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So, today we will hear something

about preprocessing. So you already heard something about

acquisition, a specific preprocessing. So

today we talk about bilateral filtering and diffusion.

OK. So, maybe

guy who is talking can define what an edge is. No. You don't know. Okay. So anybody knows

what an edge is? You already should have it. So an edge appears where you observe high

differences in intensities. So for example, at the bottom you see two ultrasound slices

of the thyroid gland. So in German it is Schildrüse. On the left side you can see a cystic part.

So it's this black circle. And here on the right side you see a nodule inside the thyroid

gland. And for example, here you would have an edge between this black cystic part and

the brighter part below. So this would be the edge. So there is a high difference between

these intensities and these intensities. You can measure these differences by the gradient.

The gradient is defined here by delta f. So these are the derivatives in x and y direction.

And yeah, you use this to determine if there is an edge or not. So here you see an example.

On the left side you see a CT slice where you see the different organs. So for example,

the liver, the kidney, for example, or the spine. And on the right side you see color

encoded the gradient norm. So you can see if you have warmer colors. So for example,

here at this part you have a high edge. So because here we have intensities between,

or we have a step or a jump between black and white. And for example, here at the liver

we have an edge between black intensities and gray intensities. So the edge is not that

strong as here outside. Okay? So this would be the gradient image where you look at to

find some edges. You can see here, we will see this image a few slides later. So usually

we want to pre-process this image because we see here some noise. So here in the liver,

for example, you see some noisy patterns. We want to smooth here inside this liver but

not smooth across the edges. Okay? So the gradient points into the direction of the

highest change in intensities. An edge is supposed to be orthogonal to the gradient.

The derivatives are highly sensitive to noise. So for example, in MATLAB you can see if you

have a function and you add some noise on it and then you compute the derivative, you

will see that in the derivative you have streak artifacts. So this is really sensitive to

noise. So you have to deal with this. And we have different discretizations for the

gradient. So for example, in MATLAB you can use forward differences. You can see here

it's defined by f is x plus 1 minus fx. Or for example, backward differences. So you

can see here we are at the position x minus 1. Also the possibility for central differences.

So here we use the right neighbor minus the left neighbor divided by 2 or the sobel operator.

So these are the possibilities how you can compute the gradient of an image. Okay. Now

we want to do some edge preserve filtering. So as I mentioned before, you had acquisition

specific pre-processing for more time in this lecture. So now we have some general methods

for different modalities. Okay. So the problem is we want to average inside a smooth region

and do not want to average across an edge. So where we think we have, for example, a

soft tissue. So we want, so we will not smooth across the soft tissue. The bilateral filtering

is now a method for edge preserve noise reduction. So we want to reduce the noise inside a region

and want to preserve edges of some structure. And what we do now is we want to average this.

We want to average image values in dependence on their geometric closeness and also the

photometric similarity of nearby pixels is considered. So what does this mean? So we

look for, we have the image. The image can be seen as a grid point. So you have your

grid points and at a specific point you get a certain intensity value. This geometric

closeness means that you just observe certain neighbors. For example, a 3 by 3 mask or a