So now we need our observation model, okay?
So we're going to use, it's, it's referred to as probabilistic PCA but
we're basically going to use PCA.
And given some possible location, we have to evaluate that
observation as saying, well, how likely is it that that's the right place?
And what it's going to mean is, how well can I generate that?
And the math that was done here, this is the math that sort of justifies this
idea of distance from face space, all right?
So, here's the idea.
Let's suppose I have some eigen vector subspace.
This is face space.
And that's written as B here.
Okay?
And so that's just shown as this plane in 3D.
So the idea being I'm only using two dimensions, let's say.
So I'm assuming everything's in a plane.
And furthermore, it's got some mean value there.
And in this plane is in face space.
All right?
And what this little ellipse shows you is, this is sort of the probabilistic
distribution of how the faces fall within that space.
And now I have some observation z here.
What this equation is saying here, is that the probability
of getting some point z, our observation, all right?
Is distributed about the mean, okay?
So then what's the most likely face?
The most likely face is the mean face.
The mean is the average, right?
That's the most like, that's the center of the normal distribution.
And what this equation says is that,
the probability sort of breaks down into two parts.
There is a distance that has to do with sort of your,
your co-variance or your variances in the face space, and
then there's a distance that's sort of out of face space.
The amount of difference from the reconstruction.
So that's written like this, where the idea is that this BB transpose, that's
the co-variance matrix sort of within face space, and then we're going to just
say some epsilon I, I is an identity matrix, epsilon is some small number.
So things get to come out of that plane a little bit.
Okay? So faces are mostly in face space and
they move away a little bit.
Now what happens is, as epsilon goes to zero, or
just as epsilon becomes small, or another way of saying that is,
that the variance within face space covers most of the faces, and
there's only a little bit of variance left.
What this says is that really,
you can think of that as this variance becomes bigger and bigger,
that the only thing that matters is how far you are out of face space.
So it's mostly a function between the point z that you've observed and
the face space that's spanned by these vectors.
So the math behind this can be written in the following way.
What this says is, that p of z given B.
So the probability of some observation given face space, is an exponential, so
it's going to be decreasing, as a function of the square of this term.
And what this term is, is it's the difference,
so z minus mew is this, okay?
Subtract off the reconstructions.
So this little blue thing is the reconstruction.
So if I subtract that off, all I get is this distance to face space.
And so what I'm going to do is,
I'm going to evaluate the distance to face space for the reconstruction.
And later when we're tracking something else,
it's going to be the distance to whatever my eigen space is.
So a question is, why is that term the reconstruction?
Why is that term the projection of z onto that plane.
Well, it's actually pretty straightforward.
So if B is D by K, all right?
B transpose is just K by K by D, all right?
So what is z minus u?
Z minus u is just the vector of the new observation minus the mean.
Okay?
So B transpose times z minus u, that's just the dot
product of all the eigen vectors with z minus u.
That's just the co-efficient vector.
So this thing right there,
the transpose z minus u, that's the co-efficient vector.
And we can call that gamma.
All right?
Well, given the coefficient vector I can reconstruct.
How do I do that?
I just multiply that, I multiply B times the coefficient vector,
that will be the, each coefficient times each of the eigen vectors summed up.
So that's the reconstruction.
So when I'm all done, this z minus mew minus the total
reconstruction gives me just the amount that you're out of that plane.
That you're out of the face space or the eigen space.
Another way of saying it is, that's how much of the signal you cannot
reconstruct, using the vectors that are just in that basis set.