On Visual Periodicity Estimation Using Singular Value Decomposition

The periodicity of an object is one of its most important visual characteristics. Recently, several low-rank/sparse matrix decomposition techniques have indicated that a relationship exists between the frequency components of the motion matrix and its decomposition components. This relationship was mostly identified based on empirical evidence without proper analysis, which led to an unclear understanding and poor utilization. This paper attempts to establish the relationship between the periodic components in the motion matrix and its singular value decomposition (SVD) components. The transformation of the periodic components of the motion matrix through QR factorization and Golub-Kahan bidiagonalization, which are the two essential steps of SVD, is thoroughly discussed and analyzed. Furthermore, the two cases of fully and partially periodic motion matrices are considered where the relationships between their frequency components and left singular vectors are established. The performance of the proposed SVD-based scheme is validated using real videos with ground truth. The results show that the proposed scheme correctly estimates the singular vectors that contain the frequency information.