Okay, ladies and gentlemen, good morning everybody for the lecture on diagnostic medical image

processing.

I wish the days back when you just press a button and things happen.

Good.

So we are currently in the section or in the chapter on image reconstruction or volume

reconstruction.

So the problem that we are considering is we have a 3D object.

We have different views from different directions, x-ray views.

And our task is compute out of the x-ray projections the original signal.

And what we are considering currently is instead of reconstructing a 3D object, we say we reconstruct

the 3D object slice by slice.

And first we consider the case that we have parallel projections

according to Baer's law.

We know how the x-ray particles are attenuated.

We have parallel projections and we have seen how the Fourier transform of the projection

is linked to the 2D Fourier transform of the object to be reconstructed.

With each projection we get the 2D Fourier transform of the slice to be reconstructed

along the line that goes through the origin and is parallel to the detector line.

That's the Fourier slice theorem.

And we all remember the Fourier slice theorem.

And then we remembered or we looked at the sampling pattern we get for the 2D Fourier

transform by each projection line and we found out that the 2D Fourier transform of the function

to be reconstructed is sampled using polar coordinates.

You remember that?

Polar coordinates.

Polar coordinates.

And then we incorporated the coordinate transform into the Fourier slice theorem and we ended

up with a filtered back projection algorithm.

So by the coordinate transform from the polar coordinates to Cartesian ones we found out

that f at the position x and y, the function we want to reconstruct, is one-half over zero

to 2 pi.

So we integrate over the whole angle range.

We have to divide by one-half because each projection ray is counted twice.

And then we have minus infinity to plus infinity.

And then we have P of s theta.

What was theta?

Theta is the rotation angle.

What is s? that's the index on the detector for a given theta.

And this here is convolved with a kernel function or multiplied h of x cosine theta plus y sine

theta minus s ds d theta.

So here we have a convolution of this function with this kernel.

This is the filter that we get if we back transform or if we compute the inverse Fourier

transform of the absolute value function.

That was the RIM filter that we have incorporated into the back projection formula due to the

coordinate transform.

This is nothing else but the absolute value of the Jacobian of the transform.

If you didn't follow up carefully the previous lectures, this might sound like Chinese to

you now, but not for you, but it might sound like German for you.

So that's the story.

That's the story.