Hello, and welcome back. I want to give you an example of a very

particular and important application of partial differential equations and

variation of formulations in image processing.

And this will put together, some of the concepts that we have just learned, in a

very, very simple fashion. So, what we are going to do is we're

going to do histogram modification with partial differential equations.

We know how to do histogram modification. We have learned that a few weeks ago.

Now I'm going to show you how to do that with partial differential equations.

The basic idea is, as we always do with partial differential equations.

We want to deform the image towards a certain goal.

In particular now, we want to change the contrast.

We want to deform the image in such a way that improves the contrast, and we can

start thinking about how to do that if we understand partial differential

equations. We want to stretch the grey values of the

image. Now, how we are going to do that?

In this particular case, we're going back into histogram modification.

We know how to do that. We learned that.

It's just a simple map that we learned a few weeks ago.

So here is how you do it with a partial differential equation, as an

illustration. Here is the partial differential

equation. You're going to deform the image,

according to its own pixel value at the given pixel, minus the number of pixels

that are great than u. So, you have an image,

you're sitting at a certain pixel, and say How am I going to change these

pixels? Very simple. You look at your own value, you look at

how many values are above you, and you deform yourself according to that, to the

difference between both of them. Now why is this doing the histogram

modification? Very simple. Remember what we talked.

Normally you deform the image equal to your goal. Now, when you get to steady

state, these becomes 0. So the image becomes equal the pixel value at, of the

image, becomes equal to the number of pixel values, above your own value.

And this is exactly, if you go back to your notes on histogram modification,

this is exactly histogram equalization. So this is a very interesting way to

achieve histogram equalization. Instead of doing a look-up table, as we

learned, we just slowly deform the image towards it.

Now, you may ask yourself, why do I have to implement a partial differential

equation that is computationally more expensive than just doing that look-up

table? There's a couple of reasons. One, pedagogically, I'm giving you these

as an example of a partial differential equation acheiving something that, we

want to achieve like, exactly the condition for histogram equalization.

Now, there is another reason. You might stop, this deformation, at any

moment. In histogram equalization, you do the

math. There is no intermediate solution.

If that was too much contrast, then you're in trouble.

There is no way, either, no way back either.

The original image, or that. Here you can start stretching your

histogram. Start moving towards histogram

equalization. But you can stop before you finish that,

if you are already satisfied with the result.

Or even if that result is better than going all the way to histogram

equalization. So those are examples of why you might

want to do a partial differential equation instead of a lookup table.

Now, of course. In the same way that you can do partial ?

equations you can try to do this contrast enhancement by solving a variational

problem. And here is an example of such a

variational problem. Again the art here is for you to start

putting terms in the variational problem that will help you to achieve your goal.

So here is one term. Remember, what we try is to optimize for this, it's

basically to minimize or to maximize, in this case, to minimize this energy.

So this is a term that normally you put it in this kind of things, and here I'm

assuming the image is between 0 and 1. It has been normalized, so the pixel

values are between 0 and 1. You say, you know, I want the image to be

nice. I don't want it just to be too far away

from the average grey values, and that you achieve by this.

On the other hand, you want contrast. So here is the contrast.

You don't want the pixel values all to be together.

If I don't have this term If I remove this term, the solution of this is a

constant value equal to half. All the pixels values are in the middle

all together. That's not what I wanted.

I want them to be nicely distributed, as expressed here, but I also wanted

contrast. And note that we have a minus sign here,

and, because I want to encourage Fix the values to get far away.

Now, I don't want them to go crazy far away, and that's why you add other types

of terms. Now, we have integrals here, which is,

where do I count these? Do I count it all around the image? Now, this is also part

of your gain? You can say, hey, I'm going to just ask for these contrasts in

certain regions of the image. So, you can do the integral only for

certain regions of the image. Or you can say, I'm going to ask for this

contrast only for certain. Ranges of pixel values.

So you can actually become very creative in, designing, this variation of

formulation, this same for this direct pd's.

Something that you couldn't do when you had to look up table.

It was like 1 to 1, from here to here, There's nothing in between.

Here, it gives you a lot of flexibility to become very, very creative.

So let's see some results of this. Here we have two examples.

Contrast enhancement here. Now the interesting thing here, you might

notice that basically, this is kind of localized.

Every region, kind of move by itself, and there no mixing of the region, and these

were achieved by playing with those eastern bound, boundaries, with those

integration boundaries, sorry. And making sure that things that have

similar pixel values, or in this case is exactly the same piece of values, kind of

moves together. So, here's an example of an artificial

image to illustrate how these variational formulations can give you contrasting

enhansements that preserve the shapes of the image.

And here is another example of a natural image.

I hope you can appreciate that through the video, that this is a very dark

image, and here you have much vivid colors, a much nicer looking image.

And this 2 examples are acheived with these PDEs, or variations and

formulations. Those are two different, and one gives

you some capabilities, the other gives you another capabilities, and playing

with the integral limits and playing with the terms inside the integrals, you can

start achieving the kind of application that you want for this type of contrast

enhancement. So I hope you enjoyed this example.

It illustrates both the concept of PD's, that we can almost achieve any condition

that we want if we run the PD to steady state, and also it illustrates how we can

design variational formulation For an extremely important application that is

Contrast Enhancement. Thank you very much.

Looking forward to seeing you in the future videos.