Statistical Problems in Detectability

PI将研究可检测性的统计问题，即回答是否可以解决给定数据集（例如，图像）的检测问题。将有两个主要组成部分：(a)确定检测任务何时可行的基本阈值是什么，以及(b)当检测问题可解决时，解决它的计算复杂性的适当顺序是什么。基于目前最先进的技术，PI建议得出更准确的结果，并比较不同的公式及其对可探测性理论的影响。拟议的工作有三个主要重点。(1)将导出检验统计量在渐近可检出率下的极限分布。这将推进可探测性理论。(2)可探测性理论将采用应用驱动模型。在许多情况下，这些应用驱动的模型是复杂的，可探测性理论的适应性和可能的泛化不是微不足道的。(3)分析了不同统计公式对可探测性理论的影响。提出的工作植根于PI之前的两个工作：(1)多尺度几何检测（MGD）项目，该项目推导了检测一系列几何对象的渐近可检测率，以及(2)多尺度显著性运行算法（MSRA）项目及其极限分布的结果。在第二个方案中，在知道MSRA中的渐近率后，推导出检验统计量的极限分布（在一种更简单的情况下），从而表征了渐近率处的可检测性。极限分布也解释了已经在模拟中证明的检测算法的鲁棒性。检测是许多图像处理应用中的一个基本问题。一些应用包括(1)低温电镜图像中的粒子检测，它在分子结构的自动重建中起着重要作用；(2)自动目标识别（ATR），它在军事和民用监视中有许多应用；(3)地貌学中的陨石坑检测，它用于地外测绘和行星年表研究。提出的理论问题是这些应用的基础。研究生也会参与其中。

The PI will study statistical problems in detectability, which is to answer whether a detection problem is solvable for a given data set (e.g., an imagery). There will be two major components: (a) what is the fundamental threshold to determine when a detection task is doable, and (b) when a detection problem is solvable, what is the adequate order of computational complexity to solve it. Based on the current state-of-the-art, the PI proposes to derive more accurate results, and to compare different formulations and their influence on the theory of detectability. The proposed works have three main thrusts. (1) Limit distributions of the test statistics at the asymptotic rate of detectability will be derived. This will advancethe detectability theory. (2) Application-driven models will be adopted in the detectability theory. In many cases, these application-driven models are complex, and the adaptation and the possible generalization of the detectability theory are not trivial. (3) Influences of different statistical formulations on the theory of detectability will be characterized. The proposed works are rooted in two of PI's prior works: (1) the project of multiscale geometric detection (MGD), which derived the asymptotic rate of detectability for detecting a range of geometric objects, and (2) the project of multiscale significance run algorithms (MSRA) and theconsequent results on limit distributions. In the second project, after knowing the asymptotic rate in MSRA, the limit distribution of the test statistic is derive (in a simpler situation), so that the detectability right at the asymptotic rate is characterized. The limit distribution also explains the robustness of the detectionalgorithm that have been demonstrated in simulations.Detection is a fundamental problem in many image processing applications. Some applications include (1) particle detection in cryo-EM images, which plays an important role in automated reconstruction of a molecular structure, (2) automatic target recognition (ATR), which has many military and civil surveillance applications, and (3) crater detection in geomorphology, which is utilized in extraterrestrial mapping and planetary chronological research. Proposed theoretical problems are fundamental in theseapplications. Graduate students will get involved.