Okay, so now with microphone. Okay. So basically it depends here at this point how you define

Omega ij as the open or the closed rectangle.

But just to be sure, you can write this is the union of the.

So the closure is always the union of the closure,

and then you can decide what to do with the lines in between,

whether you put it to the left rectangle or to the right.

That doesn't really matter.

We define anyway functions.

First of all, in an almost everywhere sense.

So these lines would be sets of measure 0 in the Lebesgue measure.

So currently we will not really, really care about it.

Okay, and then if you have this, we can say we define a function.

So that U in omega ij is constant equal to the matrix entry capital U ij.

And in the other direction, we could say we define a corresponding matrix entry by taking

the mean value in this pixel.

So we integrate and divide by the area or 3D by the volume.

Okay, so we naturally have piecewise constant functions in the pixels or in 3D in the voxels

that can be then extended from the matrix entry to the function setting or we can somehow

project the function by taking the mean value in each pixel to the discrete setting.

And then if you have different resolutions, okay, say you have one image which is like

this and maybe then you want to have it like this in a higher resolution.

So with more pixels, you have an immediate way of comparing them because both of them

correspond to a function.

And then you could say if I want to go to one of these two resolutions, I can do the

projection this way, I integrate on the pixels.

So if I have something with high resolution and I want to go for example to some lower

resolution, so maybe to these pixels, I would take the mean value in these four for example.

Or in the other way around, naturally if I have, if I want to go from this to a smaller

resolution, I would just take the same pixel value in all four of them.

Okay, that's the easy part that you don't need integration but of course if you would

have for example something like this and you want to go to a resolution like this which

is not, okay, so the new pixels will be this one, this one.

So they would intersect more than one of the other pixels, then you would take the corresponding

mean value here between here and here to interpolate.

So once you have defined that correspondence between functions and discrete images, you

don't have to worry anymore, this is then automatically defined.

So you identify with each image a piecewise constant function and if you want to go from

the function to some other resolution you just do the integration like here, okay and

of course it's not a full integration if it's a piecewise constant on some person it just

find easier formulas how to integrate.

Okay and then you can manipulate continuous or discrete images from the Bachelor Thus

dycossky.

continuous or discrete images in order to do some of the first image processing tasks of denoising.

So the typical model would be, okay you have F being sum of a clean image plus noise, and F is the given image.

Okay.

Okay, so for example in each pixel if you make a little error in measuring the gray value, you go to this model instead.

Then the first option to get a better image would be to solve the heat equation.

So for some time, and with initial value F.

So you put the noisy image as the initial value of the heat equation, and then of course the heat equation is smoothing out the events.

Or if you want to interpret it differently you could discretize, and you've also done that.