† Corresponding author. E-mail:
Project supported by the Zhejiang Provincial Welfare Technology Applied Research Project, China (Grant No. 2017C31080).
Dithering optimization techniques can be divided into the phase-optimized technique and the intensity-optimized technique. The problem with the former is the poor sensitivity to various defocusing amounts, and the problem with the latter is that it cannot enhance phase quality directly nor efficiently. In this paper, we present a multi-objective optimization framework for three-dimensional (3D) measurement by utilizing binary defocusing technique. Moreover, a binary patch optimization technique is used to solve the time-consuming issue of genetic algorithm. It is demonstrated that the presented technique consistently obtains significant phase performance improvement under various defocusing amounts.
Digital fringe projection (DFP) techniques become multiply crucial due to their measurement accuracy and flexibility. Actually, it remains difficult to obtain high accuracy and high speed simultaneously.[1–3] The conventional DFP technique uses 8-bit sinusoidal fringes. which limits the measurement speed to the projector refresh rate (typically 120 Hz).[4] In addition, measurement error is introduced, if a commercial projector is utilized without calibrating its nonlinear gamma.[5]
The binary defocusing techniques have been developed to break the speed bottleneck.[1,6,7] Moreover, nonlinear gamma can be eliminated automatically with 1-bit depth patterns. However, some additional issues are present and need handling: (i) the smaller depth range and (ii) the high-order harmonic error. To address the aforementioned issues, endeavors have been made. Su et al.[8] introduced a three-dimensional (3D) measurement system by utilizing the defocused projection of a Ronchi grating at the very beginning. Later, Lei and Zhang[9] introduced squared binary defocusing method (SBM), then Ayubi et al.[10] presented sinusoidal pulse width modulation (SPWM) technique, and Wang and Zhang[11] presented optimal pulse width modulation (OPWM) technique. The SBM technique can achieve good performance only when a projector is properly defocused. Both SPWM and OPWM technique can generate good results even with a nearly focused projector. However, SBM, SPWM, and OPWM all fail to produce a high-performance 3D measurement with a long fringe period. Since the techniques described above only modulate the patterns in one dimension, it is natural to utilize two-dimensional (2D) area modulation methods to enhance the measurement quality for wide fringe stripes.
To address challenges for wide fringe stripes, dithering,[12] also called halftoning, technique is introduced. It can significantly enhance the phase quality with wide fringe stripes, but suffer limitation when fringes are narrow.[13] Actually, dithering technique is only a simple matrix transformation therefore far from being optimal. Various methods of dithering optimization are developed for DFP systems. A phase-optimized framework was proposed to optimize the Bayer-dithering technique by Dai and Zhang.[14] And then, an intensity-optimized dithering technique[15] was also introduced. Compared with intensity-optimized technique, the phase-optimized technique can achieve better 3D shape measurement results, but it is more sensitive to defocusing amount.
Genetic algorithm (GA)[16] was first introduced by Holland and his colleagues. The GA is a non-deterministic global optimization fitting for binary fringe generation, which obtains an optimal solution through selection, crossover, and mutation stages. Lohry and Zhang[17] introduced a genetic method to optimize dithering techniques, but the time-consuming problem remained to be solved.
In this paper, we present a multi-objective optimization framework, which includes both an intensity-based objective and a phase-based objective. It effectively balances the measurement quality and sensitivity to the defocusing amount. In addition, to solve the time-consuming issue of GA, our framework abandons any attempt to optimize the whole pattern pixel by pixel. According to the property of a sinusoidal pattern, the binary pattern should be symmetric and periodical. We search a best binary patch and tile the patches together so as to present a whole pattern utilizing periodicity and symmetry. It is shown that the presented technique can consistently achieve substantial phase quality improvement with various defocusing amounts.
Some related principles of the proposed technique are introduced in Section 2. Simulations and experiments are presented in Sections 3 and 4, respectively. Finally we draw some conclusions from the present study in Section 5.
Various phase-shifting algorithms have been developed for phase analysis technique.[18] Typically, these algorithms obtained better 3D measurement results with more fringe patterns. Usually a three-step phase-shifting algorithm is adopted in high-speed metrology, with requiring a minimum number of patterns. Three 2π/3 phase-shifted fringes can be expressed as follows:
The dithering techniques used in computer graphics have been developed to represent a color depth with a limited depth.[20] To modulate the binary patterns, various dithering techniques are utilized, such as error-diffusion dithering, pattering dithering, simple thresholding, ordered dithering, and random dithering. The high accuracy of the error-diffusion dithering technique makes it extensively used in the past years. For an error-diffusion algorithm, a diffusion kernel
As discussed previously, the intensity-optimized technique tries to eliminate the intensity differences compared with the sinusoidal fringe. That is to say, a Gaussian smoothed binary fringe ought to be as close to a sinusoidal fringe as possible. The intensity-optimized problem can be described as
The optimization we propose is to take a comprehensive account of the intensity domain and the phase domain. Mathematically, the objective of optimization can be described as
To solve the time-consuming issue of GA, a binary patch is optimized, instead of the whole fringe pattern which typically is 1024 × 768. The optimized patch can then be tiled with respect to symmetry and periodicity so as to generate the whole pattern. Assuming that the sinusoidal fringe is vertical, two phenomena come into notice: i) it ought to be periodic in not only the horizontal direction but also the vertical direction; and ii) it must be symmetric in one period along the horizontal direction. The flowchart of optimization steps is shown in Fig.
The patch size is initialized by the row period
The GA considers a collection of individuals called a population. We design a population of 60 patches for each generation. Different patches can be formed with respect to the property that it is path- and origin-dependent for an error-diffusion algorithm. Among the patches, 4 patches are respectively extracted from the error-diffusion dither patterns with algorithm starting from the four vertices of the sinusoidal fringe pattern, and the other patches are randomly initialized.
Individuals are evaluated by the fitness function described in Eq. (
Individuals with good characteristics are selected as parents related to probability. The stochastic uniform selection method is adopted in this paper.
Crossover is the most important operator of the GA. A random position and a random length is chosen to speed up the process, the region which has a larger error is more likely to be chosen with a probability that is greater than 50%.
In typical GA implementations, the mutation rate is very small. We set the rate to be 1%, and the region with higher error undergoes mutating more than 50% of the time like crossover.
If the stopping criterion is not satisfied, go back to Step 3. In this paper, the stopping criterion is that the average change of the fitness is less than
Increase
Select the best individual to produce the optimal patch which minimizes Eq. (
The optimal binary patch can be tiled with respect to symmetry and periodicity so as to generate the whole pattern.
Some simulations were conducted to verify the superiority of the presented technique. In these simulations, fringe patterns (resolution 1024 × 768) with periods T = 12, 18, …, 114, 120 were used to ensure the practicability. Various defocusing amounts were simulated by using the Gaussian filters with different sizes S and standard σ, (S=[5,9,13],
In this paper, different optimization techniques are optimized by utilizing the Gaussian filter with a size of 5 × 5 and a deviation of 5/3. During our simulation, we found that the proposed technique was able to obtain fairly good results, when the weighting coefficient *
Experiments were conducted to evaluate the presented technique by utilizing a 3D measurement system equipped with a Daheng mercury camera (MER-130-30UM) of 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. According to the simulation results, we set α = 0.7 for Eq. (
The simulations were experimentally verified with a flat white board. The phase RMS error is shown in Fig.
To visually compare these different techniques, we consider a more complex statue. The fringe stripe was chosen to be T = 18 pixels in this experiment. Figure
Figure
In this work, we present a multi-objective optimization framework, which includes intensity-based objective and phase-based objective. It effectively balances the measurement quality and sensitivity to the defocusing amount. It is demonstrated that the proposed technique can significantly improve the measurement performance under various defocusing amounts.
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