The authors examine some computational aspects of determining optical flow. Both area- and curve-based approaches are discussed. Necessary and sufficient conditions are investigated for the existence and uniqueness of the smoothing spline from regularization schema prevalent. The authors discuss a variety of boundary constraints: free, Neuman, Dirichlet, and two-point boundary conditions. It is shown that both free and Neuman boundary problems are ill-conditioned, and are not appropriate for optical flow computation. This partly explains why practitioners have attested to the difficulty of computing flow velocities using such regularization scheme. Therefore, it is necessary to use either Dirichlet boundary conditions or design different regularization schema. As a common practice in early vision, a continuous problem is formulated, and a discrete version of the problem is solved instead. The authors estimate the discretization errors, and compute the resulting discrete smoothing splines. They study efficient algorithms for solving the system of linear equations for the discrete smoothing splines.