yes 4. Okay?
Let's see if you know the answer to that. Okay.
Let's explain. The answer is 2.
How do I know that? Because it's the average of these three
numbers. The smallest number that will minimize
this sum is the average of these numbers. This is a very interesting property that
in general reads as follows. if I give you a set of numbers which are, let's
call them ai, a sub i, and I'm asking you to find the number a, such that the sum
for every i of (a - ai)^2 is as small as possible.
The result of this is the average of a. So, the number I'm looking is the average
of all the numbers that I have. So you basically sum all the numbers and
divide by the total number of ai's that you have.
It's actually not very hard to prove that if you remember some of your calculus,
very early calculus. So you basically define this as a n and
you take the derivative of these, with respect to a.
You make that derivative equal to 0. You're going to find the result is
basically the average. So this is very interesting.
When we are taking in an image, let's say a 3 x 3 average and replacing
this by the average of all the nine numbers here.
What I'm actually doing is I'm replacing this pixel value by the number that
minimizes the square error. Okay?
So this is kind of the mean square error. Remember, we talk about that when we were
doing compression. A very simple operation, once again, that
you can prove just by taking derivatives. So, that's one interesting characteristic
of the mean. When we're doing image processing, we're
basically finding the value that replaces that.
This is going to be very interesting. We're going to see in a few videos ahead
that by replacing this way of measuring the error,
we can get other substitutions of this. So this is one of the properties that are
very interesting of averaging. Let's look at a different property.
So, once again remember that what we are doing is averaging in a window.
It can be 3 x 3, 5 x 5, 10 x 10. It's just a window.
So what we are actually doing here is convolution.
For those that have a bit of background in signal processing, that wouldn't be a
surprise. But I'm going to be basically explaining
at, kind of a slightly different operation for convolution.
So the basic idea is, what we are doing is we are taking our image x, y and we're
convolving it with a filter. And, let's just convolve it with what's
called a Gaussian filter. Just a Gaussian function that has average
0 and some variants. And, remember, a Gaussian function looks
something like this. Okay?
It's one of the most simple functions we know.
So that's a weighted average. So in, in a discrete fashion,
we would basically instead of just putting 1, 1, 1 every place,
we would basically replace by the weights provided by the Gaussian.
So that gives us another function which we are going to call x, y sigma, because
depends on the variants of the Gaussian, this function.
So we could actually take a regular average, but it's nicer, it's more
elegant if I explain this with a Gaussian filter.
So, while we are doing this, we are doing a weighted average.
We are replacing the pixel by a weighted average around it.
That's all what we are doing. Now,
this function is very interesting. Once again, it's x, y sigma because
depends on the variance of the Gaussian. So let's just write something interesting
here. You might remember what's called the heat
flow. If not, I'm going to t explain it to you.
I'm going to just write it down and explain it to you.
So we are going to write and I'm going to use the same notations on purpose.
So I'm going to write that the function f, is changing in time.
So, I'm going to write the derivative is changing in time according to what's
called the Laplacian of f, [SOUND] which is equal to the second
derivative of f with respect to x plus the second derivative of f with respect
to y. So the image is changing according to the
Laplacian. So I basically look at the change in the
x direction. I do the second derivative, look at the
change in the y direction. Second derivative, so, double the change,
and basically, this is the heat flow. And, if you remember some basic
properties of the heat flow, it basically talks about diffusion.
So you have heat in a room and the heat diffuses in the room.
If everything is isotropic, nothing can stop it.
It diffuses according to this equation. Now, why did I write this?
Because, the Gaussian filter actually satisfies this equation.
It's actually something very interesting and exactly actually depends on the time,
so the more you let this diffuse is equivalent to having a large sigma, a
larger variance. So once again,
the result of doing this local averaging with a Gaussian filtering is equivalent
to the result of taking the image, thinking like the pixel values, the gray
values are heat, and let it run according to the heat flow
which is these and those are equivalent operations.
So if I let it run for certain time, once again, the pixel values are heat.
It's running for certain time, that's equivalent to picking a certain variance
that depends exactly on the time and applying a Gaussian filter to the image.
So the pixel values are diffusing in the same way as the heat flow is diffusing
heat. Those two are basically equivalent
things. Once again, it's not very hard to prove
that. And if you want one other interesting
exercise replace f by this expression. So basically, take the derivatives, don't
do the derivatives according to T, do the derivatives according to sigma.
And here, do the derivatives according to x and to y, and you're going to find that
this satisfies this equation. And that's because of derivatives are
linear, so this is a linear operation. And if you're not very familiar to how to
do that, just consider that as an extra exercise for those that want to go a bit
deeper in the math for this particular video.
And now, why is this interesting is because as we are going to see in a few
weeks ahead. Somebody might ask, okay, now you just told me that local image
processing is related to heat flow. Maybe I can design a different equation
here that will do other interesting things into image processing and we're
going to see that that's possible. And again, that's going to be a big
mathematical more advance and is going to happen in a few weeks.
So we have seen a couple if interesting properties of this local average.
One is that is kind of computing the means, the value that minimizes the mean
square error in the region. And the second, that there is an
equivalence between pixel values and heat.
And what this local averaging is doing is diffusing the pixel values and that's why
also is blurring the image. If you remember, we saw that it's
blurring it, because it's diffusing them. And we know that, again, that happens
with heat. If you start with hot in one place and
cold in the other, the heat flow will diffuse the heat and
will become more uniform. Of course, if you wait for a long time,
which is equivalent to using a very large variance,
what we get is basically the mean value. So you are going to basically get the
average in the whole image. So once again, a couple of interesting
properties about the local averaging. Next, we're going to see a different type
of local operation that will actually help us not to blur the edges.
See you in the next video. Thank you.
Guillermo SapiroProfessor
