So let's look at these equations
a little more carefully.
So the brightness constancy equation.
What we're going to do is assume we know
that some pixel is moving in amount u v.
U is the amount in x, v is the amount
of y when we get to t plus 1.
Okay?
The brightness constant, constraint
equation can be written like this,
I of x,y,t that is,
the image at time t, at location x,y,
brightness if it's gray-scale, color if
it's color, is going to be the same at t
plus 1 at location x plus u, y plus v,
where u and v are the amount
the point has moved in the x direction
and the y direction respectively.
So that's called the brightness
constancy constraint and in fact I can
rewrite it like this, 0 is equal to
the x plus u, y plus v, t plus 1 image.
Minus the original i of x,y,t.
All right, that's the brightness
constancy constraint.
The second assumption was that we get
a very small amount of motion, okay?
So that basically, u and v,
let's assume they're, like,
one pixel, or part of a pixel.
Or just, things are changing smoothly.
Yes you know it's coming.
What that means is I can estimate
using the Taylor expansion here
that the value of an image displaced
from x by u and displaced from,
from y by v, is approximately,
well it's exactly here if I order terms.
It's the original value plus
the gradient x times delta x,
plus the gradient y times delta y.
U and v are delta x and
delta y, respectively,
plus some high order terms.
Right?
Remember Taylor expansion?
If you go infinite series,
you get the exact solution,
but we're just going to say
plus some high order terms.
And then when we make those high
order terms go away, poof, wow.
Isn't that great?
I've got the power right here.
All right?
Then we say that it's an approximation.
Okay, so we say that x plus u, y plus
v is approximately the original image
plus the gradient in x times delta x
plus the gradient y times delta y, or v.
