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.
Guillermo SapiroProfessor
