Stochastic Geometry and Its Applications
Stochastic Geometry and Its Applications
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DOI：
10.1046/j.1365-2818.1996.00654.x
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发表时间：
1996-09
影响因子：
2
通讯作者：
T. Mattfeldt
中科院分区：
文献类型：
--
作者：
T. Mattfeldt
在回顾过去 10-15 年定量显微镜的创新时，我们发现了两个显着的趋势：构建基于设计的体视学工具，能够对参数进行公正的估计，不受形状假设的影响；以及体视学与随机几何方法的联系。组织学切片是通过三维生物组织的染色薄片，包括曲线特征（肌肉纤维、血管……）、表面（细胞膜……）和颗粒（细胞、细胞核……）的集合。基于设计的工具提供对每单位组织体积的这些特征的长度、表面积和体积的无偏估计，以及平均颗粒体积的无偏估计。这种类型中最著名的工具是解剖器。当使用随机几何时，我们更进一步，尝试探索可能控制特征的明显几何排列的潜在原理。从随机的角度来看，这些被认为是由随机机制（广义上）生成的点、线、面或集合的空间模式。与将随机性限制为完全随机性的狭隘概念的常见偏见相反，随机几何提供了多种明确定义的替代模型，例如允许特征的吸引（聚类）和排斥，从而生成相当复杂的模式。三维显微镜（例如共焦显微镜）的可用性的增加增强了体视学与随机几何的联系，其能够直接记录坐标（例如细胞重心的坐标），可用于进一步的数据处理。随机几何方法可以从数学家的原始文章和教科书中进行研究，但这些方法分布广泛，应用科学家通常很难获得。通过将严谨性与易于理解的解释和大量示例相结合，斯托扬、肯德尔和梅克的专着在数学世界和希望将随机方法纳入研究项目的显微镜学家之间架起了一座桥梁。现已推出的第二版标志着第一版（1987 年）的成功。它不仅是前者的重印，还包括自 1987 年以来的新发展，插入新插图以及旧图形的版本，消除印刷错误和错误，以及扩展参考文献部分（截至 1996 年出版的论文）。这本书包括关于随机几何、点过程、随机闭集、随机测量、几何对象的随机过程、纤维和表面过程、随机镶嵌（马赛克）和体视学的数学基础的章节。这些章节基本上是独立的；当着手一个狭义的项目（例如纤维加工的体视学）时，读者可能会跳过本书的大部分内容并专注于一些相关的部分。这些章节基本上包括非正式的介绍和基本定义，然后是对模型和统计估计方法的主要属性的回顾。这些都配有插图、应用科学的工作示例以及通过计算机模拟过程的提示。相关的数学公式通常都给出而不带证明，极大地方便了阅读的流畅性；相反，提供了对原始文章中的证明的引用。由于这一特性，Stoyan/Kendall/Mecke 已成为该领域的标准参考书，其广泛使用从应用随机几何论文中的参考文献中可见一斑。我感谢教科书提供了……
When reviewing innovations of quantitative microscopy in the last 10–15 years, we find two striking trends: the construction of design-based stereological tools which enable the unbiased estimation of parameters, free from shape assumptions; and the connection of stereology with methods of stochastic geometry. Histological sections are stained thin slices through three-dimensional biological tissues that include collections of curvilinear features (muscle fibres, blood vessels,...), surfaces (cell membranes,...) and particles (cells, nuclei,...). The design-based tools provide unbiased estimates of the length, surface area and volume of these features per unit tissue volume, as well as unbiased estimates of the mean particle volume. The most prominent tool of this kind is the disector. When using stochastic geometry, we go a step further and try to explore the latent principles that might govern the manifest geometrical arrangement of the features. From the stochastic viewpoint, these are considered as spatial patterns of points, lines, surfaces or sets, generated by a random mechanism (in a wide sense). In contrast to a common prejudice which restricts stochasticity to the narrow concept of complete randomness, stochastic geometry offers a broad variety of well-defined alternative models, which allow, eg for attraction (clustering) and repulsion of features, and thus generation of fairly complex patterns. The linkage of stereology to stochastic geometry is enhanced by the increased availability of three-dimensional microscopy, such as confocal microscopes, which enable direct recording of coordinates (eg the coordinates of the centres of gravity of cells), that can be used for further data processing. Stochastic–geometric methods can be studied from original articles and textbooks for mathematicians, but these are widely scattered and often difficult to access by applied scientists. By combining rigour with understandable explanations and a multitude of examples, the monograph of Stoyan, Kendall and Mecke builds a bridge between the world of mathematics and microscopist who wants to incorporate a stochastic approach into a research project. The second edition which has now become available indicates the success of the first (1987). It is not just a reprint of the former, but includes new developments since 1987, insertion of new illustrations as well as edition of old figures, elimination of misprints and errors, and expansion of the References section (up to papers in press 1996).The book includes chapters on mathematical foundations of stochastic geometry, point processes, random closed sets, random measures, random processes of geometrical objects, fibre and surface processes, random tessellations (mosaics), and stereology. The chapters are largely self-contained; when embarking on a narrowly defined project (stereology of fibre processes, say), the reader may skip most of the book and concentrate on a few relevant sections. The chapters consist basically of an informal introduction and fundamental definitions, followed by a review of the main properties of the models and of statistical estimation methods. These are supplied with illustrations, worked examples from the applied sciences and hints for simulating the processes by computer. The relevant mathematical formulae are usually given without proofs, which greatly facilitates the fluency of reading; references to the proofs in the original articles are provided instead. Due to this property, the Stoyan/Kendall/Mecke has become a standard reference-book in the field, whose widespread use is evident from the references in papers on applied stochastic geometry. I am indebted to the textbook for providing …