θ is all you need: Revisiting SVD in caputuring changes in matrices

Given a series of matrices (e.g., frames in video) vary over time, figuring out the spatially changing regions (e.g., moving objects) is a critical issue both in theory and applications. In this article, we propose a change detection scheme based on E-SVD, a theory uses Givens transformation, that only determined by the rotation angle θ, to reduce the number of parameters representing a matrix after singular value decomposition (SVD) compression. Inspired by the close relationship between SVD and principal component analysis (PCA), we firstly provide the analytical dependence between θ and matrix elements when changes happen, which guarantees the theoretical rationality of our selection of target θ to efficiently capture these changes. Secondly, we present our detection scheme which is implemented to accurately locate the changing regions of a matrix spatially. The proposed methodology is verified using both simulation and empirical data, results of which show its efficiency and effectiveness. In order to clarify the realistic application of this scheme in object detection without supervision, an additional experiment is also conducted using surveillance video to further demonstrate its potential. Our findings in both theory and application give a new perspective of figuring out spatial variation in a matrix, leading to a wider usage of matrix factorization methods in the domain of unsupervised object detection.