2.5.1
Gradient-based methods advocated by [Horn and Schunck 1981] utilize the
constraints of intensity constancy, spatial and temporal coherence. Let
denote the image intensity at the point 
at time t.
A constraint is that the intensity of a moving point does not change
between time t and 
, that is, 
.
This gives the following equation of intensity constancy
This equation relates the change in image intensity at a point to the motion of the intensity pattern (the distribution of pixel intensities). Let
designate the optical flow vector at 
. Then 
is
the optical flow field we want to compute. The intensity constancy
constraint is a single equation with the two unknowns u and v
This corresponds to a straight line in the velocity space. This equation shows that only the velocity components parallel to the spatial image gradient can be recovered through the local computation. This is termed the ``aperture problem'' by [Marr 1982]. When noise is taken into consideration, the flow may be computed by minimizing
However, optical flow f thus computed under the intensity constancy constraint alone is not unique. Additional constraints must be imposed on the flow.
An important constraint is the smoothness, i.e. flow at nearby places image will be similar unless discontinuities exist there. The smoothness is imposed in a similar way as in the restoration and reconstruction discussed earlier. The movement of intensity points should present some coherence and so the flow in a neighborhood should change smoothly unless there are discontinuities. One way to impose this constraint is to minimize the squared magnitude of the gradient of the optical flow [Horn and Schunck 1981]
See [Snyder 1991] for a study of smoothness constraints for optical flow.
In the MAP-MRF framework, (2.62) corresponds to the likelihood potential due to independent Gaussian noise in the image intensities and (2.68) corresponds to the prior potential due to the prior distribution of the MRF, f. The posterior potential is obtained by combining (2.62) and (2.63) into a weighted sum
where 
is a weighting factor. Integrating it over all
gives the posterior energy
This formulation [Horn and Schunck 1981] not only provides a method for computing optical flow, but also is a pioneer work on the variational approach in low level vision which are later developed into the regularization framework [Poggio et al. 1985 ; Bertero et al. 1988]. Various MAP-MRF formulations for flow estimation can be found in e.g.
[Murray and Buxton 1987 ; Konrad and Dubois 1988a ; Black and Anandan 1990 ; Konrad and Dubois 1992 ; Heitz and Bouthemy 1993].
In discrete computation of the variational solution, it is important
that the spatial and temporal intensity partial derivatives of the input
image sequence 
, which appear in
(2.62), are consistent. They may be
estimated by averaging the four first differences taken over the
adjacent measurements [Horn and Schunck 1981]
Another formula calculates them using a three-point approximation
[Battiti et al. 1991]. The spatial derivatives of the flow 
in in
(2.68) are usually approximated by the simple
differences
