基于互信息的医学图像配准方法研究
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摘要
现代医学影像学为人们提供了多种模态的医学图像(如CT,MR,PET等),不同模态的图像提供了人体的不同信息。通过医学图像融合技术可以把不同模态图像的有用信息融合在一起,形成一幅综合反映人体信息的图像,但在图像融合前必须首先实现图像的配准。医学图像配准技术起源于二十世纪九十年代,目前已成为医学图像处理领域热点之一。本文主要对基于互信息的医学图像配准方法进行研究,该方法具有自动化程度高、配准精度高、鲁棒性强等优点,是目前研究最多的配准方法之一。本文主要的工作如下：
(1)根据刚体变换前后像素点之间的距离和角度保持不变这一性质,对二维刚体变换公式和三维刚体变换公式进行了步骤分解,通过分步计算的方式来实现刚体变换,在不影响配准结果的情况下,能够有效地降低空间变换运算的次数,降低配准时间。
(2)在前人研究基础上对互信息局部极值产生原因进行了进一步的研究,发现局部极值的产生不仅是受到插值算法的影响,还与熵本身的性质有关,并结合PV插值法和熵的性质对互信息局部极值产生原因进行了详细的分析。
(3)介绍了加权熵的概念,给出了一种权值选取方案,并把该权值加权熵引用到图像配准方法中,提出了一种基于加权熵互信息医学图像配准方法,阐述了该方法的配准原理。
(4)在二维图像配准领域,从多个方面比较了归一化互信息、Tsallis熵互信息以及加权熵互信息的配准结果,实验结果表明在采用了一些降低运算时间的技术后,加权熵互信息能够较好地避免局部极值的影响,获得正确的配准结果。
(5)在三维图像配准领域,结合主轴法和互信息方法来实现图像的配准,不仅能够有效的避免局部极值的影响,而且能够大大降低配准时间。本文首先采用Canny算子提取图像的边缘,然后使用数学形态法对提取后的边缘进行连接并获得图像的轮廓,再利用主轴法来计算轮廓的主轴和质心,获得图像的粗配准结果,最后将粗配准结果作为互信息配准方法中优化算法的初始搜索点进一步获得精确的配准结果。
Modern medical imaging has provided many medical images with different modality (such as CT, MR, PET, and so on.), which provide different information on the human body. Through the image fusion technique, we can obtain one synthetical image which combines the useful information of different medical images together to reflect the information of human body comprehensively. However, image registration must be done before image fusion. The techniques of medical image registration originated in the ninety's of twentieth century, has become one of the most important research fields of medical image processing. In this thesis, medical image registration by mutual information is investigated, and it has been accepted as one of the most accurate, highly-automated, robust, popular registration methods. The main work and contributions in the thesis are as follows:
(1) Because the distance and angle between pixels do not change in rigid transformation, we decompose the two-dimensional and three-dimensional rigid transformation formula, calculate the rigid body transformation step by step. The improved formula can effectively reduce the time-consuming of transformation and not affect the registration results.
(2) We continue to study the reasons of local extremums of mutual information based on previous studies and find that the local extremums are not only affected by the interpolation algorithm, but also affected by the nature of entropy itself. Combining the PV interpolation with entropy nature, we carried out a detailed analysis about the reasons of local extremums.
(3) The concept of weighted entropy is introduced and one kind weight selection is given. This kind of weighted entropy is refered to image registration and medical image registration by mutual information based on weighted entropy being put forward. Also we describe the registration principle of this method.
(4) In the field of two-dimensional image registration, we compare registration results of normalized mutual information, Tsallis entropy mutual information and entropy-weighted mutual information in many options. Experimental results show that entropy-weighted mutual information can be better to avoid the impact of local extremums and obtain the correct registration results when using some techniques to reduce the computing time.
(5) In the field of three-dimensional image registration, we combined the principal axis method with the mutual information method to achieve three-dimensional image registration, It is not only effectively avoid the local extremums, but also greatly reduce the registration time. First of all, we get the edge of images by Canny edge detect operator, then use mathematical morphology method to connect the edge and pick-up the figures of brain image. Secondly we use the principal axis method to compute the principal axis and the center of mass of the figures, and obtain coarse registration results. Finally, we make use of this coarse registration results as the initial search points of the optimization of the image registration by mutual information method and obtain accurate registration results.
(1)根据刚体变换前后像素点之间的距离和角度保持不变这一性质,对二维刚体变换公式和三维刚体变换公式进行了步骤分解,通过分步计算的方式来实现刚体变换,在不影响配准结果的情况下,能够有效地降低空间变换运算的次数,降低配准时间。
(2)在前人研究基础上对互信息局部极值产生原因进行了进一步的研究,发现局部极值的产生不仅是受到插值算法的影响,还与熵本身的性质有关,并结合PV插值法和熵的性质对互信息局部极值产生原因进行了详细的分析。
(3)介绍了加权熵的概念,给出了一种权值选取方案,并把该权值加权熵引用到图像配准方法中,提出了一种基于加权熵互信息医学图像配准方法,阐述了该方法的配准原理。
(4)在二维图像配准领域,从多个方面比较了归一化互信息、Tsallis熵互信息以及加权熵互信息的配准结果,实验结果表明在采用了一些降低运算时间的技术后,加权熵互信息能够较好地避免局部极值的影响,获得正确的配准结果。
(5)在三维图像配准领域,结合主轴法和互信息方法来实现图像的配准,不仅能够有效的避免局部极值的影响,而且能够大大降低配准时间。本文首先采用Canny算子提取图像的边缘,然后使用数学形态法对提取后的边缘进行连接并获得图像的轮廓,再利用主轴法来计算轮廓的主轴和质心,获得图像的粗配准结果,最后将粗配准结果作为互信息配准方法中优化算法的初始搜索点进一步获得精确的配准结果。
Modern medical imaging has provided many medical images with different modality (such as CT, MR, PET, and so on.), which provide different information on the human body. Through the image fusion technique, we can obtain one synthetical image which combines the useful information of different medical images together to reflect the information of human body comprehensively. However, image registration must be done before image fusion. The techniques of medical image registration originated in the ninety's of twentieth century, has become one of the most important research fields of medical image processing. In this thesis, medical image registration by mutual information is investigated, and it has been accepted as one of the most accurate, highly-automated, robust, popular registration methods. The main work and contributions in the thesis are as follows:
(1) Because the distance and angle between pixels do not change in rigid transformation, we decompose the two-dimensional and three-dimensional rigid transformation formula, calculate the rigid body transformation step by step. The improved formula can effectively reduce the time-consuming of transformation and not affect the registration results.
(2) We continue to study the reasons of local extremums of mutual information based on previous studies and find that the local extremums are not only affected by the interpolation algorithm, but also affected by the nature of entropy itself. Combining the PV interpolation with entropy nature, we carried out a detailed analysis about the reasons of local extremums.
(3) The concept of weighted entropy is introduced and one kind weight selection is given. This kind of weighted entropy is refered to image registration and medical image registration by mutual information based on weighted entropy being put forward. Also we describe the registration principle of this method.
(4) In the field of two-dimensional image registration, we compare registration results of normalized mutual information, Tsallis entropy mutual information and entropy-weighted mutual information in many options. Experimental results show that entropy-weighted mutual information can be better to avoid the impact of local extremums and obtain the correct registration results when using some techniques to reduce the computing time.
(5) In the field of three-dimensional image registration, we combined the principal axis method with the mutual information method to achieve three-dimensional image registration, It is not only effectively avoid the local extremums, but also greatly reduce the registration time. First of all, we get the edge of images by Canny edge detect operator, then use mathematical morphology method to connect the edge and pick-up the figures of brain image. Secondly we use the principal axis method to compute the principal axis and the center of mass of the figures, and obtain coarse registration results. Finally, we make use of this coarse registration results as the initial search points of the optimization of the image registration by mutual information method and obtain accurate registration results.
引文
[1]http://baike.baidu.com/view/110986.htm?fr=ala0_1_1
[2]罗述谦.医学图像配准技术[J].国外医学生物医学工程分册,1999,22(1)：1～8
[3]罗述谦,李响.基于最大互信息的多模态医学图像配准[J].中国图象图形学报,2000,5(7)：551-558.
[4]田捷.医学影像处理与分析[M].电子工业出版社,2003,96-114
[5]J. B. Antoine Maintz, Max A. Viergever. A survey of medical image registration[J]. Medical Image Analysis,1998,2(1):1-36.
[6]A Collignon, F Maes, D Delaere. Automated multi-modality image registration based on information theory[C]. Information Processing in Medical Imaging,1995, 263-274.
[7]Paul Viola, William M.Wells Ⅲ. Alignment by Maximization of Mutual Information[C]. Proceedings of the 5th international conference on computer vision, Washington.D.C:IEEE Computer Society,1995,16-23
[8]王海南,郝重阳,雷方元等.非刚性医学图像配准研究综述[J].计算机工程与应用,2005,11：180-184.
[9]L.G. Brown, A survey of image registration techniques[J]. ACM Computing Surveys,1992,24:325-375.
[10]P.A. vanden Elsen, E.J.D. Pol, M.A. Viergever, Medical image matching:A review with classification[J]. IEEE Engineering in Medicine and Biology. 1993,12:26-39.
[11]D.J. Hawkes, Algorithms for radiological image registration and their clinical application[J]. Journal of Anatomy.1998,193:347-361.
[12]Jay West,J.Michael Fitzpatrick,Matthew Y.Wang,etal. Comparison and evaluation of retrospective intermodality brain image registration techniques [J]. Journal of Computer Assisted Tomography,1997,21 (4):554-566
[13]戴明远.基础信息论[M].上海：同济大学出版社,2003：30-31
[14]R.V.L.Hartley.Transmission of information[J].Bell Syst.Tech.J.1928,7:535-563
[15]C.E.Shannon.The mathematical theory of communication [J]. Bell Syst. Tech. J. 1948,27:379-423
[16]C Studholme. D.L.G Hill. D.J Hawkes. An overlap invariant entropy measure of 3D medical image alignment[J]. Pattern Recognition,1999,32 (1):71-86.
[17]A Renyi. On measures of entropy and information[C]. in:Proceedings of the Fourth Berkeley Symposium on Mathematics, Statistics and Probability.1960. Vol 1:547-561.
[18]M. Capek, L. Mroz, and R. Wegenkittl. Robust and fast medical registration of 3D-multimodality data sets[J]. Proc.of Medicon 2001, Part I, pp.515-518,2001.
[19]B. Ma, A. Hero, and J. Gorman. Image registration with minimal spanning tree algorithm [C], IEEE International Conference on Image Processing (ICIP), Vancouver, BC, Canada,2000.
[20]Mark P. Wachowiak, Renata Smolikova, Georgia D. Tourassi, Similarity metrics based on nonadditive entropies for 2D-3D multimodal biomedical image registration[J]. Proceedings of SPIE,2003, Vol.5032:1090-1100
[21]C Tsallis. Possible generalization of Boltzmann-Gibbs statistics [J]. Journal of Statistical Physics,1988,52:479-487
[22]L. Borland, A. R. Plastino, and C. Tsallis, Information gain within nonextensive thermostatistics[J], Journal of Mathematical Physics 39, pp.6490-6501,1998.
[23]S. Martin, G. Morison, W. Nailon and T. Durrani. Fast and accurate image registration using Tsallis entropy and simultaneous perturbation stochastic approximation[J].IEEE,2004,40(10)
[24]Maes F, Collignon A, Vandermeulen D, et al. Multimodality image registration by maximization of mutual information[J]. IEEE Transactions on Medical Imaging, 1997,16(2):187-198
[25]Maes F, Vandermeulen D, Suetens P. Comparative evaluation of multiresolution optimization strategies for multimodality image registration by maximization of mutual information[J]. Medical Image Analysis,1999,3(4)373-386
[26]Lehmann TM, Gonner C, Spitzer K. Survey:Interpolation methods in medical image processing [J]. IEEE Transactions on Medical Imaging,1999,18(11):1049-1075
[27]Jeffrey Tsao. Interpolation Artifacts in Multimodality Image Registration Based on Maximization of Mutual Information[J]. IEEE Transactions on Medical Imaging. 2003,22(7):854-864
[28]Qi Li, Isao Sato, Yutaka Murakami. Interpolation Effects on Accuracy of Mutual Information Based Image Registration[C]. Geoscience and Remote Sensing Symposium,2006.180-183.
[29]Castleman,K.R著,朱志刚等译.数字图像处理[M].北京：电子工业出版社,2002,95-98
[30]孙德敏.工程最优化方法及应用[M].合肥：中国科学技术大学出版社,1997：111-121
[31]Meyer C, Boes JL, Kim B, et al. Demonstration of accuracy and clinical versatility of mutual information for automatic multimodality image fusion using affine and thin plate spline-warped geometric deformations[J]. Medical Image Analysis,1997,1(3):195-206
[32]Wells Wm, Viola P, Astumi H, et al. Multi-modal volume registration by maximization of mutual information [J]. Medical Image Analysis,1996,1(1):35-51
[33]Viola P, Wells WM. Alignment by maximization of mutual information[J]. International Journal of Computer Vision,1997,24(2):137-154
[34]Rouet JM, Jacq JJ, Roux C. Genetic algorithms for a robust 3D MR-CT registration[J]. IEEE Transactions on Information Technology in biomedicine,2000,4(2):126-136
[35]陈贵云,张江,自适应遗传算法在多模图像配准中的应用[J],计算技术与自动化,2007.3,26(1)：123～125
[36]Ritter N,Owens R,Copper J,et al. Registration of stereo and temporal images of the retina[J]. IEEE Transactions on Medical Imaging,1999,18(5):404-418
[37]冯林,严亮,黄德根,等.PSO和Powell混合算法在医学图像配准中的应用研究[J].北京生物医学工程,2005,24(11)：8-12.
[38]彭景林,章兢,李树涛.基于改进PV插值和混合优化算法的医学图像配准[J].电子学报,2006,5(35)：962-965
[39]高智勇,林家瑞.灰度级别对基于互信息医学图像配准方法的影响[J].北京生物医学工程,2003,22(1)：5-9.
[40]徐东,李升辉。互信息医学图像配准研究与局部极值的克服[J]。计算机工程与应用,2007,43(3)：217-219
[41]刘哲星,江桂平,张煜,基于互信息量图像配准中目标函数局部极值的成因分析与克服[J].中国医学生物医学工程学报,2003,22(6)：26-30
[42]Hua-mei Chen, Pramod K. Varshney. Mutual Information Based CT-MR Brain Image Registration Using Generalized Partial Volume Joint Histogram Estimation[J]. IEEE Trans, Medical Imaging,2003,22 (9):1111-1119.
[43]刘哲星.医学图像回溯性配准研究：(博士学位论文).广州：第一军医大学,2002.
[44]Pluim J P W, Maintz J B A, Viergever M A. Interpolation artifacts in mutual information based image registration [J]. Computer Vision and Image Understanding, 2000,77(2):211-232.
[45]叶波,陆雪松,张素等.基于互信息图像配准中的局部极值问题研究[J].计算机工程与应用,2007,43(6)：58-61.
[46]Guiasu S. Information Theory with Applications[M], McGraw-Hill, International Book Company, New York,1977
[47]Louis K. Arata, Atam P. Dhawan, Three-dimensional anatomical model-based segmentation of MR brain images through principal axes registration[J]. IEEE
transactions on biomedical engineering,1995; 42 (11):1069-1077.
[48]Alpert NM,Bradshaw D,Kennedy D,The principal axes transformation:A method for image registration[M],Jonrnal of Nuclear Medicine,1990,31:1712-1717
[49]吴锋,钱宗才,杭洽时等.基于轮廓的力矩主轴法在医学图像配准中的应用[J].第四军医大学学报.2001,22(6)：567～569
[50]金人超,王金华,宋恩民.基于主轴法和互信息的脑部MR图像配准算法[J],计算机仿真,2007.4,24(4)：61～63
[51]邹柏贤,林京壤.图像轮廓提取方法研究[J].计算机工程与应用.2008,44(25)：161～165
[52]张小琳,图像边缘检测技术综述[J,高能量密度物理,2007,3(1)：37～40
[53]Canny J.A computational approach to edge detection[J], IEEE Transactions on Pattern Analysis and Machine Intelligence,1986.11,8(6):679-698
[54]冈萨雷斯.数字图像处理[M].电子工业出版社.2007.
[55]李小红.基于LOG滤波器的图像边缘检测算法的研究[J].计算机应用与软件,2005,22(5)；107-108.
[56]张艳玲,刘桂雄,曹东,等.数学形态学基本算法及在图像预处理中的应用[J].科学技术与工程,2007,7(3)：356-359.
[2]罗述谦.医学图像配准技术[J].国外医学生物医学工程分册,1999,22(1)：1～8
[3]罗述谦,李响.基于最大互信息的多模态医学图像配准[J].中国图象图形学报,2000,5(7)：551-558.
[4]田捷.医学影像处理与分析[M].电子工业出版社,2003,96-114
[5]J. B. Antoine Maintz, Max A. Viergever. A survey of medical image registration[J]. Medical Image Analysis,1998,2(1):1-36.
[6]A Collignon, F Maes, D Delaere. Automated multi-modality image registration based on information theory[C]. Information Processing in Medical Imaging,1995, 263-274.
[7]Paul Viola, William M.Wells Ⅲ. Alignment by Maximization of Mutual Information[C]. Proceedings of the 5th international conference on computer vision, Washington.D.C:IEEE Computer Society,1995,16-23
[8]王海南,郝重阳,雷方元等.非刚性医学图像配准研究综述[J].计算机工程与应用,2005,11：180-184.
[9]L.G. Brown, A survey of image registration techniques[J]. ACM Computing Surveys,1992,24:325-375.
[10]P.A. vanden Elsen, E.J.D. Pol, M.A. Viergever, Medical image matching:A review with classification[J]. IEEE Engineering in Medicine and Biology. 1993,12:26-39.
[11]D.J. Hawkes, Algorithms for radiological image registration and their clinical application[J]. Journal of Anatomy.1998,193:347-361.
[12]Jay West,J.Michael Fitzpatrick,Matthew Y.Wang,etal. Comparison and evaluation of retrospective intermodality brain image registration techniques [J]. Journal of Computer Assisted Tomography,1997,21 (4):554-566
[13]戴明远.基础信息论[M].上海：同济大学出版社,2003：30-31
[14]R.V.L.Hartley.Transmission of information[J].Bell Syst.Tech.J.1928,7:535-563
[15]C.E.Shannon.The mathematical theory of communication [J]. Bell Syst. Tech. J. 1948,27:379-423
[16]C Studholme. D.L.G Hill. D.J Hawkes. An overlap invariant entropy measure of 3D medical image alignment[J]. Pattern Recognition,1999,32 (1):71-86.
[17]A Renyi. On measures of entropy and information[C]. in:Proceedings of the Fourth Berkeley Symposium on Mathematics, Statistics and Probability.1960. Vol 1:547-561.
[18]M. Capek, L. Mroz, and R. Wegenkittl. Robust and fast medical registration of 3D-multimodality data sets[J]. Proc.of Medicon 2001, Part I, pp.515-518,2001.
[19]B. Ma, A. Hero, and J. Gorman. Image registration with minimal spanning tree algorithm [C], IEEE International Conference on Image Processing (ICIP), Vancouver, BC, Canada,2000.
[20]Mark P. Wachowiak, Renata Smolikova, Georgia D. Tourassi, Similarity metrics based on nonadditive entropies for 2D-3D multimodal biomedical image registration[J]. Proceedings of SPIE,2003, Vol.5032:1090-1100
[21]C Tsallis. Possible generalization of Boltzmann-Gibbs statistics [J]. Journal of Statistical Physics,1988,52:479-487
[22]L. Borland, A. R. Plastino, and C. Tsallis, Information gain within nonextensive thermostatistics[J], Journal of Mathematical Physics 39, pp.6490-6501,1998.
[23]S. Martin, G. Morison, W. Nailon and T. Durrani. Fast and accurate image registration using Tsallis entropy and simultaneous perturbation stochastic approximation[J].IEEE,2004,40(10)
[24]Maes F, Collignon A, Vandermeulen D, et al. Multimodality image registration by maximization of mutual information[J]. IEEE Transactions on Medical Imaging, 1997,16(2):187-198
[25]Maes F, Vandermeulen D, Suetens P. Comparative evaluation of multiresolution optimization strategies for multimodality image registration by maximization of mutual information[J]. Medical Image Analysis,1999,3(4)373-386
[26]Lehmann TM, Gonner C, Spitzer K. Survey:Interpolation methods in medical image processing [J]. IEEE Transactions on Medical Imaging,1999,18(11):1049-1075
[27]Jeffrey Tsao. Interpolation Artifacts in Multimodality Image Registration Based on Maximization of Mutual Information[J]. IEEE Transactions on Medical Imaging. 2003,22(7):854-864
[28]Qi Li, Isao Sato, Yutaka Murakami. Interpolation Effects on Accuracy of Mutual Information Based Image Registration[C]. Geoscience and Remote Sensing Symposium,2006.180-183.
[29]Castleman,K.R著,朱志刚等译.数字图像处理[M].北京：电子工业出版社,2002,95-98
[30]孙德敏.工程最优化方法及应用[M].合肥：中国科学技术大学出版社,1997：111-121
[31]Meyer C, Boes JL, Kim B, et al. Demonstration of accuracy and clinical versatility of mutual information for automatic multimodality image fusion using affine and thin plate spline-warped geometric deformations[J]. Medical Image Analysis,1997,1(3):195-206
[32]Wells Wm, Viola P, Astumi H, et al. Multi-modal volume registration by maximization of mutual information [J]. Medical Image Analysis,1996,1(1):35-51
[33]Viola P, Wells WM. Alignment by maximization of mutual information[J]. International Journal of Computer Vision,1997,24(2):137-154
[34]Rouet JM, Jacq JJ, Roux C. Genetic algorithms for a robust 3D MR-CT registration[J]. IEEE Transactions on Information Technology in biomedicine,2000,4(2):126-136
[35]陈贵云,张江,自适应遗传算法在多模图像配准中的应用[J],计算技术与自动化,2007.3,26(1)：123～125
[36]Ritter N,Owens R,Copper J,et al. Registration of stereo and temporal images of the retina[J]. IEEE Transactions on Medical Imaging,1999,18(5):404-418
[37]冯林,严亮,黄德根,等.PSO和Powell混合算法在医学图像配准中的应用研究[J].北京生物医学工程,2005,24(11)：8-12.
[38]彭景林,章兢,李树涛.基于改进PV插值和混合优化算法的医学图像配准[J].电子学报,2006,5(35)：962-965
[39]高智勇,林家瑞.灰度级别对基于互信息医学图像配准方法的影响[J].北京生物医学工程,2003,22(1)：5-9.
[40]徐东,李升辉。互信息医学图像配准研究与局部极值的克服[J]。计算机工程与应用,2007,43(3)：217-219
[41]刘哲星,江桂平,张煜,基于互信息量图像配准中目标函数局部极值的成因分析与克服[J].中国医学生物医学工程学报,2003,22(6)：26-30
[42]Hua-mei Chen, Pramod K. Varshney. Mutual Information Based CT-MR Brain Image Registration Using Generalized Partial Volume Joint Histogram Estimation[J]. IEEE Trans, Medical Imaging,2003,22 (9):1111-1119.
[43]刘哲星.医学图像回溯性配准研究：(博士学位论文).广州：第一军医大学,2002.
[44]Pluim J P W, Maintz J B A, Viergever M A. Interpolation artifacts in mutual information based image registration [J]. Computer Vision and Image Understanding, 2000,77(2):211-232.
[45]叶波,陆雪松,张素等.基于互信息图像配准中的局部极值问题研究[J].计算机工程与应用,2007,43(6)：58-61.
[46]Guiasu S. Information Theory with Applications[M], McGraw-Hill, International Book Company, New York,1977
[47]Louis K. Arata, Atam P. Dhawan, Three-dimensional anatomical model-based segmentation of MR brain images through principal axes registration[J]. IEEE
transactions on biomedical engineering,1995; 42 (11):1069-1077.
[48]Alpert NM,Bradshaw D,Kennedy D,The principal axes transformation:A method for image registration[M],Jonrnal of Nuclear Medicine,1990,31:1712-1717
[49]吴锋,钱宗才,杭洽时等.基于轮廓的力矩主轴法在医学图像配准中的应用[J].第四军医大学学报.2001,22(6)：567～569
[50]金人超,王金华,宋恩民.基于主轴法和互信息的脑部MR图像配准算法[J],计算机仿真,2007.4,24(4)：61～63
[51]邹柏贤,林京壤.图像轮廓提取方法研究[J].计算机工程与应用.2008,44(25)：161～165
[52]张小琳,图像边缘检测技术综述[J,高能量密度物理,2007,3(1)：37～40
[53]Canny J.A computational approach to edge detection[J], IEEE Transactions on Pattern Analysis and Machine Intelligence,1986.11,8(6):679-698
[54]冈萨雷斯.数字图像处理[M].电子工业出版社.2007.
[55]李小红.基于LOG滤波器的图像边缘检测算法的研究[J].计算机应用与软件,2005,22(5)；107-108.
[56]张艳玲,刘桂雄,曹东,等.数学形态学基本算法及在图像预处理中的应用[J].科学技术与工程,2007,7(3)：356-359.