Analysis of Physics-Based Optical Flow

This chapter describes the mathematical aspects of the variational optical flow solution to preserve discontinuities in a velocity field extracted from flow visualization images. The uniqueness of the solution and the convergence of a successive approximation sequence are proven. First, the bounded variation (BV) space is briefly reviewed, and an integral representation of the measure ∇·(gu) is given to analyze the physics-based optical flow equation. Then, the general variational formulation with the L1-norm is applied to the equation, which regularizes a velocity filed while preserving its discontinuities in a velocity field. In this case, the minimization problem admits a unique solution in the BV space. Since it is difficult to directly solve the nonlinear Euler-Lagrange equation associated with the functional with the L1-norm, a successive approximate method is proposed, and a convergence analysis of the successive approximation is given. Next, the numerical algorithm is developed and simulations are conducted to evaluate the accuracy of the algorithm.