We represent a sequence of images by 
. Let a(t) and b(t) be the position
of a point belonging to a small patch of a constant brightness pattern that is moving
across the image. This can be expressed by the following equation:
which implies 
.
We differentiate the preceeding equation with respect to time
in order to obtain an equation for the velocity components 
and
:
In effect, this equation yields only the absolute value of the velocity along the direction of the image gradient, which is the reason why further constraints have to be established.
In the literature this problem is solved in mainly three different ways. One possibility is the use of second order derivatives, which has major drawbacks since these derivatives are often inaccurate and resulting flow estimates tend to be sparser. Another, more commonly used approach consists in combining flow estimates over a neighbourhood of pixels. This can be done by means of a simple model for local velocities or by the assumption of constant velocity over a small window. The method of Lucas and Kanade [LK81], presented below, makes use of this method. By means of regularisation techniques it is furthermore possible to define a global smothness constraint on the velocity field, thereby implicitely defining the latter as a solution of some functional (see for example [HS81]).
