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Okay, so good afternoon everybody. We are today finalizing the chapter on cardiac reconstruction.

And just to remind you where we are, what we learned in winter semester was how to do

reconstruction of static objects using several x-ray projections.

From an abstract point of view, this is well understood for those of you who did not attend

the winter semester lecture. I'll just let you know this is just a filtering operation on the images

and a integration over all projections for each pixel. And we can generate wonderful

images of static objects for the mathematical formulation and that's important because we will

reuse the notation later on several times. We have a point x in 3D space, we have a point u in 2D space

and we have the i-th projection ray that belongs to the pixel element u. That means if you go to

the hospital and you look at an x-ray image what you have in mind is for each pixel there was a ray

a projection ray through space that has the property that x-ray particles were propagated

along this and attenuated and for each 3D point we can tell at which point in the detector field

we end up with. And it's clear that for all the points in space that sit on this projection ray

they all end up with the same 2D coordinate in the image plane. And reconstruction of static objects

if I want to do a reconstruction of the spatial properties like the physical density at position

x in 3D space what I have to do is I have to sum over all the projection rays that basically

map the point x by the projection mapping a into the image space. And this is here characterizing

the ray this is the u the projection point and this is the filter response weighted by a weight

function depend on the ray we are considering and the spatial point. And then we consider a few

things and last week we took a look at the images which we want to use for reconstruction because

now we have an additional problem coming into the game. And this problem is that we are not

only rotating around the object but the object itself is moving. For instance here you see the

contrast agent that was injected into the coronary arteries and you see the beating heart

from various directions and the motion the motion of the heart is basically unknown. We have a non

static a dynamic object that is viewed from different directions using an x-ray system

and the task now is do a reconstruction of this vessel tree using the projection images

which are basically given by the 2D images and the calibration data that defines the mapping from

3D to 2D. And yesterday last week we said okay the heart state can be associated basically

with an indirect measurement and that's the ECG. Yeah it's the ECG which measures the electronic

activities of the heart and the idea now is select a point in the ECG that is periodically

repeating and collect the associated projections which are expected to show the same heart state

and bring them all together and do a reconstruction. And that's actually how cardiac CT systems in the

cardiology department work. You sit there you have three rotations a second the ECG is recorded in

parallel to the projections and then the ECG tells us basically which projection belongs to which heart

state and then we collect all the projection data do a standard 3D reconstruction of heart state one

heart state two heart state three and so on and we get a dynamic four-dimensional data set

3D volume that is changing over time. And in the cardiac environment using C-arm systems we cannot

use this idea because there are two things you have to remember also for the oral exam. Two things

one is that the rotation speed of a C-arm system is rather limited it takes round about four to five

seconds to rotate by 200 degrees around the patient and the second is that the contrast agent injection

is not done using the veins but using the artery or arterial access and that means that it's washed

out in a very short time period so you have to inject the contrast agent for quite a long time

if you want to repeat the acquisition several times for instance. So cardiac reconstruction

of the coronary vessels or let me say the other way around to do a reconstruction of the coronary

arteries using C-arm systems is a very very difficult task due to the two facts that I said.

Then the question is and that's also a repetition of what or a brief reminder of what we have

discovered last week. There are three types of motion cardiac motion the heart is beating

respiratory motion the patient is breathing and patient motion that means the patient is

moving around. You have to know that the patient while he is getting a stent or while he is treated