The geometric median on Riemannian manifolds with application to robust atlas estimation.
The geometric median on Riemannian manifolds with application to robust atlas estimation.
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DOI：
10.1016/j.neuroimage.2008.10.052
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发表时间：
2009-03
期刊：
影响因子：
5.7
通讯作者：
Joshi S
中科院分区：
文献类型：
--
作者：
Fletcher PT;Venkatasubramanian S;Joshi S
计算解剖学的主要目标之一是对大量图像中的解剖变异性进行统计分析。解剖形状的研究本质上与底层坐标空间的变换构造有关，它将一个解剖结构映射到另一个解剖结构。现在已经确定，在欧几里得空间中表示形状或图像的几何形状会破坏我们表示种群自然变化的能力。在我们之前的工作中，我们已经扩展了经典的统计分析技术，如平均，主成分分析和回归，到黎曼流形，这是描述解剖变异性更合适的表示。在本文中，我们将稳健估计的概念扩展到解剖学变异性的流形值表示，稳健估计是传统欧几里得数据统计分析中一个完善而强大的工具。特别地，我们扩展了几何中位数，一个经典的欧几里德空间数据中心性的鲁棒估计。我们将黎曼流形上数据的几何中位数表示为到数据点的测地线距离和的最小值。证明了非正截面曲率流形几何中值的存在唯一性，给出了正截面曲率流形几何中值唯一性的充分条件。推广了求欧几里德数据几何中位数的Weiszfeld方法，给出了求任意流形几何中位数的一种算法。我们证明了当唯一解存在时，该算法收敛于唯一解。在本文中，我们通过将该方法应用于医学图像分析中常用的各种流形来举例说明估计技术的鲁棒性。利用这种方法，我们还提出了一种基于可变形图像空间中的几何中位数的鲁棒脑图谱估计技术。
One of the primary goals of computational anatomy is the statistical analysis of anatomical variability in large populations of images. The study of anatomical shape is inherently related to the construction of transformations of the underlying coordinate space, which map one anatomy to another. It is now well established that representing the geometry of shapes or images in Euclidian spaces undermines our ability to represent natural variability in populations. In our previous work we have extended classical statistical analysis techniques, such as averaging, principal components analysis, and regression, to Riemannian manifolds, which are more appropriate representations for describing anatomical variability. In this paper we extend the notion of robust estimation, a well established and powerful tool in traditional statistical analysis of Euclidian data, to manifold-valued representations of anatomical variability. In particular, we extend the geometric median, a classic robust estimator of centrality for data in Euclidean spaces. We formulate the geometric median of data on a Riemannian manifold as the minimizer of the sum of geodesic distances to the data points. We prove existence and uniqueness of the geometric median on manifolds with non-positive sectional curvature and give sufficient conditions for uniqueness on positively curved manifolds. Generalizing the Weiszfeld procedure for finding the geometric median of Euclidean data, we present an algorithm for computing the geometric median on an arbitrary manifold. We show that this algorithm converges to the unique solution when it exists. In this paper we exemplify the robustness of the estimation technique by applying the procedure to various manifolds commonly used in the analysis of medical images. Using this approach, we also present a robust brain atlas estimation technique based on the geometric median in the space of deformable images.
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影响因子：
2.7
作者：
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通讯作者：
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