Okay, so we start today with some images to show you a bit what you can get when you do

paranormalic or similar methods.

So what you see here is really a bit of the time evolution.

So this would be a typical noisy image and here you see several time steps of paranormalic

and you see, okay, after some time the background noise of course gets much better.

The image is still reasonably sharp.

Of course you start losing also something if you look carefully here in the face and

if you go on in time you see that you go somehow to less and less details.

So you see that somehow the face is going to disappear and the camera here and then

the whole thing gets quite uniform and if you go on further in time it just goes to

a uniform gray value.

So one thing that you should always take into account is somehow what is the optimal time

to stop.

So when you are, so in this case, so among these three of course I guess everybody would

choose this one.

So we have already eliminated most of the noise but still have some of the most important

features in the images left.

So there is always a trade off, you will always lose a bit of details in the image and on

the other hand get a bit of, yeah, on the other hand somehow eliminate the noise.

Of course this basically has no noise but it also does not have much details of the

image.

So you have these problems, so somewhere you have two things balancing and there should

always be an optimal time to select when you stop this kind of PDE.

So you are not in this case interested like in the typical case of PDEs in physics or

in the natural sciences where you say okay I have a PDE model for some physical phenomenon

and I want to study for example what goes on for a long time.

So here we are never interested really in the long time, we know what happens in large

time, that is really boring, we go to constant image, we are interested more in this case

what happens in short time.

We eliminate the noise but not so much of the other things.

Okay, so here are some examples from the original paper by Perona-Malik which is maybe more

of historical value, so here you roughly see what was image processing 30 years ago when

they proposed this method.

So here they considered this thing as a denoising of this image and probably if you look at

the beam you don't even recognize a real difference between these two images.

It is maybe a bit easier if you use some kind of edge detector, so you look at where the

gradient is large and then you see in the second image much less of these small things

that are due to noise and really the main edges corresponding to the objects in the

image.

Okay, and then I wanted to show you, oops.

So this is now for comparison, this is now with total variation regularization, so the

second method in this case is not the PDE but it's the variational method but for denoising

they don't look much different.

That was called ROF denoising, I will explain it, it's just a variational method with total

variation as a regularizer.

So here this is an artificial example where we have, we know what the nice image is, so

this is always the U star in our language, then this would be the F, so this would be

U star plus noise, you see the noise which comes here and then this is what you reconstruct

with this variational method.

So you see, and particularly in this case too since the image is very nice for total