Some computational aspects of determining optical flow are studied. Necessary and sufficient conditions are investigated for the existence and uniqueness of the smoothing-spline form regularization. Free, Neuman, and Dirichlet boundary conditions are studied. It is shown that both free and Newman boundary problems are ill-conditioned, and are not appropriate for optical flow computation. Dirichlet boundary problem is discussed in more details. As a common practice in low-level vision, a continuous problem is formulated, and a discrete version of the problem is solved instead. The discretization errors are estimated and the resulting discrete smoothing splines are computed. Efficient iterative methods are studied for solving the system of linear equations for the discrete smoothing splines. The Chebyshev method is proposed for the computation. The Chebyshev method converges faster than the Gauss-Seidel and Jacobi methods, and is parallelizable.