Welcome, Diagnostic Medical Image Processing.

So we have the final countdown.

We will have one or two lectures on reconstruction,

and then we will dig into the problem of image registration.

And I will not summarize what we have done so far,

but a few items I just want to mention.

The big summary will be given tomorrow morning.

So we talked about reconstruction,

and basically we have looked at the X-ray attenuation law.

We have looked at the integral that can be associated

with each pixel in the image that we measure,

in each X-ray image.

Then we have seen the important result

that is basically stated by the Fourier slice theorem.

The Fourier slice theorem tells us how the Fourier transform

of the detector row is associated with the Fourier transform

of the function we want to reconstruct.

Then we have looked into the problem

of different coordinate systems.

If you use the Fourier transform of,

or the Fourier transforms of the projections,

you sample the 2D Fourier transform of the slice

that you want to reconstruct in terms of polar coordinates.

So we need to transform the polar coordinates

into Cartesian ones.

We incorporated this coordinate transform

into our reconstruction formula,

and it turned out that the reconstruction

can be basically decomposed into two steps.

And these two steps are high-pass filtering

of the projections and numerical integration

of the filter response that we get.

That was the big picture of analytic reconstruction methods.

And right before Christmas, we have talked also about

the algebraic reconstruction methods.

So what we did is we took the X-ray attenuation law,

we did a discretization right up front.

So in contrast to the analytic reconstruction,

we considered all the integrals and things in the continuous

until we ended up with the Fourier transform,

and then we did the discretization.

In the algebraic reconstruction case,

we looked at the integral equations discretized,

we have discretized the integrals right up front,

and then we looked at these systems of linear equations.

And these systems of linear equations

are usually in a completely different dimension

as we are used to from engineering mathematics.

You know the challenges in the first semester

have been solve a system of three equations