脑血管疾病发生在脑部血管，因颅内血液循环障碍而造成脑组织损害的一组疾病，以急性发病居多，多表现为半身不遂，言语障碍等。脑血管的精确分割是血管病变准确可视化、诊断和定量分析的关键步骤。尤其是细小的低对比度血管，通常因血液的低流速而在图像上显示不足，但这类血管对疾病的诊断意义重大。虽然目前医学影像所获得的序列切片给医生提供了直接的解剖结构相关的体数据信息，但这会导致主观解释而非定性特征，而可视化能帮助医生克服由主观因素而带来的差异性，并提高诊断阶段治疗方案的快速确定。微血管是非常小的血管，位于新陈代谢活跃的组织及器官中，负责将血液输送到身体的各个部分。微血管其半径太小而无法在医学影像数据中显示，但微血管的体系结构对健康和疾病有深远的影响。了解制约微血管的形态原理以及其分支结构对揭开其传输的方式，以及提高我们对大脑中的血管信号和功能活动之间的关系的理解是必不可少的。而微血管的结构复杂性及多样性吸引了越来越多医生和研究人员的兴趣。目前大多数血管几何形态的生成都是在物体内部构建的，而物体表面血管的分布情况以及生成方式则很少得到关注。物体内部血管构建的算法是基于传统的欧式距离的，但对于物体表面真实的血管来说，欧式距离则并非合适的选择。测地线作为空间中两点的局域最短路径，测地距离比欧氏距离更能反映空间的点的拓扑结构。而且物体表面在是三维空间中是一个二维流形，因而用刻画流形最短距离的测地线更符合物体形状的存在形式。在开展脑血管分割、几何形态生成以及流形布线方式的研究中，本论文的主要工作有：1) 提出基于Allen-Cahn方程和似然函数模型的脑血管分割模型。该模型首先用AC方程代替平均曲率来表示演化曲线的长度，提高了计算效率；将整幅图像的密度分布特性作为全局项添加到能量泛函中，并用水平集策略来驱动曲线到达血管的真实边界，提高了分割准确性。实验结果表明，该模型对低对比度血管图像的分割性能与其他模型相比得到了较好的提高和改善。2) 提出基于相场模型和多角度投影三维脑血管分割算法。该算法首先对三维体数据进行多角度最大密度投影，同时记录拥有最大值密度值的体素空间位置；之后采用相场模型对投影图进行血管像素的提取；然后将得到的血管像素点反投影到体数据中相应的血管体素空间坐标位置，将体数据中的血管体素尽可能的保留；最后将血管体素进行绘制得到血管三维立体结构。实验结果表明，该方法提高了脑血管三维血管分割的准确度，降低了仅靠有限混合模型或水平集方法的局限性。3) 提出基于局部最优图的最短布线路径的微血管几何形态生成算法。该算法基于局部最优图和Murray定律构建了每个分支节点的成本函数，结合几何量之间的平衡因子来共同确定血管的分支结构。与已有的算法相比，该算法生成的微血管结构更具有灵活性。4) 提出基于测地距离的二维流形血管生成算法。该算法采用测地距离为度量的空间血管生成算法为来生成脑皮层血管的分支结构。与已有的算法相比，本方法采用真实反映空间的点拓扑结构的测地距离为准则，更符合表面血管的布线方式。

Cerebrovascular disease referred to as a cerebrovascular accident (CVA), cerebrovascular insult (CVI), or colloquially brain attack, is the loss of brainfunction due to disturbance in the blood supply to the brain, especially when it occurs quickly. This can occur following ischemia (lack of blood flow) caused by blockage (thrombosis, arterial embolism), or a hemorrhage of central nervous system (CNS), or intracranial blood-vessels. As a result, the affected area of the brain cannot function normally, which might result in an inability to move one or more limbs on one side of the body, failure to understand or formulate speech, or an vision impairment of one side of the visual field. Therefore, segmentation and visualization of cerebral blood vessels is of great importance in three-dimensional (3D) reconstruction, quantitative analysis and computer aided diagnosis, especially for embolization of cerebral aneurysms and arteriovenous malformations (AVMs). While most of the arterial anatomy can be shown clearly in MRA images, this is not the case for thinner ones. Thinner vessels can contain low or complex flow and are poorly represented in the images. The presence of disease such as an aneurysm could cause significant vascular signal loss in the MRA image with some intensity levels approximately equal to those of background signal, thereby producing a heterogeneous intensity pattern within the aneurysm. However, segmentation is a challenging task and generally requires sophisticated algorithms and human intervention. The distribution of gray level values corresponding to one structure may vary throughout the structure and may also overlap those of another structure. These inhomogeneous subregions pose a challenge for robust vascular segmentation. This work was motivated by the need to develop a fully automatic segmentation algorithm that could reliably segment the vasculature including other regions of low or complex flow from MRA images.Microvessels refer to the smallest systems of blood vessels in a body, including those responsible for microcirculation, the system of smaller blood vessels that distribute blood within tissues. They are too small to be shown in the scanned images, but it is crucial in diagnosis of diseases. For example, stroke or transient ischemic attack (TIA), is one of the most common diseases in the world. But ‘Minor stroke, big problem’. Most people having minor strokes don't recognize the symptoms, and a large percentage fails to seek timely treatment. A condition characterized by stroke-like symptoms that generally last just a few minutes and cause no lasting impairment. TIAs are warning signs of possible serious and disabling strokes, but are probably one of the most misdiagnosed conditions. Therefore recognizing a TIA and determining its cause can reduce the risk for damage from major stroke. Microvascular morphology is one of the most prominent features of circular system. The geometrical shapes of microvasculature are extraordinarily complex. They come in various shapes and sizes, branch and match with each other to form a network. Numbers and dimensions of microvascular branches vary systematically among different locations. The architecture of microvessels has profound implications for both health and disease. Understanding the principles governing microvessel and its branching is essential for unraveling the way they spread and improving our understanding of the relationship between vascular signals and functional activity in the brain. Detailed information on microvascular network anatomy is necessary for understanding vascular formation as well as of diagnosing and staging diseases. Such complexity and variability of microvascular structure raise a growing topic interest to both doctors and researchers. At present, vascular trees are generated in three-dimensional space with some bounding box for restricting shapes, while the distribution and wiring of blood vessels on the surface are rarely mentioned. The algorithm for building vessels inside an object makes use of the traditional Euclidean distance. However, for the real blood vessels on surfaces, the Euclidean distance is not the appropriate choice. Because Euclidean space is just the simplest example of manifolds. Geodesic is considered as the shortest path between two points on a surface in a local area. Geodesic distance is a better choice for characterizing the topology of space points than the Euclidean distance. Because the concept of the surface in three-dimensional space is two-dimensional manifolds.The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows more complicated structures to be described and understood in terms of the relatively well-understood properties of Euclidean space. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. Manifolds may have additional features. One important class of manifolds is the class of differentiable manifolds. This differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity. The distance calculated in manifolds is along the surface, rather than straight-line distance in space. Thus geodesic distance which portrays the shortest distance in manifold is more consistent with the shape of object. Based on the above analysis, the main work can be summarized as follows:1) A phase-field and likelihood model is developed for segmenting blood vessels in angiograms. Its level set formulation consists of the length term, the region-based term and the regularization term. The length term is represented by the AC equation with a double well potential. The region-based term is composed of both local and global statistical information, where the local part deals with the intensity inhomogeneity, and the global part solves the low contrast problem. The regularization term ensures the stability of contour evolution. Experimental results show that the proposed method is efficient and robust, and is able to segment inhomogeneous images with an arbitrary initial contour. It outperforms other methods in detecting finer detail.2)A three-dimensional extraction of vessel networks based on multiview projection and Allen-Cahn equation.The proposed framework includes three steps: a) projection from 3D volume to 2D plane: in order to make up the small percent that vessels occupy in each slice, we project the volume data onto the 2D plane; and for the purpose of avoiding overlapping between blood vessels, maximum intensity projection is performed from multiviews instead of one or three directions; thus projection images are obtained; b) extraction on 2D plane: a new energy model proposed on two-dimensional segmentation, using phase-field and statistical information, is adopted for such extraction blood vessels from MIP images; c) projecting back from 2D plane to 3D volume: the pixels segmented from the previous step will be projected back into the volume data, and the corresponding voxels in the volume will be reserved to construct the blood vessels in three-dimensional space. Experimental results illustrate the performance of the methods is better than existing techniques.3) A geometric approach for generating microvascular trees is proposed. Inspired by Murray’s law of minimum energy hypothesis, we propose s a formalism which can capture the general features of microvascular branching. The essential structure of a microvascular tree is thereby captured by the density profile of its spanning field and by three parameters balancing the costs for material conservation, conduction time and energy minimization. These balancing factors determine a microvessels branching structure. The simulations presented here provide new insights into the constraints governing microvascular architectures. We provide an integrated explanation for anatomical and physiological scaling relationships by developing a general model for the geometry and hydrodynamics of resource distribution with specific reference to vascular network system.4) A wiring generation of blood vessels on two manifolds is proposed. Firstly, mesh sampling is based on cumulative distribution method in order to guarantee a more uniform coverage of the samples. And the geodesic distance is used to measure the distance of samples on surfaces to generate the branch structure of the brain cortex vessels. Compared with the classical space colonization algorithm, the new method adapts geodesic distance, which reflects the true topology of points on manifolds, is more in accordance with vascular structure generation on surfaces.