自然图象中轮廓检测方法研究与实现
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摘要
由于自然图象的背景复杂,在进行图象的轮廓检测时可能会有一些重要轮廓丢失,从而会给接下来对象检测和分析、对象跟踪等工作带来额外的困难。自然图象中的轮廓检测算法主要是围绕着梯度特征和曲线连续性特征进行图象的概率轮廓的检测,通过建立逻辑回归和曲线连续性模型检测和补全图象轮廓。这种轮廓检测结合补全的方法并不局限于采用某一种边缘检测算法,因此具有广泛的适用性。
以亮度梯度、颜色梯度和纹理梯度为基础,采用了用逻辑回归的方法检测对象的轮廓边缘。对图象的局部区域的象素点进行亮度、颜色和纹理的梯度对比,区域差分直方图的值作为中心点的梯度值,逻辑回归模型组合这三个梯度特征,学习到特征参数,从而判断图象中每个点是轮廓的概率。
考虑到自然图象中背景的复杂性,重要轮廓的丢失可能不可避免,利用曲线连续性模型对概率轮廓建模,通过学习轮廓的局部特征从而将丢失的轮廓补全。轮廓补全工作是从德劳奈三角剖分图开始的,它是沃罗诺伊图的偶图。约束的德劳奈三角剖分将图象的概率轮廓补全之后,通过曲线连续性模型学习到模型参数,从而判断约束的德劳奈三角剖分图中补全的轮廓哪些是真实的轮廓,哪些不是的,最后得到的轮廓是较完整的图象轮廓。曲线的连续性模型分为两种,一种是局部的,一种是全局的。曲线的局部连续性轮廓补全采用的是逻辑回归模型,曲线的全局连续性轮廓补全采用的是条件随机场模型。
在实验中选择坎尼边缘检测算法与概率轮廓检测和补全算法做比较,分别在查全率和准确率以及全度量上进行了性能对比分析,结论表明在自然图象中采用概率轮廓检测和补全算法的性能要比坎尼边缘检测算法略优。
Because of the complexity of the background in the natural images, some important boundary may be missed on detecting, which will introduce additional difficulty to the following object detecting, analysis or tracing. This work of boundary detecting in the natural images is surrounded with the gradient features and curvilinear continuity features to detect the probability of boundary in images. Boundary of images is detected and completed by constructing logistic regression and curvilinear continuity models. This method of boundary detecting combined with completion isn’t limited with any kind of edge detecting arithmetic, then it is global applicable.
Based on the brightness gradient, color gradient and texture gradient, boundary of object in the image is detected using logistic regression. It is necessary to compare the gradient of brightness, color and texture of pixels with each other in the two half areas of images, and the gradient value of the center pixel is computed from the difference of histograms of the two half. After combining the three gradient features with logistic regression model and learning parameters of these features, the probability of boundary(Pb) to every pixel in the images could be decided.
The missing of significant boundary might be impossible to avoid with regard to the complex of the background in natural images. Using the curvilinear continuity models with the probability of boundary (Pb), gaps in the Pb will be completed by learning the features of local boundary. Before the task of completion of boundary, it is important to building a Delaunay Triangulation(DT) with Pb, which is the antithetic graphics of Voronoi. Constrained Delaunay Triangulation(CDT) completes the Pb, and then parameters in the models are learned from the curvilinear continuity, finally, whether a completed edge in the CDT is a real bound or not could be fixed. As a result, the output boundary is better in its integrity. There are two curvilinear continuity models. One is local, while the other is global. The local curvilinear continuity boundary completion uses logistic regression model, while the global curvilinear continuity boundary completion uses Conditioned Random Field (CRF) model.
In the experiment, the arithmetic of detecting probability of boundary and its completion arithmetic are compared with Canny detector. The comparison aspects include ratio、precision and F-measure. The result suggests that using the arithmetic of detecting and completing probability of boundary is better than Canny edge detecting arithmetic in performance.
以亮度梯度、颜色梯度和纹理梯度为基础,采用了用逻辑回归的方法检测对象的轮廓边缘。对图象的局部区域的象素点进行亮度、颜色和纹理的梯度对比,区域差分直方图的值作为中心点的梯度值,逻辑回归模型组合这三个梯度特征,学习到特征参数,从而判断图象中每个点是轮廓的概率。
考虑到自然图象中背景的复杂性,重要轮廓的丢失可能不可避免,利用曲线连续性模型对概率轮廓建模,通过学习轮廓的局部特征从而将丢失的轮廓补全。轮廓补全工作是从德劳奈三角剖分图开始的,它是沃罗诺伊图的偶图。约束的德劳奈三角剖分将图象的概率轮廓补全之后,通过曲线连续性模型学习到模型参数,从而判断约束的德劳奈三角剖分图中补全的轮廓哪些是真实的轮廓,哪些不是的,最后得到的轮廓是较完整的图象轮廓。曲线的连续性模型分为两种,一种是局部的,一种是全局的。曲线的局部连续性轮廓补全采用的是逻辑回归模型,曲线的全局连续性轮廓补全采用的是条件随机场模型。
在实验中选择坎尼边缘检测算法与概率轮廓检测和补全算法做比较,分别在查全率和准确率以及全度量上进行了性能对比分析,结论表明在自然图象中采用概率轮廓检测和补全算法的性能要比坎尼边缘检测算法略优。
Because of the complexity of the background in the natural images, some important boundary may be missed on detecting, which will introduce additional difficulty to the following object detecting, analysis or tracing. This work of boundary detecting in the natural images is surrounded with the gradient features and curvilinear continuity features to detect the probability of boundary in images. Boundary of images is detected and completed by constructing logistic regression and curvilinear continuity models. This method of boundary detecting combined with completion isn’t limited with any kind of edge detecting arithmetic, then it is global applicable.
Based on the brightness gradient, color gradient and texture gradient, boundary of object in the image is detected using logistic regression. It is necessary to compare the gradient of brightness, color and texture of pixels with each other in the two half areas of images, and the gradient value of the center pixel is computed from the difference of histograms of the two half. After combining the three gradient features with logistic regression model and learning parameters of these features, the probability of boundary(Pb) to every pixel in the images could be decided.
The missing of significant boundary might be impossible to avoid with regard to the complex of the background in natural images. Using the curvilinear continuity models with the probability of boundary (Pb), gaps in the Pb will be completed by learning the features of local boundary. Before the task of completion of boundary, it is important to building a Delaunay Triangulation(DT) with Pb, which is the antithetic graphics of Voronoi. Constrained Delaunay Triangulation(CDT) completes the Pb, and then parameters in the models are learned from the curvilinear continuity, finally, whether a completed edge in the CDT is a real bound or not could be fixed. As a result, the output boundary is better in its integrity. There are two curvilinear continuity models. One is local, while the other is global. The local curvilinear continuity boundary completion uses logistic regression model, while the global curvilinear continuity boundary completion uses Conditioned Random Field (CRF) model.
In the experiment, the arithmetic of detecting probability of boundary and its completion arithmetic are compared with Canny detector. The comparison aspects include ratio、precision and F-measure. The result suggests that using the arithmetic of detecting and completing probability of boundary is better than Canny edge detecting arithmetic in performance.
引文
[1] Canny J . A computational approach to edge detection[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence, 1986, 8(6):679～697
[2]王植.一种基于Canny理论的自适应边缘检测方法.中国图像图形学报, 2004, 9(8):336～351
[3]韦海萍. Canny算法的改进及其硬件实现.光学技术, 2006, 32(2):437～452
[4]李宏贵,李兴国.一种基于δ函数的边缘图像检测算法[J].中国图形图像学报, 2003, 8A(2):188～192
[5] Wang Zhiqian, Jezekiel Ben-Arie. Detection and segmentation of generic shapes based on affine modeling of energy in eigenspace[J]. IEEE Transactions on Image Processing, 2001, 10(11):1621～1629
[6] L.Kessal, N.Abel, D.Demigny. Real-time image processing with dynamically reconfigurable architecture[J]. Real-Time Imaging,2003, 9(2):297～313
[7] Demigny D. On optimal linear filtering for edge detection[J]. IEEE Transaction on Image Processing, 2002, 11(7):728～737
[8]徐建华.图像处理与分析[M].北京:科学出版社,1992. 79～84
[9] Castleman K R.数字图象处理[M].朱志刚,林学闫,石定机译.北京:电子工业出版社,1998. 56～76
[10] Q.Wu and Y.Yu.Two-level image segmentation based on region and edge integration.In Proc.DICTA, 2003, 3(11):320～368
[11] W.T.Freeman,E.C.Pasztor,and O.T.Carmichael.Learning low-level vision. Int’l. J. Comp. Vision, 2000, 11(7):703～725
[12] S.Geman and D.Geman.Stochastic relaxation,gibbs distribution, and the Bayesian retoration of images.IEEE Trans.PAMI, 1984, 24(2):361～404
[13] S.Z.Li.Markov Random Field Modeling in Computer Vision. Springer-Verlag, 1995. 2～8
[14] S.C.Zhu,Y.N.Wu,and D.B.Mumford.Frame:Filters,random field and maximum entropy:Towards a unified theory for texture modeling. Int’l. J. Comp. Vision, 1998, 9(7):442～451
[15] M.Wertheimer. Laws of organization in perceptual forms(partial translation).In W.B.ELLIS, editor, A sourcebook of Gestalt Psychology. Harcourt Brace and Company, 1938. 39～47
[16] G.Kanizsa.Organization in vision:essays on gestalt perception.Praeger, 1979, 12(2):211～217
[17] R.von der Heydt,E.Peterhans,and G.Baumgartner. Illusory contours and cortical neuron responses. Science, 1984, 30(7):365～398
[18] P.J.Kellman and T.F.Shipley.A theory of visual interpolation in object perception. Cognitive Psychology, 1991, 12(2):170～231
[19] S.Belongie, J.Malik.Matching with Shape Contexts.CBAIVL, 2000, 123～144
[20] E.Borenstein and S.Ullman.Class-specific,top-down segmentation.ECCV, 2002, 24(11):581～592
[21] A.Shashua and S.Ullman.Structural saliency:the detection of globally salient structures using a locally connected network. In Proc. 2nd Int’l. Conf. Comp. Vision, 1988. 811～819
[22] T.Leung and J.Malik. Contour continuity in region based image segmentation.ECCV, 1998,23(2):107～119
[23] P.Parent and S.W.Zucker.Trace inference,curvature consistency and curve detection.IEEE Trans.PAMI,1989,8(8):18～23
[24] A.Berg, T.Berg and J.Malik. Shape matching and object recognition using low distortion correspondence.CVPR, 2005,31(9):141～148
[25] J.August, K.Siddiqi, S.Zucker. Contour fragment grouping and shared, simple occluders.Computer Vision and Image Understanding, 1999,24(8):1369～1392
[26] F.Heitger and R.von der Heydt. A computational model of neural contour processing. In Proc. 4th Int’l. Conf. Comp. Vision, 1993, 20(6):1185～1189
[27] J. Canny. A computational approach to edge detection. IEEE Trans. Pattern Analysis and Machine Intelligence, 1986, 8(7):679～698
[28] J. Rivest and P. Cavanagh. Localizing contours defined by more than one attribute. Vision Research, 1996, 36(1):53～66
[29] J. Malik, S. Belongie, T. Leung, and J. Shi. Contour and texture analysis for imagesegmentation. Int’l. Journal of Computer Vision, 2001, 43(1):7～27
[30] W. Niblack, R. Barber,W. Equitz, M. Flickner, E. Glasman, D. Petkovic, P. Yanker, C. Faloutsos, and G. Taubin. The QBIC project: Querying image by content using color, texture, and shape. Proceedings of the SPIE - The International Society for Optical Engineering v.1908(Storage and Retrieval for Image and Video Databases),1993, 1908(2):173～187
[31] C. Carson, S. Belongie, H. Greenspan, and J. Malik. Blobworld: Image segmentation using expectation-maximization and its application to image querying. IEEE Trans. Pattern Analysis and Machine Intelligence, 2002, 24(8):1026～1038
[32] M. Ruzon and C. Tomasi. Color edge detection with the compass operator. In Proc. IEEE Conf. Comput. Vision and Pattern Recognition, 1999. 160～166
[33] M. Ruzon and C. Tomasi. Corner detection in textured color images. In Proc. Int’l. Conf. Computer Vision, 1999. 1039～1045
[34] I. Fogel and D. Sagi. Gabor filters as texture discriminator. Bio. Cybernetics, 1989, 61(11):103～113
[35] J. Malik and P. Perona. Preattentive texture discrimination with early vision mechanisms. J. Opt. Soc. Am., 1990, 7(2):923～932
[36] D. Heeger and J. Bergen. Pyramid-based texture analysis/synthesis. In Proceedings of SIGGRAPH’95, 1995, 18(7):229～238
[37] J. Puzicha, T. Hofmann, and J. Buhmann. Non-parametric similarity measures for unsupervised texture segmentation and image retrieval. In Proc. IEEE Conf. Comput. Vision and Pattern Recognition, 1997. 267～272
[38] J. Puzicha, Y. Rubner, C. Tomasi, and J. Buhmann. Empirical evaluation of dissimilarity measures for color and texture. In Proc. Int’l. Conf. Computer Vision, 1999. 1165～1173
[39] C. Mallows. A note on asymptotic joint normality. Annals of Mathematical Statistics, 1972, 43(2):508～515
[40] E. Levina and P. Bickel. The Earth Mover’s distance is the Mallows distance: some insights from statistics. In Proc. Int’l. Conf. Computer Vision, 2001. II:251～256
[41] M. I. Jordan and R. A. Jacobs. Hierarchical mixtures of experts and the EM algorithm. Neural Computation, 1994, 6(8):181～214
[42] C. Chang and C. Lin. LIBSVM: a library for support vector machines, 2001. Software available at http://www.csie.ntu.edu.tw/?cjlin/libsvm
[43] J. Shewchuk. Triangle: Engineering a 2d quality mesh generator and delaunay triangulator. In First Workshop on Applied Computational Geometry, 1996. 124～133
[44] X. Ren. Learning and matching line aspects for articulated objects. In Proc. IEEE Conf. Comput. Vision and Pattern Recogn., 2007. 758～802
[45] X. Ren, A. Berg, and J. Malik. Recovering human body configurations using pairwise constraints between parts. In Proc. 10th Int’l. Conf. Computer Vision, 2005. I:824～831
[46] Ross Kindermann and J. Laurie Snell, Markov random fields and their applications. American Mathematical Society, 1980. 743～758
[47] X. He, R. Zemel, and M. Carreira-Perpinan. Multiscale conditional random fields for image labelling. In Proc. IEEE Conf. Comput. Vision and Pattern Recogn., 2004. II:695～702
[48] S. Kumar and M. Hebert. Discriminative random fields. In Proc. Int’l. J. Comp. Vision, 2006, 68(2):179～202
[49] N. Shental, A. Zomet, T. Hertz, and Y. Weiss. Learning and inferring image segmentations with the gbp typical cut algorithm. In Proc. 9th Int’l. Conf. Comp. Vision, 2003. 1243～1250
[50] J. Sun, H.-Y. Shum, and N.-N. Zheng. Stereo matching using belief propagation. In Proc. 7th Europ. Conf. Comp. Vision, 2002. 510～524
[51] G. Mori, X. Ren, A. Efros, and J. Malik. Recovering human body configurations: Combining segmentation and recognition. In CVPR, 2004, 33(10):149～158
[52] E. Borenstein and S. Ullman. Class-Specific, Top-Down Segmentation. In Proc. European Conf. Computer Vision (ECCV '02), 2002. II:109～124
[53] D. Martin, C. Fowlkes, D. Tal, and J. Malik. A Database of Human Segmented Natural Images and its Application to Evaluating Segmentation Algorithms and Measuring Ecological Statistics. ICCV, 2001, 26(2):985～992
[54] Michael Ortega, Yong Rui, Kaushik Chakrabarti, Kriengkrai Porkaew, Sharad Mehrotra, and Thomas S. Huang. Supporting Ranked Boolean Similarity Queries in MARS. IEEE Transaction on Knowledge and Data Engineering, 1998, 10(6):905～925
[55] Kaushik Chakrabarti, and Sharad Mehrotra. The Hybrid Tree: An Index Structure for High Dimensional Feature Spaces. In Proc. IEEE International Conference on Data Engineering (ICDE), 1999. 440～447
[2]王植.一种基于Canny理论的自适应边缘检测方法.中国图像图形学报, 2004, 9(8):336～351
[3]韦海萍. Canny算法的改进及其硬件实现.光学技术, 2006, 32(2):437～452
[4]李宏贵,李兴国.一种基于δ函数的边缘图像检测算法[J].中国图形图像学报, 2003, 8A(2):188～192
[5] Wang Zhiqian, Jezekiel Ben-Arie. Detection and segmentation of generic shapes based on affine modeling of energy in eigenspace[J]. IEEE Transactions on Image Processing, 2001, 10(11):1621～1629
[6] L.Kessal, N.Abel, D.Demigny. Real-time image processing with dynamically reconfigurable architecture[J]. Real-Time Imaging,2003, 9(2):297～313
[7] Demigny D. On optimal linear filtering for edge detection[J]. IEEE Transaction on Image Processing, 2002, 11(7):728～737
[8]徐建华.图像处理与分析[M].北京:科学出版社,1992. 79～84
[9] Castleman K R.数字图象处理[M].朱志刚,林学闫,石定机译.北京:电子工业出版社,1998. 56～76
[10] Q.Wu and Y.Yu.Two-level image segmentation based on region and edge integration.In Proc.DICTA, 2003, 3(11):320～368
[11] W.T.Freeman,E.C.Pasztor,and O.T.Carmichael.Learning low-level vision. Int’l. J. Comp. Vision, 2000, 11(7):703～725
[12] S.Geman and D.Geman.Stochastic relaxation,gibbs distribution, and the Bayesian retoration of images.IEEE Trans.PAMI, 1984, 24(2):361～404
[13] S.Z.Li.Markov Random Field Modeling in Computer Vision. Springer-Verlag, 1995. 2～8
[14] S.C.Zhu,Y.N.Wu,and D.B.Mumford.Frame:Filters,random field and maximum entropy:Towards a unified theory for texture modeling. Int’l. J. Comp. Vision, 1998, 9(7):442～451
[15] M.Wertheimer. Laws of organization in perceptual forms(partial translation).In W.B.ELLIS, editor, A sourcebook of Gestalt Psychology. Harcourt Brace and Company, 1938. 39～47
[16] G.Kanizsa.Organization in vision:essays on gestalt perception.Praeger, 1979, 12(2):211～217
[17] R.von der Heydt,E.Peterhans,and G.Baumgartner. Illusory contours and cortical neuron responses. Science, 1984, 30(7):365～398
[18] P.J.Kellman and T.F.Shipley.A theory of visual interpolation in object perception. Cognitive Psychology, 1991, 12(2):170～231
[19] S.Belongie, J.Malik.Matching with Shape Contexts.CBAIVL, 2000, 123～144
[20] E.Borenstein and S.Ullman.Class-specific,top-down segmentation.ECCV, 2002, 24(11):581～592
[21] A.Shashua and S.Ullman.Structural saliency:the detection of globally salient structures using a locally connected network. In Proc. 2nd Int’l. Conf. Comp. Vision, 1988. 811～819
[22] T.Leung and J.Malik. Contour continuity in region based image segmentation.ECCV, 1998,23(2):107～119
[23] P.Parent and S.W.Zucker.Trace inference,curvature consistency and curve detection.IEEE Trans.PAMI,1989,8(8):18～23
[24] A.Berg, T.Berg and J.Malik. Shape matching and object recognition using low distortion correspondence.CVPR, 2005,31(9):141～148
[25] J.August, K.Siddiqi, S.Zucker. Contour fragment grouping and shared, simple occluders.Computer Vision and Image Understanding, 1999,24(8):1369～1392
[26] F.Heitger and R.von der Heydt. A computational model of neural contour processing. In Proc. 4th Int’l. Conf. Comp. Vision, 1993, 20(6):1185～1189
[27] J. Canny. A computational approach to edge detection. IEEE Trans. Pattern Analysis and Machine Intelligence, 1986, 8(7):679～698
[28] J. Rivest and P. Cavanagh. Localizing contours defined by more than one attribute. Vision Research, 1996, 36(1):53～66
[29] J. Malik, S. Belongie, T. Leung, and J. Shi. Contour and texture analysis for imagesegmentation. Int’l. Journal of Computer Vision, 2001, 43(1):7～27
[30] W. Niblack, R. Barber,W. Equitz, M. Flickner, E. Glasman, D. Petkovic, P. Yanker, C. Faloutsos, and G. Taubin. The QBIC project: Querying image by content using color, texture, and shape. Proceedings of the SPIE - The International Society for Optical Engineering v.1908(Storage and Retrieval for Image and Video Databases),1993, 1908(2):173～187
[31] C. Carson, S. Belongie, H. Greenspan, and J. Malik. Blobworld: Image segmentation using expectation-maximization and its application to image querying. IEEE Trans. Pattern Analysis and Machine Intelligence, 2002, 24(8):1026～1038
[32] M. Ruzon and C. Tomasi. Color edge detection with the compass operator. In Proc. IEEE Conf. Comput. Vision and Pattern Recognition, 1999. 160～166
[33] M. Ruzon and C. Tomasi. Corner detection in textured color images. In Proc. Int’l. Conf. Computer Vision, 1999. 1039～1045
[34] I. Fogel and D. Sagi. Gabor filters as texture discriminator. Bio. Cybernetics, 1989, 61(11):103～113
[35] J. Malik and P. Perona. Preattentive texture discrimination with early vision mechanisms. J. Opt. Soc. Am., 1990, 7(2):923～932
[36] D. Heeger and J. Bergen. Pyramid-based texture analysis/synthesis. In Proceedings of SIGGRAPH’95, 1995, 18(7):229～238
[37] J. Puzicha, T. Hofmann, and J. Buhmann. Non-parametric similarity measures for unsupervised texture segmentation and image retrieval. In Proc. IEEE Conf. Comput. Vision and Pattern Recognition, 1997. 267～272
[38] J. Puzicha, Y. Rubner, C. Tomasi, and J. Buhmann. Empirical evaluation of dissimilarity measures for color and texture. In Proc. Int’l. Conf. Computer Vision, 1999. 1165～1173
[39] C. Mallows. A note on asymptotic joint normality. Annals of Mathematical Statistics, 1972, 43(2):508～515
[40] E. Levina and P. Bickel. The Earth Mover’s distance is the Mallows distance: some insights from statistics. In Proc. Int’l. Conf. Computer Vision, 2001. II:251～256
[41] M. I. Jordan and R. A. Jacobs. Hierarchical mixtures of experts and the EM algorithm. Neural Computation, 1994, 6(8):181～214
[42] C. Chang and C. Lin. LIBSVM: a library for support vector machines, 2001. Software available at http://www.csie.ntu.edu.tw/?cjlin/libsvm
[43] J. Shewchuk. Triangle: Engineering a 2d quality mesh generator and delaunay triangulator. In First Workshop on Applied Computational Geometry, 1996. 124～133
[44] X. Ren. Learning and matching line aspects for articulated objects. In Proc. IEEE Conf. Comput. Vision and Pattern Recogn., 2007. 758～802
[45] X. Ren, A. Berg, and J. Malik. Recovering human body configurations using pairwise constraints between parts. In Proc. 10th Int’l. Conf. Computer Vision, 2005. I:824～831
[46] Ross Kindermann and J. Laurie Snell, Markov random fields and their applications. American Mathematical Society, 1980. 743～758
[47] X. He, R. Zemel, and M. Carreira-Perpinan. Multiscale conditional random fields for image labelling. In Proc. IEEE Conf. Comput. Vision and Pattern Recogn., 2004. II:695～702
[48] S. Kumar and M. Hebert. Discriminative random fields. In Proc. Int’l. J. Comp. Vision, 2006, 68(2):179～202
[49] N. Shental, A. Zomet, T. Hertz, and Y. Weiss. Learning and inferring image segmentations with the gbp typical cut algorithm. In Proc. 9th Int’l. Conf. Comp. Vision, 2003. 1243～1250
[50] J. Sun, H.-Y. Shum, and N.-N. Zheng. Stereo matching using belief propagation. In Proc. 7th Europ. Conf. Comp. Vision, 2002. 510～524
[51] G. Mori, X. Ren, A. Efros, and J. Malik. Recovering human body configurations: Combining segmentation and recognition. In CVPR, 2004, 33(10):149～158
[52] E. Borenstein and S. Ullman. Class-Specific, Top-Down Segmentation. In Proc. European Conf. Computer Vision (ECCV '02), 2002. II:109～124
[53] D. Martin, C. Fowlkes, D. Tal, and J. Malik. A Database of Human Segmented Natural Images and its Application to Evaluating Segmentation Algorithms and Measuring Ecological Statistics. ICCV, 2001, 26(2):985～992
[54] Michael Ortega, Yong Rui, Kaushik Chakrabarti, Kriengkrai Porkaew, Sharad Mehrotra, and Thomas S. Huang. Supporting Ranked Boolean Similarity Queries in MARS. IEEE Transaction on Knowledge and Data Engineering, 1998, 10(6):905～925
[55] Kaushik Chakrabarti, and Sharad Mehrotra. The Hybrid Tree: An Index Structure for High Dimensional Feature Spaces. In Proc. IEEE International Conference on Data Engineering (ICDE), 1999. 440～447