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
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.