Let us suppose that
the image intensity is given by *I*(*x*,*y*,*t*), where the intensity is
now a function of time, *t*, as well as of *x* and *y*.

At a point
a small distance away, and a small time later, the intensity is

where the dots stand for higher order terms.

Now, suppose that part of an object is at a position (*x*,*y*) in
the image at a time *t*, and that by a time *dt* later it has
moved through a distance (*dx*,*dy*) in the image.

Furthermore, let us suppose that the intensity of that part of the object is just the same in our image before and afterwards.

Provided that we are justified in
making this assumption, we then have that

and so

However, dividing through by *dt*, we have that

as
these are the speeds the object is moving in the *x* and *y*
directions respectively. Thus, in the limit that *dt* tends to
zero, we have

which is called the *optical
flow constraint equation*.

Now, at a given pixel is just how fast the intensity
is changing with time, while and are the
spatial rates of change of intensity, *i.e.* how rapidly
intensity changes on going across the picture, so all three of
these quantities can be estimated for each pixel by considering
the images.

David Marshall 1994-1997