We develop the information geometry of scaled Gaussian distributions for which the covariance matrix exhibits a Kronecker product structure. This model and its geometry are then used to propose an online change detection (CD) algorithm for multivariate image times series (MITS). The proposed approach relies mainly on the online estimation of the structured covariance matrix under the null hypothesis, which is performed through a recursive (natural) Riemannian gradient descent. This approach exhibits a practical interest compared to the corresponding offline version, as its computational cost remains constant for each new image added in the time series. Simulations show that the proposed recursive estimators reach the Intrinsic Cram\'er-Rao bound. The interest of the proposed online CD approach is demonstrated on both simulated and real data.

翻译：我们发展了具有Kronecker积结构协方差矩阵的缩放高斯分布的信息几何，并利用该模型及其几何性质提出了一种面向多变量图像时间序列的在线变化检测算法。该方法的核心理念在于通过递归（自然）黎曼梯度下降法，在零假设下对结构化协方差矩阵进行在线估计。与相应的离线版本相比，该方法展现出实用优势，因为其计算成本随时间序列新增图像始终保持恒定。仿真实验表明，本文提出的递归估计器达到了内在克拉美-罗界。通过在仿真数据和真实数据上的实验，验证了所提在线变化检测方法的有效性。