Jump Detection In Blurred Regression Surfaces

We consider the problem of detecting jump location curves of regression surfaces when they are spatially blurred and contaminated pointwise by random noise. This problem is common in various applications, including equi-temperature surface estimation in meteorology and oceanography and edge detection in image processing. In the literature, most existing jump-detection methods are developed under the assumption that there is no blurring involved, or that the blurring mechanism described by a point spread function (psf) is completely specified. In this article, we propose four possible jump detectors, without imposing restrictive assumptions on either the psf or the true regression surface. Their theoretical and numerical properties are studied and compared. We also propose a new quantitative metric for measuring the performance of a jump detector. A data-driven bandwidth selection procedure via the bootstrap is suggested as well. This article has supplementary material online.

本文针对空间模糊且逐点受随机噪声污染的回归曲面跳跃位置曲线检测问题展开研究。该问题广泛存在于多类实际应用场景中，例如气象学与海洋学领域的等温面估计，以及图像处理中的边缘检测任务。现有研究文献中，多数已有的跳跃检测方法均基于两类假设：要么假设不存在模糊效应，要么假设由点扩散函数（point spread function, psf）描述的模糊机制完全已知。本文提出四种无需对点扩散函数或真实回归曲面施加限制性假设的跳跃检测器，并对其理论特性与数值表现展开研究与对比分析。此外，本文还提出一种用于评估跳跃检测器性能的新型量化指标，同时给出一种基于自助法（bootstrap）的数据驱动带宽选择流程。本文附带在线补充材料。