2.5.2
The quadratic smoothness in (2.63) is unable to deal with discontinuities, i.e. boundaries between regions moving differently. For the discontinuity-preserving computation of optical flow, the smoothness is more appropriately imposed as minimizing the following
where the g function satisfies a necessary condition
(2.16). When g is a truncated quadratic,
(2.68) gives the line process model
[Koch et al. 1986 ; Murray and Buxton 1987]. A more careful treatment, ``oriented
smoothness'' constraint [Nagel and Enkelmann 1986], may be used to avoid smoothing
across intensity, rather than flow, discontinuities. In
[Shulman and Herve 1989], g is chosen to be the Huber robust penalty function
[Huber 1981]. In [Black and Anandan 1993], such a robust function g is
also applied to the data term, giving 
.
Recently, Heitz and Bouthemy (1993) propose an MRF interaction model which combines constraints from both gradient based and edge based paradigms. It is assumed that motion discontinuities appear with a rather low probability when there is no intensity edge at the same location [Gamble and Poggio 1987]. This is implemented as an energy term which imposes a constraint on the interaction between motion discontinuities and intensity edges. The energy is designed to prevent motion discontinuities from appearing at points where there are no intensity edges.
The computation of optical flow discussed so far is based on the assumption of constant intensity. The assumption is valid only for very small displacement or short-range motion. For long-range motion there is a significant time gap between different frames. In this case, the analysis is usually feature based. This now requires to resolve the correspondence between features in successive frames.
