变分水平集理论及其在医学图像分割中的应用
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摘要
医学图像分割是医学图像处理中的一个关键技术,其任务就是从医学图像中提取感兴趣的目标——解剖组织,是服务于临床医生的计算机辅助诊断的一个重要组成部分。
变分水平集方法,作为一类基于无参数几何活动轮廓模型的曲线演化图像分割方法,其实质是把一个低维空间上描述的问题嵌入到高一维的空间上分析,具备处理拓扑结构变化的轮廓演化能力,数值计算上也能获得稳定的结果。
因此,变分水平集方法成为目前十分热门的一类图像处理方法,特别是很好地适用于拓扑结构复杂多变、强噪声和低对比度的医学图像分析领域。本文的重点就是围绕变分水平集方法在医学图像分割的应用来展开研究。
本文的主要工作和创新点如下:
(1)构造了一个基于能量函数模型改进的水平集分割新模型。
综合基于区域信息的Chan-Vese(C-V)模型和基于边界信息的Li的模型的各自优势构造一个新的能量函数,然后通过变分法的分析以及水平集方法的应用来实现对CT图像的分割。新模型优于传统的水平集分割方法:首先将符号距离保持项引入C-V模型使得演化过程中无需重新初始化,提高了分割速度;演化轮廓也更加平滑,且对处理弱边界和强噪声有较好的鲁棒性;迭代中可选用多种格式如中心差分格式等而不再只是迎风格式,改善了方法的柔性。接着对分割后的图像采用了数学形态学方法提取肝脏轮廓,实验结果显示改进方法能自动准确实现肝脏提取任务。
(2)提出了一种基于水平集演化方程改进的单参数调试的快速水平集演化模型。
对于含有符号距离保持项的能量函数模型所对应的演化偏微分方程(PDE),用水平集函数梯度的模代换Dirac函数并且只保留长度项系数作为唯一的调试参数,构造出无需重新初始化且具有全局优化的新的单参数演化模型。该方法降低了对初始轮廓的依赖性,加上仅一个参数需调整使得分割处理更为方便。算法中停止迭代判定式的引入总结出了模型中的单参数取值规律,可以获得所期望的理想分割精度,并且能实现图像的自动分割。实验结果表明,该方法以更快的分割速度和更好的鲁棒性可应用于的复杂医学图像分割。
(3)建立了基于数值算法改进的一类基于半隐差分的水平集分割新方法。
针对改进C-V模型的无需重新初始化的新模型,数值演化中构造一类半隐式的有限差分格式,其收敛过程是无条件稳定的。该格式与加性算子分裂(AOS)格式相比,前者在迭代中无需矩阵求逆过程,缩短每次迭代时间,而后者的分解过程中要处理复杂的矩阵方程,致使分割速度提高有限。新的分割算法利用停止迭代判定式实现自动分割,使得分割效率更为明显提高。对合成图像、医学图像和视频图像的实验结果表明,该方法迭代步数更少,分割更快更准确,甚至能够满足视频跟踪实时性的需求。新方法更好的适应性和交互性为医学图像视频识别、医学图像三维重建提供了应用参考。
(4)提出了一种从能量建模和数值算法上均有创新的基于全局变分的分割方法。
首先利用图像区域信息建立基于后验概率的能量函数项,然后参照测地活动轮廓(GAC)模型建模方式,利用图像梯度信息得到基于全局变分的能量项,综合构造出一个新的图像分割模型。理论上借用凸函数的全局最小化的优化思想证明了新模型最小化解的存在性。在数值方法上运用全变分范数(TV-norm)的对偶公式构造了一个新的迭代格式,该格式不需求解PDE,也使得演化过程无需设置初始轮廓并且成功避开了重新初始化矫正过程。实验显示该方法收敛稳定、分割准确度好、参数调整简单并且分割速度快,能更好地处理低对比度医学模糊图像。
Medical image segmentation is a key technique and extracts the objective of interest, that is, the abnormal anatomic structures from medical images. It is an important component of computer-aided diagnosis used by the clinic doctors.
The variational level set method is a class of curve evolution methods based on the geometry active contour model without parameters. The level set segmentation, essentially, embeds the geometrical evolution problem of lower dimension into higher dimension. And it is numerically stable and capable of describing the topology change of the contour. And so it can segment the medical image with complex topological changes, high noise and lower contrast.
This dissertation is deeply developed around the improvement of level set segmentation method for the complicated medical images. The main contributions can be summarized as follows:
(1) An improved level set segmentation model without re-initialization is developed in the energy function respect. Liver segmentation on computed tomography (CT) images is a challenging task due to the anatomic complexity and the imaging system noise. So, we develop a region-based level-set approach, which has many advantages over the conventional active contour models by combining C-V model and Li's model. First, the improved model can get much smoother contour by adding a signed distance preserving term to evolution PDE and has good robustness to the presence of weak boundaries and strong noise. Second, the difference scheme can be chosen freely and enhance the algorithm flexibility. Third, we can obtain accurate extracted liver image by morphological filters. Therefore, our algorithm can be applied to detect the internal malignant structure of liver image. This modified level set function speeds up the segmentation process significantly. Experimental results show that the proposed method gives automatic and accurate liver structure segmentation.
(2) A new level set method for fast segmentation based on a single parameter is presented in the evolving PDE respect. The traditional level set methods for image segmentation need inevitably too many parameter adjustment and have usually lower computationally implementation. To solve this problem, the proposed method improves the Chan-Vese (C-V) PDE model based on the Mumford- Shah Model, by adding a penalized energy term and replacing the dirac function with the norm of level function gradient, and so it constructs a new PDE model with better globe optimization and no re-initialization. Besides, only the parameter of the length term is reserved in the model and an evolution criterion is introduced for the sake of the value rules of this single parameter as well as the accuracy of segmentation and semi-automation. The experimental results of synthesized and biomedical images show that the new method is faster and more robust. Moreover, the new method has more extensive adaptability on account of the zero level set function being set anyplace freely and the single parameter adjustment convenience.
(3) A new iteration algorithm for an improved level set method is proposed. At the beginning, to the given method based on C-V model, the adding operator split (AOS) algorithm is introduced for the evolution. And then we develop new semi-implicit schemes to shorten the time of every loop without matrix-inversion. An evolutional criterion for ending segmentation is introduced during the iterating process. Experimentations for synthesized, biomedical images and video sequences show that the new approach is faster and more accurate than the traditional level set methods, and satisfy the real time requirement. Moreover, the initial level set curve can be set freely and the parameter can be adjusted conveniently, and so the proposed approach can be applied in practice more flexibly and interactivity.
(4) A fast global minimization segmentation model based on total variation is presented around function modeling and algorithm constructing. First, an Active region contour model is developed by Maximum A-posterior Probability (MAP), and then a total variation model based on gradient information is constructed by the hint of geodesic active contour (GAC) model. So a new segmentation model is given by combining the two models. We establish theorems with proofs to determine the existence of the global minimum of this active contour model. From a numerical point of view, we propose a new practical way to solve the propagation problem toward object boundaries through a dual formulation of the total variation norm. It avoids the usual drawback of initializing and re-initializing. We apply our segmentation algorithms on medical images, and the model is found the convergence steady, good adaptability, high precision and well processing to lower contrast medical image.
变分水平集方法,作为一类基于无参数几何活动轮廓模型的曲线演化图像分割方法,其实质是把一个低维空间上描述的问题嵌入到高一维的空间上分析,具备处理拓扑结构变化的轮廓演化能力,数值计算上也能获得稳定的结果。
因此,变分水平集方法成为目前十分热门的一类图像处理方法,特别是很好地适用于拓扑结构复杂多变、强噪声和低对比度的医学图像分析领域。本文的重点就是围绕变分水平集方法在医学图像分割的应用来展开研究。
本文的主要工作和创新点如下:
(1)构造了一个基于能量函数模型改进的水平集分割新模型。
综合基于区域信息的Chan-Vese(C-V)模型和基于边界信息的Li的模型的各自优势构造一个新的能量函数,然后通过变分法的分析以及水平集方法的应用来实现对CT图像的分割。新模型优于传统的水平集分割方法:首先将符号距离保持项引入C-V模型使得演化过程中无需重新初始化,提高了分割速度;演化轮廓也更加平滑,且对处理弱边界和强噪声有较好的鲁棒性;迭代中可选用多种格式如中心差分格式等而不再只是迎风格式,改善了方法的柔性。接着对分割后的图像采用了数学形态学方法提取肝脏轮廓,实验结果显示改进方法能自动准确实现肝脏提取任务。
(2)提出了一种基于水平集演化方程改进的单参数调试的快速水平集演化模型。
对于含有符号距离保持项的能量函数模型所对应的演化偏微分方程(PDE),用水平集函数梯度的模代换Dirac函数并且只保留长度项系数作为唯一的调试参数,构造出无需重新初始化且具有全局优化的新的单参数演化模型。该方法降低了对初始轮廓的依赖性,加上仅一个参数需调整使得分割处理更为方便。算法中停止迭代判定式的引入总结出了模型中的单参数取值规律,可以获得所期望的理想分割精度,并且能实现图像的自动分割。实验结果表明,该方法以更快的分割速度和更好的鲁棒性可应用于的复杂医学图像分割。
(3)建立了基于数值算法改进的一类基于半隐差分的水平集分割新方法。
针对改进C-V模型的无需重新初始化的新模型,数值演化中构造一类半隐式的有限差分格式,其收敛过程是无条件稳定的。该格式与加性算子分裂(AOS)格式相比,前者在迭代中无需矩阵求逆过程,缩短每次迭代时间,而后者的分解过程中要处理复杂的矩阵方程,致使分割速度提高有限。新的分割算法利用停止迭代判定式实现自动分割,使得分割效率更为明显提高。对合成图像、医学图像和视频图像的实验结果表明,该方法迭代步数更少,分割更快更准确,甚至能够满足视频跟踪实时性的需求。新方法更好的适应性和交互性为医学图像视频识别、医学图像三维重建提供了应用参考。
(4)提出了一种从能量建模和数值算法上均有创新的基于全局变分的分割方法。
首先利用图像区域信息建立基于后验概率的能量函数项,然后参照测地活动轮廓(GAC)模型建模方式,利用图像梯度信息得到基于全局变分的能量项,综合构造出一个新的图像分割模型。理论上借用凸函数的全局最小化的优化思想证明了新模型最小化解的存在性。在数值方法上运用全变分范数(TV-norm)的对偶公式构造了一个新的迭代格式,该格式不需求解PDE,也使得演化过程无需设置初始轮廓并且成功避开了重新初始化矫正过程。实验显示该方法收敛稳定、分割准确度好、参数调整简单并且分割速度快,能更好地处理低对比度医学模糊图像。
Medical image segmentation is a key technique and extracts the objective of interest, that is, the abnormal anatomic structures from medical images. It is an important component of computer-aided diagnosis used by the clinic doctors.
The variational level set method is a class of curve evolution methods based on the geometry active contour model without parameters. The level set segmentation, essentially, embeds the geometrical evolution problem of lower dimension into higher dimension. And it is numerically stable and capable of describing the topology change of the contour. And so it can segment the medical image with complex topological changes, high noise and lower contrast.
This dissertation is deeply developed around the improvement of level set segmentation method for the complicated medical images. The main contributions can be summarized as follows:
(1) An improved level set segmentation model without re-initialization is developed in the energy function respect. Liver segmentation on computed tomography (CT) images is a challenging task due to the anatomic complexity and the imaging system noise. So, we develop a region-based level-set approach, which has many advantages over the conventional active contour models by combining C-V model and Li's model. First, the improved model can get much smoother contour by adding a signed distance preserving term to evolution PDE and has good robustness to the presence of weak boundaries and strong noise. Second, the difference scheme can be chosen freely and enhance the algorithm flexibility. Third, we can obtain accurate extracted liver image by morphological filters. Therefore, our algorithm can be applied to detect the internal malignant structure of liver image. This modified level set function speeds up the segmentation process significantly. Experimental results show that the proposed method gives automatic and accurate liver structure segmentation.
(2) A new level set method for fast segmentation based on a single parameter is presented in the evolving PDE respect. The traditional level set methods for image segmentation need inevitably too many parameter adjustment and have usually lower computationally implementation. To solve this problem, the proposed method improves the Chan-Vese (C-V) PDE model based on the Mumford- Shah Model, by adding a penalized energy term and replacing the dirac function with the norm of level function gradient, and so it constructs a new PDE model with better globe optimization and no re-initialization. Besides, only the parameter of the length term is reserved in the model and an evolution criterion is introduced for the sake of the value rules of this single parameter as well as the accuracy of segmentation and semi-automation. The experimental results of synthesized and biomedical images show that the new method is faster and more robust. Moreover, the new method has more extensive adaptability on account of the zero level set function being set anyplace freely and the single parameter adjustment convenience.
(3) A new iteration algorithm for an improved level set method is proposed. At the beginning, to the given method based on C-V model, the adding operator split (AOS) algorithm is introduced for the evolution. And then we develop new semi-implicit schemes to shorten the time of every loop without matrix-inversion. An evolutional criterion for ending segmentation is introduced during the iterating process. Experimentations for synthesized, biomedical images and video sequences show that the new approach is faster and more accurate than the traditional level set methods, and satisfy the real time requirement. Moreover, the initial level set curve can be set freely and the parameter can be adjusted conveniently, and so the proposed approach can be applied in practice more flexibly and interactivity.
(4) A fast global minimization segmentation model based on total variation is presented around function modeling and algorithm constructing. First, an Active region contour model is developed by Maximum A-posterior Probability (MAP), and then a total variation model based on gradient information is constructed by the hint of geodesic active contour (GAC) model. So a new segmentation model is given by combining the two models. We establish theorems with proofs to determine the existence of the global minimum of this active contour model. From a numerical point of view, we propose a new practical way to solve the propagation problem toward object boundaries through a dual formulation of the total variation norm. It avoids the usual drawback of initializing and re-initializing. We apply our segmentation algorithms on medical images, and the model is found the convergence steady, good adaptability, high precision and well processing to lower contrast medical image.
引文
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