摘要
图像演化模型是目前图像处理研究中较为活跃的领域之一，广泛应用于图像平滑、去噪、分割、特征提取和匹配等方面，开展相关研究对于在图像处理领域建立严格统一的数学研究体系并将其付之应用具有重要的理论和示范意义。
论文在综述大量研究文献的基础上分析了不同图像演化模型之间的理论联系，建立了统一的理论框架，进而研究了演化模型的作用机理和相关数值算法；然后从较为简单的Gauss模型入手，利用奇点理论研究模型特征点的演化性质，从理论上揭示Gauss模型的深层结构。在一般图像特征点理论研究的基础上，本文着重讨论了两类特征点：分岔点和曲率过零点，并将模型和算法进一步应用于图像匹配和边缘检测。同时，在另一类重要的图像特征——骨架研究方面，借鉴基于能量函数的水平集思想，本文提出了一种新的基于多尺度理论的图像骨架提取算法。此外，由于Gauss模型是一类各向同性算子，因而本文还建立了新的各向异性扩散方程模型，应用于图像去噪，并使之具有较好的自适应性。论文的主要研究工作和创新性工作有以下几点：
（1）通过对图像结构相似度分析方法的改进，本文构造了一类新的非线性自适应扩散模型并应用于图像去噪，相应的算法可自适应地确定迭代步数并很好地保留图像的边缘。与传统的各向异性模型相比，该算法简单易行，可在无需人工干预的情况下，有效地去除图像噪声，同时较好地保留图像边缘信息，具有较好的自适应性。
（2）利用奇点理论论证了二维尺度图像特征点的演化状况，尤其是特征点的融合与生成内在机理，本文给出了图像深层结构的严格描述与证明。这些研究可为目前尺度空间的应用研究提供理论保证，同时本文关于特征点的性质分析与证明可为后续尺度空间在图像匹配、运动跟踪、图像分割等领域的应用提供算法基础。
（3）本文分析了两类重要的图像特征点——分岔点和曲率过零点的基本性质，并在此基础上给出了新的图像匹配算法和边缘检测算法。关于这两类特征点的研究为从多尺度角度研究图像提供了有益的尝试，并为进一步拓宽多尺度应用领域提供了切实可行的思路。
（4）通过对快速行进法进行改进并结合水平集方法，本文提出了一种提取狭长带状图像骨架的算法。利用该算法可以导出连续完整的单像素骨架，有效克服了目前提取图像骨架研究中的间断和毛刺问题。同时由于该算法具有较好的抗噪性，为处理医学图像及进一步的医疗诊断分析提供辅助手段奠定了良好的基础。
上述研究较为系统地刻画了多尺度图像典型几何特征的演化性质，并将之应用于图像匹配、边缘检测等领域，这为进一步综合深入研究图像的多尺度信息、拓展医学图像等应用领域提供了新的思路和理论基础。另外，本文研究建立在演化思想的统一框架下，这有利于从总体上把握目前不同研究模型的共性，通过分析各模型之间的内在联系和作用机理，可以互为借鉴，为构造合适的优化算法奠定基础。
Image evolution model is one of relatively active fields of image processing,widely used in image smoothing, denoising, segmentation, feature extraction andmatching, etc. Researching on this field pays an important theoretical anddemonstration significance in the field of image processing for the establishmentof a strict mathematical system and its application.
Based on large amount of research literature, we analyze the theoreticalrelationship between different images’ evolution models, and establish a unifiedtheoretical framework, and then study the mechanism of the evolution model andassociated numerical algorithms. After that, we start to study Gauss model whichis relatively simple; by using singularity theory, the evolution characristic ofmodel feature points is researched, and the deep structure of Gauss model isrevealed theoretically. Based on the research of general image feature pointstheroy, this paper focuses on two types of feature points: the bifurcation point andthe curvature zero-crossing point, and applies the model and algorithms on imagematch and edge detection. Simultaneously, in the field of another important imagefeature—skeleton research, in this paper we propose a new method for imageskeleton extraction based on multi-scale learning theory from the idea of level setbased on energy function. In addition, since the Gauss model is a class of isotropyoperator, we will construct a new anisotropic diffusion equation model used inimage denoising, so that it has better adaptability. The main research and theinnovative work are as follow:
(1) By improving the analysis method of the image structural similarity, thepaper constructs a new nonlinear diffusion adaptive equation to be used in imagedenoising. The corresponding algorithm can adaptively determine the number ofiteration steps and preserve image edges at the same time. Compared with thetraditional anisotropic model, the algorithm is simple. It can effectively removeimage noise without the manual intervention while preserves image edgeinformation better. So it has better adaptability.
(2) Demonstrating the evolution of two-dimensional scale image featurepoint using singularity theory, especially the internal mechanism of the creationand the annihilation of feature points, this paper gives the strict description andthe demonstration of the deep structure of image. These studies can provide a theoretical for current research using the scale-space, and the nature evidence andanalysis of the feature points in the paper provide some algorithm basis forapplying the scale space to image match、motion tracking、segmentation etc.
(3) The paper analyzes two kinds of important image feature points: thefundamental nature of the bifurcation point and the curvature zero-crossing point,and on this basis, we give some new image matching algorithms and edgedetection algorithms. The research about the two kinds of feature points providesa good attempt for studying image form the multi-scale perspective, and offers apractical idea for broadening further the multi-scale applications.
(4) Through improved fast marching method, presents an algorithm ofextracting the narrow band image skeleton. The algorithm can get a continuoussingle-pixel skeleton, and effectively overcome the intermittent and burr problemsappearing in the current skeleton research. Because the algorithm has good noiseimmunity, at the fields of medical image processing and medical diagnosis, it laysa good foundation.
The studies above systematically depict typical geometric evolution nature ofthe multi-scale image, and use them in the field of image matching、detection andso on, which provides new ideas and theoretical basis for further integratingin-depth study of multi-scale image and expanding the application field. Inaddition, this study builds on the unified framework of the evolution idea, whichis conducive to grasp the common of the different research models currently formoverall; and then lay the foundation for constructing a suitable optimizationalgorithm by analyzing the intrinsic link between the model and mechanism andlearning form each other.
引文
[1] Xue-Cheng Tai, Knut-Andreas Lie, Tony F. Chan, Stanley Osher(Eds.),Image Processing Based on Partial Differential Equations[M], Springer,2006.
[2] M. Kerckhove(Ed.), Scale Space and Morphology in Computer Vision[M],Springer,2001.
[3] Ron Kimmel, Nir Sochen, JoachimWeickert (Eds.), Scale Space and PDEMethods in Computer Vision[M], Springer,2005.
[4] Fiorella Sgallari, Almerico Murli, Nikos Paragios (Eds.), Scale Space andVariational Methods in Computer Vision[M], Springer,2007.
[5] Xue-Cheng, Tai Knut M rken, Marius Lysaker Knut-Andreas Lie (Eds.),Scale Space and Variational Methods in Computer Vision[M], Springer,2009.
[6] T. Lindeberg, Scale Space theory in Computer Vision[M], Kluwer Academicpublishers,1994.
[7] T. F. Chan, J. Shen, Image Processing and Analysis-Variational, PDE,Wavelet, and Statistical Methods[M], SIAM,2005.
[8] J. Sporring, M. Nielsen, L.M.J. Florack, P. Johansen(Eds.), Gaussian ScaleSpace[M], Springer,1997.
[9] L.M.J. Florack, Image Structure[M], Springer,1997.
[10] J. Weickert. Anisotropic Diffusion in Image Processing. Stuttgart[M], B. G.Teubner,1998.
[11] Stanley Osher, Nikos Paragios, Geometric Level Set Methods in Imaging,Vision, and Graphics[M], Springer,2003.
[12] Stanley Osher, Ronald Fedkiw, Level Set Methods and Dynamic ImplicitSurfaces[M], Springer,2002.
[13] G. Aubert, P. Kornprobst, Mathematical Problems in ImageProcessing(2nd)[M], Springer,2006.
[14] Fr′ed′eric Cao, Geometric Curve Evolution and Image Processing[M],Springer,2003.
[15] Yoshikazu Giga, Surface Evolution Equations_A Level Set Approach[M],Birkhauser Verlag,2006.
[16] A. Kuijper, The deep structure of Gaussian scale space images, Ph.Dthesis[D], Utrecht University,2002.
[17] L. Alvarez, F. Guichard, P.-L. Lions, and J.-M. Morel, Axioms andfundamental equations of image Processing[J], Tech. Rep., Ceremade,Universit’e Paris-Dauphine, Paris, France,1992.
[18] D. Marr, Early processing of visual information[J], Phil. Trans. Royal Soc(B),1976(27)：483:524.
[19] D. Marr, Vision. W.H. Freeman[M], New York,1982.
[20] R. Watt, Visual Processing: Computational, Psychophysical and CognitiveResearch[C]. London: Lawrence Erlbaum Associates,1988.
[21] S.G. Mallat, Multifrequency channel decompositions of images and waveletmodels[J], IEEE Trans. Acoustics, Speech and Signal Processing,1989(37):2091:2110.
[22] S.G. Mallat and S. Zhong, Characterization of signals from multi-scaleedges[J], IEEE Trans. Pattern Analysis and Machine Intell.,1972(14):710:723,1992. K. V. Mardia, Statistics of Directional Data. Academic Press,London.
[23] J. J. Koenderink, The structure of images[J], Biological Cybernetics,1984(50):363:370.
[24] L. M. J. Florack, B. M. ter Haar Romeny, J. J. Konenderink, and M. A.Viergever, Scale and the differential structure of images[J], Image andVision Computing,, July/August,1992(10):376:388.
[25] A. P. Witkin, Scale-space filtering, in Proc[C].8th Int. Joint Conf. Art.Intell.,(Karlsruhe, West Germany),1019:1022, Aug1983.
[26] J. J. Koenderink, Solid Shape[M]. MIT Press, Cambridge, Mass,1990.
[27] J. Weickert, S. Ishikawa, A. Imiya, Linear scale-space has first beenpeoposed in Japan[J], Journal of Mathematical Imaging and Vision,1999(10):237:252.
[28] T. Lindeberg, Scale-space behaviour of local extrema and blobs[J], J. ofMathematical Imaging and Vision, Mar,1992(1):65:99.
[29] T. Lindeberg, Detecting salient blob-like image structures and their scaleswith a scale-space primal sketch—A method for focus-of-attention[J], Int. J.of Computer Vision,1993,38(3):282:318.
[30] F. Mokhtarian and A. Mackworth, Scale-based description and recognitionof planar curves and two-dimensional objects[J], IEEE Trans. PatternAnalysis and Machine Intell.,1986(8):34:43.
[31] Harris, C., Stephens, M.: A combined corner and edge detector[C]. In: Proc.4th Alvey Vision Conf,1988,189:192.
[32] J. Babaud, A.P. Witkin, M. Baudin, and R. O. Duda, Uniqueness of theGaussian kernel for scale-space filtering[J], IEEE Trans. Pattern Analysisand Machine Intell.,1986,8(1):26:33.
[33] Lindeberg, T.: Scale-space theory: A basic tool for analysing structures atdifferent scales[J]. J. of Applied Statistics,1994,21(2):224:270.
[34] Lowe, D.: Distinctive image features from scale-invariant keypoints[J]. Int.J. Comput. Vision,2004,60(2):91:110.
[35] Mikolajczyk, K., Schmid, C.: Scale and affine invariant interest pointdetectors[J]. International Journal of Computer Vision,2004,60(1)63:86.
[36] Schmid, C., Mohr, R., Bauckhage, C.: Evaluation of interest pointdetectors[J]. Int. J. Comput. Vision,2000,37(2):151:172.
[37] Mikolajczyk, K., Schmid, C.: A performance evaluation of localdescriptors[J]. IEEE Transactions on Pattern Analysis&MachineIntelligence,2005,27(10):1615:1630.
[38] G. Sapiro and A. Tannenbaum, Affine invariant scale-space[J], Int. J. ofComputer Vision,1993,11(1):25:44.
[39] S. Grossberg, Neural dynamics of brightness perception: Features,boundaries, diffusion, and resonance, Percept[J]. Psychophys,1984,36(5):428:456.
[40] J. J. Clark, Singularity theory and phantom edges in scale-space[J], IEEETrans. Pattern Analysis and Machine Intell,1988,10(5):720:727.
[41] P. Perona and J. Malik, Scale-space and edge detection using anisotropicdiffusion[J], IEEE Trans. Pattern Analysis and Machine Intell,1990,12(7):629:639.
[42] Perona P, Malik J Scale-space and edge detection using anisotropicdiffusion[J]. IEEE transaction13on Pattern Analysis and MachineIntelligence,1990,12(7):629:639.
[43] D. Terzopoulos, Regularization of inverse visual problems involvingdiscontinuities[J], IEEE Trans. Pattern Analysis and Machine Intell.,1986,8:413:424.
[44]吴健,崔志明,徐婧等.基于Level Set模型的彩色脑血管图像骨架提取算法[J].计算机科学,2009,36(12):278:281.
[45] L. Alvarez, P. L. Lions, and J. M. Morel, Image Selective Smoothing andEdge Detection by Nonlinear Diffusion[J]. SIAM Journal on NumericalAnalysis,1992(129):845:866.
[46] S. OSher, J. A. Sethian, Front Propagating with Curvature Dependent Speed:Algorithms Based on Hamilton-Jacobi Formulations[J], Journal ofComputational Physics,1988,79:12:49.
[47] J. Shah, Segmentation by non-linear diffusion, in Proc[J]. IEEE Comp. Soc.Conf. on Computer Vision and Pattern Recognition,1991:202-207.
[48] J. Gomes and O. Faugeras, Reconciling Distance Functions and LevelSets[J], Journal of Visual Communication and Image Representation,2000,11(2):209:223.
[49] S. Osher, J. A. Sethian, Level Set Methods and Dynamic Implicit Surface[J],New York: Springer-Verlag,2002.
[50] H. K. Zhao, A Fast Sweeping Method for Eikonal equations[J], Mathematicsof Computation,2004,74(250):603:627.
[51] J. A. Sethian, Level Set Methods[M].Cambridge, U. K.: Cambridge Univ.Press,1996.
[52] F. Mokhtarian, M. Bober, Curvature Scale Space Representation: Theory,Applications, and MPEG-7Standardization[M], Kluwer AcademicPublishers,2003.
[53] M. Kass, A.Witkin, and D. Terzopoulos, Snakes: Active contour models[J],Internat. J. Comput.1988,1:321:331.
[54] T. Chan and L. Vese, Active Contours without Edges[J], IEEE Trans. ImageProcess.2001,10:266:277.
[55]钱惠敏，茅耀斌，王执铨.基于各向异性扩散的几种平滑算法比较及改进[J].南京理工大学学报,2007,31（5）：606:610.
[56] T. Chan, and L. Vese, Active Contours without Edges for Vector-valuedImges[J], Journal of Visual Communication and Image Representation11,2000:130~141.
[57] Nagao M. Edge Preserving Smoothing[J]. CGIP,1979,9:394-407.
[58] Rudin L I. Image,Numerical Analysis of singularities and shock filter[D].Ph.D dissertation Caltech, Pasadena, California,1987
[59]宋锦萍，职占江.图像分割方法研究[J].现代电子技术.2006,6：59-64.
[60] Osher S, Rudin L,Feature-oriented image enhancement using shockfilters[J]. SIAM J. on Numerical Analysis,1990,27:919~940.
[61] Evans L C, Spruck J. Motion of Level Sets by Mean Curvature[J]. DiffGeormetry,1991,16(9):200:257.
[62] Koepfler G, Lopez C, Morel J M.A Multiscale Algorithm for ImageSegmentation by Variationa Method[J]. SIAM journal on NumericalAnalysis,1993.
[63] Alvarez L, Guichard F, Lions P L, et al. Axioms and Fundamental Equationsof Image Processing Archive for Rational Mechanics and Analysis[J],1993,16(9):200:257.
[64] Kass M,Witkin A, Terzopoulos D. Snakes:active contour models[C].1st Int.Computer Vision Conf.,(777),1987
[65]陈一虎，叶正麟.一种改进的各向异性扩散图像去噪方法[J].计算机工程与应用,2008,44(13)：170:172.
[66] Baraldi, A., Parmiggiani, F., A Nagao-Matsuyama approach tohigh-resolution satellite image classification[J]. pp.749-758, IEEETransactions on Geoscience and Remote Sensing,1994,32(4).
[67] F. Catte, et al. Image selective smoothing and edge-detection by nonlineardiffusion[J]. SIAM Joumal on Numerical Analysis,1992,29(1):182:193.
[68]林宙辰,石青云.一个能去噪和保持真实感的各向异性扩散方程[J].计算机学报,1999,22(11):1133:1137.
[69] Zhouchen Lin, Qingyun Shi. An anisotropic diffusion PDE for noisereduction and thin edge preservation[C]. In: Proc Tenth InternationalConference on Image Analysis and Processing,Venice, Italy,1999
[70] Mannos J L, Sakrison D J. The effects of a visual fidelity criterion on theencoding of images[J]. IEEE Trans on Information Theeory,1974,20(4):525.
[71]丛舸.计算机视觉中的非线性尺度空间理论[D].中国科学院自动化研究所,北京,1997
[72] ZhouWang, AlanConradBovik. A universal image quality index[J]. IEEESignal Processing Letters,2002,9:81.
[73] ZhouWang, AlanConradBovik, HamidRahimSheikh, et al. Image qualityassessment: from error visibility to structural similarity[J]. IEEETransactions on Image Processing,2004,13(4):600.
[74]叶盛楠,苏开那,肖创柏等.基于结构信息提取的图像质量评价[J].电子学报,2008,36(5):856.
[75]杨春玲,旷开智,陈冠豪等.基于梯度的结构相似度的图像质量评价方法[J].华南理工大学学报:自然科学版,2006,34(9):22.
[76]李庆扬，数值分析基础教程.[M]北京:高等教育出版社,2001.
[77] V. I. Arnold, editor. Dynamical Systems V: Bifurcation Theory andCatastrophe Theory[J], volume5of Encyclopaedia of MathematicalSciences. Springer-Verlag, Berlin,1994.
[78] Zhihui Yang, Wenjuan Cui, Mengmeng Zhang. Deep Structure of GaussianScale Space. IEEE International Conference on Computer Science andSoftware Engineering. Pp.381-384,2008
[79] MengMeng Zhang, Zhihui Yang, Yang Yang, Yan Sun. BifurcationProperties of Image Gaussian Scale-space Model, will be published by theinternational journal Advanced Science Letter，May2012.
[80] D.P.L. Castrigiano, Catastrophe Theory[M], Westview Press,2004.
[81] J.J. Clark, Singularity theory and phantom edges in scale space[J], IEEETransactions on PAMI,1988, Vol.10(5):720:726.
[82] J. Damon, Local Morse theory for Gaussian blurred functions[J], inSporring et al.,1997,147:162.
[83] L.M.J. Florack, A.H. Salden, B.M.ter Haar Romeny, J.J. Koenderink,andM.A[J]. Viergever, Nonlinear scale-space, Image and VisionComputing,1995,3(4):279:294.
[84] M. Golubitsky, D.G. Schaeffer, Singularities and Groups in BifurcationTheory[J], Math. Sci.51, Springer-Verlag, New York, Appl1995,1.
[85] L.D. Griffin, A.C.F. Colchester, Superficial and deep structure in lineardiffusion scale space: isophotoes，critical points and separatrices[J], Imageand vision Computing,1995,13(7):543:557.
[86] R. Hummel, R. moniot, Reconstructions from aero crossing in scale space[J],IEEE Trans. om acoustics, speech and signal processing,1989,37(2):2111:2130.
[87] T. Iijima, Basic theory on normalization of a pattern(in case of typicalone-dimensional pattern)[J], Bulletin of Electrical Laboratory,1962,26:368–388.
[88] J. J. Koenderink. Image structure. In M. A. Viergever and A. Todd-Pokropek,editors, Mathematics and Computer Science in Medical Imaging, NATOASI Series[J]. F: Computer and Systems Sciences, Springer-Verlag,1987,39:67:104.
[89] A. Kuijper and L.M.J. Florack, Using Catastrophe Theory to DeriveTrees from Images[J], Journal of Mathematical Imaging and Vision,2005,23:219:238.
[90] A. Kuijper and L.M.J. Florack, M.A. Viergever, Scale space hierarchy[J],Journal of Mathematical Imaging and Vision,2003,18:169:189.
[91] M. Lillholm, M. Nielsen, Feature-based image analysis[J], InternationalJournal of Computer Vision,2003,52(2/3):73:95.
[92] T. Lindeberg, Feature detection with automatic scale selection[J],InternationalJournal of Computer Vision,1998,30(2):79:116.
[93] J. Mather, Stability of C∞mappings:Finitely determined map terms[J],Inst.Hautes E′tudes Sci. Publ. Math,1968,36:127:156.
[94] B. Platel, F.M.W. Kanters, L.M.J. Florack, etc. Using Multiscale top pointsin image matching[C], International Conference on Image Processing,389:392,2004.
[95] J. Sporring, M. Nielsen, L. Florack, P. Johansen(Eds.)[J], Gaussianscale-space theory, Kluwer Academic Publishers,1997.
[96] Han Chunming, Guo Huadong, Wang Changlin et a1. A New MultiscaleEdge Detection Techinique[J]. Tan Qulin Geoscience and Remote SensingSymposium. IGARSS'02.IEEE International,2002(6):3402:3404.
[97] Demigny D. An Optimal Linear Filtering for EdgeDetection[J]. IEEETransactions on Image Processing,2002,11(7):728:737.
[98]张萌萌，杨扬，杨志辉，李夏，白慧慧.改进的基于单尺度的医学图像边缘检测[J].《太原理工大学学报》，2011,42(4):329:333.
[99]李夏.尺度理论框架下的水平集方法研究及其应用[D].北方工业大学.2011
[100] Mengmeng Zhang, Zhihui Yang, Xia Li, Yang Yang. A novelzero-crossing edge detection method based on multi-scale space theory.2010IEEE10th International Conference on Signal Processing (ICSP),pp1036-1039. October2010.
[101] P.perona, J.Malik.Scale Space and Edge Detection Using AnisotropicDiffusion.Pro.IEEE Comp.Soc.Workshop on computer Vision[J], IEEEcomputer Society Press, Washington,1996:16:22.
[102] D.Marr, and E.C.Hildreth,“Theory of Edge Detection”[M], Proceedingsof the Royal Society of London B.207,1980:187:217.
[103]张玲华,朱幼莲.用零交叉方法对图像进行边界检测的研究与实现[J].南京邮电大学学报, Sep,1997,17(3):46:60.
[104] Robert s L. Machine perception of three dimensional solids. in: Tippet t J.ed. Opt ical an d Elect rooptical Information Processing[J]. Cambridge:M IT1965:159:197.
[105] Yuille, Alan L.; Poggio, Tomaso A.; Scaling Theorems for Zero CrossingsPattern Analysis and Machine Intelligence[J], IEEE Transactions,1986,PAMI-8(1):15:25.
[106] W. E. L. Crimson and E. C. Hildreth."Comments on 'digital step edgesfrom zero crossings of second directional derivatives'."[J]IEEE Trans.Pattern Anal. Machine Intell, Jan1985, PAMI-7:121:129.
[107] Thom, Rene. Structural Stability And Morphogenesis[M]. WestviewPress. Jan.1994
[108] E. Balmachnova, L.M.J. Florack, B. Platel and F.M.W. Kanters. Stabilityof Top-Points in Scale Space[C], Scale Space Methods in Computer Vision.Proceedings of the5th international conference on Scale Space2005,Germany, April2005:62:72.
[109] F.M.W. Kanters, B. Platel, L.M.J. Florack, and B.M. ter Haar Romeny.Content based image retrieval using multiscale top points[C]. InProceedings of the4th international conference on Scale Space Methods inComputer Vision (Isle of Skye,UK,), June2003:33:43.
[110] F.M.W. Kanters, L.M.J. Florack, B. Platel, and B.M. ter Haar Romeny.Image reconstruction from multiscale critical points[C]. In Proceedings ofthe4th international conference on Scale Space Methods in ComputerVision (Isle of Skye, UK,), June2003:464:478.
[111]张少辉，沈晓蓉，范耀祖。一种基于图像特征点提取及匹配的方法[J].北京航空航天大学学报,2008,34(5):516:519.
[112] Pal S K,King R A.Image enhancement using smoothing with fuzzy sets.IEEE Transactions on Systems[J],Man and Cybemetics,1981,11(7):494:501.
[113] Pal S K,King R A. On edge detection of X-ray images using fuzzy sets[J].IEEE Trans Patt Analand Machine Intell.,1983,5(1):69:77.
[114] Pal S K,Pal N R. A review of images segmentation techniques[J]. PatternRecognition.1993,26(9):1227:1294.
[115]赵世亮。基于小波域的模糊增强算法[J]。微处理机，2010，第2期：71:74.
[116]高丽，令晓明。一种基于模糊增强的多尺度边缘检测[J]。兰州交通大学学报；2008,27(4)：106:108.
[117]王文娟，韩峰，崔桐。一种基于模糊增强的Canny边缘检测方法[J]。内蒙古工业大学学报,2008,27(1):65:68.
[118]史卉萍，耿国华，周明全等。基于模糊集的图像增强[J]。微计算机信息,2008,24(8):291:292.
[119]董鸿燕，王磊，李吉成，沈振康。基于拉普拉斯金字塔分解的多尺度边缘检测[J]。光电工程,2007,34(7):135:140.
[120]廖志武.2-D骨架提取算法研究进展[J].四川师范大学学报（自然科学版）,2009,32(5):676:689.
[121]朱付平,田捷.基于Level Set方法的医学图像分割[J].软件学报,13(9):1866:1872.
[122] H. Blum, Biological Shape and Visual Science[J]. Journal of TheoreticalBiology,2002,38:205-287.
[123]徐婧.基于Level Set模型的脑血管图像骨架提取研究[D],苏州大学2009.
[124]陆东莹,庄天戈.基于Level Set方法的低对比度医学图像分割[J].上海交大学学报,2006,40(8).
[125]赵春江.图像目标识别中若干关键技术研究[D].上海交通大学.2006.
[126]赵春江,施文康,邓勇.具有鲁棒性的图像骨架提取方法[J].计算机应用,25(6):1305-1306.2005
[127]庞宵.信息熵蚁群算法在特征提取和图像识别中的应用[D],2008,
[128]鲍征烨,周卫平,舒华忠.一种基于水平集的骨架提取方法[J].生物医学工程研究,2007,26(2):187:190.
[129] Hongbo Jiang, Wenping Liu, Dan Wang et al. Connectivity-BasedSkeleton Extraction in Wireless Sensor Networks[J]. IEEE Transactions onParallel and Distributed Systems,2010,21(5):710:721.
[130] Tagliasacchi, A, Zhang, H,Cohen-Or, Curve Skeleton Extraction fromIncomplete Point Cloud[J]. ACM Transactions on Graphics,2009,28(7):1:9.
[131] Au, OKC, Tai, CL, Chu, HK et al. Skeleton extraction by meshcontraction[J]. ACM Transactions on Graphics,2008,27(3):1:10.
[132] Torsello A. Correcting curvature-density effects in the Hamilton-Jacobiskeleton [J]. IEEE Transactions on Image Processing,2006,15(4):877:891.
[133] Jianning Xu. A generalized discrete morphological skeleton transformwith multiple structuring elements for the extraction of structural shapecomponents[J]. IEEE Transactions on Image Processing,2003,12(12):1677:1686.
[134] Cornea, N.D.,Silver, D.,Min, Curve-Skeleton Properties, Applications,and Algorithms[J]. IEEE transactions on visualization and computergraphics,2007,13(3):530:548.
[135] F. Kanters, T. Denton, A. Shokoufandeh, etc., Combing different types ofscale space interest points using canonical sets, LNCS4485,2007:374:385.