Mean field models for large data–clustering problems

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September 2020, 15(3): 463-487. doi: 10.3934/nhm.2020027

Mean field models for large data–clustering problems

Michael Herty ^1, , Lorenzo Pareschi ^2,, and Giuseppe Visconti ^1,

1.	RWTH Aachen University, Institut für Geometrie und Praktische Mathematik, Templergraben 55, 52062 Aachen, Germany
2.	University of Ferrara, Mathematics and Computer Science Department, Via Machiavelli 35, 44121 Ferrara, Italy

^* Corresponding author: Lorenzo Pareschi

Received July 2019 Revised March 2020 Published September 2020

We consider mean-field models for data–clustering problems starting from a generalization of the bounded confidence model for opinion dynamics. The microscopic model includes information on the position as well as on additional features of the particles in order to develop specific clustering effects. The corresponding mean–field limit is derived and properties of the model are investigated analytically. In particular, the mean–field formulation allows the use of a random subsets algorithm for efficient computations of the clusters. Applications to shape detection and image segmentation on standard test images are presented and discussed.

Keywords: Data clustering, opinion dynamic, mean field equations, image segmentation, shape detection.

Mathematics Subject Classification: Primary: 82C40, 94A08; Secondary: 68U10.

Citation: Michael Herty, Lorenzo Pareschi, Giuseppe Visconti. Mean field models for large data–clustering problems. Networks & Heterogeneous Media, 2020, 15 (3) : 463-487. doi: 10.3934/nhm.2020027

References:

[1]	G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha and J. Kim, et al., Vehicular traffic, crowds, and swarms.: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005. doi: 10.1142/S0218202519500374. Google Scholar
[2]	G. Albi and L. Pareschi, Binary interaction algorithms for the simulation of flocking and swarming dynamics, Multiscale Model. Simul., 11 (2013), 1-29. doi: 10.1137/120868748. Google Scholar
[3]	G. Albi and L. Pareschi, Modeling of self-organized systems interacting with a few individuals: From microscopic to macroscopic dynamics, Appl. Math. Lett., 26 (2013), 397-401. doi: 10.1016/j.aml.2012.10.011. Google Scholar
[4]	G. Albi, L. Pareschi, G. Toscani and M. Zanella, Recent advances in opinion modeling: Control and social influence, in Active Particles, Vol. 1, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 2017, 49–98. doi: 10.1007/978-3-319-49996-3_2. Google Scholar
[5]	G. Albi, L. Pareschi and M. Zanella, Opinion dynamics over complex networks: Kinetic modeling and numerical methods, Kinet. Relat. Models, 10 (2017), 1-32. doi: 10.3934/krm.2017001. Google Scholar
[6]	P. Arbeláez, M. Maire, C. Fowlkes and J. Malik, Contour detection and hierarchical image segmentation, IEEE Trans. Pattern Anal. Mach. Intell., 33 (2011), 898-916. doi: 10.1109/TPAMI.2010.161. Google Scholar
[7]	V. D. Blondel, J. M. Hendrickx and J. N. Tsitsiklis, On Krauses's multi-agent consensus model with state-dependent connectivity, IEEE Trans. Automat. Control, 54 (2009), 2586-2597. doi: 10.1109/TAC.2009.2031211. Google Scholar
[8]	V. D. Blondel, J. M. Hendrickx and J. N. Tsitsiklis, Continuous time average-preserving opinion dynamics with opinion-dependent communications, SIAM J. Control Optim., 48 (2010), 5214-5240. doi: 10.1137/090766188. Google Scholar
[9]	D. Borra and T. Lorenzi, Asymptotic analysis of continuous opinion dynamics models under bounded confidence, Commun. Pure Appl. Anal., 12 (2013), 1487-1499. doi: 10.3934/cpaa.2013.12.1487. Google Scholar
[10]	L. Boudin, R. Monaco and F. Salvarani, Kinetic model for multidimensional opinion formation, Phys. Rev. E, 81 (2010), 9pp. doi: 10.1103/PhysRevE.81.036109. Google Scholar
[11]	L. Boudin and F. Salvarani, Opinion dynamics: Kinetic modelling with mass media, application to the Scottish independence referendum, Phys. A, 444 (2016), 448-457. doi: 10.1016/j.physa.2015.10.014. Google Scholar
[12]	J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 519-539. doi: 10.1142/S0218202511005131. Google Scholar
[13]	C. Canuto, F. Fagnani and P. Tilli, An Eulerian approach to the analysis of rendez-vous algorithms, IFAC Proceedings Volumes, 41 (2008), 9039-9044. doi: 10.3182/20080706-5-KR-1001.01526. Google Scholar
[14]	C. Canuto, F. Fagnani and P. Tilli, An Eulerian approach to the analysis of Krause's consensus models, SIAM J. Control Optim., 50 (2012), 243–265. doi: 10.1137/100793177. Google Scholar
[15]	J. A. Carrillo, Y.-P. Choi, C. Totzeck and O. Tse, An analytical framework for consensus-based global optimization method, Math. Models Methods Appl. Sci., 28 (2018), 1037-1066. doi: 10.1142/S0218202518500276. Google Scholar
[16]	J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Boston, MA, 2010, 297–336. doi: 10.1007/978-0-8176-4946-3_12. Google Scholar
[17]	L.-C. Chen, P. G., I. Kokkinos, K. Murphy and A. Loddon Yuille, DeepLab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected CRFs, IEEE Trans. Pattern Anal. Machine Intelligence, 40 (2018), 834-848. doi: 10.1109/TPAMI.2017.2699184. Google Scholar
[18]	I. D. Couzin, J. Krause, N. R. Franks and S. A. Levin, Effective leadership and decision-making in animal groups on the move, Nature, 433 (2005), 513-516. doi: 10.1038/nature03236. Google Scholar
[19]	F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842. Google Scholar
[20]	P. Degond and S. Motsch, A macroscopic model for a system of swarming agents using curvature control, J. Stat. Phys., 143 (2011), 685-714. doi: 10.1007/s10955-011-0201-3. Google Scholar
[21]	F. Dietrich, S. Martin and M. Jungers, Transient cluster formation in generalized Hegselmann-Krause opinion dynamics, 2016 European Control Conference (ECC), Aalborg, Denmark, 2016. doi: 10.1109/ECC.2016.7810339. Google Scholar
[22]	J. C. Dittmer, Consensus formation under bounded confidence, Nonlinear Anal., 47 (2001), 4615-4622. doi: 10.1016/S0362-546X(01)00574-0. Google Scholar
[23]	B. Düring, P. Markowich, J.-F. Pietschmann and M.-T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 3687-3708. doi: 10.1098/rspa.2009.0239. Google Scholar
[24]	S. Gould, R. Fulton and D. Koller, Decomposing a scene into geometric and semantically consistent regions, 2009 IEEE 12th International Conference on Computer Vision, Kyoto, Japan, 2009. doi: 10.1109/ICCV.2009.5459211. Google Scholar
[25]	R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence: Models, analysis and simulation, J. Artifical Societies Social Simulation, 5 (2002). Google Scholar
[26]	J. M. Hendrickx, Graphs and Networks for the Analysis of Autonomous Agent Systems, Ph.D thesis, Ecole Polytechnique de Louvain, 2008. Google Scholar
[27]	P.-E. Jabin and S. Motsch, Clustering and asymptotic behavior in opinion formation, J. Differential Equations, 257 (2014), 4165-4187. doi: 10.1016/j.jde.2014.08.005. Google Scholar
[28]	J. Kennedy and R. Eberhart, Particle swarm optimization, Proc. Internat. Conference Neural Networks, Perth, Australia, 1995. doi: 10.1109/ICNN.1995.488968. Google Scholar
[29]	X. Liu, Y. Qiao, X. Chen, J. Miao and L. Duan, Color image segmentation based on modified Kuramoto model, Procedia Comp. Sci., 88 (2016), 245-258. doi: 10.1016/j.procs.2016.07.432. Google Scholar
[30]	J. Lorenz, A stabilization theorem for dynamics of continuous opinions, Phys. A, 355 (2005), 217-223. doi: 10.1016/j.physa.2005.02.086. Google Scholar
[31]	J. MacQueen, Some methods for classification and analysis of multivariate observations, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability, Vol. Ⅰ: Statistics, Univ. California Press, Berkeley, CA, 1967, 281-297. Google Scholar
[32]	D. Martin, C. Fowlkes, D. Tal and J. Malik, A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics, Proc. Eighth IEEE Internat. Conference Comp. Vision, Vancouver, Canada, 2001. doi: 10.1109/ICCV.2001.937655. Google Scholar
[33]	K. H. Memon, S. Memon, M. A. Qureshi, M. B. Alvi, D. Kumar and R. A. Shah, Kernel possibilistic fuzzy $c$-means clustering with local information for image segmentation, Internat. J. Fuzzy Syst., 21 (2019), 321-332. doi: 10.1007/s40815-018-0537-9. Google Scholar
[34]	S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947. doi: 10.1007/s10955-011-0285-9. Google Scholar
[35]	S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621. doi: 10.1137/120901866. Google Scholar
[36]	A. Nedić and B. Touri, Multi-dimensional Hegselmann-Krause dynamics, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), Maui, HI, 2012. doi: 10.1109/CDC.2012.6426417. Google Scholar
[37]	A. V. Novikov and E. N. Benderskaya, Oscillatory neural networks based on the Kuramoto model for cluster analysis, Pattern Recognition Image Anal., 24 (2014), 365-371. doi: 10.1134/S1054661814030146. Google Scholar
[38]	G. Oliva, D. La Manna, A. Fagiolini and R. Setola, Distributed data clustering via opinion dynamics, Internat. J. Distributed Sensor Networks, 11 (2015). doi: 10.1155/2015/753102. Google Scholar
[39]	S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. Google Scholar
[40]	L. Pareschi and G. Toscani, Interacting Multiagent Systems. Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2013. Google Scholar
[41]	R. Pinnau, C. Totzeck, O. Tse and S. Martin, A consensus-based model for global optimization and its mean-field limit, Math. Models Methods Appl. Sci., 27 (2017), 183-204. doi: 10.1142/S0218202517400061. Google Scholar
[42]	G. Puppo, M. Semplice, A. Tosin and G. Visconti, Kinetic models for traffic flow resulting in a reduced space of microscopic velocities, Kinet. Relat. Models, 10 (2017), 823-854. doi: 10.3934/krm.2017033. Google Scholar
[43]	J. A. Sethian, Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge Monographs on Applied and Computational Mathematics, 3, Cambridge University Press, Cambridge, 1999. Google Scholar
[44]	P. Shan, Image segmentation method based on K-mean algorithm, EURASIP J. Image Video Processing, 81 (2018). doi: 10.1186/s13640-018-0322-6. Google Scholar
[45]	L. Tian, L. Han and J. Yue, Research on image segmentation based on clustering algorithm, Internat. J. Signal Process. Image Process. Pattern Recognition, 9 (2016), 1-12. doi: 10.14257/ijsip.2016.9.2.01. Google Scholar
[46]	L. Zhang, Y. Gao, Y. Xia, K. Lu, J. Shen and R. Ji, Representative discovery of structure cues for weakly-supervised image segmentation, IEEE Transac. Multimedia, 16 (2014), 470-479. doi: 10.1109/TMM.2013.2293424. Google Scholar
[47]	X. Zheng, Q. Lei, R. Yao, Y. Gong and Q. Yin, Image segmentation based on adaptive $K$-means algorithm, EURASIP J. Image Video Processing, 68 (2018). doi: 10.1186/s13640-018-0309-3. Google Scholar

show all references

References:

[1]	G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha and J. Kim, et al., Vehicular traffic, crowds, and swarms.: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005. doi: 10.1142/S0218202519500374. Google Scholar
[2]	G. Albi and L. Pareschi, Binary interaction algorithms for the simulation of flocking and swarming dynamics, Multiscale Model. Simul., 11 (2013), 1-29. doi: 10.1137/120868748. Google Scholar
[3]	G. Albi and L. Pareschi, Modeling of self-organized systems interacting with a few individuals: From microscopic to macroscopic dynamics, Appl. Math. Lett., 26 (2013), 397-401. doi: 10.1016/j.aml.2012.10.011. Google Scholar
[4]	G. Albi, L. Pareschi, G. Toscani and M. Zanella, Recent advances in opinion modeling: Control and social influence, in Active Particles, Vol. 1, Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, 2017, 49–98. doi: 10.1007/978-3-319-49996-3_2. Google Scholar
[5]	G. Albi, L. Pareschi and M. Zanella, Opinion dynamics over complex networks: Kinetic modeling and numerical methods, Kinet. Relat. Models, 10 (2017), 1-32. doi: 10.3934/krm.2017001. Google Scholar
[6]	P. Arbeláez, M. Maire, C. Fowlkes and J. Malik, Contour detection and hierarchical image segmentation, IEEE Trans. Pattern Anal. Mach. Intell., 33 (2011), 898-916. doi: 10.1109/TPAMI.2010.161. Google Scholar
[7]	V. D. Blondel, J. M. Hendrickx and J. N. Tsitsiklis, On Krauses's multi-agent consensus model with state-dependent connectivity, IEEE Trans. Automat. Control, 54 (2009), 2586-2597. doi: 10.1109/TAC.2009.2031211. Google Scholar
[8]	V. D. Blondel, J. M. Hendrickx and J. N. Tsitsiklis, Continuous time average-preserving opinion dynamics with opinion-dependent communications, SIAM J. Control Optim., 48 (2010), 5214-5240. doi: 10.1137/090766188. Google Scholar
[9]	D. Borra and T. Lorenzi, Asymptotic analysis of continuous opinion dynamics models under bounded confidence, Commun. Pure Appl. Anal., 12 (2013), 1487-1499. doi: 10.3934/cpaa.2013.12.1487. Google Scholar
[10]	L. Boudin, R. Monaco and F. Salvarani, Kinetic model for multidimensional opinion formation, Phys. Rev. E, 81 (2010), 9pp. doi: 10.1103/PhysRevE.81.036109. Google Scholar
[11]	L. Boudin and F. Salvarani, Opinion dynamics: Kinetic modelling with mass media, application to the Scottish independence referendum, Phys. A, 444 (2016), 448-457. doi: 10.1016/j.physa.2015.10.014. Google Scholar
[12]	J. A. Cañizo, J. A. Carrillo and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21 (2011), 519-539. doi: 10.1142/S0218202511005131. Google Scholar
[13]	C. Canuto, F. Fagnani and P. Tilli, An Eulerian approach to the analysis of rendez-vous algorithms, IFAC Proceedings Volumes, 41 (2008), 9039-9044. doi: 10.3182/20080706-5-KR-1001.01526. Google Scholar
[14]	C. Canuto, F. Fagnani and P. Tilli, An Eulerian approach to the analysis of Krause's consensus models, SIAM J. Control Optim., 50 (2012), 243–265. doi: 10.1137/100793177. Google Scholar
[15]	J. A. Carrillo, Y.-P. Choi, C. Totzeck and O. Tse, An analytical framework for consensus-based global optimization method, Math. Models Methods Appl. Sci., 28 (2018), 1037-1066. doi: 10.1142/S0218202518500276. Google Scholar
[16]	J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Boston, MA, 2010, 297–336. doi: 10.1007/978-0-8176-4946-3_12. Google Scholar
[17]	L.-C. Chen, P. G., I. Kokkinos, K. Murphy and A. Loddon Yuille, DeepLab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected CRFs, IEEE Trans. Pattern Anal. Machine Intelligence, 40 (2018), 834-848. doi: 10.1109/TPAMI.2017.2699184. Google Scholar
[18]	I. D. Couzin, J. Krause, N. R. Franks and S. A. Levin, Effective leadership and decision-making in animal groups on the move, Nature, 433 (2005), 513-516. doi: 10.1038/nature03236. Google Scholar
[19]	F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control, 52 (2007), 852-862. doi: 10.1109/TAC.2007.895842. Google Scholar
[20]	P. Degond and S. Motsch, A macroscopic model for a system of swarming agents using curvature control, J. Stat. Phys., 143 (2011), 685-714. doi: 10.1007/s10955-011-0201-3. Google Scholar
[21]	F. Dietrich, S. Martin and M. Jungers, Transient cluster formation in generalized Hegselmann-Krause opinion dynamics, 2016 European Control Conference (ECC), Aalborg, Denmark, 2016. doi: 10.1109/ECC.2016.7810339. Google Scholar
[22]	J. C. Dittmer, Consensus formation under bounded confidence, Nonlinear Anal., 47 (2001), 4615-4622. doi: 10.1016/S0362-546X(01)00574-0. Google Scholar
[23]	B. Düring, P. Markowich, J.-F. Pietschmann and M.-T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 465 (2009), 3687-3708. doi: 10.1098/rspa.2009.0239. Google Scholar
[24]	S. Gould, R. Fulton and D. Koller, Decomposing a scene into geometric and semantically consistent regions, 2009 IEEE 12th International Conference on Computer Vision, Kyoto, Japan, 2009. doi: 10.1109/ICCV.2009.5459211. Google Scholar
[25]	R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence: Models, analysis and simulation, J. Artifical Societies Social Simulation, 5 (2002). Google Scholar
[26]	J. M. Hendrickx, Graphs and Networks for the Analysis of Autonomous Agent Systems, Ph.D thesis, Ecole Polytechnique de Louvain, 2008. Google Scholar
[27]	P.-E. Jabin and S. Motsch, Clustering and asymptotic behavior in opinion formation, J. Differential Equations, 257 (2014), 4165-4187. doi: 10.1016/j.jde.2014.08.005. Google Scholar
[28]	J. Kennedy and R. Eberhart, Particle swarm optimization, Proc. Internat. Conference Neural Networks, Perth, Australia, 1995. doi: 10.1109/ICNN.1995.488968. Google Scholar
[29]	X. Liu, Y. Qiao, X. Chen, J. Miao and L. Duan, Color image segmentation based on modified Kuramoto model, Procedia Comp. Sci., 88 (2016), 245-258. doi: 10.1016/j.procs.2016.07.432. Google Scholar
[30]	J. Lorenz, A stabilization theorem for dynamics of continuous opinions, Phys. A, 355 (2005), 217-223. doi: 10.1016/j.physa.2005.02.086. Google Scholar
[31]	J. MacQueen, Some methods for classification and analysis of multivariate observations, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability, Vol. Ⅰ: Statistics, Univ. California Press, Berkeley, CA, 1967, 281-297. Google Scholar
[32]	D. Martin, C. Fowlkes, D. Tal and J. Malik, A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics, Proc. Eighth IEEE Internat. Conference Comp. Vision, Vancouver, Canada, 2001. doi: 10.1109/ICCV.2001.937655. Google Scholar
[33]	K. H. Memon, S. Memon, M. A. Qureshi, M. B. Alvi, D. Kumar and R. A. Shah, Kernel possibilistic fuzzy $c$-means clustering with local information for image segmentation, Internat. J. Fuzzy Syst., 21 (2019), 321-332. doi: 10.1007/s40815-018-0537-9. Google Scholar
[34]	S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, J. Stat. Phys., 144 (2011), 923-947. doi: 10.1007/s10955-011-0285-9. Google Scholar
[35]	S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Review, 56 (2014), 577-621. doi: 10.1137/120901866. Google Scholar
[36]	A. Nedić and B. Touri, Multi-dimensional Hegselmann-Krause dynamics, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), Maui, HI, 2012. doi: 10.1109/CDC.2012.6426417. Google Scholar
[37]	A. V. Novikov and E. N. Benderskaya, Oscillatory neural networks based on the Kuramoto model for cluster analysis, Pattern Recognition Image Anal., 24 (2014), 365-371. doi: 10.1134/S1054661814030146. Google Scholar
[38]	G. Oliva, D. La Manna, A. Fagiolini and R. Setola, Distributed data clustering via opinion dynamics, Internat. J. Distributed Sensor Networks, 11 (2015). doi: 10.1155/2015/753102. Google Scholar
[39]	S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2. Google Scholar
[40]	L. Pareschi and G. Toscani, Interacting Multiagent Systems. Kinetic Equations and Monte Carlo Methods, Oxford University Press, 2013. Google Scholar
[41]	R. Pinnau, C. Totzeck, O. Tse and S. Martin, A consensus-based model for global optimization and its mean-field limit, Math. Models Methods Appl. Sci., 27 (2017), 183-204. doi: 10.1142/S0218202517400061. Google Scholar
[42]	G. Puppo, M. Semplice, A. Tosin and G. Visconti, Kinetic models for traffic flow resulting in a reduced space of microscopic velocities, Kinet. Relat. Models, 10 (2017), 823-854. doi: 10.3934/krm.2017033. Google Scholar
[43]	J. A. Sethian, Level Set Methods and Fast Marching Methods. Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge Monographs on Applied and Computational Mathematics, 3, Cambridge University Press, Cambridge, 1999. Google Scholar
[44]	P. Shan, Image segmentation method based on K-mean algorithm, EURASIP J. Image Video Processing, 81 (2018). doi: 10.1186/s13640-018-0322-6. Google Scholar
[45]	L. Tian, L. Han and J. Yue, Research on image segmentation based on clustering algorithm, Internat. J. Signal Process. Image Process. Pattern Recognition, 9 (2016), 1-12. doi: 10.14257/ijsip.2016.9.2.01. Google Scholar
[46]	L. Zhang, Y. Gao, Y. Xia, K. Lu, J. Shen and R. Ji, Representative discovery of structure cues for weakly-supervised image segmentation, IEEE Transac. Multimedia, 16 (2014), 470-479. doi: 10.1109/TMM.2013.2293424. Google Scholar
[47]	X. Zheng, Q. Lei, R. Yao, Y. Gong and Q. Yin, Image segmentation based on adaptive $K$-means algorithm, EURASIP J. Image Video Processing, 68 (2018). doi: 10.1186/s13640-018-0309-3. Google Scholar

Figure 1. Trend to the steady–state of the one–dimensional Hegselmann–Krause model (1) with $ n = 100 $ agents equally spaced at initial time and non–symmetric interactions (top row) and of the mean–field model (12) computed with Algorithm 1 (bottom row) up to final time $ T = 20 $. Left panels show the case for $ \epsilon_1 = 0.5 $, the right panels show the case for $ \epsilon_1 = 0.15 $

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Figure 2. Evolution in time of the first moment (left) and of the second moment (right) for the two values of the bounded confidence level $ \epsilon_1 = 0.5 $ (dashed lines) and $ \epsilon_1 = 0.15 $ (solid lines)

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Figure 3. Left: trend to the steady–state of the mean–field model (12) computed with Algorithm 1 with $ N = 2\times10^4 $, $ M = 2 $, $ \epsilon_1 = 0.15 $ and up to final time $ T = 20 $. Right: energy decay of the mean–field model (12) for several values of interacting particles $ M $

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Figure 4. Particle solution (left plots) with $ N = 10000 $ and kinetic density (right plots). Results are provided at time $ t = 4 $ (top row) and final time $ T = 50 $ (bottom row). The bounded confidence level is $ \epsilon_1 = 0.15 $

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Figure 5. Evolution in time of the two–dimensional first moments (left) and second moments (right) for the bounded confidence level $ \epsilon_1 = 0.15 $

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Figure 6. Top row: particles and kinetic density at initial time (left plot) and at equilibrium (right plot). Bottom row: at left, analysis of the distances between clusters in $ x $ (blue line with circle markers) and $ c $ direction (red line with triangle markers); at right, plot of the marginals. Confidence levels are $ \epsilon_1 = 0.15 $ and $ \epsilon_2 = 0.025 $

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Figure 7. Particle and kinetic density at equilibrium with confidence levels $ \epsilon_1 = 0.15 $, $ \epsilon_2 = 0.1 $ (left) and $ \epsilon_1 = 1 $, $ \epsilon_2 = 0.025 $ (right)

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$ 10\% $ of a uniform additive noise. Top left panel shows the initial condition. We show clusters obtained with bounded confidence values $ \epsilon_1 = 0.06 $ (top right), $ \epsilon_1 = 0.08 $ (bottom left) and $ \epsilon_1 = 0.1 $ (bottom right)">Figure 8. Shape detection of the letter "A" initialized with $ 10\% $ of a uniform additive noise. Top left panel shows the initial condition. We show clusters obtained with bounded confidence values $ \epsilon_1 = 0.06 $ (top right), $ \epsilon_1 = 0.08 $ (bottom left) and $ \epsilon_1 = 0.1 $ (bottom right)

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$ 5.5\% $ of a Gaussian additive noise. Top left panel shows the initial condition. We show clusters obtained with bounded confidence values $ \epsilon_1 = 0.05 $ (top right), $ \epsilon_1 = 0.0675 $ (bottom left) and $ \epsilon_1 = 0.08 $ (bottom right)">Figure 9. Shape detection of the letter "A" initialized with $ 5.5\% $ of a Gaussian additive noise. Top left panel shows the initial condition. We show clusters obtained with bounded confidence values $ \epsilon_1 = 0.05 $ (top right), $ \epsilon_1 = 0.0675 $ (bottom left) and $ \epsilon_1 = 0.08 $ (bottom right)

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Figure 10. Left panel: initial image of $ 4096 $ pixels with four regions with different gray intensity. Middle panel: red dots show the positions of the clusters at equilibrium. Right panel: segmentation of the initial image in two regions

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6]">Figure 11. Image segmentation of $ 174\times73 $ gray scale image taken by the data–set [6]

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24]">Figure 12. Image segmentation of $ 93\times93 $ gray scale image taken by the data–set [24]

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32]">Figure 13. Image segmentation of $ 128\times94 $ gray scale image taken by the data–set [32]

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32]">Figure 14. Image segmentation of $ 67\times67 $ gray scale image taken by the data–set [32]

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32]">Figure 15. Image segmentation of $ 132\times106 $ gray scale image taken by the data–set [32]

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Table 1. Number of clusters and errors depending on the bounded confidence value and the percentage of the noise when uniformly distributed

$ \alpha = 5\% $			$ \alpha = 7.5\% $			$ \alpha = 10\% $
$ \epsilon_1 $	$ \mathcal{E} $	$ \tilde{n} $	$ \epsilon_1 $	$ \mathcal{E} $	$ \tilde{n} $	$ \epsilon_1 $	$ \mathcal{E} $	$ \tilde{n} $
0.03	1.25e-02	30	0.03	3.47e-02	51	0.06	2.64e-02	11
0.04	4.10e-03	16	0.05	1.21e-02	14	0.07	1.48e-02	8
0.05	4.00e-03	12	0.07	7.70e-03	8	0.08	1.12e-02	8
0.06	4.60e-03	9	0.09	7.90e-03	8	0.09	1.63e-02	5
0.07	5.40e-03	8	0.11	1.66e-02	3	0.10	1.63e-02	5

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$ \alpha = 5\% $			$ \alpha = 7.5\% $			$ \alpha = 10\% $
$ \epsilon_1 $	$ \mathcal{E} $	$ \tilde{n} $	$ \epsilon_1 $	$ \mathcal{E} $	$ \tilde{n} $	$ \epsilon_1 $	$ \mathcal{E} $	$ \tilde{n} $
0.03	1.25e-02	30	0.03	3.47e-02	51	0.06	2.64e-02	11
0.04	4.10e-03	16	0.05	1.21e-02	14	0.07	1.48e-02	8
0.05	4.00e-03	12	0.07	7.70e-03	8	0.08	1.12e-02	8
0.06	4.60e-03	9	0.09	7.90e-03	8	0.09	1.63e-02	5
0.07	5.40e-03	8	0.11	1.66e-02	3	0.10	1.63e-02	5

Table 2. Number of clusters and errors depending on the bounded confidence value and the percentage of the noise when normally distributed

$ \alpha = 5\% $			$ \alpha = 5.5\% $			$ \alpha = 6\% $
$ \epsilon_1 $	$ \mathcal{E} $	$ \tilde{n} $	$ \epsilon_1 $	$ \mathcal{E} $	$ \tilde{n} $	$ \epsilon_1 $	$ \mathcal{E} $	$ \tilde{n} $
0.05	4.44e-02	23	0.05	4.73e-02	24	0.05	6.37e-02	30
0.06	1.36e-02	11	0.06	2.62e-02	13	0.06	4.16e-02	16
0.065	6.40e-03	9	0.065	1.63e-02	11	0.07	2.12e-02	9
0.07	6.70e-03	7	0.0675	7.40e-03	10	0.075	9.70e-03	7
0.08	8.50e-03	7	0.07	8.00e-03	9	0.08	9.20e-03	7
0.09	1.00e-02	6	0.08	9.80e-03	7	0.085	1.10e-02	5

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$ \alpha = 5\% $			$ \alpha = 5.5\% $			$ \alpha = 6\% $
$ \epsilon_1 $	$ \mathcal{E} $	$ \tilde{n} $	$ \epsilon_1 $	$ \mathcal{E} $	$ \tilde{n} $	$ \epsilon_1 $	$ \mathcal{E} $	$ \tilde{n} $
0.05	4.44e-02	23	0.05	4.73e-02	24	0.05	6.37e-02	30
0.06	1.36e-02	11	0.06	2.62e-02	13	0.06	4.16e-02	16
0.065	6.40e-03	9	0.065	1.63e-02	11	0.07	2.12e-02	9
0.07	6.70e-03	7	0.0675	7.40e-03	10	0.075	9.70e-03	7
0.08	8.50e-03	7	0.07	8.00e-03	9	0.08	9.20e-03	7
0.09	1.00e-02	6	0.08	9.80e-03	7	0.085	1.10e-02	5

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