标题： 膨胀伯努利网络中最长运行的渐近收敛速率 显示英文标题

标题： Asymptotic convergence rate of the longest run in an inflating Bernoulli net

摘要： 在图像检测中，一个问题是测试该集合虽然主要由均匀分布的点组成，但也是否包含一小部分从某些（先验未知的）曲线中采样的点，例如，一个具有$C^{\alpha}$-范数不超过$\beta$的曲线。 一种方法是通过计数多尺度多各向异性条带中的成员关系来分析数据，这涉及一个深入研究连接许多连续“显著”节点的路径长度的算法。 在本文中，我们开发了该算法的数学形式，并分析了最长显著序列长度的统计特性。 推导出了收敛速率。 利用渗流理论和随机图论，我们提出了一种名为伪树模型的新概率模型。 基于伪树模型的渐近结果，我们进一步研究了“膨胀”伯努利网络中最长显著序列的长度。 我们发现显著节点的概率参数$p$起着重要作用：存在一个阈值$p_c$，在$p<p_c$和$p>p_c$的情况下，观察到显著序列长度有非常不同的渐近行为。 我们将我们的结果应用于底层曲线特征的检测，并论证我们在理论上实现了最低可能的可检测强度。 显示英文摘要

摘要： In image detection, one problem is to test whether the set, though mostly consisting of uniformly scattered points, also contains a small fraction of points sampled from some (a priori unknown) curve, for example, a curve with $C^{\alpha}$-norm bounded by $\beta$. One approach is to analyze the data by counting membership in multiscale multianisotropic strips, which involves an algorithm that delves into the length of the path connecting many consecutive "significant" nodes. In this paper, we develop the mathematical formalism of this algorithm and analyze the statistical property of the length of the longest significant run. The rate of convergence is derived. Using percolation theory and random graph theory, we present a novel probabilistic model named pseudo-tree model. Based on the asymptotic results for pseudo-tree model, we further study the length of the longest significant run in an "inflating" Bernoulli net. We find that the probability parameter $p$ of significant node plays an important role: there is a threshold $p_c$, such that in the cases of $p<p_c$ and $p>p_c$, very different asymptotic behaviors of the length of the significant are observed. We apply our results to the detection of an underlying curvilinear feature and argue that we achieve the lowest possible detectable strength in theory.