When we were talking about histogram modification, either equalization or
matching, we were replacing the grey value of a pixel only based on its actual
grey value. And we weren't looking at the surrounding
area of the image. What we're going to do next is we're
going to do spatial operations. So, we're going to basically replace the
pixel value at the given pixel depending on what's happening around the pixel.
So, there's going to be influence of the neighborhood.
So if we have a complete image, we're going to look at the pixel and we see
here magnifications of that, or let us observe here.
We're going to decide what to replace the pixel value here with depending on what's
happening around it. So, it's not going to be any more just
okay, I decide how am I going to change depending on my own pixel value. We're
going to say how these pixel's going to change depending on neighborhood pixels.
And the simplest of those operations is basically averaging.
So, we are going to replace the pixel here by the average of let's say, a three
by three neighborhood around it. Of course, we have to normalize.
If we are averaging nine pixels, we have to divide by nine.
We can also do weighted averages, so we can average in a sense that, yourself,
the pixel its self, waits four.
And those that are in four neighborhood, remember we talked about four and eight
neighborhood structures, those are, are in four will count double and those that
are in diagonal, we just count single. And then, we basically align sixteen
things, 1 + 2 is 3, + 1 is 4, + 2 is 6, and so on.
So, we will have to normalize by sixteen. So this is just an average, this is a
weighted average. Let's see their facts of this in a real
image. What we have here is an image, and this
is a good image because it has structures of different sizes.
And what we are going to see for each one of these other images is that local
average, and we take neighborhoods of different sizes.
We go from three to thirteen. So this is 3, 5, 9, 15, and 35.
So we go from three to 35. The, the larger the neighborhood, the
more pixels we are averaging. And we see the effect of that.
Here when we average by three we don't see much of a change, there is some
change here within the defined structure, but let's just jump to the 35.
So this is a window of 35, a neighborhood of 35 and we see that a lot of things get
blurred and I'm going to explain in a second why, give you the intuition behind
that. So a lot of the structure gets blurred.
If we are using a neighborhood of only 15, less things get blurred.
We see that here, for example, this is much easier to distinguish the bars here
than it is here. And, that depends on the distance between
the objects. So, why are we getting blur?
Because here, for example, this point. Imagine that we have a 35 by 35 window
around it. So, we end up averaging the block of the
pixel with the grey around it, and then it gets a bit brighter.
So this was very dark, black. This is bright, and if we're averaging
pixels around it, the average basically goes up for this pixel.
And when we're averaging here, because some of the pixels in the neighbor, in
the neighborhood are dark, the average here would basically go down.
And then, we're clarifying this blurring effect.
Let me just illustrate with that one dimensional function.
If I'm basically averaging this with this, and this, clearly the pixel value
of this goes down because of the effect of this.
So, this will be replaced by something around here.
This one, if I'm averaging with this one and maybe
with a pixel here, then it goes slightly up.
And then, this perfectly sharp profile becomes a bit of a smooth profile because
I'm averaging things on the side. We're going to see later towards the
second part of this course, how to basically average without rating these
blurring effect. But this is just to illustrate a very
simple operation which is local averaging,
and how neighbors, and basically the size of the neighbors affect what's happening
on the pixel. Now, you might wonder, is this operation
important? Yes, it is.
Although, it's going to blur. The basic idea is that we are not going
to average with a window size of 35. We need to design window sizes that are
appropriate for the particular problem. For example, if we have this image and we
want to get rid of some of the very small objects in the image.
We, maybe they're noise or maybe they're basically stars or particles that we
don't care about them too much. Then, we can do a local filter of this
image. In this case, it was used a 15 by 15
image. We're blurring the image but we're
somehow getting rid of these small particles.
So, a small particle here will be basically average with a lot of black
pixels around it. And then, it's value will go so much
down, so much close to zero then when I threshold I get basically only the
regions of interest, the larger and the very bright regions.
So, we first blur and then we threshold. And we got rid of all these which we
consider noise and we only get the really important objects in scene.
So this very simple operation is very powerful for tasks like this.
More interesting, this very simple averaging operation has really
interesting mathematical properties that I'm going to explain next.
And those are important because they're going to help us to develop other type of
averaging operations that, for example, do not blur edges.
And we're going to do that explanation of the mathematical properties and
relationships that this simple averaging has with other structures in the next
video. Thank you.
Guillermo SapiroProfessor
