Now having discussed the first fundamental challenge in biological EM, which is how do we preserve native structure within the ultra high vacuum of the electron microscope. Let's proceed to the next one, which is how do we obtain three dimensional information from the projections? Starting with a 3D object whose structure we would like to know. If we record projection images from different directions, for instance, we have four, five, six, seven, eight, nine different directions that we're recording. If we record projections from all these different directions, then given this set of projection images. If we know how they're related one to another. In other words, if we know which perspective they're giving of the 3D object, we can take that same set of projection images and we can calculate what is the structure of an object in 3D that must have existed to give rise to all of these different projection images. Conceptually, the way it's done is you can imagine that the structure in each projection is. Back projected through that volume and their sum gives you the structure of that 3D object. Now just to get a feel for how how that might work, imagine we had an object with three major components. Maybe they're domains of a protein or large densities within a cell or maybe they're three atoms. So here's our object of three components and now let's imagine that we record a projection image in this direction. Now what we'll see in the image as we project it down onto our detector is that we'll have a density there and a density there and a density there. So this projection image will have these three spots aligned. Now let's suppose that we record another projection image in this direction. Okay. Now what we'll see on a detector is a density there and there and then a density over here where this projection would come to. Now finally, let's imagine that we recorded yet another image from this direction. And so we got this projection image with the density there and there and there. Now given these three projection images, we could start with the first one, which told us that there was a density approximately there and here and then one a little bit closer to it. And so we smear those densities through the reconstruction volume like that. And what we know is that there's three components and they're somewhere along these lines, but we don't know where along those lines those components are. But now we take our second projection image and assuming that we know what the relative orientation between this projection and this projection is. This allows us to say, okay, well there was a density there, there and there in this direction. And so now we have really nine candidate positions in the reconstruction volume where our three components might be found. And we can't tell which one is the correct one yet, but if we take our third projection image. And again, assuming we know what direction it represents compared to the other two and it had a density here and here and here and then we back project this through the reconstruction volume. Now when we look into the reconstruction volume, we find that there are three positions wou, that are the intersections of three lines. So here, here and here. And if I had drawn my lines just right, you would see that these positions match well the original object. It doesn't quite match, because I, I couldn't draw them exactly right, but the concept is clear. The locations where all of the projection images have density rise above the noise of all the other artefactual intersections here and reveal themselves as the position of the real components. So, even with just three projections, these three positions now are the strongest and you can see that's where the components were. But imagine if you had a 100 projection images from all around this object, then you could imagine that this back projection procedure would be even more powerful, because the locations of the real components would now be the intersection of 100 lines, whereas all the other intersections would become two, three, four lines. And so the real positions would become clear. So this is the basic idea of 3D reconstruction by back projection. Now while that description of the back projection might be helpful conceptually. In practice, 3D reconstructions are calculated in reciprocal space usually. And in order to understand that, we need to talk about the projection theorem. The theorem is this. Imagine that you have some 3D object and then you record a projection of that object from say, the, from above. This will produce a two-dimensional image of your object. So this is a two-dimensional projection image of the object. Now you could calculate the two dimensional Fourier transform of that image. And the result would be a two-dimensional Fourier transform, which I'll draw with some kind of circular pattern. So this is a two-dimensional Fourier transform. And remember that this is full of pixels, a bunch of amplitudes and phases. Each pixel represents one structure factor present in the two-dimensional image. Assuming we had known the structure of our original 3D object, we could have calculated a three-dimensional Fourier transform of that object. And if we had done that, we would have gotten a three-dimensional array of amplitudes and phases. All right. So, as three-dimensional array of amplitudes and phases, each pixel having an amplitude and phase of one of the 3D structure factors that are part of this 3D object. Okay. And now we can ask the question, what is the relationship between the amplitudes and phases in this two dimension Fourier transform of a projection image? And the amplitudes and phases in this three dimensional Fourier transform of the object. What is the relationship between these numbers and these numbers? And the answer is that these numbers are a central section through the three-dimensional Fourier transform of the object. In other words, the data on this plane through the 3D Fourier transform has the amplitudes and phases as these. A central section is just a section through a volume that goes through the origin in that volume and so the projection theorem states that the Fourier transform of a projection of an object has the same amplitudes and phases as a central slice through the three-dimensional Fourier transform of that same object. Now the reason this is so powerful is that it tells us how to solve the structure of an object by recording projections. The way you do it is you record projection images from lots of different directions. So here, we've drawn one direction. Let's say, we could record another projection image of the object in this direction. In which case, we would obtain a different two-dimensional image of that object. Now that also has a Fourier transform and we can calculate its Fourier transform. Now what we would find is that the amplitude's in phases in that Fourier transform would be the same as the amplitudes and phases in a different cross section of the three dimensional Fourier transform. So, it's a different central section. And if we were to repeat this and record another image say, from this other direction, we would get a third projection image. And its transform might give us the data say of a central section. This one will be a little tougher to draw, but a central section through the object here. [SOUND] If we could take a projection image say, in that direction from the object and get a completely different projection of the object from its, from behind, for instance. That might give us the data for a central section that crosses through the object like this, for instance. And so the strategy for obtaining a three-dimensional reconstruction is to record projection images from a large variety of angles, calculate the two dimensional Fourier transforms of each of those images. And then populate a three-dimensional Fourier transform of our object with all of that data. And we just need to know how to insert the transforms as central slices into this volume. And if we record enough projections, pretty soon we will have populated most of the data we need in this 3D Fourier space. We then interpolate from the various central sections that were measured in the experiment onto a regular coordinate system, a regular lattice, usually a Cartesian coordinate system. Then we can do an inverse Fourier transform in 3D to recover the structure of our object of interest.
