{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T10:58:13Z","timestamp":1770289093981,"version":"3.49.0"},"reference-count":59,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2018,3,30]],"date-time":"2018-03-30T00:00:00Z","timestamp":1522368000000},"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>Image segmentation is a significant step in image analysis and computer vision. Many entropy based approaches have been presented in this topic; among them, Tsallis entropy is one of the best performing methods. However, 1D Tsallis entropy does not consider make use of the spatial correlation information within the neighborhood results might be ruined by noise. Therefore, 2D Tsallis entropy is proposed to solve the problem, and results are compared with 1D Fisher, 1D maximum entropy, 1D cross entropy, 1D Tsallis entropy, fuzzy entropy, 2D Fisher, 2D maximum entropy and 2D cross entropy. On the other hand, due to the existence of huge computational costs, meta-heuristics algorithms like genetic algorithm (GA), particle swarm optimization (PSO), ant colony optimization algorithm (ACO) and differential evolution algorithm (DE) are used to accelerate the 2D Tsallis entropy thresholding method. In this paper, considering 2D Tsallis entropy as a constrained optimization problem, the optimal thresholds are acquired by maximizing the objective function using a modified chaotic Bat algorithm (MCBA). The proposed algorithm has been tested on some actual and infrared images. The results are compared with that of PSO, GA, ACO and DE and demonstrate that the proposed method outperforms other approaches involved in the paper, which is a feasible and effective option for image segmentation.<\/jats:p>","DOI":"10.3390\/e20040239","type":"journal-article","created":{"date-parts":[[2018,3,30]],"date-time":"2018-03-30T12:43:48Z","timestamp":1522413828000},"page":"239","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["2D Tsallis Entropy for Image Segmentation Based on Modified Chaotic Bat Algorithm"],"prefix":"10.3390","volume":"20","author":[{"given":"Zhiwei","family":"Ye","sequence":"first","affiliation":[{"name":"School of Computer Science, Hubei University of Technology, Wuhan 430068, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Juan","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Computer Science, Hubei University of Technology, Wuhan 430068, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mingwei","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Remote Sensing and Information Engineering, Wuhan University, Wuhan 430079, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinlu","family":"Zong","sequence":"additional","affiliation":[{"name":"School of Computer Science, Hubei University of Technology, Wuhan 430068, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lingyu","family":"Yan","sequence":"additional","affiliation":[{"name":"School of Computer Science, Hubei University of Technology, Wuhan 430068, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Computer Science, Hubei University of Technology, Wuhan 430068, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,3,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1016\/j.patrec.2017.05.011","article-title":"Deep learning with geodesic moments for 3D shape classification","volume":"105","author":"Luciano","year":"2017","journal-title":"Pattern Recognit. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"678","DOI":"10.1109\/23.289373","article-title":"Medical image segmentation by a constraint satisfaction neural network","volume":"38","author":"Chen","year":"2016","journal-title":"IEEE Trans. Nucl. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1016\/j.cageo.2016.12.001","article-title":"An interactive image segmentation method for lithological boundary detection: A rapid mapping tool for geologists","volume":"100","author":"Vasuki","year":"2017","journal-title":"Comput. Geosci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1155\/2017\/3295769","article-title":"Modified Discrete Grey Wolf Optimizer Algorithm for Multilevel Image Thresholding","volume":"2017","author":"Li","year":"2017","journal-title":"Comput. Intell. Neurosci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1016\/j.eswa.2017.06.021","article-title":"An optimal Color Image Multilevel Thresholding Technique Using Grey-Level Co-occurrence Matrix","volume":"87","author":"Pare","year":"2017","journal-title":"Expert Syst. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.rse.2016.12.011","article-title":"Region merging using local spectral angle thresholds: A more accurate method for hybrid segmentation of remote sensing images","volume":"190","author":"Yang","year":"2017","journal-title":"Remote Sens. Environ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1016\/j.ijleo.2018.02.062","article-title":"New Result on Maximum Entropy Threshold Image Segmentation Based on P System","volume":"163","author":"Wang","year":"2018","journal-title":"Optik"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2966","DOI":"10.1109\/TGRS.2009.2014364","article-title":"Unsupervised Synthetic Aperture Radar Image Segmentation Using Fisher Distributions","volume":"47","author":"Galland","year":"2009","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_9","first-page":"944","article-title":"An adaptive thresholding algorithm based on grayscale wave transformation for industrial inspection images","volume":"37","author":"Wei","year":"2011","journal-title":"Acta Autom. Sin."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1016\/S0167-8655(98)00057-9","article-title":"An iterative algorithm for minimum cross entropy thresholding","volume":"19","author":"Li","year":"1998","journal-title":"Pattern Recognit. Lett."},{"key":"ref_11","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_12","doi-asserted-by":"crossref","first-page":"8707","DOI":"10.1016\/j.eswa.2015.07.025","article-title":"Tsallis entropy based multilevel thresholding for colored satellite image segmentation using evolutionary algorithms","volume":"42","author":"Bhandari","year":"2015","journal-title":"Expert Syst. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1016\/j.knosys.2016.12.021","article-title":"A less-greedy two-term Tsallis Entropy Information Metric approach for decision tree classification","volume":"120","author":"Wang","year":"2017","journal-title":"Knowl. Based Syst."},{"key":"ref_14","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_15","doi-asserted-by":"crossref","unstructured":"Khader, M., Schiavi, E., and Hamza, A.B. (2017). A Multicomponent Approach to Nonrigid Registration of Diffusion Tensor Images, Kluwer Academic Publishers.","DOI":"10.1007\/s10489-016-0833-8"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1213","DOI":"10.1016\/j.optlaseng.2010.06.011","article-title":"Medical image registration using stochastic optimization","volume":"48","author":"Mohamed","year":"2010","journal-title":"Opt. Lasers Eng."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1016\/0031-3203(92)90034-G","article-title":"Thresholding of digital images using two-dimensional entropies","volume":"25","author":"Brink","year":"1992","journal-title":"Pattern Recognit."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1149","DOI":"10.1016\/j.patcog.2003.10.008","article-title":"A thresholding method based on two-dimensional Renyi\u2019s entropy","volume":"37","author":"Sahoo","year":"2004","journal-title":"Pattern Recognit."},{"key":"ref_19","first-page":"127","article-title":"Modification of two-dimensional entropy thresholding method and its fast iterative algorithm","volume":"23","author":"Wu","year":"2010","journal-title":"Pattern Recognit. Artif. Intell."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"5234","DOI":"10.1016\/j.ijleo.2014.05.003","article-title":"Two-dimensional Otsu\u2019s thresholding segmentation method based on grid box filter","volume":"125","author":"Guo","year":"2014","journal-title":"Optik Int. J. Light Electron Opt."},{"key":"ref_21","first-page":"1","article-title":"Automatic Change Detection Method of Multitemporal Remote Sensing Images Based on 2D-Otsu Algorithm Improved by Firefly Algorithm","volume":"2015","author":"Huang","year":"2015","journal-title":"J. Sens."},{"key":"ref_22","first-page":"414","article-title":"A two-dimensional image segmentation method based on genetic algorithm and entropy","volume":"131","author":"Ishak","year":"2016","journal-title":"Optik Int. J. Light Electron Opt."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Zhou, C., Tian, L., Zhao, H., and Zhao, K. (2015, January 8\u201312). A method of Two-Dimensional Otsu image threshold segmentation based on improved Firefly Algorithm. Proceedings of the IEEE International Conference on Cyber Technology in Automation, Control, and Intelligent Systems, Shenyang, China.","DOI":"10.1109\/CYBER.2015.7288151"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"4788","DOI":"10.1109\/TIP.2013.2277832","article-title":"Multilevel Image Thresholding Based on 2D Histogram and Maximum Tsallis Entropy\u2014A Differential Evolution Approach","volume":"22","author":"Sarkar","year":"2013","journal-title":"IEEE Trans. Image Process."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Jiang, S., Wang, D., and Liu, C. (2015, January 25\u201326). Two Dimension Threshold Image Segmentation Based on Improved Artificial Fish-Swarm Algorithm. Proceedings of the International Conference on Chemical, Material and Food Engineering (CMFE-2015), Kunming, China.","DOI":"10.2991\/cmfe-15.2015.156"},{"key":"ref_26","unstructured":"Yang, X.S. (2010). Nature-Inspired Metaheuristic Algorithms, Luniver Press."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.artmed.2017.10.002","article-title":"Chaotic genetic algorithm and Adaboost ensemble metamodeling approach for optimum resource planning in emergency departments","volume":"84","author":"Yousefi","year":"2017","journal-title":"Artif. Intell. Med."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Feng, Y., Yang, J., Wu, C., Lu, M., and Zhao, X.J. (2016). Solving 0\u20131 knapsack problems by chaotic monarch butterfly optimization algorithm with Gaussian mutation. Memet. Comput., 1\u201316.","DOI":"10.1007\/s12293-016-0211-4"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.knosys.2017.10.011","article-title":"Chaotic Dynamic Weight Particle Swarm Optimization for Numerical Function Optimization","volume":"139","author":"Chen","year":"2018","journal-title":"Knowl. Based Syst."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Sayed, G.I., Khoriba, G., and Haggag, M.H. (2018). A novel chaotic salp swarm algorithm for global optimization and feature selection. Appl. Intell., 1\u201320.","DOI":"10.1007\/s10489-018-1158-6"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"666","DOI":"10.1016\/j.energy.2015.12.096","article-title":"Economic dispatch using chaotic bat algorithm","volume":"96","author":"Adarsh","year":"2016","journal-title":"Energy"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"503","DOI":"10.1016\/j.asoc.2017.02.005","article-title":"Multilevel Thresholding based on Chaotic Darwinian Particle Swarm Optimization for Segmentation of Satellite Images","volume":"55","author":"Suresh","year":"2017","journal-title":"Appl. Soft Comput."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.neucom.2017.07.059","article-title":"A band selection method for airborne hyperspectral image based on chaotic binary coded gravitational search algorithm","volume":"273","author":"Wang","year":"2017","journal-title":"Neurocomputing"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Mlakar, U., Brest, J., and Fister, I. (2016, January 6\u20139). A study of chaotic maps in differential evolution applied to gray-level image thresholding. Proceedings of the IEEE Symposium Series on Computational Intelligence (SSCI), Athens, Greece.","DOI":"10.1109\/SSCI.2016.7850256"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"3349","DOI":"10.1007\/s00500-015-1726-1","article-title":"Chaotic cuckoo search","volume":"20","author":"Wang","year":"2016","journal-title":"Soft Comput."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2015\/813507","article-title":"Cuckoo Search Algorithm with Chaotic Maps","volume":"2015","author":"Wang","year":"2015","journal-title":"Math. Probl. Eng."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1016\/j.applthermaleng.2016.05.103","article-title":"A novel hybrid chaotic ant swarm algorithm for heat exchanger networks synthesis","volume":"104","author":"Zhang","year":"2016","journal-title":"Appl. Therm. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/j.apenergy.2017.05.029","article-title":"Parameter estimation of photovoltaic cells using an improved chaotic whale optimization algorithm","volume":"200","author":"Oliva","year":"2017","journal-title":"Appl. Energy"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/j.engappai.2016.01.034","article-title":"A new optimization algorithm based on chaotic maps and golden section search method","volume":"50","author":"Koupaei","year":"2016","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1080\/0305215X.2016.1191837","article-title":"Cost optimization of reinforced concrete cantilever retaining walls under seismic loading using a biogeography-based optimization algorithm with Levy flights","volume":"49","author":"Aydogdu","year":"2016","journal-title":"Eng. Optim."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/j.cpc.2017.05.009","article-title":"A particle swarm optimization algorithm with random learning mechanism and Levy flight for optimization of atomic clusters","volume":"219","author":"Yan","year":"2017","journal-title":"Comput. Phys. Commun."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"248","DOI":"10.1016\/j.asoc.2016.02.018","article-title":"An enhanced particle swarm optimization with levy flight for global optimization","volume":"43","author":"Jensi","year":"2016","journal-title":"Appl. Soft Comput."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.asoc.2017.06.044","article-title":"An efficient modified grey wolf optimizer with L\u00e9vy flight for optimization tasks","volume":"60","author":"Heidari","year":"2017","journal-title":"Appl. Soft Comput."},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Gray, R.M. (2011). Entropy and Information Theory, Springer.","DOI":"10.1007\/978-1-4419-7970-4"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. Stat. Phys."},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Ram\u00edrez-Reyes, A., Hern\u00e1ndez-Montoya, A., Herrera-Corral, G., and Dom\u00ednguez-Jim\u00e9nez, I. (2016). Determining the Entropic Index q of Tsallis Entropy in Images through Redundancy. Entropy, 18.","DOI":"10.3390\/e18080299"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"732","DOI":"10.1109\/83.841949","article-title":"Thresholding using two-dimensional histogram and fuzzy entropy principle","volume":"9","author":"Cheng","year":"1999","journal-title":"IEEE Trans. Image Process."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"841","DOI":"10.3390\/e13040841","article-title":"Optimal Multi-Level Thresholding Based on Maximum Tsallis Entropy via an Artificial Bee Colony Approach","volume":"13","author":"Zhang","year":"2011","journal-title":"Entropy"},{"key":"ref_49","unstructured":"Shih, C.J., Teng, T.L., and Chen, S.K. (2013, January 12\u201314). A Niche-Related Particle Swarm Meta-Heuristic Algorithm for Multimodal Optimization. Proceedings of the 2nd International Conference on Intelligent Technologies and Engineering Systems (ICITES2013), Kaohsiung, Taiwan."},{"key":"ref_50","doi-asserted-by":"crossref","unstructured":"Yang, X.S. (2010, January 12\u201314). A new metaheuristic bat-inspired algorithm. Proceedings of the Nature Inspired Cooperative Strategies for Optimization (NICSO), Canterbury, UK.","DOI":"10.1007\/978-3-642-12538-6_6"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Yang, X.S., and Deb, S. (2009). Cuckoo search via Levy flights. Proceedings of Nature & Biologically Inspired Computing, Coimbatore, India, 9\u201311 December 2009, IEEE.","DOI":"10.1109\/NABIC.2009.5393690"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1504\/IJBIC.2013.055093","article-title":"Bat algorithm: literature review and applications","volume":"5","author":"Yang","year":"2013","journal-title":"Int. J. Bio-Inspir. Comput."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"1229","DOI":"10.1016\/j.cnsns.2013.08.017","article-title":"Chaos optimization algorithms based on chaotic maps with different probability distribution and search speed for global optimization","volume":"19","author":"Yang","year":"2014","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Kovacic, I., and Brennan, M.J. (2011). The Duffing Equation: Nonlinear Oscillators and Their Behaviour, John Wiley & Sons.","DOI":"10.1002\/9780470977859"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"728","DOI":"10.1073\/pnas.1318616111","article-title":"Evidence of Levy walk foraging patterns in human hunter\u2013gatherers","volume":"111","author":"Raichlen","year":"2014","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"630","DOI":"10.1109\/TNET.2011.2120618","article-title":"On the Levy-walk nature of human mobility","volume":"19","author":"Rhee","year":"2011","journal-title":"IEEE\/ACM Trans. Netw."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"3069","DOI":"10.1016\/S0167-8655(03)00166-1","article-title":"Image segmentation by three-level thresholding based on maximum fuzzy entropy and genetic algorithm","volume":"24","author":"Tao","year":"2003","journal-title":"Pattern Recognit. Lett."},{"key":"ref_58","first-page":"720","article-title":"Piecewise Logistic Chaotic Map and It s Performance Analysis","volume":"37","author":"Fan","year":"2009","journal-title":"Acta Electron. Sin."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"382","DOI":"10.1016\/j.physa.2013.08.055","article-title":"The role of fluctuating modes of autocorrelation in crude oil prices","volume":"393","author":"An","year":"2014","journal-title":"Phys. A Stat. Mech. Its Appl."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/4\/239\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:59:06Z","timestamp":1760194746000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/4\/239"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,3,30]]},"references-count":59,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2018,4]]}},"alternative-id":["e20040239"],"URL":"https:\/\/doi.org\/10.3390\/e20040239","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,3,30]]}}}