The next example will be imaging and mathematical imaging has applications of course in digital

photography and also microscopy, medical imaging and astronomical applications.

And before we start with imaging problems we first have to talk about images, modeling

of images.

There are two kind of models you can pick.

The first one is the discrete image model and the discrete image model is that you say

that a picture consists of one times n2 pixels and an image, a grayscale image can be written

as a matrix U in R n1 times n2.

So that means for example this is a very low resolution image of n1 is equal to 3 and n2

is equal to 2 pixels and each pixel has one as a grayscale number so maybe 1 is almost

white and 2 is a bit darker, 3 is a lot darker, I'll have to put this on top, it's 2 and maybe

4, now it's not very quite dark.

So this has, I'll put this on top in red so it's easier, you can see easier.

So each grayscale number corresponds to some kind of brightness or darkness of a pixel

and a color image can be similarly modeled.

Color image has RGB values, so red, green and blue values and then this color image

is now a tensor n1 times n2 times 3 so the same structure but in 3 copies so there's

a red one, a green one and a blue layer but of course it has the same form so again 6

pixels for blue, 6 pixels for green and 6 pixels for red and now each color gets its

own pixel value or in other words each pixel has 3 components red, green and blue so for

example 1 here and 2 here and maybe 0 here or something like that so this would be a

mixture of red and green with more green so probably something like orange.

So that's how you can discretely define images, you can also think of moving images which

are movies and movies are then just, well let's say color movies, this is now an object

n times n2 pixels, 3 colors and then m time steps.

So as you can see this gets quite large and that's why raw video data is a lot of data

with short video being, you know, having gigabytes or terabytes of data and that is of course

why you need image compression in order to make this smaller.

But we're not talking about compression, we're just talking about images so a discrete image

could be thought of as being an object like that if it's a movie or this if it's an RGB

image or just this if it's a grayscale image.

And the second interpretation of images is the continuous interpretation and you could

also say that an image is a function u from some domain so you would usually say a rectangle,

a rectangular domain but of course that's a restriction that you can drop if you want

to, to r so that's a grayscale.

So what you're saying is, well this is your domain and attached to each point, so we're

not talking about pixels here, you know this is a very, this is a fine function and at

each point it has some label, a grayscale label, for example this could be a sharp,

I can't really draw but you know, you could have something like that.

This drawing is at least ideally a function.

For each point that I can look at there will be some discrete grayscale number, of course

this is realized discretely but it's a good mental model number, so this would be a grayscale

thing or you could have three dimensions if you want to have a color image and that works

as well.

So the difference is that you're starting with a discrete model here and you're starting

with a continuous model here, maybe you're thinking all images are discrete and of course

you're right, there are no continuous images, this is a mathematical ideal model that you

won't see in reality but it's a good mental model nevertheless because if you're thinking

of images in different resolutions it's easier to start with a continuous version of some

image and think of the different resolutions as being down sampled version of the same