We study anomaly detection in images under a fixed-camera environment and propose a \emph{doubly smoothed} (DS) density estimator that exploits spatial structure to improve estimation accuracy. The DS estimator applies kernel smoothing twice: first over the value domain to obtain location-wise classical nonparametric density (CD) estimates, and then over the spatial domain to borrow information from neighboring locations. Under appropriate regularity conditions, we show that the DS estimator achieves smaller asymptotic bias, variance, and mean squared error than the CD estimator. To address the increased computational cost of the DS estimator, we introduce a grid point approximation (GPA) technique that reduces the computation cost of inference without sacrificing the estimation accuracy. A rule-of-thumb bandwidth is derived for practical use. Extensive simulations show that GPA-DS achieves the lowest MSE with near real-time speed. In a large-scale case study on underground mine surveillance, GPA-DS enables remarkable sub-image extraction of anomalous regions after which a lightweight MobileNet classifier achieves $\approx$99\% out-of-sample accuracy for unsafe act detection.

翻译：本文研究固定摄像头环境下的图像异常检测问题，提出一种利用空间结构提升估计精度的双重平滑密度估计器。该估计器实施两次核平滑操作：首先在值域上进行平滑以获取逐点经典非参数密度估计，随后在空间域上进行二次平滑以融合相邻位置的信息。在适当正则性条件下，我们证明该估计器相较于经典非参数密度估计具有更小的渐近偏差、方差与均方误差。为应对双重平滑估计器增加的计算成本，我们提出网格点近似技术，在不牺牲估计精度的前提下显著降低推断计算成本。文中推导了便于实际应用的经验带宽选择准则。大量仿真实验表明，网格点近似-双重平滑方法能以近实时速度实现最低均方误差。在地下矿井监控的大规模案例研究中，该方法实现了异常区域的显著子图像提取，随后通过轻量级MobileNet分类器在不安全行为检测中达到约99%的样本外准确率。