{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T12:48:36Z","timestamp":1775479716827,"version":"3.50.1"},"reference-count":56,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,12,21]],"date-time":"2021-12-21T00:00:00Z","timestamp":1640044800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Masi entropy is a popular criterion employed for identifying appropriate threshold values in image thresholding. However, with an increasing number of thresholds, the efficiency of Masi entropy-based multi-level thresholding algorithms becomes problematic. To overcome this, we propose a novel differential evolution (DE) algorithm as an effective population-based metaheuristic for Masi entropy-based multi-level image thresholding. Our ME-GDEAR algorithm benefits from a grouping strategy to enhance the efficacy of the algorithm for which a clustering algorithm is used to partition the current population. Then, an updating strategy is introduced to include the obtained clusters in the current population. We further improve the algorithm using attraction (towards the best individual) and repulsion (from random individuals) strategies. Extensive experiments on a set of benchmark images convincingly show ME-GDEAR to give excellent image thresholding performance, outperforming other metaheuristics in 37 out of 48 cases based on cost function evaluation, 26 of 48 cases based on feature similarity index, and 20 of 32 cases based on Dice similarity. The obtained results demonstrate that population-based metaheuristics can be successfully applied to entropy-based image thresholding and that strengthening both exploitation and exploration strategies, as performed in ME-GDEAR, is crucial for designing such an algorithm.<\/jats:p>","DOI":"10.3390\/e24010008","type":"journal-article","created":{"date-parts":[[2021,12,21]],"date-time":"2021-12-21T09:50:43Z","timestamp":1640080243000},"page":"8","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["A Grouping Differential Evolution Algorithm Boosted by Attraction and Repulsion Strategies for Masi Entropy-Based Multi-Level Image Segmentation"],"prefix":"10.3390","volume":"24","author":[{"given":"Seyed Jalaleddin","family":"Mousavirad","sequence":"first","affiliation":[{"name":"Computer Engineering Department, Hakim Sabzevari University, Sabzevar 96179-76487, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2487-7950","authenticated-orcid":false,"given":"Davood","family":"Zabihzadeh","sequence":"additional","affiliation":[{"name":"Computer Engineering Department, Hakim Sabzevari University, Sabzevar 96179-76487, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8781-7993","authenticated-orcid":false,"given":"Diego","family":"Oliva","sequence":"additional","affiliation":[{"name":"Departamento de Innovaci\u00f3n Basada en la Informaci\u00f3n y el Conocimiento, Universidad de Guadalajara, CUCEI, Guadalajara 44430, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6493-0408","authenticated-orcid":false,"given":"Marco","family":"Perez-Cisneros","sequence":"additional","affiliation":[{"name":"Departamento de Innovaci\u00f3n Basada en la Informaci\u00f3n y el Conocimiento, Universidad de Guadalajara, CUCEI, Guadalajara 44430, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gerald","family":"Schaefer","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Loughborough University, Loughborough LE11 3TT, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,12,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/j.cmpb.2019.04.016","article-title":"A novel framework for MR image segmentation and quantification by using MedGA","volume":"176","author":"Rundo","year":"2019","journal-title":"Comput. Methods Programs Biomed."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"408","DOI":"10.1016\/j.ins.2014.10.005","article-title":"Dynamic-context cooperative quantum-behaved particle swarm optimization based on multilevel thresholding applied to medical image segmentation","volume":"294","author":"Li","year":"2015","journal-title":"Inf. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1817","DOI":"10.1177\/0021998316662133","article-title":"Characterization, synthetic generation, and statistical equivalence of composite microstructures","volume":"51","author":"Sanei","year":"2017","journal-title":"J. Compos. Mater."},{"key":"ref_4","unstructured":"Mousavirad, S., Akhlaghian, F., and Mollazade, K. (2011, January 16\u201317). Classification of rice varieties using optimal color and texture features and BP neural networks. Proceedings of the 7th Iranian Conference on Machine Vision and Image Processing, Tehran, Iran."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1277","DOI":"10.1016\/0031-3203(93)90135-J","article-title":"A review on image segmentation techniques","volume":"26","author":"Pal","year":"1993","journal-title":"Pattern Recognit."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Minaee, S., Boykov, Y.Y., Porikli, F., Plaza, A.J., Kehtarnavaz, N., and Terzopoulos, D. (2021). Image segmentation using deep learning: A survey. IEEE Trans. Pattern Anal. Mach. Intell., Available online: https:\/\/ieeexplore.ieee.org\/abstract\/document\/9356353.","DOI":"10.1109\/TPAMI.2021.3059968"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Mousavirad, S.J., Schaefer, G., Oliva, D., and Hinojosa, S. (2021, January 10\u201314). HCS-BBD: An effective population-based approach for multi-level thresholding. Proceedings of the Genetic and Evolutionary Computation Conference Companion, Lille, France.","DOI":"10.1145\/3449726.3463149"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1007\/s42044-018-0022-5","article-title":"A multilevel image thresholding using the animal migration optimization algorithm","volume":"2","author":"Farshi","year":"2019","journal-title":"Iran J. Comput. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"15273","DOI":"10.1007\/s11042-020-10432-4","article-title":"Multilevel image thresholding with multimodal optimization","volume":"80","author":"Farshi","year":"2021","journal-title":"Multimed. Tools Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"10685","DOI":"10.1007\/s00521-020-04820-y","article-title":"A novel equilibrium optimization algorithm for multi-thresholding image segmentation problems","volume":"33","author":"Chang","year":"2021","journal-title":"Neural Comput. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"115106","DOI":"10.1016\/j.eswa.2021.115106","article-title":"An efficient method to minimize cross-entropy for selecting multi-level threshold values using an improved human mental search algorithm","volume":"182","author":"Esmaeili","year":"2021","journal-title":"Expert Syst. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Mousavirad, S.J., Schaefer, G., and Ebrahimpour-Komleh, H. (2019, January 10\u201313). A benchmark of population-based metaheuristic algorithms for high-dimensional multi-level image thresholding. Proceedings of the IEEE Congress on Evolutionary Computation, Wellington, New Zealand.","DOI":"10.1109\/CEC.2019.8790273"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Mousavirad, S.J., and Ebrahimpour-Komleh, H. (2015, January 1\u20133). Entropy based optimal multilevel thresholding using cuckoo optimization algorithm. Proceedings of the 11th International Conference on Innovations in Information Technology, Dubai, United Arab Emirates.","DOI":"10.1109\/INNOVATIONS.2015.7381558"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Mousavirad, S.J., Schaefer, G., and Korovin, I. (2020, January 8\u201312). High-dimensional multi-level image thresholding using self-organizing migrating algorithm. Proceedings of the Genetic and Evolutionary Computation Conference Companion, Cancun, Mexico.","DOI":"10.1145\/3377929.3398125"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s00530-020-00716-y","article-title":"A hybrid firefly and particle swarm optimization algorithm applied to multilevel image thresholding","volume":"27","author":"Farshi","year":"2021","journal-title":"Multimed. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"17197","DOI":"10.1007\/s11042-018-7034-x","article-title":"A generalized Masi entropy based efficient multilevel thresholding method for color image segmentation","volume":"78","author":"Shubham","year":"2019","journal-title":"Multimed. Tools Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1016\/0734-189X(85)90125-2","article-title":"A new method for gray-level picture thresholding using the entropy of the histogram","volume":"29","author":"Kapur","year":"1985","journal-title":"Comput. Vision, Graph. Image Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1016\/S0031-3203(96)00065-9","article-title":"Threshold selection using Renyi\u2019s entropy","volume":"30","author":"Sahoo","year":"1997","journal-title":"Pattern Recognit."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1109\/18.61115","article-title":"Divergence measures based on the Shannon entropy","volume":"37","author":"Lin","year":"1991","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1059","DOI":"10.1016\/j.patrec.2004.03.003","article-title":"Image thresholding using Tsallis entropy","volume":"25","author":"Esquef","year":"2004","journal-title":"Pattern Recognit. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"35543","DOI":"10.1007\/s11042-020-10467-7","article-title":"A leader Harris hawks optimization for 2-D Masi entropy-based multilevel image thresholding","volume":"80","author":"Naik","year":"2021","journal-title":"Multimed. Tools Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/j.physleta.2005.01.094","article-title":"A step beyond Tsallis and R\u00e9nyi entropies","volume":"338","author":"Masi","year":"2005","journal-title":"Phys. Lett. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"345","DOI":"10.3233\/FI-2020-1887","article-title":"A survey on nature-inspired medical image analysis: A step further in biomedical data integration","volume":"171","author":"Rundo","year":"2020","journal-title":"Fundam. Inform."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"33573","DOI":"10.1007\/s11042-019-08117-8","article-title":"Masi entropy based multilevel thresholding for image segmentation","volume":"78","author":"Khairuzzaman","year":"2019","journal-title":"Multimed. Tools Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"7793","DOI":"10.1007\/s12652-020-02506-w","article-title":"IFODPSO-based multi-level image segmentation scheme aided with Masi entropy","volume":"12","author":"Chakraborty","year":"2020","journal-title":"J. Ambient. Intell. Humaniz. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3058","DOI":"10.1007\/s00034-018-0993-3","article-title":"A water cycle algorithm-based multilevel thresholding system for color image segmentation using Masi entropy","volume":"38","author":"Kandhway","year":"2019","journal-title":"Circuits Syst. Signal Process."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.infrared.2019.03.010","article-title":"A context sensitive Masi entropy for multilevel image segmentation using moth swarm algorithm","volume":"98","author":"Bhandari","year":"2019","journal-title":"Infrared Phys. Technol."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Jia, H., Peng, X., Song, W., Oliva, D., Lang, C., and Li, Y. (2019). Masi entropy for satellite color image segmentation using tournament-based L\u00e9vy multiverse optimization algorithm. Remote Sens., 11.","DOI":"10.3390\/rs11080942"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"22613","DOI":"10.1007\/s11042-019-7506-7","article-title":"Spatial context cross entropy function based multilevel image segmentation using multi-verse optimizer","volume":"78","author":"Kandhway","year":"2019","journal-title":"Multimed. Tools Appl."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Wunnava, A., Naik, M.K., Panda, R., Jena, B., and Abraham, A. (2020). A differential evolutionary adaptive Harris hawks optimization for two dimensional practical Masi entropy-based multilevel image thresholding. J. King Saud-Univ.-Comput. Inf. Sci., in press.","DOI":"10.1016\/j.asoc.2020.106526"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1007\/s00521-020-05013-3","article-title":"A fused contextual color image thresholding using cuttlefish algorithm","volume":"33","author":"Bhandari","year":"2021","journal-title":"Neural Comput. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"213130","DOI":"10.1109\/ACCESS.2020.3040177","article-title":"A logistic chaotic barnacles mating optimizer with Masi entropy for color image multilevel thresholding segmentation","volume":"8","author":"Li","year":"2020","journal-title":"IEEE Access"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1023\/A:1008202821328","article-title":"Differential evolution\u2013a simple and efficient heuristic for global optimization over continuous spaces","volume":"11","author":"Storn","year":"1997","journal-title":"J. Glob. Optim."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"514","DOI":"10.1016\/j.neucom.2019.12.141","article-title":"Fitness distance correlation and mixed search strategy for differential evolution","volume":"458","author":"Li","year":"2021","journal-title":"Neurocomputing"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Mousavirad, S.J., and Rahnamayan, S. (2019, January 19\u201321). Differential Evolution Algorithm Based on a Competition Scheme. Proceedings of the 14th International Conference on Computer Science and Education, Toronto, ON, Canada.","DOI":"10.1109\/ICCSE.2019.8845065"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"106761","DOI":"10.1016\/j.asoc.2020.106761","article-title":"MTDE: An effective multi-trial vector-based differential evolution algorithm and its applications for engineering design problems","volume":"97","author":"Taghian","year":"2020","journal-title":"Appl. Soft Comput."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"10799","DOI":"10.1007\/s00521-018-3395-3","article-title":"Post hoc analysis of sport performance with differential evolution","volume":"32","author":"Fister","year":"2018","journal-title":"Neural Comput. Appl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"8682124","DOI":"10.1155\/2019\/8682124","article-title":"A differential evolution-oriented pruning neural network model for bankruptcy prediction","volume":"2019","author":"Tang","year":"2019","journal-title":"Complexity"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1016\/j.renene.2020.09.109","article-title":"Wind speed forecasting based on variational mode decomposition and improved echo state network","volume":"164","author":"Hu","year":"2021","journal-title":"Renew. Energy"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1186\/s13662-017-1461-2","article-title":"Wavelets optimization method for evaluation of fractional partial differential equations: An application to financial modelling","volume":"2018","author":"Ara","year":"2018","journal-title":"Adv. Differ. Equ."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Mousavirad, S.J., Rahnamayan, S., and Schaefer, G. (2020, January 19\u201324). Many-level image thresholding using a center-based differential evolution algorithm. Proceedings of the Congress on Evolutionary Computation, Glasgow, UK.","DOI":"10.1109\/CEC48606.2020.9185679"},{"key":"ref_42","unstructured":"MacQueen, J. (1967, January 7). Some methods for classification and analysis of multivariate observations. Proceedings of the 5th Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA, USA."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1363","DOI":"10.1016\/j.asoc.2010.04.008","article-title":"A clustering-based differential evolution for global optimization","volume":"11","author":"Cai","year":"2011","journal-title":"Appl. Soft Comput."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1007\/s00500-004-0377-4","article-title":"A population-based algorithm-generator for real-parameter optimization","volume":"9","author":"Deb","year":"2005","journal-title":"Soft Comput."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"902","DOI":"10.1109\/TCSI.2005.846216","article-title":"A hybrid evolutionary programming method for circuit optimization","volume":"52","author":"Damavandi","year":"2005","journal-title":"IEEE Trans. Circuits Syst. I Regul. Pap."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.advengsoft.2016.01.008","article-title":"The whale optimization algorithm","volume":"95","author":"Mirjalili","year":"2016","journal-title":"Adv. Eng. Softw."},{"key":"ref_47","unstructured":"Martin, D., Fowlkes, C., Tal, D., and Malik, J. (2001, January 7\u201314). A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics. Proceedings of the 8th International Conference on Computer Vision, Vancouver, BC, Canada."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Rahnamayan, S., and Wang, G.G. (2009, January 18\u201321). Center-based sampling for population-based algorithms. Proceedings of the IEEE Congress on Evolutionary Computation, Trondheim, Norway.","DOI":"10.1109\/CEC.2009.4983045"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Yang, X.S. (2010). Firefly algorithm, stochastic test functions and design optimisation. arXiv.","DOI":"10.1504\/IJBIC.2010.032124"},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Yang, X.S. (2010). A new metaheuristic bat-inspired algorithm. Nature Inspired Cooperative Strategies for Optimization, Springer.","DOI":"10.1007\/978-3-642-12538-6_6"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/j.knosys.2015.07.006","article-title":"Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm","volume":"89","author":"Mirjalili","year":"2015","journal-title":"Knowl.-Based Syst."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.1007\/s00521-015-1920-1","article-title":"Dragonfly algorithm: A new meta-heuristic optimization technique for solving single-objective, discrete, and multi-objective problems","volume":"27","author":"Mirjalili","year":"2016","journal-title":"Neural Comput. Appl."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"2378","DOI":"10.1109\/TIP.2011.2109730","article-title":"FSIM: A feature similarity index for image quality assessment","volume":"20","author":"Zhang","year":"2011","journal-title":"IEEE Trans. Image Process."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Csurka, G., Larlus, D., Perronnin, F., and Meylan, F. (2013, January 9\u201313). What is a good evaluation measure for semantic segmentation?. Proceedings of the British Machine Vision Conference, Bristol, UK.","DOI":"10.5244\/C.27.32"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/j.swevo.2011.02.002","article-title":"A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms","volume":"1","author":"Derrac","year":"2011","journal-title":"Swarm Evol. Comput."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1166\/jmihi.2014.1217","article-title":"Interactive image segmentation based on graph cuts and automatic multilevel thresholding for brain images","volume":"4","author":"Touria","year":"2014","journal-title":"J. Med. Imaging Health Inform."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/1\/8\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:50:35Z","timestamp":1760169035000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/1\/8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12,21]]},"references-count":56,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1]]}},"alternative-id":["e24010008"],"URL":"https:\/\/doi.org\/10.3390\/e24010008","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,12,21]]}}}