including selection, crossover and mutation.[25] The solutions are called chromosomes. Chromosomes are reproduced by crossover and mutation to create a new generation.[26]
To solve the time-consuming problem, we search an optimal binary patch, not the full-size pattern. The optimal binary patch is then tiled together according to periodicity and symmetry to generate a full-size pattern. Assuming that the sinusoidal fringe is vertical, the binary fringe should be symmetric about the horizontal direction and periodic along both horizontal and vertical directions. Figure
Simulations were implemented to evaluate the superiority of the proposed technique with a wide range of fringe periods. In these simulations, fringe patterns (resolution 1024 × 768) with fringe periods T = 18, 36, …, 90, 108 were used to ensure practicability. Various amounts of defocusing were simulated by adopting Gaussian filter with different values of size G and standard deviation σ (G = 5∼17, and σ = G/3). A three-step phase-shifting algorithm is used to calculate the phase map, and the phase RMS error was achieved between the phase acquired from the ideal sinusoidal patterns and that from the blurred binary patterns. To compare the difference, the phase errors obtained from the i-opt[17] fringes and those from the p-opt[18] fringes were also evaluated. The two optimization methods are both optimized on condition that G = 5, and σ = G/3.
The desired ideal sinusoidal pattern and binary patterns obtained by adopting the optimization methods are shown in Fig.
Figure
Figure
Experiments were conducted to verify the proposed technique using a 3D shape measurement system including a Daheng mercury camera (MER-130-30UM) with an 8-mm focal length megapixel lens (Computar M0814-MP2) and a projector (Esonic HD-720P). The projector has a resolution of 1280 × 720, and the image has a resolution of 1024 × 768. Figure
First, an experiment was conducted to measure a flat white board by using the p-opt technique, the i-opt technique and the proposed technique. The phase errors were obtained between the phases recovered from the ideal phase and defocused binary patterns. The ideal phase was obtained from the measurement results of sinusoidal fringes with 18-step phase-shifting algorithm. The phase RMS errors achieved by different optimization techniques with various amounts of defocusing are shown in Fig.
![]() | Fig. 7. Comparisons of experimental results from p-opt technique, i-opt technique, and proposed technique for (a) nearly-focused case, (b) slightly defocused case, and (c) severely defocused case. |
To visually compare these different techniques, a more complex 3D statue is also measured. Fringe period is set to be T = 36 pixels in this experiment. The binary fringe patterns utilized in the experiment are shown in Fig.
![]() | Fig. 8. Three defocused patterns obtained by utilizing p-opt technique, i-opt technique, and proposed technique, respectively. |
![]() | Table 1.
Phase RMS error (in unit rad) of measurement results. . |
In this paper, a genetic optimization framework is proposed to produce high-quality binary fringe patterns in the frequency domain. The harmonic noise can be easily eliminated in our framework based on frequency domain. We find that the proposed technique can consistently obtain substantial measurement performance improvement for various amounts of defocusing. It is verified that the proposed technique can obtain pretty good phase performance and it ensures the robustness to different amounts of defocusing.
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