This research is motivated by studying image processing algorithms through a topological lens. The images we focus on here are those that have been segmented by digital Jordan curves as a means of image compression. The algorithms of interest are those that continuously morph one digital image into another digital image. Digital Jordan curves have been studied in a variety of forms for decades now. Our contribution to this field is interpreting the set of digital Jordan curves that can exist within a given digital plane as a finite topological space. Computing the topological complexity of this space determines the minimal number of continuous motion planning rules required to transform one image into another, and determining the motion planners associated to topological complexity provides the specific algorithms for doing so. The main result of Section 3 is that our space of digital Jordan curves is connected, hence, its topological complexity is finite. To build up to that, we use Section 2 to prove some results about paths and distance functions that are obvious in Hausdorff spaces, yet surprisingly elusive in $T_0$ spaces. We end with Section 4, in which we study applications of these results. In particular, we prove that our interpretation of the space of digital Jordan curves is the only topologically correct interpretation. This article is an adaptation of the author's Ph.D. dissertation.
Topological analysis of the connectome of digital reconstructions of neural microcircuits
最近的一份出版物提供了一个新皮层微电路的网络图，该微电路包括31,000个神经元之间的800万个连接（H。Markram等人，Reconstruction and simulation of neocortical microcircuitry，Cell，163（2015）no.2，456-492）。由于传统的图论方法可能不足以理解这样一个生物网络的巨大复杂性，我们探讨了代数拓扑学的方法是否可以提供一个新的视角，其结构和功能组织。结构拓扑分析表明，有向图表示的微电路中的神经元之间的连接显着偏离不同品种的随机图。特别地，包含在$10^7$顺序中的有向图简化了具有全对全有向连接的神经元群。其中一些单形包含多达8个神经元，使它们成为有史以来报道的最极端的神经元簇集基序。在微电路中模拟神经元活动的功能拓扑分析揭示了新的时空指标，提供了一个有效的分类功能反应的定性不同的刺激。这项研究代表了第一次代数拓扑分析的结构连接和连接为基础的时空活动在生物现实的神经微电路。研究中使用的方法显示出在网络科学中更普遍应用的前景。
Mix-GENEO: A Flexible Filtration for Multiparameter Persistent Homology Detects Digital Images
School of Mathematics, Jilin University（吉林大学数学学院）；School of Artificial Intelligence, Jilin University（吉林大学人工智能学院）；School of Mathematical Science, Heilongjiang University（黑龙江大学数学科学学院）
