Convergence of Object Focused Simultaneous Estimation of Optical Flow and State Dynamics

Abstract

The purpose of this study is to prove the convergence of the simultaneous estimation of the optical flow and object state (SEOS) method. The SEOS method utilizes dynamic object parameter information when calculating optical flow in tracking a moving object within a video stream. Optical flow estimation for the SEOS method requires the minimization of an error function containing the object's physical parameter data. When this function is discretized, the Euler-Lagrange equations form a system of linear equations. The system is arranged such that its property matrix is positive definite symmetric, proving the convergence of the Gauss-Seidel iterative methods. The system of linear equations produced by SEOS can alternatively be resolved by Jacobi iterative schemes. The positive definite symmetric property is not sufficient for Jacobi convergence. The convergence of SEOS for a block diagonal Jacobi is proved by analysing the Euclidean norm of the Jacobi matrix. In this paper, we also investigate the use of SEOS for tracking individual objects within a video sequence. The illustrations provided show the effectiveness of SEOS for localizing objects within a video sequence and generating optical flow results.