Abstract

An improved moth-flame optimization (IMFO) algorithm is proposed to increase the location accuracy of a vision measurement system. This algorithm can optimize the initial pose parameters by improving a series of random solutions to the required precision. A measurement experiment system of space manipulator is designed to precision test. The IMFO algorithm is evaluated on 23 benchmark functions and measurement experiments for pose, and the results are verified by a comparative study with self-adaptive differential evolution (SaDE), moth-flame optimization (MFO), and proactive particle swarm optimization (PPSO). The statistical results of the benchmark functions show that the IMFO algorithm can provide very promising and competitive results. Additionally, the experimental results of pose measurement show that the accuracy of the IMFO algorithm is approximately twice higher than that of other three algorithms. All in all, the experiments indicate that the IMFO algorithm has a good optimization ability to complete the visual identification accurately.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  17. S. Mirjalili, S. M. Mirjalil, and A. Lewis, “Grey wolf optimizer,” Adv. Eng. Softw. 69(3), 46–61 (2014).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  20. S. J. Zhang, X. B. Cao, and F. Zhang, “Monocular vision-based iterative pose estimation algorithm from corresponding feature points,” Inf. Sci. 53(8), 1682–1696 (2010).
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    [Crossref]
  22. M. Molga and C. Smutnicki, “Test functions for optimization needs,” http://www.zsd.ict.pwr.wroc.pl/files/docs/functions.pdf (2005).
  23. X. S. Yang, “Test Problems in Optimization,” Mathematics 2(2), 63–86 (2010).
  24. A. K. Qin, V. L. Huang, and P. N. Suganthan, “Differential evolution algorithm with strategy adaptation for global numerical optimization,” IEEE Trans. Evol. Comput. 13(2), 398–417 (2009).
    [Crossref]
  25. A. Tangherloni, L. Rundo, and M. Nobile, “Proactive particles in swarm optimization: a settings-free algorithm for real-parameter single objective optimization problems,” in Congress on Evolutionary Computation (IEEE, 2017), pp. 1940–1947.
  26. V. Lepetit, F. Moreno-Noguer, and P. Fua, “EPnP: An Accurate O(n) Solution to the PnP Problem,” Int. J. Comput. Vis. 81(2), 155–166 (2009).
    [Crossref]

2018 (1)

2017 (3)

B. S. Yildiz and A. R. Yildiz, “Moth-flame optimization algorithm to determine optimal machining parameters in manufacturing processes,” Mater. Test. 59(5), 425–429 (2017).
[Crossref]

B. S. Yildiz, “Natural frequency optimization of vehicle components using the interior search algorithm,” Mater. Test. 59(5), 456–458 (2017).
[Crossref]

R. Cheng, M. Q. Li, Y. Tian, X. Zhang, S. Yang, Y. Jin, and X. Yao, “A Benchmark Test Suite for Evolutionary Many-objective Optimization,” Comp. Intell. Syst. 3(1), 67–81 (2017).
[Crossref]

2016 (5)

S. Mirjalili and A. Lewis, “The whale optimization algorithm,” Adv. Eng. Softw. 95, 51–67 (2016).
[Crossref]

S. Mirjalili, “SCA: A sine cosine algorithm for solving optimization problems,” Knowl. Base. Syst. 96, 120–133 (2016).
[Crossref]

D. Li, G. Lu, and Y. Y. Shao, “A novel camera calibration technique based on differential evolution particle swarm optimization algorithm,” Neurocomputing 174, 456–465 (2016).
[Crossref]

Q. Y. Wang, Z. Y. Wang, Z. J. Yao, J. Forrest, and W. Zhou, “An improved measurement model of binocular vision using geometrical approximation,” Meas. Sci. Technol. 27(12), 125013 (2016).
[Crossref]

W. S. Jiang and Z. Y. Wang, “Calibration of visual model for space manipulator with a hybrid LM-GA algorithms,” Mech. Syst. Signal Process. 66(1), 399–409 (2016).
[Crossref]

2015 (4)

C. M. Tsai, Y. C. Fang, and C. T. Lin, “Application of genetic algorithm on optimization of laser beam shaping,” Opt. Express 23(12), 15877–15887 (2015).
[Crossref] [PubMed]

S. Kenneth, “Metaheuristics-the metaphor exposed,” Int. Trans. Oper. Res. 22(1), 3–18 (2015).
[Crossref]

Y. Hong, G. Ren, and E. Liu, “Non-iterative method for camera calibration,” Opt. Express 23(18), 23992–24003 (2015).
[Crossref] [PubMed]

S. Mirjalili, “Moth-flame optimization algorithm: a novel nature-inspired heuristic paradigm,” Knowl. Base. Syst. 89, 228–249 (2015).
[Crossref]

2014 (1)

S. Mirjalili, S. M. Mirjalil, and A. Lewis, “Grey wolf optimizer,” Adv. Eng. Softw. 69(3), 46–61 (2014).
[Crossref]

2013 (1)

D. Li and J. D. Tian, “An accurate calibration method for a camera with telecentric lenses,” Opt. Lasers Eng. 51(5), 538–541 (2013).
[Crossref]

2012 (1)

F. Zhou, Y. Cui, B. Peng, and Y. Wang, “A novel optimization method of camera parameters used for vision measurement,” Opt. Laser Technol. 44(6), 1840–1849 (2012).
[Crossref]

2010 (2)

S. J. Zhang, X. B. Cao, and F. Zhang, “Monocular vision-based iterative pose estimation algorithm from corresponding feature points,” Inf. Sci. 53(8), 1682–1696 (2010).

X. S. Yang, “Test Problems in Optimization,” Mathematics 2(2), 63–86 (2010).

2009 (3)

A. K. Qin, V. L. Huang, and P. N. Suganthan, “Differential evolution algorithm with strategy adaptation for global numerical optimization,” IEEE Trans. Evol. Comput. 13(2), 398–417 (2009).
[Crossref]

V. Lepetit, F. Moreno-Noguer, and P. Fua, “EPnP: An Accurate O(n) Solution to the PnP Problem,” Int. J. Comput. Vis. 81(2), 155–166 (2009).
[Crossref]

A. R. Yildiz, “A novel hybrid immune algorithm for global optimization in design and manufacturing,” Robo. Com-Int. Manuf. 25(2), 261–270 (2009).

2001 (1)

J. Digalakis and K. Margaritis, “On benchmarking functions for genetic algorithms,” Int. J. Comput. Math. 77(4), 481–506 (2001).
[Crossref]

1997 (1)

D. H. Wolpert and W. G. Macready, “No free lunch theorems for optimization,” IEEE Trans. Evol. Comput. 1(1), 67–82 (1997).
[Crossref]

Alvarez-Alvarez, G.

Arreola, M.

Briones, E.

Briones, J.

Cao, X. B.

S. J. Zhang, X. B. Cao, and F. Zhang, “Monocular vision-based iterative pose estimation algorithm from corresponding feature points,” Inf. Sci. 53(8), 1682–1696 (2010).

Cheng, R.

R. Cheng, M. Q. Li, Y. Tian, X. Zhang, S. Yang, Y. Jin, and X. Yao, “A Benchmark Test Suite for Evolutionary Many-objective Optimization,” Comp. Intell. Syst. 3(1), 67–81 (2017).
[Crossref]

Cui, Y.

F. Zhou, Y. Cui, B. Peng, and Y. Wang, “A novel optimization method of camera parameters used for vision measurement,” Opt. Laser Technol. 44(6), 1840–1849 (2012).
[Crossref]

Digalakis, J.

J. Digalakis and K. Margaritis, “On benchmarking functions for genetic algorithms,” Int. J. Comput. Math. 77(4), 481–506 (2001).
[Crossref]

Fang, Y. C.

Forrest, J.

Q. Y. Wang, Z. Y. Wang, Z. J. Yao, J. Forrest, and W. Zhou, “An improved measurement model of binocular vision using geometrical approximation,” Meas. Sci. Technol. 27(12), 125013 (2016).
[Crossref]

Fua, P.

V. Lepetit, F. Moreno-Noguer, and P. Fua, “EPnP: An Accurate O(n) Solution to the PnP Problem,” Int. J. Comput. Vis. 81(2), 155–166 (2009).
[Crossref]

Gonzalez, N.

Hong, Y.

Huang, V. L.

A. K. Qin, V. L. Huang, and P. N. Suganthan, “Differential evolution algorithm with strategy adaptation for global numerical optimization,” IEEE Trans. Evol. Comput. 13(2), 398–417 (2009).
[Crossref]

Jiang, W. S.

W. S. Jiang and Z. Y. Wang, “Calibration of visual model for space manipulator with a hybrid LM-GA algorithms,” Mech. Syst. Signal Process. 66(1), 399–409 (2016).
[Crossref]

Jin, Y.

R. Cheng, M. Q. Li, Y. Tian, X. Zhang, S. Yang, Y. Jin, and X. Yao, “A Benchmark Test Suite for Evolutionary Many-objective Optimization,” Comp. Intell. Syst. 3(1), 67–81 (2017).
[Crossref]

Kenneth, S.

S. Kenneth, “Metaheuristics-the metaphor exposed,” Int. Trans. Oper. Res. 22(1), 3–18 (2015).
[Crossref]

Lepetit, V.

V. Lepetit, F. Moreno-Noguer, and P. Fua, “EPnP: An Accurate O(n) Solution to the PnP Problem,” Int. J. Comput. Vis. 81(2), 155–166 (2009).
[Crossref]

Lewis, A.

S. Mirjalili and A. Lewis, “The whale optimization algorithm,” Adv. Eng. Softw. 95, 51–67 (2016).
[Crossref]

S. Mirjalili, S. M. Mirjalil, and A. Lewis, “Grey wolf optimizer,” Adv. Eng. Softw. 69(3), 46–61 (2014).
[Crossref]

Li, D.

D. Li, G. Lu, and Y. Y. Shao, “A novel camera calibration technique based on differential evolution particle swarm optimization algorithm,” Neurocomputing 174, 456–465 (2016).
[Crossref]

D. Li and J. D. Tian, “An accurate calibration method for a camera with telecentric lenses,” Opt. Lasers Eng. 51(5), 538–541 (2013).
[Crossref]

Li, M. Q.

R. Cheng, M. Q. Li, Y. Tian, X. Zhang, S. Yang, Y. Jin, and X. Yao, “A Benchmark Test Suite for Evolutionary Many-objective Optimization,” Comp. Intell. Syst. 3(1), 67–81 (2017).
[Crossref]

Lin, C. T.

Liu, E.

Lu, G.

D. Li, G. Lu, and Y. Y. Shao, “A novel camera calibration technique based on differential evolution particle swarm optimization algorithm,” Neurocomputing 174, 456–465 (2016).
[Crossref]

Macready, W. G.

D. H. Wolpert and W. G. Macready, “No free lunch theorems for optimization,” IEEE Trans. Evol. Comput. 1(1), 67–82 (1997).
[Crossref]

Margaritis, K.

J. Digalakis and K. Margaritis, “On benchmarking functions for genetic algorithms,” Int. J. Comput. Math. 77(4), 481–506 (2001).
[Crossref]

Mirjalil, S. M.

S. Mirjalili, S. M. Mirjalil, and A. Lewis, “Grey wolf optimizer,” Adv. Eng. Softw. 69(3), 46–61 (2014).
[Crossref]

Mirjalili, S.

S. Mirjalili, “SCA: A sine cosine algorithm for solving optimization problems,” Knowl. Base. Syst. 96, 120–133 (2016).
[Crossref]

S. Mirjalili and A. Lewis, “The whale optimization algorithm,” Adv. Eng. Softw. 95, 51–67 (2016).
[Crossref]

S. Mirjalili, “Moth-flame optimization algorithm: a novel nature-inspired heuristic paradigm,” Knowl. Base. Syst. 89, 228–249 (2015).
[Crossref]

S. Mirjalili, S. M. Mirjalil, and A. Lewis, “Grey wolf optimizer,” Adv. Eng. Softw. 69(3), 46–61 (2014).
[Crossref]

Moreno-Noguer, F.

V. Lepetit, F. Moreno-Noguer, and P. Fua, “EPnP: An Accurate O(n) Solution to the PnP Problem,” Int. J. Comput. Vis. 81(2), 155–166 (2009).
[Crossref]

Peng, B.

F. Zhou, Y. Cui, B. Peng, and Y. Wang, “A novel optimization method of camera parameters used for vision measurement,” Opt. Laser Technol. 44(6), 1840–1849 (2012).
[Crossref]

Qin, A. K.

A. K. Qin, V. L. Huang, and P. N. Suganthan, “Differential evolution algorithm with strategy adaptation for global numerical optimization,” IEEE Trans. Evol. Comput. 13(2), 398–417 (2009).
[Crossref]

Ren, G.

Ruiz-Cruz, R.

Shao, Y. Y.

D. Li, G. Lu, and Y. Y. Shao, “A novel camera calibration technique based on differential evolution particle swarm optimization algorithm,” Neurocomputing 174, 456–465 (2016).
[Crossref]

Simon, J.

Suganthan, P. N.

A. K. Qin, V. L. Huang, and P. N. Suganthan, “Differential evolution algorithm with strategy adaptation for global numerical optimization,” IEEE Trans. Evol. Comput. 13(2), 398–417 (2009).
[Crossref]

Tian, J. D.

D. Li and J. D. Tian, “An accurate calibration method for a camera with telecentric lenses,” Opt. Lasers Eng. 51(5), 538–541 (2013).
[Crossref]

Tian, Y.

R. Cheng, M. Q. Li, Y. Tian, X. Zhang, S. Yang, Y. Jin, and X. Yao, “A Benchmark Test Suite for Evolutionary Many-objective Optimization,” Comp. Intell. Syst. 3(1), 67–81 (2017).
[Crossref]

Tsai, C. M.

Wang, Q. Y.

Q. Y. Wang, Z. Y. Wang, Z. J. Yao, J. Forrest, and W. Zhou, “An improved measurement model of binocular vision using geometrical approximation,” Meas. Sci. Technol. 27(12), 125013 (2016).
[Crossref]

Wang, Y.

F. Zhou, Y. Cui, B. Peng, and Y. Wang, “A novel optimization method of camera parameters used for vision measurement,” Opt. Laser Technol. 44(6), 1840–1849 (2012).
[Crossref]

Wang, Z. Y.

Q. Y. Wang, Z. Y. Wang, Z. J. Yao, J. Forrest, and W. Zhou, “An improved measurement model of binocular vision using geometrical approximation,” Meas. Sci. Technol. 27(12), 125013 (2016).
[Crossref]

W. S. Jiang and Z. Y. Wang, “Calibration of visual model for space manipulator with a hybrid LM-GA algorithms,” Mech. Syst. Signal Process. 66(1), 399–409 (2016).
[Crossref]

Wolpert, D. H.

D. H. Wolpert and W. G. Macready, “No free lunch theorems for optimization,” IEEE Trans. Evol. Comput. 1(1), 67–82 (1997).
[Crossref]

Yang, S.

R. Cheng, M. Q. Li, Y. Tian, X. Zhang, S. Yang, Y. Jin, and X. Yao, “A Benchmark Test Suite for Evolutionary Many-objective Optimization,” Comp. Intell. Syst. 3(1), 67–81 (2017).
[Crossref]

Yang, X. S.

X. S. Yang, “Test Problems in Optimization,” Mathematics 2(2), 63–86 (2010).

Yao, X.

R. Cheng, M. Q. Li, Y. Tian, X. Zhang, S. Yang, Y. Jin, and X. Yao, “A Benchmark Test Suite for Evolutionary Many-objective Optimization,” Comp. Intell. Syst. 3(1), 67–81 (2017).
[Crossref]

Yao, Z. J.

Q. Y. Wang, Z. Y. Wang, Z. J. Yao, J. Forrest, and W. Zhou, “An improved measurement model of binocular vision using geometrical approximation,” Meas. Sci. Technol. 27(12), 125013 (2016).
[Crossref]

Yildiz, A. R.

B. S. Yildiz and A. R. Yildiz, “Moth-flame optimization algorithm to determine optimal machining parameters in manufacturing processes,” Mater. Test. 59(5), 425–429 (2017).
[Crossref]

A. R. Yildiz, “A novel hybrid immune algorithm for global optimization in design and manufacturing,” Robo. Com-Int. Manuf. 25(2), 261–270 (2009).

Yildiz, B. S.

B. S. Yildiz and A. R. Yildiz, “Moth-flame optimization algorithm to determine optimal machining parameters in manufacturing processes,” Mater. Test. 59(5), 425–429 (2017).
[Crossref]

B. S. Yildiz, “Natural frequency optimization of vehicle components using the interior search algorithm,” Mater. Test. 59(5), 456–458 (2017).
[Crossref]

Zhang, F.

S. J. Zhang, X. B. Cao, and F. Zhang, “Monocular vision-based iterative pose estimation algorithm from corresponding feature points,” Inf. Sci. 53(8), 1682–1696 (2010).

Zhang, S. J.

S. J. Zhang, X. B. Cao, and F. Zhang, “Monocular vision-based iterative pose estimation algorithm from corresponding feature points,” Inf. Sci. 53(8), 1682–1696 (2010).

Zhang, X.

R. Cheng, M. Q. Li, Y. Tian, X. Zhang, S. Yang, Y. Jin, and X. Yao, “A Benchmark Test Suite for Evolutionary Many-objective Optimization,” Comp. Intell. Syst. 3(1), 67–81 (2017).
[Crossref]

Zhou, F.

F. Zhou, Y. Cui, B. Peng, and Y. Wang, “A novel optimization method of camera parameters used for vision measurement,” Opt. Laser Technol. 44(6), 1840–1849 (2012).
[Crossref]

Zhou, W.

Q. Y. Wang, Z. Y. Wang, Z. J. Yao, J. Forrest, and W. Zhou, “An improved measurement model of binocular vision using geometrical approximation,” Meas. Sci. Technol. 27(12), 125013 (2016).
[Crossref]

Adv. Eng. Softw. (2)

S. Mirjalili and A. Lewis, “The whale optimization algorithm,” Adv. Eng. Softw. 95, 51–67 (2016).
[Crossref]

S. Mirjalili, S. M. Mirjalil, and A. Lewis, “Grey wolf optimizer,” Adv. Eng. Softw. 69(3), 46–61 (2014).
[Crossref]

Comp. Intell. Syst. (1)

R. Cheng, M. Q. Li, Y. Tian, X. Zhang, S. Yang, Y. Jin, and X. Yao, “A Benchmark Test Suite for Evolutionary Many-objective Optimization,” Comp. Intell. Syst. 3(1), 67–81 (2017).
[Crossref]

IEEE Trans. Evol. Comput. (2)

D. H. Wolpert and W. G. Macready, “No free lunch theorems for optimization,” IEEE Trans. Evol. Comput. 1(1), 67–82 (1997).
[Crossref]

A. K. Qin, V. L. Huang, and P. N. Suganthan, “Differential evolution algorithm with strategy adaptation for global numerical optimization,” IEEE Trans. Evol. Comput. 13(2), 398–417 (2009).
[Crossref]

Inf. Sci. (1)

S. J. Zhang, X. B. Cao, and F. Zhang, “Monocular vision-based iterative pose estimation algorithm from corresponding feature points,” Inf. Sci. 53(8), 1682–1696 (2010).

Int. J. Comput. Math. (1)

J. Digalakis and K. Margaritis, “On benchmarking functions for genetic algorithms,” Int. J. Comput. Math. 77(4), 481–506 (2001).
[Crossref]

Int. J. Comput. Vis. (1)

V. Lepetit, F. Moreno-Noguer, and P. Fua, “EPnP: An Accurate O(n) Solution to the PnP Problem,” Int. J. Comput. Vis. 81(2), 155–166 (2009).
[Crossref]

Int. Trans. Oper. Res. (1)

S. Kenneth, “Metaheuristics-the metaphor exposed,” Int. Trans. Oper. Res. 22(1), 3–18 (2015).
[Crossref]

Knowl. Base. Syst. (2)

S. Mirjalili, “SCA: A sine cosine algorithm for solving optimization problems,” Knowl. Base. Syst. 96, 120–133 (2016).
[Crossref]

S. Mirjalili, “Moth-flame optimization algorithm: a novel nature-inspired heuristic paradigm,” Knowl. Base. Syst. 89, 228–249 (2015).
[Crossref]

Mater. Test. (2)

B. S. Yildiz and A. R. Yildiz, “Moth-flame optimization algorithm to determine optimal machining parameters in manufacturing processes,” Mater. Test. 59(5), 425–429 (2017).
[Crossref]

B. S. Yildiz, “Natural frequency optimization of vehicle components using the interior search algorithm,” Mater. Test. 59(5), 456–458 (2017).
[Crossref]

Mathematics (1)

X. S. Yang, “Test Problems in Optimization,” Mathematics 2(2), 63–86 (2010).

Meas. Sci. Technol. (1)

Q. Y. Wang, Z. Y. Wang, Z. J. Yao, J. Forrest, and W. Zhou, “An improved measurement model of binocular vision using geometrical approximation,” Meas. Sci. Technol. 27(12), 125013 (2016).
[Crossref]

Mech. Syst. Signal Process. (1)

W. S. Jiang and Z. Y. Wang, “Calibration of visual model for space manipulator with a hybrid LM-GA algorithms,” Mech. Syst. Signal Process. 66(1), 399–409 (2016).
[Crossref]

Neurocomputing (1)

D. Li, G. Lu, and Y. Y. Shao, “A novel camera calibration technique based on differential evolution particle swarm optimization algorithm,” Neurocomputing 174, 456–465 (2016).
[Crossref]

Opt. Express (3)

Opt. Laser Technol. (1)

F. Zhou, Y. Cui, B. Peng, and Y. Wang, “A novel optimization method of camera parameters used for vision measurement,” Opt. Laser Technol. 44(6), 1840–1849 (2012).
[Crossref]

Opt. Lasers Eng. (1)

D. Li and J. D. Tian, “An accurate calibration method for a camera with telecentric lenses,” Opt. Lasers Eng. 51(5), 538–541 (2013).
[Crossref]

Robo. Com-Int. Manuf. (1)

A. R. Yildiz, “A novel hybrid immune algorithm for global optimization in design and manufacturing,” Robo. Com-Int. Manuf. 25(2), 261–270 (2009).

Other (3)

Y. X. Zhao and L. Q. Liu, Emerging Meta-Heuristic Optimization Method (Beijing: Science Press, 2013) (in Chinese).

M. Molga and C. Smutnicki, “Test functions for optimization needs,” http://www.zsd.ict.pwr.wroc.pl/files/docs/functions.pdf (2005).

A. Tangherloni, L. Rundo, and M. Nobile, “Proactive particles in swarm optimization: a settings-free algorithm for real-parameter single objective optimization problems,” in Congress on Evolutionary Computation (IEEE, 2017), pp. 1940–1947.

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Figures (11)

Fig. 1
Fig. 1 Measurement system.
Fig. 2
Fig. 2 Coordinate transformation from WCS to CCS.
Fig. 3
Fig. 3 Each moth is assigned to a flame
Fig. 4
Fig. 4 Logarithmic spiral and space around a flame
Fig. 5
Fig. 5 Position updating in IMFO.
Fig. 6
Fig. 6 Flow chart of the IMFO algorithm
Fig. 7
Fig. 7 Convergence graphs of the algorithms on some test functions
Fig. 8
Fig. 8 Typical 2D representations of objective function
Fig. 9
Fig. 9 A robot vision measurement system.
Fig. 10
Fig. 10 Relative errors of rotation and translation of different algorithms
Fig. 11
Fig. 11 Measurement results of different algorithms

Tables (4)

Tables Icon

Table 1 Results of unimodal benchmark functions

Tables Icon

Table 2 Results of multi-modal benchmark functions

Tables Icon

Table 3 Results of composite benchmark functions

Tables Icon

Table 4 Comparison of results for vision measurement

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

{ u- u 0 x c = f x z c v- v 0 y c = f y z c
[ x c y c z c ]=R[ x w y w z w ]+T
R=( cosγcosβ sinγcosα+cosγsinβsinα sinγsinα+cosγsinβcosα sinγcosβ cosγcosα+sinγsinβsinα cosγsinα+sinγsinβcosα sinβ cosβsinα cosβcosα )
{ u= u 0 + f x cosγcosβ x w +(sinγcosα+cosγsinβsinα) y w +(sinγsinα+cosγsinβcosα) z w + t x sinβ x w +(cosβsinα) y w +cosβcosα z w + t z v= v 0 + f y sinγcosβ x w +(cosγcosα+sinγsinβsinα) y w +(cosγsinα+sinγsinβcosα) z w + t y sinβ x w +(cosβsinα) y w +cosβcosα z w + t z
{ u i = u 0 + f x cosγcosβ x wi +(sinγcosα+cosγsinβsinα) y wi +(sinγsinα+cosγsinβcosα) z wi + t x sinβ x wi +(cosβsinα) y wi +cosβcosα z wi + t z v i = v 0 + f y sinγcosβ x wi +(cosγcosα+sinγsinβsinα) y wi +(cosγsinα+sinγsinβcosα) z wi + t y sinβ x wi +(cosβsinα) y wi +cosβcosα z wi + t z
FindX=[α,β,γ, t x , t y , t z ] minf(X)= i=1 m || p i p ^ i ( f x , f y , u 0 , v 0 ,R,T, B i ) || 2
M=[ m 11 m 12 m 1d m 21 m 22 m 2d m n1 m n2 m nd ]andΟΜ=[ I( m 11 , m 12 ,, m 1d ) I( m 21 , m 22 ,, m 2d ) I( m n1 , m n2 ,, m nd ) ]=[ O M 1 O M 2 O M n ]
F=[ F 11 F 12 F 1d F 21 F 22 F 2d F n1 F n2 F nd ]andΟF=[ I( F 11 , F 12 ,, F 1d ) I( F 21 , F 22 ,, F 2d ) I( F n1 , F n2 ,, F nd ) ]=[ O F 1 O F 2 O F n ]
MFO=(I,K,T)
S( M i , F j )= D i e bt cos(2πt)+ F j
D i =| F j - M i |
flame_no=round( NG* N1 G max )
D ω =| C 1 F ω G M G |, D ε =| C 2 F ε G M G |, D η =| C 3 F η G M G |
M 1 = F ω G A 1 D ω , M 2 = F ε G A 2 D ε , M 3 = F η G A 3 D η
M G+1 = M 1 + M 2 + M 3 3
A i =2a r 1i a C i =2 r 2i
M i G+1 ={ D i e bt cos(2πt)+ F j G p0.5 D i e bl cos(2πl)+ F j G* p<0.5
O(IMFO)=O( G max (O(Quicksort)+O(positionupdate))) =O( G max ( n 2 +n×d))=O( G max n 2 + G max nd)
X i,j 0 = X j 0 -l t j +rand(0,1)(u t j +l t j ),i=1,2,,n,j=1,2,,6
{ F G =sort( M G ),O F G =sort(O M G )ifG=1 F G =sort( M G1 , M G ),O F G =sort(O M G1 ,O M G )otherwise
R err = q q ture q ture ×100%and T err = T T ture T ture ×100%
E d =| D i,j c D i,j w |