基于偏微分方程的医学图像增强与分割方法研究
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摘要
医学图像分析是临床影像诊断和手术计划必不可少的重要工具。现代成像技术的快速发展,在提高目标分辨率的同时导致数据量和复杂度显著增加,进而对图像处理算法的性能和自动化程度提出更高的要求。图像增强与图像分割是医学图像分析的基本步骤,也是病灶定位和量化评估的必要环节,因而在整个计算机辅助诊疗系统中发挥着不可或缺的作用。
偏微分方程(PDE)方法以其坚实的数学基础和灵活的开放框架,有利于融合其它理论,开发新模型以满足不同成像模态、不同应用背景图像分析需求,已成为医学图像处理领域最流行的方法之一。作为偏微分方程图像分析理论的基石,非线性扩散、主动轮廓线模型以及水平集方法,在医学图像增强滤波以及分割中得到广泛应用。
本文主要针对当前医学图像增强滤波和分割中的若干难题,开展偏微分方程医学图像处理算法研究,为计算机辅助诊疗系统的开发奠定核心技术基础。具体工作如下：
1)在超声图像相关斑点噪声抑制方面,通过引入S型流导函数改进传统的噪声各项异性扩散模型SRAD,提出了新的斑点噪声各项异性扩散模型NSRAD。基本思想是结合不同区域图像特点,采用分段式扩散系数设计：即扩散系数在同质区域取值趋近常数1,在过渡区域为急剧下降函数,在强边界区域则几乎为O。受益于S型流导函数,新的NSRAD模型与原SRAD相比,具有更好的噪声抑制平滑能力,更好的细节与弱边界保留能力、以及更强的锐化强边界的能力。此外,扩散系数的同质区域尺度控制比率与速度控制系数可以调节,能够解决超声图像中存在的不同尺度同质区域的问题,使模型能够满足不同的应用要求。
2)针对传统多相Chan-Vese模型由于单纯依靠区域灰度均值的速度函数构造方法,容易引起波阵面错乱,可能导致各演化曲线被同一目标吸引,而产生错误结果的问题。本文在Chan-Vese模型的基础上提出了一种层级分裂的Chan-Vese主动轮廓线模型。基本思想是在基本模型基础上,加入层级分裂的思想,即在上层分割结果的基础上得到并标注子图像,根据需要,提取进一步需要分割的子图像进行更深层次的分割,直到分割结果满足不同层次的应用要求时终止。该模型除了保持了原Chan-Vese模型的所有优点之外,还具有自己独特的优点：第一,下级的子分割,仅仅在上一级分割后得到的子区域而不是在整个图像区域进行,演化曲线有了区域约束,解决了多相Chan-Vese模型中各相曲线同时演化时、由于没有相互约束、可能被同一目标所吸引,导致不能得到正确分割结果的缺点；第二,由于该模型在每次分割中只需考虑一个水平集函数,采用层级计算,相比于多相水平集方法更容易实现；第三,通过层级处理技术,逐步深入分割以满足不同层次的应用要求。
3)模糊聚类分割属于低层次图像处理技术,通常需要后续的边界连接等人工处理,不利于图像处理的自动化分析。而主动轮廓线是能够融合上层先验知识的封闭光滑的演化曲线。因此本文提出一种结合模糊聚类与主动轮廓线的模型,应用于图像,特别是医学图像分割。其基本思想是：耦合由模糊聚类提供的先验统计信息基础上计算得到的区域信息和边界信息,并增加一个人工约束项作为互斥项,保证提高曲线收敛能力的同时,避免区域重迭或者象素被漏掉的情况,从而建立起耦合边界与区域信息的最小化目标函数。通过极速下降法,求得最小化目标函数所对应的偏微分方程,并利用水平集技术求解。新模型具有如下优点：第一,解决了传统的模糊聚类分割存在边界不连续,而需要后续连接边界处理的问题；第二,水平集技术的利用,使新模型具有自由处理拓扑变形的能力；第三,新模型是在最小能量驱动下的光滑曲线演化,减少了模糊聚类分割中存在的孤立噪声点引起的伪区域；第四,该模型除了可以用于图像分割之外还可以应用于图像跟踪。
4)在分析血管图像微分检测原理的基础上,提出一种改进的GVF扩散3D血管图像增强滤波新方法,并且把最大通量法融入水平集框架,进行血管图像分割。新方法推导定义出新的血管增强滤波似然函数,具有如下优点：第一,摒弃了血管径向方向的特征值,从而对轴向密度变化不敏感,有利于保持轴向密度；第二,通过加入抑制因子,抑制边界重叠与模糊,锐化了边缘；第三,不需要使用参数,简化了计算并有利于实现。此外,新方法采用梯度向量场未归一化进行计算,与已存在的梯度向量场归一化方法相比,不仅比较完整的保留了血管的网络拓扑结构,而且克服了原梯度向量场归一化方法,在血管半径估计方面的不足,取得了更满意的增强滤波与分割结果。
本文的研究,旨在为偏微分方程在医学图像处理中的研究和应用添砖加瓦,主要在非线性扩散图像增强滤波,曲线演化图像分割,以及比较棘手的3D血管图像增强滤波与分割等内容上,做了更深一步的研究和扩充。文中提出的各种新模型和新方法,在为研究者提供广阔的视野以及良好的应用前景上具有重要的意义。
The rapid development of modern imaging techniques improves the resolution of the target, but leads to a significant increase in the amount of data and complexity simultaneously, which puts forward higher requirements on the performance and automaticity of image processing algorithms. As the basic steps to medical image analysis, image enhancement and segmentation are the necessary parts of lesion localization and quantitative assessment thus play an indispensable role in the computer-aided diagnosis and treatment system.
For the solid mathematical foundation and the flexible open framework, Partial Differential Equations (PDE) is conducive to integrating other theories and developing new models to meet the needs of different imaging modes and image analysis in different application background, which has become one of the most popular methods in the field of medical image processing. As a footstone of PDE image analysis theory, nonlinear diffusion, the active contour model and level set method have been widely used in image filtering and segmentation in medical image enhancement.
This dissertation concentrates on some difficulties in medical image enhancement and segmentation to carry out PDE research on medical image processing and lay the basis of the core technology for the development of computer-aided diagnosis system. The main work and innovations of this dissertation are listed as follows:
1) In the area of speckle noise suppression for ultrasound image, a new speckle reducing anisotropic diffuse (NSRAD) is proposed by introducing a sigmoid function to the diffusion coefficient in the traditional SRAD. The basic route of NSRAD is to combine the image characteristics of different regions using segmented diffusion coefficient, in which the diffusion coefficient approaches to a constant number one in the homogeneous regions, and declines rapidly in the transition regions, then approaches to zero in the strong boundary regions. Benefited by the sigmoid function in the new diffusion function, the NSRAD has better noise-suppressing ability, as well as the abilities of retaining details and weak edges, even sharp the strong boundary, compared to the traditional SRAD. Moreover, the control ratio of the homogeneous regions and the speed control coefficient are adjustable, which well meet different application requirements by solving the problem of different homogeneous regions scales in the ultrasound image.
2) The traditional multi-phase Chan-Vese model may generate error results as it solely relies on the speed function constructed by the regional mean gray, which may easily cause wave front disorder and lead several evolution curves attracting to the same goal. To solve this problem, a new hierarchical splitting Chan-Vese active contour method for image segmentation is presented. Its basic route is to add the hierarchical splitting segmentation to the basic Chan-Vese model. In detail, the segmentation results are obtained and the sub-images are marked by the upper level segmentation, and then further segmented by extracting the sub-images that need further segmentation. The whole process is terminated when the segmentation results meet different levels of application requirements. The model has all the advantages of the original Chan-Vese model, as well as the following advantages:Firstly, the lower level segmentation is only performed on the sub-image obtained by the upper level segmentation rather than the entire image area. This solves the problem that the segmentation results are incorrect due to the lack of mutual constraints in the multi-phase Chan-Vese model when several curves evolution attract to the same goal. Secondly, only one level set function is considered at each split computing. Accordingly, the mode is easier to achieve compared to the multiphase level set method. Thirdly, segmentation can been deepen gradually to meet different levels of application requirements by the hierarchical method.
3) Fuzzy classification of segmentation is a low-level image processing techniques and performs weakly in the automated image processing due to its requirement of subsequent manual boundary connections. However, the active contour is a closed smooth evolution curve, which can fuse the upper prior knowledge. Thus, in this dissertation, a new model is presented to integrate the fuzzy clustering and active contour for image segmentation, especially for medical image segmentation. The basic idea is as follows:by coupling the region information and the boundary one which are calculated by the priori statistical information gotten from the fuzzy clustering, and adding an artificial force not only to increase the robustness and the convergence rate by imposing the idea of mutually exclusive propagating curves but also to constrain regions not overlap and no pixels not assigned to any region, the minimum objective function coupling region and boundary one is established. The PDE corresponding to the defined objective function minimization is established by a gradient descent method. And a level set approach is used to solve the PDE system. The new model has the following advantages:firstly, the new model solves the problem that the boundary requires subsequent connections in the traditional fuzzy classification segmentation for its discontinuity boundary. Secondly, the new model has the ability to freely deal with the topological deformation due to the level set method. Thirdly, the new model is the curves evolution driven by the minimum energy, which can reduce the pseudo-region caused by the isolated noise points. Fourthly, the model can be used not only for image segmentation but also for image tracking.
4) Based on the analysis of vascular image differential detection principle, an improved GVF diffusion3D blood vessel image enhancement filter is presented, which fuses maximum flux method into the level set framework for vascular image segmentation. The new blood vessels likelihood function is derived and defined in the new method. It has the following advantages:Firstly, the new blood vessels likelihood function is insensitive to the changes in the axial density by abandoning the eigenvalues for the blood vessels radial direction, which well preserves the axial density. Secondly, the new blood vessels likelihood function can inhibit the overlapped and vague border, and sharpen the edge by adding the inhibitory factor. Thirdly, the new blood vessels likelihood function does not require any parameter, which is conducive to simplify implementation. Moreover, the new method calculated by non-normalized gradient vector field is better than the existing method calculated by the corresponding normalized vector field. The new method can achieve better effect in the image enhance filter, which can well keep the whole topology of the3D blood vessel network, and overcome the deficiency in estimating the blood vessel radius than the existing one.
This dissertation contributes to the application domains of PDE method in image processing and provides new techniques for nonlinear diffusion, curve evolution method and the intractable3D blood vessel enhancement filtering and segmentation. The new models and methods proposed in this dissertation widen the research horizon of contemporary and have a promising future.
偏微分方程(PDE)方法以其坚实的数学基础和灵活的开放框架,有利于融合其它理论,开发新模型以满足不同成像模态、不同应用背景图像分析需求,已成为医学图像处理领域最流行的方法之一。作为偏微分方程图像分析理论的基石,非线性扩散、主动轮廓线模型以及水平集方法,在医学图像增强滤波以及分割中得到广泛应用。
本文主要针对当前医学图像增强滤波和分割中的若干难题,开展偏微分方程医学图像处理算法研究,为计算机辅助诊疗系统的开发奠定核心技术基础。具体工作如下：
1)在超声图像相关斑点噪声抑制方面,通过引入S型流导函数改进传统的噪声各项异性扩散模型SRAD,提出了新的斑点噪声各项异性扩散模型NSRAD。基本思想是结合不同区域图像特点,采用分段式扩散系数设计：即扩散系数在同质区域取值趋近常数1,在过渡区域为急剧下降函数,在强边界区域则几乎为O。受益于S型流导函数,新的NSRAD模型与原SRAD相比,具有更好的噪声抑制平滑能力,更好的细节与弱边界保留能力、以及更强的锐化强边界的能力。此外,扩散系数的同质区域尺度控制比率与速度控制系数可以调节,能够解决超声图像中存在的不同尺度同质区域的问题,使模型能够满足不同的应用要求。
2)针对传统多相Chan-Vese模型由于单纯依靠区域灰度均值的速度函数构造方法,容易引起波阵面错乱,可能导致各演化曲线被同一目标吸引,而产生错误结果的问题。本文在Chan-Vese模型的基础上提出了一种层级分裂的Chan-Vese主动轮廓线模型。基本思想是在基本模型基础上,加入层级分裂的思想,即在上层分割结果的基础上得到并标注子图像,根据需要,提取进一步需要分割的子图像进行更深层次的分割,直到分割结果满足不同层次的应用要求时终止。该模型除了保持了原Chan-Vese模型的所有优点之外,还具有自己独特的优点：第一,下级的子分割,仅仅在上一级分割后得到的子区域而不是在整个图像区域进行,演化曲线有了区域约束,解决了多相Chan-Vese模型中各相曲线同时演化时、由于没有相互约束、可能被同一目标所吸引,导致不能得到正确分割结果的缺点；第二,由于该模型在每次分割中只需考虑一个水平集函数,采用层级计算,相比于多相水平集方法更容易实现；第三,通过层级处理技术,逐步深入分割以满足不同层次的应用要求。
3)模糊聚类分割属于低层次图像处理技术,通常需要后续的边界连接等人工处理,不利于图像处理的自动化分析。而主动轮廓线是能够融合上层先验知识的封闭光滑的演化曲线。因此本文提出一种结合模糊聚类与主动轮廓线的模型,应用于图像,特别是医学图像分割。其基本思想是：耦合由模糊聚类提供的先验统计信息基础上计算得到的区域信息和边界信息,并增加一个人工约束项作为互斥项,保证提高曲线收敛能力的同时,避免区域重迭或者象素被漏掉的情况,从而建立起耦合边界与区域信息的最小化目标函数。通过极速下降法,求得最小化目标函数所对应的偏微分方程,并利用水平集技术求解。新模型具有如下优点：第一,解决了传统的模糊聚类分割存在边界不连续,而需要后续连接边界处理的问题；第二,水平集技术的利用,使新模型具有自由处理拓扑变形的能力；第三,新模型是在最小能量驱动下的光滑曲线演化,减少了模糊聚类分割中存在的孤立噪声点引起的伪区域；第四,该模型除了可以用于图像分割之外还可以应用于图像跟踪。
4)在分析血管图像微分检测原理的基础上,提出一种改进的GVF扩散3D血管图像增强滤波新方法,并且把最大通量法融入水平集框架,进行血管图像分割。新方法推导定义出新的血管增强滤波似然函数,具有如下优点：第一,摒弃了血管径向方向的特征值,从而对轴向密度变化不敏感,有利于保持轴向密度；第二,通过加入抑制因子,抑制边界重叠与模糊,锐化了边缘；第三,不需要使用参数,简化了计算并有利于实现。此外,新方法采用梯度向量场未归一化进行计算,与已存在的梯度向量场归一化方法相比,不仅比较完整的保留了血管的网络拓扑结构,而且克服了原梯度向量场归一化方法,在血管半径估计方面的不足,取得了更满意的增强滤波与分割结果。
本文的研究,旨在为偏微分方程在医学图像处理中的研究和应用添砖加瓦,主要在非线性扩散图像增强滤波,曲线演化图像分割,以及比较棘手的3D血管图像增强滤波与分割等内容上,做了更深一步的研究和扩充。文中提出的各种新模型和新方法,在为研究者提供广阔的视野以及良好的应用前景上具有重要的意义。
The rapid development of modern imaging techniques improves the resolution of the target, but leads to a significant increase in the amount of data and complexity simultaneously, which puts forward higher requirements on the performance and automaticity of image processing algorithms. As the basic steps to medical image analysis, image enhancement and segmentation are the necessary parts of lesion localization and quantitative assessment thus play an indispensable role in the computer-aided diagnosis and treatment system.
For the solid mathematical foundation and the flexible open framework, Partial Differential Equations (PDE) is conducive to integrating other theories and developing new models to meet the needs of different imaging modes and image analysis in different application background, which has become one of the most popular methods in the field of medical image processing. As a footstone of PDE image analysis theory, nonlinear diffusion, the active contour model and level set method have been widely used in image filtering and segmentation in medical image enhancement.
This dissertation concentrates on some difficulties in medical image enhancement and segmentation to carry out PDE research on medical image processing and lay the basis of the core technology for the development of computer-aided diagnosis system. The main work and innovations of this dissertation are listed as follows:
1) In the area of speckle noise suppression for ultrasound image, a new speckle reducing anisotropic diffuse (NSRAD) is proposed by introducing a sigmoid function to the diffusion coefficient in the traditional SRAD. The basic route of NSRAD is to combine the image characteristics of different regions using segmented diffusion coefficient, in which the diffusion coefficient approaches to a constant number one in the homogeneous regions, and declines rapidly in the transition regions, then approaches to zero in the strong boundary regions. Benefited by the sigmoid function in the new diffusion function, the NSRAD has better noise-suppressing ability, as well as the abilities of retaining details and weak edges, even sharp the strong boundary, compared to the traditional SRAD. Moreover, the control ratio of the homogeneous regions and the speed control coefficient are adjustable, which well meet different application requirements by solving the problem of different homogeneous regions scales in the ultrasound image.
2) The traditional multi-phase Chan-Vese model may generate error results as it solely relies on the speed function constructed by the regional mean gray, which may easily cause wave front disorder and lead several evolution curves attracting to the same goal. To solve this problem, a new hierarchical splitting Chan-Vese active contour method for image segmentation is presented. Its basic route is to add the hierarchical splitting segmentation to the basic Chan-Vese model. In detail, the segmentation results are obtained and the sub-images are marked by the upper level segmentation, and then further segmented by extracting the sub-images that need further segmentation. The whole process is terminated when the segmentation results meet different levels of application requirements. The model has all the advantages of the original Chan-Vese model, as well as the following advantages:Firstly, the lower level segmentation is only performed on the sub-image obtained by the upper level segmentation rather than the entire image area. This solves the problem that the segmentation results are incorrect due to the lack of mutual constraints in the multi-phase Chan-Vese model when several curves evolution attract to the same goal. Secondly, only one level set function is considered at each split computing. Accordingly, the mode is easier to achieve compared to the multiphase level set method. Thirdly, segmentation can been deepen gradually to meet different levels of application requirements by the hierarchical method.
3) Fuzzy classification of segmentation is a low-level image processing techniques and performs weakly in the automated image processing due to its requirement of subsequent manual boundary connections. However, the active contour is a closed smooth evolution curve, which can fuse the upper prior knowledge. Thus, in this dissertation, a new model is presented to integrate the fuzzy clustering and active contour for image segmentation, especially for medical image segmentation. The basic idea is as follows:by coupling the region information and the boundary one which are calculated by the priori statistical information gotten from the fuzzy clustering, and adding an artificial force not only to increase the robustness and the convergence rate by imposing the idea of mutually exclusive propagating curves but also to constrain regions not overlap and no pixels not assigned to any region, the minimum objective function coupling region and boundary one is established. The PDE corresponding to the defined objective function minimization is established by a gradient descent method. And a level set approach is used to solve the PDE system. The new model has the following advantages:firstly, the new model solves the problem that the boundary requires subsequent connections in the traditional fuzzy classification segmentation for its discontinuity boundary. Secondly, the new model has the ability to freely deal with the topological deformation due to the level set method. Thirdly, the new model is the curves evolution driven by the minimum energy, which can reduce the pseudo-region caused by the isolated noise points. Fourthly, the model can be used not only for image segmentation but also for image tracking.
4) Based on the analysis of vascular image differential detection principle, an improved GVF diffusion3D blood vessel image enhancement filter is presented, which fuses maximum flux method into the level set framework for vascular image segmentation. The new blood vessels likelihood function is derived and defined in the new method. It has the following advantages:Firstly, the new blood vessels likelihood function is insensitive to the changes in the axial density by abandoning the eigenvalues for the blood vessels radial direction, which well preserves the axial density. Secondly, the new blood vessels likelihood function can inhibit the overlapped and vague border, and sharpen the edge by adding the inhibitory factor. Thirdly, the new blood vessels likelihood function does not require any parameter, which is conducive to simplify implementation. Moreover, the new method calculated by non-normalized gradient vector field is better than the existing method calculated by the corresponding normalized vector field. The new method can achieve better effect in the image enhance filter, which can well keep the whole topology of the3D blood vessel network, and overcome the deficiency in estimating the blood vessel radius than the existing one.
This dissertation contributes to the application domains of PDE method in image processing and provides new techniques for nonlinear diffusion, curve evolution method and the intractable3D blood vessel enhancement filtering and segmentation. The new models and methods proposed in this dissertation widen the research horizon of contemporary and have a promising future.
引文
[1]Duncan J S, Ayache N. Medical image analysis:progress over two decades and the challenges ahead. IEEE Transactions on Pattern Analysis and Machine Intelligence,2000,22(1):85-105
[2]Ritter F, Boskamp T, Homeyer A. Medical image analysis. IEEE Pulse,2011, 6(2):60-70
[3]Suri J S, Singh S, Reden L. Computer vision and pattern recognition techniques for 2-D and 3-D MR cereral cortical segmentation (Part I):A state-of-the-art review. Pattern Analysis & Applications,2002,5(1):46-76
[4]Suri J S. Computer vision pattern recognition and image processing in left ventricle segmentation:the last 50 years. Pattern Analysis & Applications, 2000,3(3):209-242
[5]Black M, Sapiro G, Marimont D, et al. Robustanisotropic diffusion. IEEE Transactions on Image Processing,1998,3(7):421-432
[6]Perona P, Malik J. Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence,1990,7(12): 629-639
[7]Caselles V, Catte F, Coll T, et al. A geometric model for active contours in Image Processing. Numerische Mathematik,1993,66(1):1-31
[8]Caselles V, Kimmel R, Sapiro G. Geodesic active contours. International Journal of Computer Vision,1997,22(1):61-79
[9]Kichenassamy S, Kumar A, Olver P, et al. Gradient flows and geometric Active contour models. In:5th International Conference on Computer Vision, Boston, USA,1995,810-815
[10]肖昌炎.基于偏微分方程的医学图像恢复和分割方法研究：[上海交通大学博士论文].上海：上海交通大学,2005,40-59
[11]Osher S, Sethian J. Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics,1988,1(79):12-49
[12]Sapiro G. Geometric partial differential equations and image analysis世界图书出版公司,2003
[13]Weickert J. A review of nonlinear diffusion filtering. Scale-space theory in computer vision, Lecture Notes in computer science,1997, (1252):3-28
[14]Guy C, Ffytche D. An introduction to the principles of medical imaging. Imperial College. London Press,2005,267-307
[15]Gillespie J E, Jackson A. MRI and CT of the brain. London, Arnold press,2000
[16]Duncan J, Ayache N. Medical image analysis:Progress over two decades and the challenges ahead. IEEE Transactions on Pattern Analysis and Machine Intelligence,2000,1(22):85-106
[17]McInerney T, Terzopoulos D. Deformable models in medical image analysis:A survey. Medical Image Analysis,1996,2(1):91-108
[18]Gallagher N C, Wise G L. A theoretical analysis of properties of the median filters. IEEE Transactions on Acoustics Speech, Signal Processing,1981,29(1): 1136-1141
[19]Brejl M, Sonka M. Object localization and border detection criteria design in edge-based image segmentation:Automated learning from examples. IEEE. Transactions on Medical Imaging,2000,10(19):973-985
[20]Roberts L G. Machine perception of three dimensional solids. Optical and Electro-Optical Information Processing,1965,159-197
[21]Sobel I, Feldman G. A 3x3 isotropic gradient operator for image processing. Pattern Classification and Scene Analysis,1973,271-272
[22]Prewitt J M S. Object enhancement and extraction in picture processing and psy-chopictorics. Academic Press, New York,1970,75-150
[23]王强,胡维平,陆志敏,等Hough变换实时检测算法研究.计算机工程与设计,2001,22(3)：76-80
[24]王晓明,顾晓东,刘健.基于张量的图像边缘检测及滤波.中国图形图像学报.2002,7(8)：780-782
[25]Cheriet M, Said J N, Suen C Y. A recursive thresholding technique for image segmentation. IEEE Transactions on Image Processing,1998,6(7):918-921
[26]Lie W N. Automatic target segmentation by locally adaptive image thresh-olding. IEEE Transactions on Image Processing,1995,7(4):1036-1041
[27]R L de Queiroz, Z Fan, Tran T D. Optimizing block-thresholding segmentation for multiplayer compression of compound images. IEEE Transactions on Image Processing,2000,9(9):1461-1471
[28]Sonka M, Hlavac V, Boyle R. Image processing analysis and machine vision. London:Chapman & Hall,1999
[29]Wu H S, Barba J, Gil J A. A parametric fitting algorithm for segmentation of cell image. IEEE Transactions on Biomedical Engineering,1998,3(45): 400-410
[30]Gabor D. Information theory in electron microscopy. Laboratory Investigation. 1965,14(6):801-807
[31]Jain A K. Partial differential equations and finite difference methods in image processing. Part I:Image representation. Journal of Optimization Theory and Applications.1977,23(1):65-91
[32]Witkin A P. Scale space filtering. In:Proceedings of the 8th International Joint Conference on Artificial Intelligence, Karlsruhe, Germany,1983,1019-1021
[33]Koenderink J J. The structure of images. Biological Cybernetics,1984,50(5): 363-370
[34]Rudin L, Osher S, Fatemi E. Nonlinear total variation based noise removal algorithms. Physica D,1992,1-4(60):259-268
[35]Hummel A. Representations based on zero-crossing in scale-space. IEEE Computer Vision and Pattern Recognition, Morgan Kaufmann Publishers Inc, 1986,204-209
[36]Osher S, Sethian J A. Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics,1988,79(1):12-49
[37]Czerwinski, R N, Jones D L, JR O'Brien W R. Line and boundary detection in speckle images. IEEE Transactions on Image Processing,1998,12(7): 1700-1714
[38]Czerwinski R N, Jones D L, JR O'Brien W R. Detection of lines and boundaries in speckle images-application to medical ultrasound. IEEE Transactions on Image Processing,1999,2(18):126-136
[39]Pathak S D, Chalana V, Haynor D R, et al. Edge-guided boundary delineation in prostate ultrasound images. IEEE Transactions on Medical Imaging,2000, 12(19):1211-1219
[40]Xiao C Y, Zhang S, Chen Y Z. A diffusion stick method for speckle suppression in ultrasonic images. Pattern Recognition Letters,2004,16(22): 1867-1877
[41]Yu Y and Acton S. Speckle reducing anisotropic diffusion. IEEE Transactions on Image Processing,2002,11(11):1260-1270
[42]Lee J S. Speckle analysis and smoothing of synthetic aperture radar images. Computer Graphics and Image Processing,1981,1(17):24-32
[43]Lee J S. Digital image enhancement and noise filtering by using local statistics. IEEE Transactions on Pattern Analysis and Machine Intelligence,1980, PAMI-2(2):165-168
[44]Lee J S. Refined filtering of image noise using local statistics. Computer Graphics and Image Processing,1981,4(15):380-389
[45]Frost S, Stiles J A. Shanmuggam K S, et al. A model for radar images and its application for adaptive digital filtering of multiplicative noise. IEEE Transactions on Pattern Analysis and Machine Intelligence,1982,4(2): 157-165
[46]Nagao M. Edge pressing smoothing. Computer Graphics and Image Processing, 1979,4(9):394-407
[47]Koenderink J J. The structure of image. Biological Cybernetics,1984,50(5): 363-370
[48]Kass M, witkin A, Terzoppoulous D. Snakes:Actives contour models. International Journal of Computer Vision,1988,1(4):321-331
[49]高薪,刘来福,黄海洋.基于PDE和几何曲率流驱动扩散的图形分析与处理.数学进展,2003,32(3)：285-293
[50]Caselles, V, Morel, J. Introduction to the partions and geometry-driven difffusion in image processing and analysis. IEEE Transactions on Image Processing,1998,7(3):269-272
[51]Siddiqi K, Lanziere Y, Tannenbaum A. Area and length minimizing flows for shape segmentation. IEEE Transactions on Image Processing,1998,7(3): 433-443
[52]Mumford D, Shah J. Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics,1989, (42):577-685
[53]Chan T, Vese L. Active contours without edges. IEEE Transactions on Image Processing,2001,2(10):141-151
[54]Paragios N, Deriche R. Coupled geodesic active regions for image segmentation:A level set approach. In:European Conference in Computer Vision, Dublin, Ireland,2000, (2):224-240
[55]Paragios N, Deriche R. Geodesic active regions for supervised texture segmentation. International Journal of Computer Vision,2002,223-247
[56]Osher S, Sethian J A. Fronts propagating with curvature-dependent speed: algorithms based on Hamitlton-Jacobi formulations. Journal of Computational Physics,1988,7(9):12-49
[57]Sethian J A. Mathematical curvature and the evolution of fronts. Communication of Physics,1985,4(101):185-197
[58]Sethian J A. A review of numerical algorithms for hypersurfaces moving with curvature dependent speed. Journal of Diffrential Geometry,1989, (31): 131-167
[59]陈武凡,李超,陈和晏.空域中退化图像恢复的有效算法.计算机学报,1999,22(12)：1267-1271
[60]陈武凡.一阶与二阶常微分方程系统的隐式和半隐式解法.国防科技大学学报,1987,1,10-18
[61]高鑫,刘来福.基于水平集曲率的图像滤噪与增强.北京师范大学学报,2001,37(1)：5-9
[62]高鑫,刘来福,黄海洋.基于PDE和几何曲率流驱动扩散的图像分析与处理.数学进展,2003,32(3)：285-294
[63]马峰,唐泽圣,夏绍玮.多尺度几何活动曲线及MR图像边界提取.计算机学报,2000,23(8)：829-834
[64]于洪川,王德军,唐泽圣,等.水平集演化中的奇点与改进的窄带算法.软件学报,2003,14(4)：811-817
[65]朱付平,田捷,林瑶,等.基于Level Set方法的医学图像分割.软件学报,2002,13(9)：1866-187
[66]夏利民,谷士文,沈新权,等.基于水平集模型的3D表面重建.小型微型计算机系统,2001,22(12)：1445-1448
[67]李俊,杨新,施鹏飞.基于Mumford-Shah模型的快速水平集图像分割方法.计算机学报,2002,25(11)：1175-1183
[68]王峥,杨新,李俊,等.水平集方法中曲线跟踪算法的改进.上海交通大学学报,2002,36(12)：1833-1836
[69]肖昌炎,赵永明,张素,等.水平集方法中符号距离函数快速重构.信号处理,2003,19(6)：551-555
[70]林亚忠,程跃斌,陈武凡.一种水平集分割的快速算法.南方医科大学学报,2006,26(6)：764-766
[71]金大年,杨丰,陈武凡.基于贝叶斯分类的水平集MR图像分割方法.中国医学物理学杂志,2005,22(4)：568-571
[72]Chen G, Yang K, Chen R, et al. A new study on coupling with shock filters and anisotropic diffusion for image enhancement and denosing. ACTA Laser Biology Sinica,2008,17(6):829-833
[73]Jin Z M, Yang X P. Analysis of a hyperbolic parabolic type system for image inpainting. Acta Mathematica Sinica, English Series,2012,8(28):1663-1676
[74]Zheng-Meng Jin, Xiao-Ping Yang. Analysis of a hyperbolic parabolic type system for image inpainting. Journal of Mathematical Analysis and Applications,2010,3(62):415-426
[75]Xu D, Zhou W, Shu H, et al. Level-set based bone removal and cerebral vessel detection in CTA, Chinese Journal of Electronics,2008,17(2):325-329
[76]刘国才,王耀南.多层Mumford-Shah向量值图像分割、去噪与重建模型.自动化学报,2007,33(6)：602-607
[77]刘国才,王耀南.基于水平集逐层迭代算法的多层Mumford-Shah图像分割、去噪与重建模型.自动化学报,2006,32(4)：534-54
[78]刘国才,王耀南.基于知识的多层Mumford-Shah向量值图像分割模型.自动化学报,2009,35(4)：356-363
[79]王耀南,刘国才,全惠敏.基于多层Mumford-Shah向量模型的彩色视乳头图像杯盘重建、分割与度量.生物医学工程学报,2007,24(6)：97-105
[80]李灿飞,王耀南,刘国才A New Splitting Active Contours Framework Based on Chan-Vese Piecewise Smooth Model自动化学报,2008,34(6)：659-664.
[81]程建刚,田捷,何余良,等.基于非线性扩散滤波的指纹增强算法.自动化学报,2004,30(6)：854-862
[82]李波,苏志勋,刘秀平.基于Lp范数的局部自适应偏微分方程图像恢复.自动化学报,2008,34(8)：849-853
[83]李灿飞,王耀南,肖昌炎,等.用于超声斑点噪声滤波的各向异性扩散新模型.自动化学报,2012,38(3)：412-419.
[84]周雪,胡卫明.融合颜色和增量形状先验的目标轮廓跟踪.自动化学报,2009,35(11)：1394-1402
[85]刘博,黄剑华,唐降龙,等.结合全局概率密度差异与局部灰度拟合的超声图像分割.自动化学报,2010,36(7)：951-959
[86]郑钰辉,张建伟,陈允杰,等.加权型曲率保持PDE图像滤波方法.自动化学报,2011,37(10)：1175-1182
[87]马少贤,江成顺.基于四阶偏微分方程的盲图像恢复模型.中国图象图形学报,2010,15(1)：26-30
[88]唐磊,赵春霞,王鸿南,等.路面图像增强的多偏微分方程融合法.中国图象图形学报,2008,13(9)：1661-1666
[89]张曈,朱虹,张然,等.复小波域维纳滤波与偏微分扩散相结合的图像去噪 方法.中国图象图形学报,2009,14(5)：848-852
[90]崔丽娜,潘振宽,魏伟波,等.基于PM模型的曲面去噪变分水平集方法.中国图象图形学报,2010,15(3)：361-366
[91]魏颖,李军,徐心和.基于圆形约束C-V水平集的肺部CT图像病灶分割.中国图象图形学报,2008,13(6)：1148-1152
[92]詹天明,韦志辉,张建伟,等.脑MR图像分割和偏移场矫正的耦合水平集模型.中国图象图形学报,2011,16(11)：2017-2023
[93]董建园,郝重阳,齐敏.基于策略演化水平集的医学图像快速分割.中国图象图形学报,2009,14(8)：1689-1695
[94]Zhang W G, Zhang Q, Yang C S. Edge detection with multiscale products for SAR image despeckling. Electronics Letters,2012,4(48):211-212
[95]Kuan D T, Sawchuk A, Strand T C, et al. Adaptive restoration of images with speckle. Signal Processing & Analysis,1987,3(35):373-383
[96]Frost A V S, Stiles J A, Shanmugan K S, et al. A model for radar images and its application to adaptive digital filtering of multiplicative noise. IEEE Transactions on Pattern Analysis and Machine Intelligence,1982, (PAMI-4): 157-165
[97]Hao X, Gao S, Gao X. A novel multiscale nonlinear threshold methodfor ultrasonic speckle suppressing. IEEE Transactions on Medical Image,1999, 18(9):787-794
[98]Karaman M, Kutay M A, Bozdagi G. An adaptive speckle suppression filter for medical ultrasonic imaging. IEEE Transactions on on Medical Imaging,1995, 2(14):283-292
[99]Yu Y, Acton S T. Edge detection in ultrasound imagery using the instantaneous coefficient of variation. IEEE Transactions on Image Proccessing,12(13): 1640-1655
[100]Santiago A F, Gonzalo V S F. Marcos M F, et al. Automatic noise estimation in images using local statistics, Additive and multiplicative cases. Image and Vision Computing,2009,6(27):756-770
[101]Sakrison D. On the role of observer and a distortion measure in image transmission. IEEE Transactions on Communications,1977,11(25):1251-1267
[102]Wang Z, Bovik A. A universal quality index. IEEE Signal Processing Letters, 2002,3(9):81-84
[103]Wang Z, Bovik A C, Sheikh H R, et al. Image quality assessment:From error visibility to structural similarity. IEEE Transactions on Image Processing,2004, 4(13):600-612
[104]Terzopoulos D, Witkin A. Constraints on deformable models:Recovering shape and non-rigid motion. Artificial Intelligence,1988,1(36):91-123
[105]Caselles V, Kimmel R, Sapiro G. Geodesic active contours. In:IEEE International Conference on Computer Vision, Boston, USA,1995,694-699
[106]Ali S, Madabhushi A. An integrated region-, boundary-, shape-based active contour for multiple object overlap resolution in histological imagery. IEEE Transactions on Medical Imaging,2012,7(31):1448-1460
[107]H Xiao, Xu C, Prince J L. A topology preserving level set method for geometric deformable models. IEEE Transactions on Pattern Analysis and Machine Intelligence,2003,25(6):755-768
[108]Jasjit S Suri. Two-dimensional fast magnetic resonance brain segmentation. IEEE Engineering in Medicine and Biology,2001,20(4):84-95
[109]Sundaramoorthi G, Yezzi A, Mennucci AC. Coarse-to-fine segmentation and tracking using sobolev active contours. IEEE Transactions on Pattern Analysis and Machine Intelligence,2008,5(30):851-864
[110]Vese L, Chan T F. A multiphase level set framework for image segmentation using Mumford and Shah model. International Journal of Computer Vision, 2002,50(3):271-293
[111]Geomes J, Faugeras O. Level sets and distance functions. In:Proceedings of the 6th European Conference on Computer Vision, Berlin, Geman, Springer Verlag Press,2000, (1842):588-602
[112]Sussman M, Smereka P, Osher S. A level set approach for computing solutions to incompressible two-phase flow. Journal of Computational Physics,1994, 114(1):146-159
[113]Zadeh L A, Fuzzy sets, information and control,1965, (8):338-353
[114]Pham D L. Robust fuzzy segmentation of magnetic resonance images. In: Proceedings of the 14th IEEE Symposium on Computer-Based Medical Systems, IEEE press,2001,127-131
[115]Balafar M A, Ramli A R, Saripan M I, et al. New multi-scale medical image segmentation based on fuzzy c-mean (FCM). In:IEEE conference on innovative technologies in intelligent systems and industrial applications, IEEE Press,2008,66-70
[116]Balafar M A, Ramli A R, Saripan M. I, et al. Medical image segmentation using anisotropic filter, user interaction and fuzzy C-mean (FCM). In: Proceedings of the 4th international conference on intelligent computing, Springer press,2008, (15):169-176
[117]Roy S, Agarwal H, Carass A. Fuzzy c-means with variable compactness. Proceedings the 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro,2008,452-455
[118]Sarkar S, Boyer K L. On optimal infinite impulse response edge detection filter. IEEE Transactions on Pattern Analysis and Machine Intelligence,1991,13(11): 1154-1170
[119]Malladi R, Sethian J A, Vemur B C. Shape modeling with front propagation:a level set approach. IEEE Transactions on Pattern Analysis and Machine Intelligence,1995,2(17):158-175
[120]Siddiqi K, Lauziere Y B, Tannenbaum A, et al. Area and length minimizing flows for shape segmentation. IEEE Transactions on Image Processing,1998, (7):433-443
[121]付忠良.图像阈值选取方法-Otsu方法的推广.计算机应用,2000,20(5)：37-39
[122]Yan C X, Sang N, Zhang T X. Local entropy-based transition region extraction and thresholding. Pattern Recognition Letters,2003,24,2935-2941
[123]李俊.基于曲线演化的图像分割方法及应用研究.[上海交通大学博士学位论文].上海：上海交通大学.2001：1-10,15-37,57-64
[124]Gath I, Geva A B. Unsupervised optimal fuzzy clustering. IEEE transactions on Pattern Analysis and Machine Intelligence,1989,7(11):773-780
[125]Zhao H K, Chan T, Merriman B, et al. A variational level set approach to multiphase motion. Journal of Computational Physics,1996,1(127):179-195
[126]Samson C, L. Blanc-Feraud, G. Aubert, et al. A level set method for image classification. In:International Conference Scale-Space Theories in Computer Vision, Corfu, Greece,1999,306-317
[127]Yan W, Li B, Fairhurst M. Robust road modeling and tracking using condensation. IEEE Transactions on Intelligent Transportation Systems,2008, 4(9):570-579
[128]Wu B F, Lin C T, Chen Y-L. Dynamic calibration and occlusion handling algorithms for lane tracking. Industrial Electronics, IEEE Transactions on, 2009,5(56):1757-1773
[129]Metaxas D, Terzopoulos D. Constrained deformable superquadrics and nonrigid motion tracking. Computer Vision and Pattern Recognition,1991, 337-343
[130]Siew W C, Kah P S, Li M A. Lips contour detection and tracking using watershed region-based active contour model and modified H∞. IEEE Transactions on Circuits and Systems for Video Technology,2012,6(22): 869-874
[131]Meyer F, Bouthemy P. Region-based tracking using affine motion models in long image sequences. CVGIP:Image Understanding,1994,2(60):119-140
[132]Kass M, Witkin A, Terzopoulos D. Snakes:active contour models. International Journal of Computer Vision,1998,1(4):321-332
[133]Yue F, Erdem AT, tekalp A M. Tracking visible boundary of objects using occlusion adaptive motion snake. IEEE Transactions on Image Processing, 2000,12(9):2051-2060
[134]Cohen D. On active contour models and balloons. CVGIP:Image Understanding,1991,2(53):211-218
[135]Caselles V, Coll B. Snakes in movement. Society for Industrial and Applied Mathematics:Journal of Numerical Analysis,1996,6(33):2445-2456
[136]Bascle B, Bouthemy P, Deriche R, et al. Tracking complex primitives in an Image sequence. In:Proceedings of the 12th IAPR International Conference on Computer Vision & Image Processing,1994, (1):426-431
[137]Ivins J, Porrill J. Constrained active region models for fast tracking in color Image sequences, image and vision computing,1998,72(1):54-71
[138]Paragios N, R Deriche. Unifying boundary and region based information for geodesic active tracking. In:IEEE Conference Computer Vision and Pattern Recognition, Fort Collins, Colorado,1999, (2):300-305
[139]Stoel B C, Stolk J M. Optimization and standardization of lung densitometry in the assessment of pulmonary emphysema. Investigative Radiology,2004, 39(11):681-688
[140]Bauer C, Bischof H, Beichel R. Segmentation of airways based on gradient vector flow. International Workshop on Pulmonary Image Analysis, Medical Image Computing and Computer Assisted Intervention,2009,191-201
[141]Bauer C, Bischof H. A novel approach for detection of tubular objects and its application to medical image analysis. DAGM Symposium on Pattern Recognition,2008,163-172
[142]Florack L M J, ter Haar Romeny B M, Koenderink J J, et al. Scale and the differential structure of images. Image and Vision Computing,1992,6(10): 376-388
[143]Koenderink J J. The structure of images. Biological Cybernetics,1984,5(50): 363-370
[144]Frangi A F, Niessen W J, Vincken K L, et al. Multiscale vessel enhancement filtering. Medical Image Computing and Computer-Assisted Intervention, Lecture Notes in Computer Science, Berlin, German, Springer press,1998, (1496):130-137
[145]Xiao C, Staring M, Shamonin D, et al. A strain energy method for 3D vessel enhancement with application to pulmonary CT images. Medical Image Analysis,2011,1(15):112-124
[146]Pock T, Beichel R, Bischof H. A novel robust tube detection filter for 3d centerline extraction. International Journal of Computer Vision,2003,2-3(55): 155-179
[147]Bennink H E, Assen H C, Wee R, et al. A novel 3D multi-scale lineness filter for vessel detection. Medical Image Computing and Computer-Assisted Intervention, Springer press,2007,10(Part2):436-443
[148]Sato Y, Nakajima S, Shiraga N, et al. Three-dimensional multi-scale line filter for segmentation and visualization of curvilinear structures in medical images. Medical Image Analysis,1998,2(2):143-168
[149]Xu C, Prince J L. Snakes, shapes, and gradient vector flow. IEEE Transactions on on Image Processing,1998,7(3):359-369
[150]Sato Y, Nakajima S, Shiraga N, et al. Three-dimensional multi-scale line filter for segmentation and visualization of curvilinear structures in medical images. Medical Image Analysis,1998,2(2):143-168
[151]Krissian K, Malandain G, Ayache N, et al. Model-based detection of tubular structures in 3D images. Computer Vision and Image Understanding,2000, 2(80):130-171
[152]http://lmi.bwh.harvard.edu/research/vascular/SyntheticVessels/SyntheticVessel Images.html
[153]Vasilevskiy A, Siddiqi K. Flux maximizing geometric flows. IEEE Transactions on Pattern Analysis and Machine Intelligence,2002,12(24): 1565-1578
[2]Ritter F, Boskamp T, Homeyer A. Medical image analysis. IEEE Pulse,2011, 6(2):60-70
[3]Suri J S, Singh S, Reden L. Computer vision and pattern recognition techniques for 2-D and 3-D MR cereral cortical segmentation (Part I):A state-of-the-art review. Pattern Analysis & Applications,2002,5(1):46-76
[4]Suri J S. Computer vision pattern recognition and image processing in left ventricle segmentation:the last 50 years. Pattern Analysis & Applications, 2000,3(3):209-242
[5]Black M, Sapiro G, Marimont D, et al. Robustanisotropic diffusion. IEEE Transactions on Image Processing,1998,3(7):421-432
[6]Perona P, Malik J. Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence,1990,7(12): 629-639
[7]Caselles V, Catte F, Coll T, et al. A geometric model for active contours in Image Processing. Numerische Mathematik,1993,66(1):1-31
[8]Caselles V, Kimmel R, Sapiro G. Geodesic active contours. International Journal of Computer Vision,1997,22(1):61-79
[9]Kichenassamy S, Kumar A, Olver P, et al. Gradient flows and geometric Active contour models. In:5th International Conference on Computer Vision, Boston, USA,1995,810-815
[10]肖昌炎.基于偏微分方程的医学图像恢复和分割方法研究：[上海交通大学博士论文].上海：上海交通大学,2005,40-59
[11]Osher S, Sethian J. Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics,1988,1(79):12-49
[12]Sapiro G. Geometric partial differential equations and image analysis世界图书出版公司,2003
[13]Weickert J. A review of nonlinear diffusion filtering. Scale-space theory in computer vision, Lecture Notes in computer science,1997, (1252):3-28
[14]Guy C, Ffytche D. An introduction to the principles of medical imaging. Imperial College. London Press,2005,267-307
[15]Gillespie J E, Jackson A. MRI and CT of the brain. London, Arnold press,2000
[16]Duncan J, Ayache N. Medical image analysis:Progress over two decades and the challenges ahead. IEEE Transactions on Pattern Analysis and Machine Intelligence,2000,1(22):85-106
[17]McInerney T, Terzopoulos D. Deformable models in medical image analysis:A survey. Medical Image Analysis,1996,2(1):91-108
[18]Gallagher N C, Wise G L. A theoretical analysis of properties of the median filters. IEEE Transactions on Acoustics Speech, Signal Processing,1981,29(1): 1136-1141
[19]Brejl M, Sonka M. Object localization and border detection criteria design in edge-based image segmentation:Automated learning from examples. IEEE. Transactions on Medical Imaging,2000,10(19):973-985
[20]Roberts L G. Machine perception of three dimensional solids. Optical and Electro-Optical Information Processing,1965,159-197
[21]Sobel I, Feldman G. A 3x3 isotropic gradient operator for image processing. Pattern Classification and Scene Analysis,1973,271-272
[22]Prewitt J M S. Object enhancement and extraction in picture processing and psy-chopictorics. Academic Press, New York,1970,75-150
[23]王强,胡维平,陆志敏,等Hough变换实时检测算法研究.计算机工程与设计,2001,22(3)：76-80
[24]王晓明,顾晓东,刘健.基于张量的图像边缘检测及滤波.中国图形图像学报.2002,7(8)：780-782
[25]Cheriet M, Said J N, Suen C Y. A recursive thresholding technique for image segmentation. IEEE Transactions on Image Processing,1998,6(7):918-921
[26]Lie W N. Automatic target segmentation by locally adaptive image thresh-olding. IEEE Transactions on Image Processing,1995,7(4):1036-1041
[27]R L de Queiroz, Z Fan, Tran T D. Optimizing block-thresholding segmentation for multiplayer compression of compound images. IEEE Transactions on Image Processing,2000,9(9):1461-1471
[28]Sonka M, Hlavac V, Boyle R. Image processing analysis and machine vision. London:Chapman & Hall,1999
[29]Wu H S, Barba J, Gil J A. A parametric fitting algorithm for segmentation of cell image. IEEE Transactions on Biomedical Engineering,1998,3(45): 400-410
[30]Gabor D. Information theory in electron microscopy. Laboratory Investigation. 1965,14(6):801-807
[31]Jain A K. Partial differential equations and finite difference methods in image processing. Part I:Image representation. Journal of Optimization Theory and Applications.1977,23(1):65-91
[32]Witkin A P. Scale space filtering. In:Proceedings of the 8th International Joint Conference on Artificial Intelligence, Karlsruhe, Germany,1983,1019-1021
[33]Koenderink J J. The structure of images. Biological Cybernetics,1984,50(5): 363-370
[34]Rudin L, Osher S, Fatemi E. Nonlinear total variation based noise removal algorithms. Physica D,1992,1-4(60):259-268
[35]Hummel A. Representations based on zero-crossing in scale-space. IEEE Computer Vision and Pattern Recognition, Morgan Kaufmann Publishers Inc, 1986,204-209
[36]Osher S, Sethian J A. Fronts propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics,1988,79(1):12-49
[37]Czerwinski, R N, Jones D L, JR O'Brien W R. Line and boundary detection in speckle images. IEEE Transactions on Image Processing,1998,12(7): 1700-1714
[38]Czerwinski R N, Jones D L, JR O'Brien W R. Detection of lines and boundaries in speckle images-application to medical ultrasound. IEEE Transactions on Image Processing,1999,2(18):126-136
[39]Pathak S D, Chalana V, Haynor D R, et al. Edge-guided boundary delineation in prostate ultrasound images. IEEE Transactions on Medical Imaging,2000, 12(19):1211-1219
[40]Xiao C Y, Zhang S, Chen Y Z. A diffusion stick method for speckle suppression in ultrasonic images. Pattern Recognition Letters,2004,16(22): 1867-1877
[41]Yu Y and Acton S. Speckle reducing anisotropic diffusion. IEEE Transactions on Image Processing,2002,11(11):1260-1270
[42]Lee J S. Speckle analysis and smoothing of synthetic aperture radar images. Computer Graphics and Image Processing,1981,1(17):24-32
[43]Lee J S. Digital image enhancement and noise filtering by using local statistics. IEEE Transactions on Pattern Analysis and Machine Intelligence,1980, PAMI-2(2):165-168
[44]Lee J S. Refined filtering of image noise using local statistics. Computer Graphics and Image Processing,1981,4(15):380-389
[45]Frost S, Stiles J A. Shanmuggam K S, et al. A model for radar images and its application for adaptive digital filtering of multiplicative noise. IEEE Transactions on Pattern Analysis and Machine Intelligence,1982,4(2): 157-165
[46]Nagao M. Edge pressing smoothing. Computer Graphics and Image Processing, 1979,4(9):394-407
[47]Koenderink J J. The structure of image. Biological Cybernetics,1984,50(5): 363-370
[48]Kass M, witkin A, Terzoppoulous D. Snakes:Actives contour models. International Journal of Computer Vision,1988,1(4):321-331
[49]高薪,刘来福,黄海洋.基于PDE和几何曲率流驱动扩散的图形分析与处理.数学进展,2003,32(3)：285-293
[50]Caselles, V, Morel, J. Introduction to the partions and geometry-driven difffusion in image processing and analysis. IEEE Transactions on Image Processing,1998,7(3):269-272
[51]Siddiqi K, Lanziere Y, Tannenbaum A. Area and length minimizing flows for shape segmentation. IEEE Transactions on Image Processing,1998,7(3): 433-443
[52]Mumford D, Shah J. Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics,1989, (42):577-685
[53]Chan T, Vese L. Active contours without edges. IEEE Transactions on Image Processing,2001,2(10):141-151
[54]Paragios N, Deriche R. Coupled geodesic active regions for image segmentation:A level set approach. In:European Conference in Computer Vision, Dublin, Ireland,2000, (2):224-240
[55]Paragios N, Deriche R. Geodesic active regions for supervised texture segmentation. International Journal of Computer Vision,2002,223-247
[56]Osher S, Sethian J A. Fronts propagating with curvature-dependent speed: algorithms based on Hamitlton-Jacobi formulations. Journal of Computational Physics,1988,7(9):12-49
[57]Sethian J A. Mathematical curvature and the evolution of fronts. Communication of Physics,1985,4(101):185-197
[58]Sethian J A. A review of numerical algorithms for hypersurfaces moving with curvature dependent speed. Journal of Diffrential Geometry,1989, (31): 131-167
[59]陈武凡,李超,陈和晏.空域中退化图像恢复的有效算法.计算机学报,1999,22(12)：1267-1271
[60]陈武凡.一阶与二阶常微分方程系统的隐式和半隐式解法.国防科技大学学报,1987,1,10-18
[61]高鑫,刘来福.基于水平集曲率的图像滤噪与增强.北京师范大学学报,2001,37(1)：5-9
[62]高鑫,刘来福,黄海洋.基于PDE和几何曲率流驱动扩散的图像分析与处理.数学进展,2003,32(3)：285-294
[63]马峰,唐泽圣,夏绍玮.多尺度几何活动曲线及MR图像边界提取.计算机学报,2000,23(8)：829-834
[64]于洪川,王德军,唐泽圣,等.水平集演化中的奇点与改进的窄带算法.软件学报,2003,14(4)：811-817
[65]朱付平,田捷,林瑶,等.基于Level Set方法的医学图像分割.软件学报,2002,13(9)：1866-187
[66]夏利民,谷士文,沈新权,等.基于水平集模型的3D表面重建.小型微型计算机系统,2001,22(12)：1445-1448
[67]李俊,杨新,施鹏飞.基于Mumford-Shah模型的快速水平集图像分割方法.计算机学报,2002,25(11)：1175-1183
[68]王峥,杨新,李俊,等.水平集方法中曲线跟踪算法的改进.上海交通大学学报,2002,36(12)：1833-1836
[69]肖昌炎,赵永明,张素,等.水平集方法中符号距离函数快速重构.信号处理,2003,19(6)：551-555
[70]林亚忠,程跃斌,陈武凡.一种水平集分割的快速算法.南方医科大学学报,2006,26(6)：764-766
[71]金大年,杨丰,陈武凡.基于贝叶斯分类的水平集MR图像分割方法.中国医学物理学杂志,2005,22(4)：568-571
[72]Chen G, Yang K, Chen R, et al. A new study on coupling with shock filters and anisotropic diffusion for image enhancement and denosing. ACTA Laser Biology Sinica,2008,17(6):829-833
[73]Jin Z M, Yang X P. Analysis of a hyperbolic parabolic type system for image inpainting. Acta Mathematica Sinica, English Series,2012,8(28):1663-1676
[74]Zheng-Meng Jin, Xiao-Ping Yang. Analysis of a hyperbolic parabolic type system for image inpainting. Journal of Mathematical Analysis and Applications,2010,3(62):415-426
[75]Xu D, Zhou W, Shu H, et al. Level-set based bone removal and cerebral vessel detection in CTA, Chinese Journal of Electronics,2008,17(2):325-329
[76]刘国才,王耀南.多层Mumford-Shah向量值图像分割、去噪与重建模型.自动化学报,2007,33(6)：602-607
[77]刘国才,王耀南.基于水平集逐层迭代算法的多层Mumford-Shah图像分割、去噪与重建模型.自动化学报,2006,32(4)：534-54
[78]刘国才,王耀南.基于知识的多层Mumford-Shah向量值图像分割模型.自动化学报,2009,35(4)：356-363
[79]王耀南,刘国才,全惠敏.基于多层Mumford-Shah向量模型的彩色视乳头图像杯盘重建、分割与度量.生物医学工程学报,2007,24(6)：97-105
[80]李灿飞,王耀南,刘国才A New Splitting Active Contours Framework Based on Chan-Vese Piecewise Smooth Model自动化学报,2008,34(6)：659-664.
[81]程建刚,田捷,何余良,等.基于非线性扩散滤波的指纹增强算法.自动化学报,2004,30(6)：854-862
[82]李波,苏志勋,刘秀平.基于Lp范数的局部自适应偏微分方程图像恢复.自动化学报,2008,34(8)：849-853
[83]李灿飞,王耀南,肖昌炎,等.用于超声斑点噪声滤波的各向异性扩散新模型.自动化学报,2012,38(3)：412-419.
[84]周雪,胡卫明.融合颜色和增量形状先验的目标轮廓跟踪.自动化学报,2009,35(11)：1394-1402
[85]刘博,黄剑华,唐降龙,等.结合全局概率密度差异与局部灰度拟合的超声图像分割.自动化学报,2010,36(7)：951-959
[86]郑钰辉,张建伟,陈允杰,等.加权型曲率保持PDE图像滤波方法.自动化学报,2011,37(10)：1175-1182
[87]马少贤,江成顺.基于四阶偏微分方程的盲图像恢复模型.中国图象图形学报,2010,15(1)：26-30
[88]唐磊,赵春霞,王鸿南,等.路面图像增强的多偏微分方程融合法.中国图象图形学报,2008,13(9)：1661-1666
[89]张曈,朱虹,张然,等.复小波域维纳滤波与偏微分扩散相结合的图像去噪 方法.中国图象图形学报,2009,14(5)：848-852
[90]崔丽娜,潘振宽,魏伟波,等.基于PM模型的曲面去噪变分水平集方法.中国图象图形学报,2010,15(3)：361-366
[91]魏颖,李军,徐心和.基于圆形约束C-V水平集的肺部CT图像病灶分割.中国图象图形学报,2008,13(6)：1148-1152
[92]詹天明,韦志辉,张建伟,等.脑MR图像分割和偏移场矫正的耦合水平集模型.中国图象图形学报,2011,16(11)：2017-2023
[93]董建园,郝重阳,齐敏.基于策略演化水平集的医学图像快速分割.中国图象图形学报,2009,14(8)：1689-1695
[94]Zhang W G, Zhang Q, Yang C S. Edge detection with multiscale products for SAR image despeckling. Electronics Letters,2012,4(48):211-212
[95]Kuan D T, Sawchuk A, Strand T C, et al. Adaptive restoration of images with speckle. Signal Processing & Analysis,1987,3(35):373-383
[96]Frost A V S, Stiles J A, Shanmugan K S, et al. A model for radar images and its application to adaptive digital filtering of multiplicative noise. IEEE Transactions on Pattern Analysis and Machine Intelligence,1982, (PAMI-4): 157-165
[97]Hao X, Gao S, Gao X. A novel multiscale nonlinear threshold methodfor ultrasonic speckle suppressing. IEEE Transactions on Medical Image,1999, 18(9):787-794
[98]Karaman M, Kutay M A, Bozdagi G. An adaptive speckle suppression filter for medical ultrasonic imaging. IEEE Transactions on on Medical Imaging,1995, 2(14):283-292
[99]Yu Y, Acton S T. Edge detection in ultrasound imagery using the instantaneous coefficient of variation. IEEE Transactions on Image Proccessing,12(13): 1640-1655
[100]Santiago A F, Gonzalo V S F. Marcos M F, et al. Automatic noise estimation in images using local statistics, Additive and multiplicative cases. Image and Vision Computing,2009,6(27):756-770
[101]Sakrison D. On the role of observer and a distortion measure in image transmission. IEEE Transactions on Communications,1977,11(25):1251-1267
[102]Wang Z, Bovik A. A universal quality index. IEEE Signal Processing Letters, 2002,3(9):81-84
[103]Wang Z, Bovik A C, Sheikh H R, et al. Image quality assessment:From error visibility to structural similarity. IEEE Transactions on Image Processing,2004, 4(13):600-612
[104]Terzopoulos D, Witkin A. Constraints on deformable models:Recovering shape and non-rigid motion. Artificial Intelligence,1988,1(36):91-123
[105]Caselles V, Kimmel R, Sapiro G. Geodesic active contours. In:IEEE International Conference on Computer Vision, Boston, USA,1995,694-699
[106]Ali S, Madabhushi A. An integrated region-, boundary-, shape-based active contour for multiple object overlap resolution in histological imagery. IEEE Transactions on Medical Imaging,2012,7(31):1448-1460
[107]H Xiao, Xu C, Prince J L. A topology preserving level set method for geometric deformable models. IEEE Transactions on Pattern Analysis and Machine Intelligence,2003,25(6):755-768
[108]Jasjit S Suri. Two-dimensional fast magnetic resonance brain segmentation. IEEE Engineering in Medicine and Biology,2001,20(4):84-95
[109]Sundaramoorthi G, Yezzi A, Mennucci AC. Coarse-to-fine segmentation and tracking using sobolev active contours. IEEE Transactions on Pattern Analysis and Machine Intelligence,2008,5(30):851-864
[110]Vese L, Chan T F. A multiphase level set framework for image segmentation using Mumford and Shah model. International Journal of Computer Vision, 2002,50(3):271-293
[111]Geomes J, Faugeras O. Level sets and distance functions. In:Proceedings of the 6th European Conference on Computer Vision, Berlin, Geman, Springer Verlag Press,2000, (1842):588-602
[112]Sussman M, Smereka P, Osher S. A level set approach for computing solutions to incompressible two-phase flow. Journal of Computational Physics,1994, 114(1):146-159
[113]Zadeh L A, Fuzzy sets, information and control,1965, (8):338-353
[114]Pham D L. Robust fuzzy segmentation of magnetic resonance images. In: Proceedings of the 14th IEEE Symposium on Computer-Based Medical Systems, IEEE press,2001,127-131
[115]Balafar M A, Ramli A R, Saripan M I, et al. New multi-scale medical image segmentation based on fuzzy c-mean (FCM). In:IEEE conference on innovative technologies in intelligent systems and industrial applications, IEEE Press,2008,66-70
[116]Balafar M A, Ramli A R, Saripan M. I, et al. Medical image segmentation using anisotropic filter, user interaction and fuzzy C-mean (FCM). In: Proceedings of the 4th international conference on intelligent computing, Springer press,2008, (15):169-176
[117]Roy S, Agarwal H, Carass A. Fuzzy c-means with variable compactness. Proceedings the 5th IEEE International Symposium on Biomedical Imaging: From Nano to Macro,2008,452-455
[118]Sarkar S, Boyer K L. On optimal infinite impulse response edge detection filter. IEEE Transactions on Pattern Analysis and Machine Intelligence,1991,13(11): 1154-1170
[119]Malladi R, Sethian J A, Vemur B C. Shape modeling with front propagation:a level set approach. IEEE Transactions on Pattern Analysis and Machine Intelligence,1995,2(17):158-175
[120]Siddiqi K, Lauziere Y B, Tannenbaum A, et al. Area and length minimizing flows for shape segmentation. IEEE Transactions on Image Processing,1998, (7):433-443
[121]付忠良.图像阈值选取方法-Otsu方法的推广.计算机应用,2000,20(5)：37-39
[122]Yan C X, Sang N, Zhang T X. Local entropy-based transition region extraction and thresholding. Pattern Recognition Letters,2003,24,2935-2941
[123]李俊.基于曲线演化的图像分割方法及应用研究.[上海交通大学博士学位论文].上海：上海交通大学.2001：1-10,15-37,57-64
[124]Gath I, Geva A B. Unsupervised optimal fuzzy clustering. IEEE transactions on Pattern Analysis and Machine Intelligence,1989,7(11):773-780
[125]Zhao H K, Chan T, Merriman B, et al. A variational level set approach to multiphase motion. Journal of Computational Physics,1996,1(127):179-195
[126]Samson C, L. Blanc-Feraud, G. Aubert, et al. A level set method for image classification. In:International Conference Scale-Space Theories in Computer Vision, Corfu, Greece,1999,306-317
[127]Yan W, Li B, Fairhurst M. Robust road modeling and tracking using condensation. IEEE Transactions on Intelligent Transportation Systems,2008, 4(9):570-579
[128]Wu B F, Lin C T, Chen Y-L. Dynamic calibration and occlusion handling algorithms for lane tracking. Industrial Electronics, IEEE Transactions on, 2009,5(56):1757-1773
[129]Metaxas D, Terzopoulos D. Constrained deformable superquadrics and nonrigid motion tracking. Computer Vision and Pattern Recognition,1991, 337-343
[130]Siew W C, Kah P S, Li M A. Lips contour detection and tracking using watershed region-based active contour model and modified H∞. IEEE Transactions on Circuits and Systems for Video Technology,2012,6(22): 869-874
[131]Meyer F, Bouthemy P. Region-based tracking using affine motion models in long image sequences. CVGIP:Image Understanding,1994,2(60):119-140
[132]Kass M, Witkin A, Terzopoulos D. Snakes:active contour models. International Journal of Computer Vision,1998,1(4):321-332
[133]Yue F, Erdem AT, tekalp A M. Tracking visible boundary of objects using occlusion adaptive motion snake. IEEE Transactions on Image Processing, 2000,12(9):2051-2060
[134]Cohen D. On active contour models and balloons. CVGIP:Image Understanding,1991,2(53):211-218
[135]Caselles V, Coll B. Snakes in movement. Society for Industrial and Applied Mathematics:Journal of Numerical Analysis,1996,6(33):2445-2456
[136]Bascle B, Bouthemy P, Deriche R, et al. Tracking complex primitives in an Image sequence. In:Proceedings of the 12th IAPR International Conference on Computer Vision & Image Processing,1994, (1):426-431
[137]Ivins J, Porrill J. Constrained active region models for fast tracking in color Image sequences, image and vision computing,1998,72(1):54-71
[138]Paragios N, R Deriche. Unifying boundary and region based information for geodesic active tracking. In:IEEE Conference Computer Vision and Pattern Recognition, Fort Collins, Colorado,1999, (2):300-305
[139]Stoel B C, Stolk J M. Optimization and standardization of lung densitometry in the assessment of pulmonary emphysema. Investigative Radiology,2004, 39(11):681-688
[140]Bauer C, Bischof H, Beichel R. Segmentation of airways based on gradient vector flow. International Workshop on Pulmonary Image Analysis, Medical Image Computing and Computer Assisted Intervention,2009,191-201
[141]Bauer C, Bischof H. A novel approach for detection of tubular objects and its application to medical image analysis. DAGM Symposium on Pattern Recognition,2008,163-172
[142]Florack L M J, ter Haar Romeny B M, Koenderink J J, et al. Scale and the differential structure of images. Image and Vision Computing,1992,6(10): 376-388
[143]Koenderink J J. The structure of images. Biological Cybernetics,1984,5(50): 363-370
[144]Frangi A F, Niessen W J, Vincken K L, et al. Multiscale vessel enhancement filtering. Medical Image Computing and Computer-Assisted Intervention, Lecture Notes in Computer Science, Berlin, German, Springer press,1998, (1496):130-137
[145]Xiao C, Staring M, Shamonin D, et al. A strain energy method for 3D vessel enhancement with application to pulmonary CT images. Medical Image Analysis,2011,1(15):112-124
[146]Pock T, Beichel R, Bischof H. A novel robust tube detection filter for 3d centerline extraction. International Journal of Computer Vision,2003,2-3(55): 155-179
[147]Bennink H E, Assen H C, Wee R, et al. A novel 3D multi-scale lineness filter for vessel detection. Medical Image Computing and Computer-Assisted Intervention, Springer press,2007,10(Part2):436-443
[148]Sato Y, Nakajima S, Shiraga N, et al. Three-dimensional multi-scale line filter for segmentation and visualization of curvilinear structures in medical images. Medical Image Analysis,1998,2(2):143-168
[149]Xu C, Prince J L. Snakes, shapes, and gradient vector flow. IEEE Transactions on on Image Processing,1998,7(3):359-369
[150]Sato Y, Nakajima S, Shiraga N, et al. Three-dimensional multi-scale line filter for segmentation and visualization of curvilinear structures in medical images. Medical Image Analysis,1998,2(2):143-168
[151]Krissian K, Malandain G, Ayache N, et al. Model-based detection of tubular structures in 3D images. Computer Vision and Image Understanding,2000, 2(80):130-171
[152]http://lmi.bwh.harvard.edu/research/vascular/SyntheticVessels/SyntheticVessel Images.html
[153]Vasilevskiy A, Siddiqi K. Flux maximizing geometric flows. IEEE Transactions on Pattern Analysis and Machine Intelligence,2002,12(24): 1565-1578