一种新的图像分割算法
|
摘要
图像工程是近几年发展起来的一门学科,它的研究内容非常丰富,根据抽象程度和研究方法的不同可分为三个层次:图像处理、图像分析和图像理解。图像处理的目的之一是图像识别,而图像分割与测量是图像识别工作的基础。图像分割是将图像分成一些有意义的区域,然后对这些区域进行描述,相当于提取出某些目标区域图像的特征,判断图像中是否有感兴趣的目标。
有学者提出了一种免疫网络算法在数据分析中的新设计思想,近来又有学者结合粗糙集提出了一种新的分割算法,该算法在人脑MRI图像分割实验中得到了很好的分割效果,但由于FCM算法本身的缺陷(对区域间的连通性不确定),在具体应用中总会出现一些问题。
论文基于免疫遗传分类算法提出了一种通过无教师学习方法确定聚类数和聚类中心的免疫遗传聚类算法;接着,在一些学者研究工作的基础上,针对粗糙集中属性约简算法复杂度高的问题,提出了一种改进算法,降低了属性约简算法的复杂度;最后把无教师学习的免疫遗传算法和粗糙集结合得到了一种改进的分割算法,从而可以很好的避免前述算法需要事先设置聚类数和聚类中心、对区域的连通性不确定的缺陷,提高分割准确度,而且免疫遗传算法本身具有聚类速度快,精确度高的优点。经实验验证,提高了聚类的速度,对图像的分割更细致、更准确,提高了分割算法的效率。
Image engineering is a subject that has been developed in recent years, and it has many contents. According to the degree of abstract and the investigate methods, the research on it can be divided into three levels: image processing, image analysis and image comprehension. One of image processing goals is the pattern recognition, but the image segmentation and the survey are the pattern recognition work foundations. The image segmentation is that the image is divided into to the significance some regions, then carried on the description to these regions, equal in withdrawing the characteristic of certain target sector images, at last judged image whether it has the interested goal.
Some scholars proposed one new kind design concept of immunity network algorithm in the data analysis, recently also other scholars had gather the rough collection to propose one new kind of segmentation algorithm, this algorithm obtaining the very good segmentation effect in the human brain MRI image segmentation experiment, but because of FCM algorithm itself flaw (to connectivity between region indefinite), the algorithm has some problems in the general application.
The paper based on the immunity heredity classification algorithm proposed one kind studies the method determination cluster number and the cluster center immunity heredity cluster algorithm through the non-teacher; Then, at in some scholar research work foundation, in view of the rough centralism attribute reduction algorithm order of complexity high question, proposed one kind of improvement algorithm, reducing the attribute reduction algorithm order of complexity; Finally the immunity genetic algorithm which the non-teacher studies builds up roughly gathers obtained one kind of improvement segmentation algorithm, avoids the forecited algorithm needing to establish the cluster number and the cluster center, to the region connective indefinite flaw beforehand, and enhances the segmentation accuracy.Moreover immunity genetic algorithm itself has the cluster speed to be quick and high precision merit.Confirmed after the experiment, the new algorithm enhances the cluster speed, is more careful to the image segmentation, accurately, and increases the segmentation algorithm efficiency.
有学者提出了一种免疫网络算法在数据分析中的新设计思想,近来又有学者结合粗糙集提出了一种新的分割算法,该算法在人脑MRI图像分割实验中得到了很好的分割效果,但由于FCM算法本身的缺陷(对区域间的连通性不确定),在具体应用中总会出现一些问题。
论文基于免疫遗传分类算法提出了一种通过无教师学习方法确定聚类数和聚类中心的免疫遗传聚类算法;接着,在一些学者研究工作的基础上,针对粗糙集中属性约简算法复杂度高的问题,提出了一种改进算法,降低了属性约简算法的复杂度;最后把无教师学习的免疫遗传算法和粗糙集结合得到了一种改进的分割算法,从而可以很好的避免前述算法需要事先设置聚类数和聚类中心、对区域的连通性不确定的缺陷,提高分割准确度,而且免疫遗传算法本身具有聚类速度快,精确度高的优点。经实验验证,提高了聚类的速度,对图像的分割更细致、更准确,提高了分割算法的效率。
Image engineering is a subject that has been developed in recent years, and it has many contents. According to the degree of abstract and the investigate methods, the research on it can be divided into three levels: image processing, image analysis and image comprehension. One of image processing goals is the pattern recognition, but the image segmentation and the survey are the pattern recognition work foundations. The image segmentation is that the image is divided into to the significance some regions, then carried on the description to these regions, equal in withdrawing the characteristic of certain target sector images, at last judged image whether it has the interested goal.
Some scholars proposed one new kind design concept of immunity network algorithm in the data analysis, recently also other scholars had gather the rough collection to propose one new kind of segmentation algorithm, this algorithm obtaining the very good segmentation effect in the human brain MRI image segmentation experiment, but because of FCM algorithm itself flaw (to connectivity between region indefinite), the algorithm has some problems in the general application.
The paper based on the immunity heredity classification algorithm proposed one kind studies the method determination cluster number and the cluster center immunity heredity cluster algorithm through the non-teacher; Then, at in some scholar research work foundation, in view of the rough centralism attribute reduction algorithm order of complexity high question, proposed one kind of improvement algorithm, reducing the attribute reduction algorithm order of complexity; Finally the immunity genetic algorithm which the non-teacher studies builds up roughly gathers obtained one kind of improvement segmentation algorithm, avoids the forecited algorithm needing to establish the cluster number and the cluster center, to the region connective indefinite flaw beforehand, and enhances the segmentation accuracy.Moreover immunity genetic algorithm itself has the cluster speed to be quick and high precision merit.Confirmed after the experiment, the new algorithm enhances the cluster speed, is more careful to the image segmentation, accurately, and increases the segmentation algorithm efficiency.
引文
[1].de Castro L N, Von Zuben F J.aiNet: Artificial Immune Network for Data Analysis. In H.A. Abbass H A, Sarker R A, and Newton C S(eds). Data Minning: A Heuristic Approach. USA: Idea Group Publishing. , 2001.231-259.
[2].张东波,王耀南.FCM聚类算法和粗糙集在医疗图像分割中的应用[J].仪器仪表学报,2006,27(12):1683-1687.
[3].Gonzalez R C,Woods R E,Digital Image Processing,3rd. ed.,Addisonwesley,1992
[4].章毓晋,图像分割[M],北京:科学出版社,2001
[5].Smith S M,Brady J M . SUSAN-a new approach to low level image processing[J]. International Journal of Computer Vision. 1997,23(1):48-78
[6].Canny J F. A computational approach to edge detection[J]. IEEE Trans. on Pattern Analysis Machine Intelligence. 1986,8(6): 679-698.
[7].L.Feng,C.Y.Suen,Y.Y.Tang,L.H.Yang. Edge extraction of images by reconstruction using wavelet decomposition details at different resolution levels[J].Pattern Recognition and Artifical Intelligence. 2000,14(6)779-793.
[8].P.Moon, G.De Jager. An heuristic graph searching algorithm to find the boundary of apple images[C].IEEE Proceedings of Communications and Signal Processing, University of Cape Town,South African,September 11,1992:233-238
[9].Zhang Q,et al.Wavelet networks[J]. IEEE Trans.on Neural Networks,1992,3(6):889-898
[10].H.Choi,R.Baraniuk. Multiscale image segmentation using Wavelet-Domain Hidden Markov Models[J]. IEEE Trans. Image Process. 2001,10(9):1309-1321.
[11].A.X.Falcao, J.K.Udupa, F.K.Miyazawa. An ultra-fast user-steered image segmentation paradigm:live wire on the fly[J]. IEEE Trans. Med. Imaging , 2000,19(1),55-62
[12].Goshtasby A. Design and Recovery of 2-D and 3-D Shapes Using Rational Gaussian Curves and Surfaces [J]. International Journal of Computer Vision, 1993, 10(3): 233-256.
[13].Wu M F, Sheu H T. Representation of 3D Surfaces by Two Variable Fourier Descriptors. [J] IEEE Trans on Pattern Analysis and Machine Intelligence, 1998, 20(8): 858-863.
[14].Nalwa V S, Binford T O. On Detecting Edges [J]. IEEE Trans on Pattern Analysis and Machine lntelligence,1986, 8(6): 699-711.
[15].Cohen L D. On Active Contours and Balloons [J]. Computer Vision, Graphics and Image Processing: Image Understanding, 1991, 53(2): 211-218.
[16].Xu C Y, Prince J L. Snakes, Shapes, and Gradient Vector Flow [J]. IEEE Trans on Image Processing, 1998, 7(3): 359-369.
[17].Pavlidis T, Liow Y. Integrating Region Growing and Edge Detection [J]. IEEE Trans on Pattern Analysis and Machine Intelligence, 1990, 12(3): 225-233.
[18].Matalas L, Benjamin R, Kitney R. An Edge Detection Technique Using the Facet Model and Parameterized Relaxation Labeling [J]. IEEE Trans on Pattern Analysis and Machine Intelligence, 1997, 19(4): 328-341.
[19].Doyle W. Operation Useful for Similarity-Invariant Pattern Recognition [J]. Assoc. Comput. Mach., 1962, 9: 259-267.
[20].Lee S, Chung S. A Comparative Performance Study of Several Global Thresholding Techniques for Segmentation [J]. Computer Vision, Graphics, and Image Processing, 1990, 52: 171一190.
[21].Ostu W H. Threshold Selection Method from Gray Histogram [J], IEEE trans. On System, Man, and Cybernetics, 1979, 9: 62-66.
[22].Kittler J, Illingworth J. Minimum Error Thresholding [J]. Pattern Recognition, 1986, 19: 41-47.
[23].Kapur J N, Sahoo P K, Wong A K C. A New Method for Gray Level Picture Thresholding Using the Entropy Histogram [C]. CVGIP, 1985, 29: 273-285.
[24] Pal N R, Pal S K. Entropy Thresholding [J]. Signal Processing,1989,16(2): 97-108.
[25].Pal N R, Pal S K. Entropy: A New Definition and Its Applications [J]. IEEE Trans, Systems, Man and Cybernetics, 1991, 21: 1260-1270.
[26].Pal N R, Pal S K. A Review on Image Segmentation Technique [J]. Pattern Recognition, 1994, 26(9): 1277-1294.
[27].LIANG-KAI HUANG, MAO-JIUN J,WANG, Image Thresholding by Minimizing the Measures of Fuzziness [J]. Pattern Recognition, 1995, 28(1): 41-51.
[28].Xueqin Li, Zhiwei Zhao, Cheng H D. Fuzzy Entropy Threshold Approach to Breast Cancer Detection [J]. Information Sciences, 1995, 4(1): 49-56.
[29].Bhandari D, Pal N R, Majunder D D. Fuzzy Divergence, Probability Measure of Fuzzy Events and Image Thresholding [J]. Pattern Recognition Letters, 1992, 13: 857-867.
[30].RajaPakse J, Giedd J, Rapoporrt J. Statistical to Segmentation of Single-Channel Cerebral MR Images [J]. IEEE trans on Medical Imaging, 1997, 16(2): 176-186.
[31].Wells W, Crimson W, ICikin S, Jolesz F, Adaptive Segmentation of MRI Data[J]. IEEE Trans on Medical Imaging, 1996, 15(4): 429-442.
[32].Blanz W E, Gish S L. A Connectionist Classifier Architecture Applied to Image Segmentation [C]. Proc of the 10th International conference on Pattern Recognition, New Jersey, USA, 1990, 272-277.
[33].Babaguchi N, Yarmada K, Kisc K, Tezuka T. Connectionist Model Binarization [C]. Proc of the 10th International conference on Pattern Recognition, New Jersey, USA, 1990, 51-56.
[34].Ghosh A, Pal N R, Pal S K. Image Segmentation Using A Neural Network [J]. Biological Cybernetics. 1991, 66(2): 151-158.
[35].Shah J. Parameter Estimation, Multi-Scale Representation and Algorithms for Energy-Minimizing Segmentation [C]. Proc of the 10th International conference on Pattern Recognition, New Jersey, USA, 1990, 815-819.
[36].边肇祺,张学工.模式识别.第二版.北京,清华大学出版社,2000,338.
[37].Hunt J E and Cooke D E.An adaptive,distributed learning systems based on the immune system.In Proc. of the IEEE International Conference on Systems,Man and Cybernetics,1995,2494-2499.
[38].Nassroui O,Gonzalez F,and Dasgupta D.The Fuzzy Artificial Immune system:Motivations,Basic Concepts,and Application to Clustering and Web Profiling.Proc. of the 2002 IEEE International Conference on Computational Intelligence,12-17 May 2002,1:711-716.
[39].谭志扬.人工免疫算法在Flow-shop问题中的应用[J],计算机工程与应用,2002.14:98—99
[40].Pawlak Z.Rough sets,decision algorithms and Bayes’theorem.European Journal of Operational Research, 2002,136:181—189.
[41].Pawlak Z.Rough sets,International Journal of Computer and Information Sciences, 1982,11:341—356.
[42].Skowron A.The relationship between the rough set theory and evidence theory. Bulletin of Polish Academy of Science:Mathematics,1989,37:87—90.
[43].张文修,梁怡.不确定性推理原理[M].西安:西安交通大学出版社,1996.
[44].胡新生.群体遗传结构的理解[J].林业科学,2002,38(2):119-128.
[45].Andrew J Worth. The internet brain segmentation repository (IBSR)[EB/OL]. http://www.cma.mgh.Harvard.edu/ibsr.
[46].Pawlak Z.Rough sets:Theoretical aspects of reasoning about data.Boston,Kluwer Academic Publishers,1991.
[47].Skowron A,Rauszer C.The discernibility matrices and functions in information systems.In: Slowinski R.(ed) Intelligent Decision Support: Handbook of Applications and Advance of the Rough Sets Theory, Kluwer AcademyPublishers,1992,331-362.
[48].SEI IM S Z,ISMAIL M K-means type algorithm[J].IEEE Transactions on Pattern Analysis and Ma—chine Intelligence,1994,6(1):81—87.
[49].BEZDEK J C . Pattern recognition with fuzzy objective function algorithms[M].New York:Plenum Press,1981.
[50].李广强,赵洪伦,靳慧.并行混合免疫遗传算法及其应用[J] .计算机工程与应用,2005.(3):31—33.
[51].Rafael C.Gonzalez,Richard E.Woods,Steven L.Eddins.数字图像处理[M].北京,电子工业出版社,2005,285-320.
[52].William K.Pratt.数字图像处理[M].北京,机械工业出版社,2005,374-401.
[53].贾永红.计算机图像处理与分析[M].武汉,武汉大学出版社,2001,117-139.
[54].洪俊,杨淑莹,任翠池.基于图像分割的伪并行免疫遗传算法聚类设计[J].天津理工大学学报,2006,22(5):83-85.
[55].Rafael C.Gonzalez,Richard E.Woods,Steven L.Eddins.数字图像处理[M].北京,电子工业出版社,2005,285-320.
[56].李广强,赵洪伦,靳慧.并行混合免疫遗传算法及其应用[J].计算机工程与应用.2005,(3):31—33.
[57].张文修,仇国芳.基于粗糙集的不确定决策[M].北京,清华大学出版社,2005,12-55.
[58].李德毅.不确定性人工智能[M].北京,国防工业出版社,2005,108-117.
[59].刘少辉.Rough集高效算法的研究[J].计算机学报,2003,26(5)524-529.
[60].苗夺谦,胡桂荣.知识约简的一种启发算法[J].计算机研究与发展,1999,36(6)681-684.
[61].常梨云.一种基于Rough Set理论的属性约简及规则提取方法[J].软件学报,1999,10(11)1207-1211.
[62].唐彬.启发式属性约简算法完备性和规则发现算法的研究[J].计算机工程与应用,2003,30:191-194.
[63].唐建国.粗糙集理论中的求核与约简[J].控制与决策,2003,18(4):449-452.
[2].张东波,王耀南.FCM聚类算法和粗糙集在医疗图像分割中的应用[J].仪器仪表学报,2006,27(12):1683-1687.
[3].Gonzalez R C,Woods R E,Digital Image Processing,3rd. ed.,Addisonwesley,1992
[4].章毓晋,图像分割[M],北京:科学出版社,2001
[5].Smith S M,Brady J M . SUSAN-a new approach to low level image processing[J]. International Journal of Computer Vision. 1997,23(1):48-78
[6].Canny J F. A computational approach to edge detection[J]. IEEE Trans. on Pattern Analysis Machine Intelligence. 1986,8(6): 679-698.
[7].L.Feng,C.Y.Suen,Y.Y.Tang,L.H.Yang. Edge extraction of images by reconstruction using wavelet decomposition details at different resolution levels[J].Pattern Recognition and Artifical Intelligence. 2000,14(6)779-793.
[8].P.Moon, G.De Jager. An heuristic graph searching algorithm to find the boundary of apple images[C].IEEE Proceedings of Communications and Signal Processing, University of Cape Town,South African,September 11,1992:233-238
[9].Zhang Q,et al.Wavelet networks[J]. IEEE Trans.on Neural Networks,1992,3(6):889-898
[10].H.Choi,R.Baraniuk. Multiscale image segmentation using Wavelet-Domain Hidden Markov Models[J]. IEEE Trans. Image Process. 2001,10(9):1309-1321.
[11].A.X.Falcao, J.K.Udupa, F.K.Miyazawa. An ultra-fast user-steered image segmentation paradigm:live wire on the fly[J]. IEEE Trans. Med. Imaging , 2000,19(1),55-62
[12].Goshtasby A. Design and Recovery of 2-D and 3-D Shapes Using Rational Gaussian Curves and Surfaces [J]. International Journal of Computer Vision, 1993, 10(3): 233-256.
[13].Wu M F, Sheu H T. Representation of 3D Surfaces by Two Variable Fourier Descriptors. [J] IEEE Trans on Pattern Analysis and Machine Intelligence, 1998, 20(8): 858-863.
[14].Nalwa V S, Binford T O. On Detecting Edges [J]. IEEE Trans on Pattern Analysis and Machine lntelligence,1986, 8(6): 699-711.
[15].Cohen L D. On Active Contours and Balloons [J]. Computer Vision, Graphics and Image Processing: Image Understanding, 1991, 53(2): 211-218.
[16].Xu C Y, Prince J L. Snakes, Shapes, and Gradient Vector Flow [J]. IEEE Trans on Image Processing, 1998, 7(3): 359-369.
[17].Pavlidis T, Liow Y. Integrating Region Growing and Edge Detection [J]. IEEE Trans on Pattern Analysis and Machine Intelligence, 1990, 12(3): 225-233.
[18].Matalas L, Benjamin R, Kitney R. An Edge Detection Technique Using the Facet Model and Parameterized Relaxation Labeling [J]. IEEE Trans on Pattern Analysis and Machine Intelligence, 1997, 19(4): 328-341.
[19].Doyle W. Operation Useful for Similarity-Invariant Pattern Recognition [J]. Assoc. Comput. Mach., 1962, 9: 259-267.
[20].Lee S, Chung S. A Comparative Performance Study of Several Global Thresholding Techniques for Segmentation [J]. Computer Vision, Graphics, and Image Processing, 1990, 52: 171一190.
[21].Ostu W H. Threshold Selection Method from Gray Histogram [J], IEEE trans. On System, Man, and Cybernetics, 1979, 9: 62-66.
[22].Kittler J, Illingworth J. Minimum Error Thresholding [J]. Pattern Recognition, 1986, 19: 41-47.
[23].Kapur J N, Sahoo P K, Wong A K C. A New Method for Gray Level Picture Thresholding Using the Entropy Histogram [C]. CVGIP, 1985, 29: 273-285.
[24] Pal N R, Pal S K. Entropy Thresholding [J]. Signal Processing,1989,16(2): 97-108.
[25].Pal N R, Pal S K. Entropy: A New Definition and Its Applications [J]. IEEE Trans, Systems, Man and Cybernetics, 1991, 21: 1260-1270.
[26].Pal N R, Pal S K. A Review on Image Segmentation Technique [J]. Pattern Recognition, 1994, 26(9): 1277-1294.
[27].LIANG-KAI HUANG, MAO-JIUN J,WANG, Image Thresholding by Minimizing the Measures of Fuzziness [J]. Pattern Recognition, 1995, 28(1): 41-51.
[28].Xueqin Li, Zhiwei Zhao, Cheng H D. Fuzzy Entropy Threshold Approach to Breast Cancer Detection [J]. Information Sciences, 1995, 4(1): 49-56.
[29].Bhandari D, Pal N R, Majunder D D. Fuzzy Divergence, Probability Measure of Fuzzy Events and Image Thresholding [J]. Pattern Recognition Letters, 1992, 13: 857-867.
[30].RajaPakse J, Giedd J, Rapoporrt J. Statistical to Segmentation of Single-Channel Cerebral MR Images [J]. IEEE trans on Medical Imaging, 1997, 16(2): 176-186.
[31].Wells W, Crimson W, ICikin S, Jolesz F, Adaptive Segmentation of MRI Data[J]. IEEE Trans on Medical Imaging, 1996, 15(4): 429-442.
[32].Blanz W E, Gish S L. A Connectionist Classifier Architecture Applied to Image Segmentation [C]. Proc of the 10th International conference on Pattern Recognition, New Jersey, USA, 1990, 272-277.
[33].Babaguchi N, Yarmada K, Kisc K, Tezuka T. Connectionist Model Binarization [C]. Proc of the 10th International conference on Pattern Recognition, New Jersey, USA, 1990, 51-56.
[34].Ghosh A, Pal N R, Pal S K. Image Segmentation Using A Neural Network [J]. Biological Cybernetics. 1991, 66(2): 151-158.
[35].Shah J. Parameter Estimation, Multi-Scale Representation and Algorithms for Energy-Minimizing Segmentation [C]. Proc of the 10th International conference on Pattern Recognition, New Jersey, USA, 1990, 815-819.
[36].边肇祺,张学工.模式识别.第二版.北京,清华大学出版社,2000,338.
[37].Hunt J E and Cooke D E.An adaptive,distributed learning systems based on the immune system.In Proc. of the IEEE International Conference on Systems,Man and Cybernetics,1995,2494-2499.
[38].Nassroui O,Gonzalez F,and Dasgupta D.The Fuzzy Artificial Immune system:Motivations,Basic Concepts,and Application to Clustering and Web Profiling.Proc. of the 2002 IEEE International Conference on Computational Intelligence,12-17 May 2002,1:711-716.
[39].谭志扬.人工免疫算法在Flow-shop问题中的应用[J],计算机工程与应用,2002.14:98—99
[40].Pawlak Z.Rough sets,decision algorithms and Bayes’theorem.European Journal of Operational Research, 2002,136:181—189.
[41].Pawlak Z.Rough sets,International Journal of Computer and Information Sciences, 1982,11:341—356.
[42].Skowron A.The relationship between the rough set theory and evidence theory. Bulletin of Polish Academy of Science:Mathematics,1989,37:87—90.
[43].张文修,梁怡.不确定性推理原理[M].西安:西安交通大学出版社,1996.
[44].胡新生.群体遗传结构的理解[J].林业科学,2002,38(2):119-128.
[45].Andrew J Worth. The internet brain segmentation repository (IBSR)[EB/OL]. http://www.cma.mgh.Harvard.edu/ibsr.
[46].Pawlak Z.Rough sets:Theoretical aspects of reasoning about data.Boston,Kluwer Academic Publishers,1991.
[47].Skowron A,Rauszer C.The discernibility matrices and functions in information systems.In: Slowinski R.(ed) Intelligent Decision Support: Handbook of Applications and Advance of the Rough Sets Theory, Kluwer AcademyPublishers,1992,331-362.
[48].SEI IM S Z,ISMAIL M K-means type algorithm[J].IEEE Transactions on Pattern Analysis and Ma—chine Intelligence,1994,6(1):81—87.
[49].BEZDEK J C . Pattern recognition with fuzzy objective function algorithms[M].New York:Plenum Press,1981.
[50].李广强,赵洪伦,靳慧.并行混合免疫遗传算法及其应用[J] .计算机工程与应用,2005.(3):31—33.
[51].Rafael C.Gonzalez,Richard E.Woods,Steven L.Eddins.数字图像处理[M].北京,电子工业出版社,2005,285-320.
[52].William K.Pratt.数字图像处理[M].北京,机械工业出版社,2005,374-401.
[53].贾永红.计算机图像处理与分析[M].武汉,武汉大学出版社,2001,117-139.
[54].洪俊,杨淑莹,任翠池.基于图像分割的伪并行免疫遗传算法聚类设计[J].天津理工大学学报,2006,22(5):83-85.
[55].Rafael C.Gonzalez,Richard E.Woods,Steven L.Eddins.数字图像处理[M].北京,电子工业出版社,2005,285-320.
[56].李广强,赵洪伦,靳慧.并行混合免疫遗传算法及其应用[J].计算机工程与应用.2005,(3):31—33.
[57].张文修,仇国芳.基于粗糙集的不确定决策[M].北京,清华大学出版社,2005,12-55.
[58].李德毅.不确定性人工智能[M].北京,国防工业出版社,2005,108-117.
[59].刘少辉.Rough集高效算法的研究[J].计算机学报,2003,26(5)524-529.
[60].苗夺谦,胡桂荣.知识约简的一种启发算法[J].计算机研究与发展,1999,36(6)681-684.
[61].常梨云.一种基于Rough Set理论的属性约简及规则提取方法[J].软件学报,1999,10(11)1207-1211.
[62].唐彬.启发式属性约简算法完备性和规则发现算法的研究[J].计算机工程与应用,2003,30:191-194.
[63].唐建国.粗糙集理论中的求核与约简[J].控制与决策,2003,18(4):449-452.