Hello, it's great to have you back. This is week 6, and the topic of this

week is partial differential equations in image processing.

Some of you might remember, some of you might know what partial differential

equations are. Some of you might not remember when you

studied that, maybe a long time ago, maybe just a few years ago.

But don't worry because every other week this is going to be a very self

containing week. Before I give you a bit more of details

of what we are going to learn this week let me try to explain to you what we mean

by partial differential equations what is kind of this new area.

Relatively new area in imagine processing but certainly new for us in this class.

So far we have been considering images at discrete objects in the computer.

If we look for example at the still image.

We already talked it looks continuous to us but actually it is a connection of

pixels so its like this great two dimensional array of pixels as we have

represented here now they are so close to each other we already talked about

resolution they're so close to each other that it looks like a continuous image,

but actually it is a discreet object. It's representing the computer as a

discreet object. The same happens with movies.

We already know that movies are discrete object.

They look continuous to us, like this movie that we are watching here, because

the sampling in time is basically very fast, and that's why our perception

perceives this as a continuous object, but we already know that this is a

discrete object. 30 frames per second, 24 frames per

second. So images and videos are discreet both in

space. In time and also at the gray levels.

They look continuous to us but they are discreet objects.

Now that leads us to what we have been doing most of these five previous weeks

using tools from discreet representations from discreet mathematics because we have

discreet objects in the computer. So what's different when we start talking

about partial differential equations. What's different is that we're going to

start considering continuous objects. So what we consider discreet objects the

way that videos and images are represented in the computer we see for

example, sums. We never saw the sign of an integral we

always talk about sums we talked about discreet operations.

Also, every time we saw kind of a continuous object a derivative which is a

toll from calculus is a continuous object then we immediately discretize it.

Remember, we talked about, for example, it's derivative.

Derivative indicates direction, and we say, okay, let's do plus one and minus

one here. So immediately we redefine its discreet

counterpart because we want to be in a discreet wall.

We want to be in a discreet space. The area of partial differential

equations says forget about that the area of partial differential equations has a

completely different approach and says images are continuous objects do not

treat them as discreet images anymore that's just an artifact of computer

representations. Treat them as continuous objects.

And then you basically are going to be talking about image processing that's

iterations of infinite decimal operators, things that happen at very, very small

scale we iterate them. And when we are iterating them, actually,

we get these partial differential equations, but once again if you don't

remember exactly what they are, we are going to explain that in the next videos.

Don't worry about that, the key concept is here.

Images are not discreet objects, images are continuous.

So go and gather up all of your algorithms that are in your continuous

domain. Treat images as continuous objects, and

then you can do partial differential equations you can also differentiate

geometry, tools that are from continuous mathematics.

All of a sudden, they are valid. They're powerful for basically discreet

image processing, but then you ask yourself, wait a second.

You gather up them, but my images are still discreet objects in my computer.

But here comes to the rescue, numerical analysis.

Numerical analysis is exactly the area that says how do I implement continuous

algorithms? How do I implement continuous mathematics

in discrete domains like a computer? So you go and develop algorithms with

tools of continuous mathematics and then. Numerical analysis comes to the rescue

once we need to implement those algorithms in the computer.

So it's kind of a different paradigm. It's not better, its not worse.

It's different than the paradigm that we were used to before, when we consider

images from the very beginning as discreet objects.

Now, why?

Why is this happening now? What's going on?

Why do we move from this, from this completely discreet, that basically if we

look historically, has mostly dominated image and video processing for years and

years, until basically about ten years ago, and of course there are a number of

reasons why these continuous tools appear in image processing one is computers we

can more powerful and then numerical algorithms to implement this continuous

math that were impossible to have in a computer we can actually do able in even

small and personal computers of course every.

New tools can because some people moved into the area and we should never forget

about the influence of people. So a lot of people in the last ten years

or so that were interested in continuous mathematics also became interested in

image processing and they brought their tools, their expertise into image

processing. Now we're going to see, as I said we're

going to see examples. I am going to provide you the background

but. What is it bringing to us?

It's going to bring a number of things. It brings new concepts.

It brings accuracy. This is a very important thing, because

we are in continuous domain, and then we implement by numerical,

algorithms. The accuracy will depend on the

implementation, not on the design of the algorithm.

The algorithm is designed in the continuous domain, so there is no

intrinsic accuracy. Depending how much I'm willing to invest,

computation and resources for example, in the implementation that will determine

the actual accuracy of the algorithm. It's not intrinsic to the algorithm.

When I define a derivative at plus minus, plus one minus one,

I'm done. I have defined the accuracy of my

algorithm. Now if I define the continuum and, and

then say do the derivative the best you can,

I leave the door open to very high accuracy techniques.

Another thing that was broad, that we are not going to discuss a lot in this class,

and certainly not in this week, is very formal analysis.

A lot of the techniques that came from the area of partial differential

equations to image processing have very formal analysis.

You can prove theorems, you can prove that what your doing is right even before

your going and implementing the algorithm.

So it's one of the most mathematical areas in image and video processing.

I'm not saying its the only one, but it's one of the most mathematical areas.

And the consequences of this is that partial differential equation tools

brought a lot of state of the al-, algorithms.

But I think one of the most important things, and I want to leave you before we

go into details with this take home message, is that there are new tools and

new books in the bookshelf. So when you bring a new theory into an

area, you say, okay all these new frameworks

are now allowed. And it always good to have more tools to

solve real problems, as we are going to see this week and a bit next week.

So, what we're going to do this week is I'm going to give you the tools to

understand the underlying and the simplest possible tools to understand the

area of partial differential equations in image processing.

We're going to filter in examples that is going to show you to understand.

Remember, last week we discussed active counters, that's one example of the use

of partial differential equations in image processing as we are going to see

very soon in one of the future videos. And we are also going to talk next week

about imaging painting and some of the algorithms we are going to subscribe

there are based on partial differential equations.

As I said, is one of the most mathematical areas that we're going to

discuss during these nine weeks of classes in image and video processing.

But there's nothing to worry, because I'm going to introduce you to the basic

concepts and the basic tools that you need to understand the fundamental

concepts behind this area of partial differential equations in image

processing. And we're going to start learning about

those concepts right in the next video. Thank you.