Statistical Problems in Detectability

PI将研究可检测性中的统计问题，这是为了回答对于给定的数据集(例如，图像)检测问题是否可解。将有两个主要部分：(A)确定检测任务何时可做的基本阈值是什么，以及(B)当检测问题可解时，计算复杂性的适当顺序是什么。基于目前的技术水平，PI建议得出更准确的结果，并比较不同的配方及其对可检测性理论的影响。拟议的工程有三个主要推动力。(1)求出了检验统计量在渐近可检测率下的极限分布。这将促进可探测性理论的发展。(2)可检测性理论将采用应用驱动模型。在许多情况下，这些应用程序驱动的模型是复杂的，可检测性理论的适应性和可能的泛化并不是微不足道的。(3)刻画不同统计公式对可探测性理论的影响。这些工作源于PI以前的两个工作：(1)多尺度几何检测(MGD)计划，它推导了检测一系列几何目标的渐近速度；(2)多尺度重要性运行算法(MSRA)及其关于极限分布的相应结果。在第二个方案中，在已知MSRA中的渐近率之后，(在更简单的情况下)导出了检验统计量的极限分布，从而刻画了在渐近率下的可检测性。极限分布也解释了检测算法的稳健性，这已经在仿真中得到了证明。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)低温电磁图像中的粒子检测，这在自动重建分子结构方面发挥着重要作用；(2)自动目标识别，具有许多军事和民用监视应用；以及(3)地貌学中的陨石坑探测，用于地外测绘和行星年代学研究。提出的理论问题是这些应用中的基本问题。研究生将参与其中。

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.