Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics
Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics
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随机场偏移集的预期欧拉特性的边界修正及其在天体物理学中的应用
DOI：
10.2307/1427930
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发表时间：
1995
影响因子：
1.2
通讯作者：
Keith J. Worsley
中科院分区：
文献类型：
--
作者：
Keith J. Worsley
在天体物理学和医学中出现的某些图像被建模为固定区域内的平滑随机场，实验者对“峰值”的数量感兴趣，或者更一般地说，这种图像中存在的“热点”的拓扑结构。本文研究了随机场偏移集的欧拉特征，偏移集是图像超过一个固定阈值的点的集合，欧拉特征是偏移集中连通分量的个数减去“洞”的个数.对于高阈值，欧拉特性是峰值数量的度量。Adler（1981）研究了偏移集的几何，他给出了两个偏移集特征的期望，称为DT（微分拓扑）和IG（积分几何）特征，只要偏移集不接触区域的边界，它们就等于欧拉特征。Worsley（1995）发现了一个边界修正，它给出了二维和三维欧拉特征线本身的期望值。证明使用Hadwiger（1959）给出的欧拉特征线的表示。本文的目的是对任意维数给出一个一般结果。证明采取了不同的方法，并使用了莫尔斯理论的代表。结果适用于最近发现的宇宙微波背景辐射的异常，被认为是宇宙的创造的残余。
Certain images arising in astrophysics and medicine are modelled as smooth random fields inside a fixed region, and experimenters are interested in the number of ‘peaks', or more generally, the topological structure of ‘hot-spots' present in such an image. This paper studies the Euler characteristic of the excursion set of a random field; the excursion set is the set of points where the image exceeds a fixed threshold, and the Euler characteristic counts the number of connected components in the excursion set minus the number of ‘holes'. For high thresholds the Euler characteristic is a measure of the number of peaks. The geometry of excursion sets has been studied by Adler (1981) who gives the expectation of two excursion set characteristics, called the DT (differential topology) and IG (integral geometry) characteristics, which equal the Euler characteristic provided the excursion set does not touch the boundary of the region. Worsley (1995) finds a boundary correction which gives the expectation of the Euler characteristic itself in two and three dimensions. The proof uses a representation of the Euler characteristic given by Hadwiger (1959). The purpose of this paper is to give a general result for any number of dimensions. The proof takes a different approach and uses a representation from Morse theory. Results are applied to the recently discovered anomalies in the cosmic microwave background radiation, thought to be the remnants of the creation of the universe.