Abstract The image segmentation based on entropy is analyzed and reviewed including one-dimensional maximum entropy, minimum cross entropy, maximum cross entropy and so on.The relations of Shannon entropy, Tsallis entropy and Renyi entropy are analyzed and commented, and the performance of two dimensional (high dimension) entropy and spatial entropy is also appraised. In conclusion, it points out the future research direction, such as the computational efficiency of the high-dimensional entropy model and one-dimensional entropy and other theories integrated.
[1] Pun T.A New Method for Grey-Level Picture Thresholding Using the Entropy of the Histogram.Signal Processing,1980,2(3): 223-237 [2] Kapur J N,Shahoo P K,Wong A K C.A New Method for Gray-Level Picture Thresholding Using the Entropy of the Histogram.Computer Vision,Graphics and Image Processing,1985,29(3): 273-285 [3] Abutaleb A S.Automatic Thresholding of Gray-Level Pictures Using Two-Dimensional Entropy.Computer Vision,Graphics and Image Processing,1989,47(1): 22-32 [4] Pal N R,Pal S K.A Review on Image Segmentation Techniques.Pattern Recognition,1993,26(9): 1277-1294 [5] Sahoo P K,Soltani S,Wong A K C,et al.A Survey of Thresholding Techniques.Computer Vision,Graphics and Image Processing,1988,41(2): 233-260 [6] Pal N R,Pal S K.Entropic Thresholding.Signal Process,1989,16(2): 97-108 [7] Wong A K C,Sahoo P K.A Gray-Level Threshold Selection Method Based on Maximum Entropy Principle.IEEE Trans on Systems,Man and Cybernetics,1989,19(4): 422-429 [8] Ternigan M E,Dastous F.Entropy-Based Texture Analysis in the Spatial Frequency Domain.IEEE Trans on Patter Analysis and Machine Intelligence,1984,6(2): 237-243 [9] Brink A D.Thresholding of Digital Images Using of Two-Dimensional Entropies.Pattern Recognition,1992,25(8): 803-808 [10] Johannsen G,Bille J.A Threshold Selection Method Using Information Measures // Proc of the 6th International Conference on Pattern Recognition.Munich,Germany,1982: 140-143 [11] de Albuquerque M P,Esquef I A,Gesualdi Mello A R,et al.Image Thresholding Using Tsallis Entropy.Pattern Recognition Letters,2004,25(9): 1059-1065 [12] Wang Shitong,Chung F L.Note on the Equivalence Relationship between Renyi-Entropy Based and Tsallis-Entropy Based Image Thresholding.Pattern Recognition Letters,2005,26(14): 2309-2312 [13] Li C H,Lee C K.Segmentation of Die Patterns Using Minimum Cross Entropy // Proc of the International Conference on Industrial Electronics,Control,Instrumentation and Automation.San Diego,USA,1992,II: 721-724 [14] Li C H,Lee C K.Minimum Cross Entropy Thresholding.Pattern Recognition,1993,26(4): 617-625 [15] Xue Jinghao,Zhang Yujin,Lin Xinggang.Image Thresholding Based on Maximum Between-Class Posterior Cross Entropy.Journal of Image and Graphics,1994,4(2): 110-114 (in Chinese) (薛景浩,章毓晋,林行刚.基于最大类间后验交叉熵的阈值化分割算法.中国图像图形学报,1994,4(2): 110-114) [16] Cao Jiannong.Image Segmentation of Maximum Cross Entropy Based on Histogram Reconstruction.Journal of Computer Application,2011,31(12): 3373-3377 (in Chinese) (曹建农.基于直方图重构的极大交叉熵图像分割方法.计算机应用,2011,31(12): 3373-3377) [17] Al-Attas R.Multi-Level Minimum Cross Entropy Thresholding Using Gamma Distribution.Master Dissertation.Riyadh,Kingdom of Saudi Arabia: King Saud University,2007 [18] Li C,Tam P.An Iterative Algorithm Cross Entropy Thresholding.Pattern Recognition Letter,1998,19(8): 771-776 [19] Al-Attas R,El-Zaart A.Thresholding of Medical Images Using Minimum Cross Entropy // Proc of the Kuala Lumpur International Conference on Biomedical Engineering.Kuala Lumpur,Malaysia,2006: 312-315 [20] Al-Osaimi G,El-Zaart A.Minimum Cross Entropy Thresholding for SAR Images // Proc of the 3rd International Conference on Information and Communication Technologies: From Theory to Applications.Damascus,Syria,2008: 1-6 [21] Wu Jue,Chung A C S.Markov Random Field Energy Minimization via Iterated Cross Entropy with Partition Strategy // Proc of the IEEE International Conference on Acoustics,Speech and Signal Processing.Honolulu,USA,2007,I: 457-460 [22] Tupin F,Sigelle M,Maitre H.Definition of a Spatial Entropy and Its Use for Texture Discrimination // Proc of the International Conference on Image Processing.Vancouver,Canada,2000,1: 725-728 [23] Cao Jiannong.Image Segmentation Based on Boltzmann Entropy.Application Research of Computers,2011,28(10): 3920-3924 (in Chinese) (曹建农.基于玻耳兹曼熵分析的图像分割方法研究.计算机应用研究,2011,28(10): 3920-3924) [24] Sezgin M,Kocaeli G,Sankur B.Survey over Image Thresholding Techniques and Quantitative Performance Evaluation.Journal of Electronic Imaging,2004,13(1): 146-165 [25] Chang C,Du Y,Wang J,et al.Survey and Comparative Analysis of Entropy and Relative Entropy Thresholding Techniques.Vision,Image and Signal Processing,2006,153(6): 837-850 [26] Shannon C E.A Mathematical Theory of Communication.Bell System Technical Journal,1948,27(3): 379-423 [27] Shannon C E,Weaver W.The Mathematical Theory of Communication.Urbana,USA: University of Illinois Press,1949 [28] Pal N R,Sankar K.Pal Entropy: A New Definition and Its Applications.IEEE Trans on Systems,Man and Cybernetics,1991,21(5): 1260-1270 [29] Gonzalez R C,Woods R E.Digital Image Processing.3rd Edition.New York,USA: Addison-Wesley,1992 [30] Pun T.Entropic Thresholding: A New Approach.Signal Processing,1981,16(3): 210-239 [31] Dainty J C,Shaw R.Image Sciences.New York,USA: Academic,1974 [32] Kullback S.Information Theory and Statistics.New York,USA: Wiley,1959 [33] Renyi A.A Diary on Information Theory.Budapest,Hungary: Akadememiai Kiado,1984 [34] Sahoo P,Wilkins C,Yeager J.Threshold Selection Using Renyi’s Entropy.Pattern Recognition,1997,30(1): 71-84 [35] Kittler J,Illingworth J.Minimum Error Thresholding.Pattern Recognition,1986,19(1): 41-47 [36] Cho S,Haralick R,Yi S.Improvement of Kittler and Illingworths Minimum Error Thresholding.Pattern Recognition,1989,22(5): 609-617 [37] Otsu N.A Threshold Selection Method from Gray-Level Histogram.IEEE Trans on Systems,Man and Cybernetics,1979,9(1): 62-66 [38] Pal N R.On Minimum Cross-Entropy Thresholding.Pattern Recognition,1996,29(4): 575-580 [39] Lu Jun,Wang Runsheng.An Improved Approach to Minimum Cross-Entropy Thresholding.Computer Engineering and Science,1999,21(4): 69-73 (in Chinese) (陆 军,王润生 一种改进的最小互熵门限法.计算机工程与科学,1999,21(4): 69-73) [40] Jaynes E T.Information Theory and Statistical Mechanics.Physical Reviews,1957,106(4): 620-630 [41] Jaynes E T.On the Rationale of Maximum-Entropy Methods.Proc of IEEE,1982,70(9): 939-952 [42] Shore J E,Johnson R W.Axiomatic Derivation of the Principle of Maximum Entropy and the Principle of Minimum Cross Entropy.IEEE Trans on Information Theory,1980,26(1): 26-37 [43] Kapur J N.Maximum Entropy Models in Science and Engineering.New Delhi,USA: Wiley Eastern,1989 [44] Wong A K C,Sahoo P K.A Gray-Level Threshold Selection Method Based on Maximum Entropy Principle.IEEE Trans on Systems,Man and Cybernetics,1989,19(4): 866-871 [45] Zhu Wei,Xu Yuru,Qin Zaibai.New Thresholding Method Using Two-Dimensional Tsallis Entropy.Computer Engineering and Application,2007,43(27): 54-58 (in Chinese) (朱 炜,徐玉如,秦再白.一种新的基于二维Tsallis熵的阈值方法.计算机工程与应用,2007,43(27): 54-58) [46] Feng Duan,Feng Shaotong.The World of Entropy.Beijing,China: Science Press,2005 (in Chinese) (冯 端,冯少彤.溯源探幽——熵的世界.北京:科学出版社,2005) [47] Tsallis C.Possible Generalization of Boltzman-Gibbs Statistics.Journal of Statistical Physics,1988,52(1/2): 479-487 [48] Skilling J.Classic Maximum Entropy // Skillings J,ed.Maximum Entropy and Bayesian Methods,Dordrecht,Netherland.Kluwer Academic,1988: 45-52 [49] Brink A D,Pendock N E.Minimum Cross-Entropy Threshold Selection.Pattern Recognition,1996,29(1): 179-188 [50] Cao Jiannong.Image Segmentation Based on Boltzmann Entropy.Postdoctoral Research Report.Xian,China: Xian Institute of Surveying and Mapping,2010 (in Chinese) (曹建农 基于玻耳兹曼熵分析的图像分割方法研究.博士后研究工作报告.西安:西安测绘研究所,2010) [51] Sahoo P K,Arora G.Image Thresholding Using Two-Dimensional Tsallis-Havrda-Charvat Entropy.Pattern Recognition Letters,2006,27(6): 520-528 [52] Tang Yinggan,Di Qiuyan,Guan Xinping.Method for Thresholding Image Segmentation Based on Minimum Tsallis-Cross Entropy.Chinese Journal of Scientific Instrument,2008,29(9): 1868-1872 (in Chinese) (唐英干,邸秋艳,关新平,等.基于最小Tsallis交叉熵的阈值图像分割方法.仪器仪表学报,2008,29(9): 1868-1872) [53] Xue Jinghao,Zhang Yujin,Lin Xinggang.Image Segmentation Algorithms Based on Cross Entropy and Fuzzy Divergence.Acta Electronica Sinica,1999,10(10): 131-134 (in Chinese) (薛景浩,章毓晋,林行刚.图像分割中的交叉熵和模糊散度算法.电子学报,1999,10(10): 131-134) [54] Malik N A,Lim J S.Properties of Two-Dimensional Maximum Entropy Power Spectral Estimation // Proc of the IEEE International Conference on Acoustics,Speech and Signal Processing.Cambridge,USA: Massachusetts Institute of Technology,1982,VII: 788-798 [55] Liu Jianzhang.A Fast Algorithms of Image Thresholding Based on Two-Dimensional Entropy.Pattern Recognition and Artificial Intelligence,1991,4(3): 46-53 (in Chinese) (刘健庄.基于二维熵的图像阈值选择快速算法.模式识别与人工智能,1991,4(3): 46-53) [56] Chen W T,Wen C H,Yang C W.A Fast Two-Dimensional Entropic Thresholding Algorithm.Pattern Recognition,1994,27(7): 885-893 [57] Gong Jian,Li Liyuan,Chen Weinan.A Fast Two Dimensional Entropic Thresholding Method.Journal of Southeast University,1996,26(4): 31-36 (in Chinese) (龚 坚,李立源,陈维南.二维熵阈值分割的快速算法.东南大学学报,1996,26(4): 31-36) [58] Zhang Yijun,Wu Xuejing,Xia Liangzheng.A Fast Recurring Algorithm for Two-Dimensional Entropic Thresholding for Image Segmentation.Pattern Recognition and Artificial Intelligence,1997,10(3): 259-264 (in Chinese) (张毅军,吴雪菁,夏良正.二维图像阈值分割的快速递推算法.模式识别与人工智能,1997,10(3): 259-264) [59] Liu Jingnan,Chen Congyan,Yu Lingling.A Fast Two-Dimensional Entropy Threshold Method.Application Research of Computers,2002,19(1): 67-68,70 (in Chinese) (刘京南,陈从颜,余玲玲,等.一种快速二维熵阈值分割算法.计算机应用研究,2002,19(1): 67-68,70) [60] Mei Rong,Jiang Changsheng,Chen Mou.2D Minimum Cross Entropy Based on Genetic Algorithm for Dynamic Image Segmentation.Electronics Optics & Control,2005,12(1): 30-34 (in Chinese) (梅 蓉,姜长生,陈 谋.基于遗传算法的二维最小交叉熵的动态图像分割.电光与控制,2005,12(1): 30-34) [61] Qiao Weiwei,Wu Chengmao.Two-Dimensional Thresholding Segmentation Method Based on Maximum Inter-Class Cross Entropy.Journal of Northwest University: Natural Science Edition,2008,38(3): 374-378 (in Chinese) (乔韡韡,吴成茂.二维最大类间交叉熵阈值分割法.西北大学学报:自然科学版,2008,38(3): 374-378) [62] Fu Xiaowei,Ding Mingyue,Zhou Chengping,et al.Multi-Threshold Image Segmentation with Improved Quantum-Inspired Genetic Algorithm.Proc of SPIE,2009,7495: 749518 [63] Imtiaz H,Fattah S A.A Face Recognition Scheme Based on Spectral Domain Feature Extraction // Proc of the 6th International Conference on Electrical & Computer Engineering.Wuhan,China,2010: 514-517 [64] Hao Y,Qiu-sheng W,Hai-wen Y.An Improved Image Segmentation Algorithm and Measurement Methods for Asphalt Mixtures // Proc of the 5th IEEE International Conference on Cybernetics and Intelligent Systems.Singapore,Singapore,2011: 36-41 [65] Hao Jiasheng,Shen Yi,Xu Hongbing,et al.A Region Entropy Based Objective Evaluation Method for Image Segmentation // Proc of the International Instrumentation and Measurement Technology Conference.Singapore,Singapore,2009: 363-367 [66] Shi Hang,Zou Yuexian,Wang Yiyan,et al.A Robust Traffic Parameter Extraction Method Using Texture and Entropy // Proc of the IEEE Intelligent Vehicles Symposium.Xi’an,China,2009: 237-241 [67] Quweider M K.An Expanded Histogram Approach for Multilevel Image Thresholding // Proc of the 20th International Conference on Electronics Communications and Computers.Puebla,Mexico,2010: 109-113 [68] Chang Faliang,Liu Jing,Qiao Yizheng.Self-Adaptive Threshold Segmentation for Color Image Using Two-Dimensional Entropy Method Based on Genetic Algorithm.Control and Decision,2005,20(6): 674-678 (in Chinese) (常发亮,刘 静,乔谊正.基于遗传算法的彩色图像二维熵多阈值自适应分割.控制与决策,2005,20(6): 674-678) [69] Xu Dong,Tang Xudong.Multi-Threshold Image Segmentation Based on Two-Dimensional Tsallis // Proc of the 3rd IEEE International Conference on Computer Science and Information Technology.Beijing,China,2010,VI: 1-5 [70] Sin C E,Leung C K.Image Segmentation by Changing Template Block by Block.Electrical and Electronic Technology,2001,1: 302-305 [71] Tian Xiaoguang,Hou Xiaorong.A Tsallis-Entropy Image Thresholding Method Based on Two-Dimensional Histogram Oblique Segmentation // Proc of the WASE International Conference on Information Engineering.Taiyuan,China,2009,I: 164-168 [72] Li Gang,Fan Xiaoping,Li Yan.The Thresholding Methods Based on Two-Dimensional Non-Extensive Entropy // Proc of the 1st International Congress on Image and Signal Processing.Sanya,China,2008,III: 729-733 [73] Zhang Yufei,Wu Yiquan.Wu Shihua,et al.Multilevel Thresholding Based on Exponential Cross Entropy and Niche Chaotic Particle Swarm Optimization.Lecture Notes in Electrical Engineering,2011,121: 617-624 [74] Amani A A ,Ali E Z .Image Segmentation Using Minimum Cross-Entropy Thresholding // Proc of the IEEE International Conference on Systems,Man and Cybernetics.San Antonio,USA,2009: 1776-1781 [75] Fu Xiaowei,Ding Mingyue.A Multi-Threshold Image Segmentation Method with Adaptive Fuzzy Entropy // Proc of the 2nd International Conference on Future Information Technology and Management Engineering.Sanya,China,2009: 171-174 [76] Tian Junwei,Huang Yongxuan,Yu Yalin.A Fast FCM Cluster Multi-Threshold Image Segmentation Algorithm Based on Entropy Constraint.Pattern Recognition and Artificial Intelligence,2008,21(2): 221-226 (in Chinese) (田军委,黄永宣,于亚琳.基于熵约束的快速FCM 聚类多阈值图像分割算法.模式识别与人工智能,2008,21(2): 221-226) [77] Jurio A,Pagola M,Bustince H.Ignorance-Based Fuzzy Clustering Algorithm.International Journal of Computational Intelligence and Applications,2010,9(3): 225-239 [78] Lin Aiying,Wu Lili,Zheng Baozhou,et al.The Combination of Local Fuzzy-Entropy-Based Transition Region Extraction with Otsu Thresholding Method for Image Segmentation // Proc of the 2nd International Congress on Image and Signal.Tianjing,China,2009: 1-4 [79] Gerbrands J J.Segmentation of Noise Images.Ph.D Dissertation.Delft,Netherlands: Delft University,1988 [80] Zhang Y J,Gerbrands J J.Transition Region Determination Based Thresholding.Pattern Recognition Letters,1991,12(1): 2935-2941
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