Determining optical flow for irregular domains by minimizing quadratic functionals of a certain class

Snyder (1989) has recently classified all smoothness terms which involve first-order derivatives of the flowfield u(x, t) and of the image grey-value function g(x, t). The physically plausible smoothness terms belonging to this class are known from the work of Horn and Schunck (1981) and Nagel (1987). In this paper we discuss the possibilities of approximating the solutions to the minimization problems of Horn & Schunk (1981) and Nagel (1987). In particular, it is shown that these solutions exist, are unique, and depend continuously on the input data. These properties make it possible, while taking into consideration arbitrary models of the grey-value function, to approximate efficiently the (weak) solutions of the associated boundary-value problems in irregularly shaped domains (with a “sufficiently smooth” boundary) using finite elements. Experiments with image sequences from synthetic as well as outdoor scenes show how the orientation dependency of the smoothness term in Nagel's approach influences the results.