Open Access
Add to BibSonomy
Abstract
For precision inspection of free-form surface parts using non-contact measurement methods, the registration between the actual measurement model and the ideal design model is necessary.The traditional iterative closure point (ICP) method requires good initial parameters to obtain the global optimal transformation matrix, which is difficult to guarantee in the actual detection process. In order to improve the accuracy and robustness of free-form surface precision inspection, an Improved Whale Optimization Algorithm (IWOA) is proposed in this study.This algorithm can solve the required registration parameters by constantly updating the population. A measurement experimental system is designed to test the accuracy of blade registration. The performance of IWOA is evaluated by the actual measurement experiment, and the results are verified by a comparative study with Whale Optimization Algorithm (WOA), Lévy flight trajectory-based Whale Optimization Algorithm (LWOA), and Adaptive Whale Optimization Algorithm (AWOA). The surface registration errors are 0.1711mm for IWOA, 2.0015 mm for WOA, 1.2656 mm for LWOA, 2.8132 mm for AWOA and 2.1537 mm for ICP. The results show that the accuracy of IWOA is more than 7 times higher than other four algorithms. In general, the experiments indicate that IWOA has a good registration ability and can meet the needs of industrial measurement.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Free-form surface parts have been widely used in aerospace, automotive, biomedical engineering and other fields. In the surface inspection, with the increasing requirements of industry to improve product precision, non-contact measurement technology has become a popular choice for precision inspection [1,2]. Non-contact measurement is based on a design model, namely a computer-aided design (CAD) model. In the process, the machined parts are first scanned to generate the measurement model containing a great deal of three-dimensional (3D) points. The measurement model is then converted to align with the CAD model. Finally, the part error of the misaligned area is displayed. It should be noted that the design and measurement models are located in different coordinate systems. Therefore, an optimal transformation matrix is required for the conversion, and this process is called surface registration [3].
The purpose of surface registration is to match the measurement model and CAD model as closely as possible in a common coordinate system. At present, the ICP proposed by Besl et al. is the best-known registration method. The method is simple to calculate, but it is sensitive to the initial position. A good initial estimate is absolutely necessary to find the accurate solution for ICP. Fortunately, the heuristic algorithm can be used to overcome the above shortcomings efficiently. Although there are many ICP variants studied by many scholars [4–6], some aspects of the surface registration process still require to be further tested in actual applications. In recent years, many types of heuristic algorithms have been widely used to solve the registration problems, such as Genetic Algorithm (GA) [7], Particle Swarm Optimization (PSO) algorithm [8], Differential Evolution (DE) algorithm [9], Grey Wolf Optimizer (GWO) algorithm [10], and so on. Note that the heuristic algorithm is not dependent on the initial position, and has been recognized and applied in almost all fields of science, engineering and industry for its intuitive and effective solution to complex computational problems.
Recently, a novel heuristic algorithm called Whale Optimization Algorithm (WOA) [11] is proposed to mimic the predation behavior of humpback whales. For the original WOA, some scholars have made improvements to enhance the performance of WOA [12–15]. Ling et al. introduced an Lévy flight strategy into WOA and proposed an Lévy flight trajectory-based Whale Optimization Algorithm (LWOA) [16]. Compared with some famous heuristic algorithms, the experimental results illustrate that LWOA has higher efficiency in solving multimodal problems. However, its convergence speed is lower than other algorithms for some problems. Sun et al. proposed an Adaptive Whale Optimization Algorithm (AWOA) [17], which improves convergence accuracy and local optimization capabilities by introducing adaptive weight. Compared with basic WOA, AWOA performs better in solving unimodal problems, unfortunately, it is easy to fall into the local optimum for a few multimodal problems. In the aforementioned WOA variants, it is difficult to achieve a fast convergence rate and therefore to avoid the local optimum simultaneously. It is undeniable that many WOA improvements can be used to enhance the performance of original WOA. However, the results of these efforts are still not satisfactory in addressing many challenges. Therefore, it is necessary to further study the improvement of whale optimization algorithm for some specific problems.
In order to enhance the performance of WOA and further balance the exploitation and exploration capabilities, an Improved Whale Optimization Algorithm (IWOA) is proposed and used in surface registration. In the proposed IWOA, there are two main highlights, the first is to use the Sobol sequence to initialize the population to make it more evenly distributed in the search space, thereby improving the search ability of the algorithm. The second is the updating strategy of whale position,which can increase the diversity of the population and improve the ability of the algorithm to jump out of the local optimum. In order to evaluate the performance of the proposed algorithm, the actual surfaces are tested by experiment, and the results show that the proposed algorithm is more feasible and effective.
The rest of this paper is organized as follows. The objective function for the surface registration problem and the basic operations of the WOA algorithm are introduced in Section 2. The details of the proposed algorithm IWOA are presented in Section 3. The experimental results of simulation experiment and the practical application of registration are given in section 4. Eventually, Conclusions and some future prospects are provided in Section 5.
2. Mathematical description of surface registration and WOA algorithm
2.1 Surface registration
In the surface registration, the measurement model is established in measurement coordinate system (MCS), while the CAD model is located in design coordinate system (DCS). To realize the registration of these two models, it is necessary to establish the transformation relationship between MCS and DCS.
Two sets of point clouds, namely measurement model $P = \{ {p_i} \in {R^3},{\kern 1pt} {\kern 1pt} {\kern 1pt} i = 1,2, \cdots ,N{p}\} $ and CAD model $\textrm{Q} = \{ {q_i} \in {R^3},{\kern 1pt} {\kern 1pt} {\kern 1pt} i = 1,2, \cdots ,{Nq}\} $ are required to complete the surface registration, where Np and Nq are the number of points in each model. The solution to the registration problem is to find an optimal transformation T consisting of a rotation matrix R and a translation vector t to align the two models as closely as possible.
Q is transformed from P, the distance between T(pi) and qi should be zero for all points i, where T(pi)=Rpi+T. However, so many error factors including quantization error and other factors would make the distance not zero. The closer this distance difference is to zero, the more accurate the registration effect has. Therefore, the problem of surface registration is transformed into an optimization for minimizing the distance between T(pi) and qi. According to Euclidean distance [18], the “point-to-point” metric is used to formulate the objective function, which can be expressed as:
As can be seen from Eq. (1), T contains six parameters which are three rotation angles α, β, γ for R and three translation component tx, ty, tz for T. Obviously, the function f is a six-Dimensional (6-D) nonlinear minimization problem, then the surface registration problem is equivalent to solving the optimization equation, and its optimal solution X* satisfies f (X*) = 0. In addition, by minimizing Eq. (1), the transformation parameters can be optimized to obtain a higher registration accuracy.
2.2 Whale optimization algorithm
WOA is a heuristic optimization algorithm to simulate the hunting behavior of humpback whales in the ocean, which is proposed by Seyedali Mirjalili and Andrew Lewis. Like any other evolutionary algorithm, WOA starts with a population of M d-Dimensional parameter vectors representing the candidate solutions. Use G denotes the subsequent generations in the evolutionary process of WOA, where $\textrm{G} = 1,2, \cdots ,{G_{\max }}.$ Then, the population at the current generation can be adopted as
where Xi(G) is the i-th vector of the current population, which can be written as ${X_i}(G) = [{x_{i,1}}(G),{x_{i,2}}(G), \cdots ,{x_{i,j}}(G)],{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j = 1,2, \cdots ,d.$In order to obtain better search results, the value of each parameter in the problem to be solved should be within a certain range. The initial population (at G=0) should cover the entire search space as much as possible by uniformly randomizing individuals within the search space constrained by the prescribed minimum and maximum bounds: ${X_{\min }} = \{{{x_{1,\min }},{x_{2,\min }}, \cdots ,{x_{d,\min }}} \}$ and ${X_{\max }} = \{{{x_{1,\max }},{x_{2,\max }}, \cdots ,{x_{d,\max }}} \}.$ Hence, the initial value of the j-th parameter in the i-th individual can be written as
2.2.1 Encircling prey
Humpback whales can share the information of prey currently searched and then approach them. Since there is no prior knowledge of the global optimal solution in the search space before the optimization problem was solved, the WOA algorithm assumes that the current optimal solution of the problem to be solved will be the prey. After the location of the prey is defined, other whales will swim towards the prey to update their positions. The mathematical model of the behavior in this stage can be expressed as
where G represents the current iteration; X* indicates the position vector of prey, which is updated in each iteration; X is the position vector; A and C are coefficient vectors, which can be calculated by where u represents linearly decreased from 2 to 0 during iteration; r is a random number between 0 and 1.2.2.2 Bubble-net feeding maneuver
Two methods are designed, namely the encircling mechanism and spiral updating position, to simulate the bubble-net behavior of humpback whales.
The shrinking encircling mechanism is realized by linearly decreasing the value of parameter u from 2 to 0. Then u can be expressed as $\textrm{u} = 2 - G \times (2/{G_{\max }}),$ where Gmax is the maximum number of iterations. It is worth noting that the scope of A is also narrowed by u. Figure 1 (a) shows the possible positions from (X, Y) towards (X*, Y*), which can be achieved by 0≤A≤1 in 2D space.
In the spiral updating position method, the distance between whale and prey is calculated as shown in Fig. 1(b), and a spiral mathematical model is established to simulate the helix-shaped movement behavior of humpback whales around prey as follows
where $D^{\prime} = |{X^\ast }(G) - X(G)|$ refers to the distance of the i-th whale to the prey (best solution obtained so far); b represents a constant to define the shape of the logarithmic spiral; l indicates a random number between -1 and 1. As can be seen from Fig. 1(b), l=-1 is the nearest position to the prey, while l=1 shows the farthest.
Fig. 1. Bubble-net search mechanism implemented. (a) Shrinking encircling mechanism. (b) Spiral updating position.
It is a remarkable fact that humpback whales swim to the prey in a shrinking circle and along a spiral-shaped path simultaneously. In view of this concurrent behavior, It is believed that there is an equal possibility to choose between either the shrinking encircling mechanism or the spiral mode to update the position of whales during optimization. The updated formula is as follows
2.2.3 Search for prey
In addition to the bubble net search strategy, humpback whales also search for prey randomly. In other words, the same method based on the variation of vector A can also be used to search for prey. As a matter of fact, humpback whales can randomly search for prey based on the position of each other. Therefore, the search agent is forced to stay away from a reference whale by setting |A|≥1. In contrast to the exploitation phase, the position of a search agent in the exploration is updated based on a randomly selected search agent instead of the best search agent found so far. |A|≥1 emphasizes exploration and makes WOA algorithm perform a global search. The mathematical model is expressed as follows
where Xrand is a random whale (a random position vector) chosen from the current population.3. Proposed IWOA algorithm based on surface registration
Aiming at the problem that WOA has a low convergence speed and is easy to fall into the local optimal solution, an Improved Whale Optimization Algorithm (IWOA) is proposed for surface registration. This algorithm has two improvements compared with original WOA.
The first improvement is to introduce the Sobol sequence [21,22] in the initial population. The original WOA algorithm only uses the rand function to initialize the initial population. Different random mappings make the population have different spatial distributions, and the result directly affects the convergence performance and search efficiency of the algorithm. The Sobol sequence has a quasi-randomness. Therefore, the initial population of Sobol sequence mapping should be more uniformly distributed in the search space than the conventional method, so as to improve the search ability of the algorithm. Scatter plot of rand function and Sobol sequence are depicted in Fig. 2.
The second improvement is to increase the location update mechanism of the whale. After the Humpback whales find prey, they use a logarithmic spiral motion to capture the optimal solution. In the process, the optimal individual X* is used as the navigation coordinate of the spiral motion. Although this can speed up the convergence speed in the later period, it makes the population diversity decline and is easy to fall into a local optimal solution. The sine-cosine mechanism [23] is introduced here to control the movement area of the population individuals, to avoid individuals from getting closer to the local minimum area, and thus to improve the ability of the algorithm to jump out of the local optimum. The calculation of predation by the sine and cosine spiral motion is shown in Eqs. (11) and (12), respectively.
Figure 3 shows that the sine and cosine capture prey for global search and local optimization in different regions of the same space. The sine global search reduces the blind spots in the optimization of the cosine local exploitation and avoids the loss of the potential optimal solution, while the cosine local exploitation makes up for the slow convergence speed of the sine global search and improves the efficiency of the algorithm. By introducing a greedy mechanism, the solutions produced by searching the prey of sine and cosine are compared, and the optimal solution is retained. The crossover optimization of sine and cosines complements each other, which promotes the rapid spread of individual information in the population, so that the humpback whale finally converges to the same optimal solution. On the one hand, it prevents the algorithm from premature and improves the accuracy of the solution, and what is more, it can accelerate the convergence speed and improve the solving efficiency.
To sum up, the detailed process of the proposed IWOA for surface registration is shown in Fig. 4, and the steps are described as follows:
Step 1: Set the population size N, the maximum iterations number Gmax, decision variables ${x_1},{x_2}, \cdots ,{x_d}$ and constraints. The Sobol sequence is used to initialize the whale population in the feasible region space. Then, the initial position value of the j-th parameter in the i-th individual is can be expressed as
Step 2: Since the parameter vectors are likely to be changed over different generations, the boundary of the search space needs to be modified according to the range of surface registration parameters, That is, if the obtained parameter is between the minimum and maximum, no adjustment is required. If the minimum value is overpassed, come back to the minimum edge; If the maximum value is exceeded, return to the maximum boundary. The mathematical expression can be written as
Step 4: For each humpback whale, update the positions of the population based on parameters p and A. When p <0.5, and if |A|<1, Eq. (5) is used to update the individual position of the current prey population, otherwise, the position is updated by Eq. (10), and when p ≥ 0.5, both Eqs. (11) and (12) are used to cross-update the position.
Step 5: Calculate the optimal value in the current iteration, and set termination criterion G > Gmax or $|{f(X(G))} |< \varepsilon ,$ where G represents the number of current iterations; ε is the permissible error, and ε>0. When the criterion is satisfied, the IWOA is terminated and the current solution is regarded as the optimal one. Otherwise, set G = G+1 and return to Step 2 to continue the iteration. Finally, the optimized parameters ${X^ \ast } = [{\alpha ^ \ast },{\beta ^ \ast },{\gamma ^ \ast },t_x^ \ast ,t_y^ \ast ,t_z^ \ast ]$ and the surface registration function value f (X*) are outputted.
4. Surface registration experiment
To evaluate the performance of the proposed IWOA algorithm. The experiment is carried out by means of simulation and actual measurement. The simulation experiment is to illustrate the applicability of IWOA in surface registration, and the actual measurement experiment is to illustrate the robustness and effectiveness of the proposed IWOA in practical application. All the mentioned algorithms are implemented with MATLAB R2016b and are run on the same Intel Core i7-1065G7 CPU @ 1.30GHZ with 16.00 GB RAM in a Windows 10 operating system.
4.1 Simulation experiment
In this section, two test models with different geometric structures from the Stanford University Computer Graphics Laboratory are selected for simulation experiments, which are bunny and turbine blade as shown in Fig. 5. A preset transformation matrix is used to simulate the complex initial position. In the experiment, these two test models are CAD models, and the model obtained by fixed rotation and translation of the design model is regarded as the measurement model. The ICP algorithm is a classic algorithm, which can be applied to ICP for surface registration. Therefore, this article mainly discusses whether the heuristic algorithm is suitable for surface registration.
To show the performance of the proposed IWOA algorithm, IWOA is compared with WOA and some known WOA variants algorithms LWOA and AWOA on two simulate test registration functions. For fair comparison, each algorithm has the same setting conditions, where the population size and the maximum number of iterations are set to be 30 and 1000, respectively. According to the number of iterations, the convergence diagrams of four heuristic algorithms for bunny and turbine blade test registration functions are drawn, as illustrated in Fig. 6. In the two functions, the convergence speed and solution accuracy of the proposed IWOA algorithm are significantly better than other three algorithms. This is because IWOA has an effective location update mechanism.
Fig. 6. Registration function values obtained by different algorithms. (a) Bunny function. (b) Blade function.
Through 500 iterative simulation experiments, the optimal rabbit objective function values of IWOA, WOA, LWOA and AWOA are obtained as 0, 4.9572E-10, 4.8682E-28, 4.8187E-07, respectively. The optimal value of turbine blade objective function of IWOA, WOA, LWOA and AWOA is 0, 2.057E-07, 1.8883E-27 and 2.266E-07. In order to better see the registration results of the four heuristic algorithms, the registration times of these algorithm are increased, as shown in Table 1. Figure 6 and Table 1 show that compared with the other three heuristic algorithms, IWOA algorithm has high accuracy and fast convergence speed. The IWOA algorithm proposed in this paper can achieve high precision registration of point cloud and CAD model.
Table 1. The registration results of simulations
There are inevitably some measurement noise and errors in the real measurement point clouds. Therefore, Gaussian noise of ${\sigma ^2} = 0.025$ is added to the model after rotation and translation to simulate the actual measurement model. The models with Gaussian noise are displayed in Fig. 7, where the green part represents the ideal measurement model and the yellow area denotes the Gaussian noise model. The registration results of the noise model and the CAD model are shown in Table 2.
Table 2. The registration results with Gaussian noise
It can be seen from Table 2 that the proposed algorithm is superior to the existing algorithms in accuracy, but does not have an advantage in the registration time. The accuracy results can illustrate the robustness and applicability of the proposed algorithm to solve the registration problem.
4.2 Actual measurement experiment
Before measuring the object, the topography characteristics of object to be measured are observed, such as surface material gloss, size, etc. If the surface reflection is too strong or the color is too dark, it is not conducive to measurement, and the developer needs to be sprayed evenly on the surface. Then paper reference points are pasted on the object surface to determine the spatial position of object and improve the measurement accuracy. The obtained object three-dimensional data is obtained through the software ATOS professionalV8, and the process is shown in Fig. 8.
To verify that the proposed IWOA algorithm can be well applied in surface registration, an experimental system for evaluating precision inspection is established, as shown in Fig. 9. The scanning equipment adopts ATOS III triple scan, which is mainly composed of two high-resolution industrial CCD cameras on the left and right and a central grating projection unit. Using structured light, the projection unit projects a group of grating fringes with phase information onto the surface of the measurement workpiece. The two cameras perform synchronous measurement, and the object three-dimensional data is obtained by using the measurement principle of stereo camera. This device has automatic reference point recognition and splicing technology, which aligns the measurement data at different positions and angles in the same coordinate system to obtain the scan result of the complete part.The workpiece to be measured is an aeroengine blade with a size of 140 mm×80 mm×30 mm, which is fixed on a shock-proof platform.
There are two models in surface registration, and their initial positions are shown in Fig. 10. The CAD model is represented in red, and the measurement model is displayed in blue. In practical application, CloudCompare software is used to preprocess the data of the measurement model to remove noise points and outliers as much as possible. The purpose of surface registration is to convert the two models into a unified coordinate system. Here, the algorithm is used to convert the measurement model into CAD model as carefully as possible, and then the registration error is calculated. Finally, the color-coded error map is displayed with visual effect. The process of using surface registration to conduct precision inspection is shown in Fig. 11. The box in the dotted line represents the inspection system of the proposed IWOA algorithm.
In order to better illustrate the robustness and effectiveness of the proposed IWOA algorithm in surface registration, a quantitative comparison with the classic ICP algorithm is added, as shown in Table 3.
Table 3. Comparison for surface registration results
The surface registration errors are 0.1711mm for IWOA, 2.0015 mm for WOA, 1.2656 mm for LWOA, 2.8132 mm for AWOA and 2.1537 mm for ICP, respectively. The registration time is better than other algorithms due to the complicated initial value and the effective update mechanism of IWOA. In practical application, the population size and maximum number of iterations are 30 and 500, respectively, which are used for heuristic algorithm. The results shown in Table 3 indicate that the proposed algorithm compared with the other three algorithms. According to Eq. (1), the problem of surface registration can be transformed into a complex function with six dimensions. Note the difference between the ranges of variables. Since the first three variables in the objective function are angles, the range of their values is (-90 °, 90 °) [24]. Considering the practical cases, the translation parameters are set based on the geometry of the model, the translation values are set with the greatest model sizes [−0.01×MS, 0.01×MS], where MS is the largest size (length, width, and height) of the model. The proposed IWOA still shows a best overall performance. Figure 12 shows the current optimal value of the function value obtained by different heuristic algorithms with the increase of the number of iterations. As can be seen, the convergence speed and solving accuracy of the proposed IWOA are obviously much better than the existing WOA, LWOA and AWOA. The accuracy of IWOA algorithm meets the requirements of industrial precision inspection [25–27], which indicates that the proposed algorithm is effective in solving surface registration problems.
As shown in Fig. 12 and Table 3, the proposed algorithm shows the best overall performance compared with other known algorithms. It can be seen from the Fig. 12 that the convergence of IWOA algorithm in solving the surface registration problem is the best among the four heuristic algorithms.
In the actual measurement experiment, the RMSE of surface registration during the IWOA optimization process is constantly updated, and the RMSE in the current iteration is calculated as
According to Table 3, the RMSE [28,29] of surface registration can be obtained as 0.17111 mm for IWOA, 2.0015 mm for WOA, 1.2656 mm for LWOA, 2.8132 mm for AWOA and 2.1537 mm for ICP. The registration times of IWOA, WOA, LWOA, AWOA and ICP are 7.5694 s, 8.7935 s, 15.6223 s, 12.5158 s and 8.6419 s, respectively. It can be concluded that the accuracy of IWOA is much higher than other four algorithms.It should be noted that in the actual surface registration experiment, IWOA has advantages over some known algorithms mainly because of the complicated initial value and effective update mechanism.
4.3 Additional experiment
Inspired by the reviewers, experiments are carried out on the ideal equations of optical free-form surfaces to illustrate the applicability of the proposed algorithm to free-form optics. In order to verify the registration algorithm, rotate around the x, y and z axes on the simulated surface of Fig. 13(c), respectively, and translate -20 mm, 0 mm, and 10 mm along thex, y and z directions. The resulting optical free-form surface with rotation and translation is shown in Fig. 14(a). Adjusted by the improved IWOA algorithm, the surface 14(a) becomes 14(b), which is very close to the design surface. The difference between them is shown in Fig. 14(c). The difference between x and y coordinates is less than 1e-14 mm. The results show that the IWOA coordinate matching algorithm proposed in this paper is better than other algorithms with a difference of more than 1e-5 mm in x, y and z coordinates [19,30]. It is shown that the proposed algorithm is suitable for optical free-form surfaces.
Fig. 13. Optical free-form surface with 54 × 51 points. (a) Design optical free-form surface. (b) Measurement noise. (c) Surface with measurement noise.
Fig. 14. Demonstration of coordinate matching IWOA algorithm. (a) Optical free-form surface with coordinate shifts and rotation. (b) Surface after the IWOA adjustment. (c) Difference between the optical free-form surface with IWOA adjustment and the design surface.
5. Conclusions
In this paper, an improved WOA algorithm named IWOA is proposed to solve the registration problem in surface measurement. In order to evaluate the performance of this algorithm, the simulation experiment and the practical application of blade registration are tested. In addition, the experimental results of IWOA algorithm are compared with WOA, LWOA and AWOA for further verification. In the first test phase, two test models and add Gaussian noise to the models after rotation and translation to simulate the real measurement model. which are used to test the performance of IWOA. The results show that IWOA has good accuracy and convergence in both model registration functions. Its performance is much better than the other three heuristic algorithms. In addition, the superiority of the registration results of IWOA in the noise model shows that the proposed algorithm is very robust.
In the second registration stage, a surface measurement experiment system is established to illustrate the advantages of IWOA. The accuracy of IWOA algorithm is evaluated by surface registration error value, average deviation and standard deviation. The experimental results show that the registration errors of IWOA,WOA, LWOA, AWOA and ICP are 0.1711mm, 2.0015 mm, 1.2656 mm, 2.8132 mm and 2.1537 mm, respectively. The second test phase also indicates that the accuracy of IWOA algorithm is the best among all other algorithms. In addition, inspired by the reviewers, experiments are carried out on the ideal equation of optical free-form surface. The experimental results show that the proposed algorithm is suitable for optical free-form surface.
As for the future work, two research directions can be suggested. Firstly, we will develop binary and multi-objective versions of the IWOA. Secondly, the IWOA will be applied in other fields, such as face recognition and medical aspects, etc.
Funding
Ministry of Industry and Information Technology of the People's Republic of China (MJ-2018-J-70).
Acknowledgments
We would like to thank Jingjing Fan, Jihu Wang, Yizhang Wang and Liqun Ma from China Changcheng Institute of Metrology and Measurement for their involvement and support.
Disclosures
The authors declare that there are no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but can be obtained by the authors upon reasonable request.
References
1. R. Rantoson, H. Nouira, N. Anwer, and C. Mehdi-Souzani, “Novel automated methods for coarse and fine registrations of point clouds in high precision metrology,” Int J Adv Manuf Technol 81(5-8), 795–810 (2015). [CrossRef]
2. A. A Daniel, V. M. Brenda, and D.U. Rufino, “Null-screen design for highly freeform surface testing,” Opt. Express 28(24), 36706–36722 (2020). [CrossRef]
3. Y. T. Xing, C. Li, and Y. Liu, “Fabrication of high-precision freeform surface on die steel by ultrasonic-assisted slow tool servo,” Opt. Express 29(3), 3708–3723 (2021). [CrossRef]
4. J. L. Yang, H. D. Li, and Y. D. Jia, “Go-ICP: Solving 3D Registration Efficiently and Globally Optimally,” International Conference on Computer Vision.1457–1464 (2013).
5. O. Choi, M. G. Park, and Y. B. Hwang, “Iterative k-closest point algorithms for colored point cloud registration,” Sensors 20(18), 5331 (2020). [CrossRef]
6. J. Zhang, Y. Yao, and B. Deng, “Fast and robust iterative closest point,” IEEE Trans. Pattern Anal. Mach. Intell. 99, 1–10 (2021). [CrossRef]
7. C. K. Chow, H. T. Tsui, and T. Lee, “Surface registration using a dynamic genetic algorithm,” Pattern Recognition 37(1), 105–117 (2004). [CrossRef]
8. Z. Xu, Y Cai, and H. Li, “A point cloud registration algorithm based on normal vector and particle swarm optimization,” Measurement and Control 53(3), 265–275 (2020). [CrossRef]
9. T. F. Li, Q. K. Pan, and L Gao, “Differential evolution algorithm-based range image registration for free-form surface parts quality inspection,” Swarm and Evolutionary Computation 36, 106–123 (2017). [CrossRef]
10. Y. Feng, J. Tang, and B. Su, “Point cloud registration algorithm based on the grey wolf optimizer,” IEEE Access 8, 143375–143382 (2020). [CrossRef]
11. S. Mirjalili and A. Lewis, “The whale optimization algorithm,” Advances in Engineering Software 95, 51–67 (2016). [CrossRef]
12. M. Abdel-Basset, G. Manogaran, and D. El-Shahat, “A hybrid whale optimization algorithm based on local search strategy for the permutation flow shop scheduling problem,” Future Generation Computer Systems 85, 129–145 (2018). [CrossRef]
13. H. Chen, C. Yang, and A. A. Heidari, “An efficient double adaptive random spare reinforced whale optimization algorithm,” Expert Systems with Applications 154, 113018 (2020). [CrossRef]
14. M. K. Hassan, A. I. E Desouky, and S. M. Elghamrawy, “A hybrid real-time remote monitoring framework with NB-WOA algorithm for patients with chronic diseases,” Future Generation Computer Systems 93, 77–95 (2019). [CrossRef]
15. A. S. Mohammed, V. Shukla, and A. C. Pandey, “Enhancing sentiment analysis using enhanced whale optimisation algorithm,” Int. J. Intell. Inf. Database Syst. 13(2), 208–230 (2020). [CrossRef]
16. Y. Ling, Y. Q. Zhou, and Q. F. Luo, “Levy flight trajectory-based whale optimization algorithm for global optimization,” IEEE Access 5, 6168–6186 (2017). [CrossRef]
17. W. Sun and C. Zhang, “Analysis and forecasting of the carbon price using multi-resolution singular value decomposition and extreme learning machine optimized by adaptive whale optimization algorithm,” Applied Energy 231, 1371 (2018). [CrossRef]
18. F. Tombari, S. Salti, and L. Di Stefano, “Performance evaluation of 3D keypoint detectors,” Int J Comput Vis 102(1-3), 198–220 (2013). [CrossRef]
19. Y. Zhang, H. N. Cheng, and R. Wu, “Data processing for point-based in situ metrology of freeform optical surface,” Opt. Express 25(12), 13414–13424 (2017). [CrossRef]
20. J. Yao, A. Anderson, and J. P. Rolland, “Point-cloud noncontact metrology of freeform optical surfaces,” Opt. Express 26(8), 10242–10265 (2018). [CrossRef]
21. A. Navid, S. Khalilarya, and M. Abbasi, “Diesel engine optimization with multi-objective performance characteristics by non-evolutionary nelder-mead algorithm: sobol sequence and latin hypercube sampling methods comparison in DoE process,” Fuel 228, 349–367 (2018). [CrossRef]
22. I. M. Sobol, “The distribution of points in a cube and the approximate evaluation of integrals,” USSR Computational Mathematics and Mathematical Physics 7(4), 784–802 (1969). [CrossRef]
23. S. Mirjalili, “SCA: A sine cosine algorithm for solving optimization problems,” Knowledge-Based Systems 96, 120–133 (2016). [CrossRef]
24. L. Silva, O. R. P. Bellon, and K. L. Boyer, “Precision range image registration using a robust surface interpenetration measure and enhanced genetic algorithms,” IEEE Trans. Pattern Anal. Machine Intell. 27(5), 762–776 (2005). [CrossRef]
25. P. Bergstrm and O. Edlund, “Robust registration of surfaces using a refined iterative closest point algorithm with a trust region approach,” Numer Algor 74(3), 1–25 (2016). [CrossRef]
26. S. von Enzberg and A. Al-Hamadi, “A multiresolution approach to model-based 3-D surface quality inspection,” IEEE Trans. Ind. Inf. 12(4), 1498–1507 (2016). [CrossRef]
27. V. Mehrad, D. Xue, and P. Gu, “Robust localization to align measured points on the manufactured surface with design surface for freeform surface inspection,” Computer-Aided Design 53, 90–103 (2014). [CrossRef]
28. Z. Y. Wang, Z. M. Liu, and X. T. Xia, “Measurement error and uncertainty evaluation,” Beijing: Science Press. 2008.
29. W. S. Jiang and Z. Y. Wang, “Calibration of visual model for space manipulator with a hybrid LM-GA algorithms,” Mechanical Systems and Signal Processing 66-67(1), 399–409 (2016). [CrossRef]
30. X. Jiang, X. Zhang, and P. J. Scott, “Template Matching of freeform surfaces based on orthogonal distance fitting for precision metrology,” Meas. Sci. Technol. 21(4), 045101 (2010). [CrossRef]
