Now, we will give you a basic introduction to tomography. To help us with this, we will use both Manuel Guizar-Sicairos from Paul Scherrer Institute and Jakob Sauer Jorgensen from University of Manchester. They will explain some of the same things in slightly different ways, and we hope that this will help you get a better grasp of some of the problems that can be a little bit more technical and complex in understanding. You will also experience that there are some mentions of mathematics and you will see some formulas that are fundamental to 3D reconstruction, but don't worry, we will try to rephrase some of these things in simple words, and if you need more details then you have the full honors track where they're much more discussion of the mathematical operations required. But in the basic track we will try to keep this at a minimal level. Now, at first let's hear a little bit about what are the actual data? How do they come about through these projections? The shadow images that you need in order to do a 3D reconstruction. In tomography, we measure an object by acquiring projection images, and we acquire projection images all around the object. The simplest setup we normally work with initially that's called the parallel beam 2D setup and that's what's illustrated here. So, instead of a 3D object, we're looking at a 2D and we're looking at an one-dimensional projections acquired from all around the sample. So, this is easy to describe and analyze, so it's a good starting point. It's also nice because it's actually still used today in some large-scale synchrotrons facilities we have a parallel beam setup so that it's useful, but it's also a simple starting point to describe tomography in general. You recognize probably here that this is a quick summary of what you've heard about before. Now, Manuel will take us into a bit deeper understanding of how then to come from these shadow projections to the 3D reconstruction. The basic idea for X-ray computed tomography is that you try to image sections inside of the material. So, we want to have a three-dimensional picture inside of the material, but the only information that you have is projections. So, you only have information of the accumulated interaction of the X-rays as they go through the sample and in the end you measure a shadow projection of the image. So, in order to be able to reconstruct, this is by the way equivalent to having line integrals from different orientations on this object. So, how do you come from that kind of shadow imaging to know what is inside the object? This is the basics of computed tomography. So, in the end the object is rotated by a 180 degrees and the shadow of the object is collected for each of these angles, and then these angles are assembled into what we call sinograms. So, for each vertical slice of the object if the object is rotating in this direction for each vertical slice, we will obtain one-dimensional shadow as you can see in the slide, and then for each angle we will put this in one line of the sinogram that you can also see. The sinogram is this wavy function that actually is fully described by sines and cosines for each point. You heard Manuel mentioning the sinogram. So, try to remember this word. This represents the projection data in a slightly different way as a function of angle, and this is an important tool that is used in 3D reconstruction so, try to take note of that word, the sinogram. Now, let's try to have another a bit more simple examples showing the relation between the actual object and the projections that we measure so that you can see how these are related to each other. Let's get back to Jakob. We have a simple square, and we take projections in three directions and you'll see the projections look, you can tell from the projections that it's a square. You have it looking a bit like a square here. It's a bit pointy here, and it's something in between here. So, from each of the projections you can get some idea of what it is you're looking at, but you need all of them and put them together to be able to see that it was actually a square. Hopefully now you have an understanding of how the projections relates to the object. Now I suggest that we take a moment to do a small number of quizzes and see if you can guess how for example a projection comes about from a certain object or vice-versa.
