Now in order to understand the data collection strategies for tootie crystallography, and the image processing that follows. We need to understand what the foray transform of a two dimensional crystal looks like in 3D. And to understand that let's begin by thinking about the Fourier transform of an asymmetric object. For instance here this image of a duck.

Well the image of the duck, of let's imagine that it's pixellated. So here are the various pixels in the image.

Obviously this a very corse pixelation in the, and the image is higher resolution than my pixels. But for the sake of explanation, each pixel would have a particular density value for that pixel. Some are obviously black, some are obviously white. >> And let's imagine that there was, say, ten by ten pixels in this real space image. And so, as explained in the lectures on Fourier transforms, the Fourier transform of such an image will have approximately five times ten series of amplitudes associated with phases. So the same number of values, 100 numbers basically representing it in reciprocal space. There's one extra column here representing a zero spatial frequency column, but to a first approximation, it's the same number of values in the transform and the real space image. And so we have a series of pixels here. Notice that the transform has two fold symmetry. So the unique part of the transform is just one half of it. Okay, and so it's full of pixels as well and each pixel has both an amplitude and a face. And so each of these pixels is going to have a substantial amplitude. It's an important amplitude in phase that's required to specify the image of the duck. So I'll just put a check mark in each of these pixels, because they're all required. Now let's think about what would happen, though, if instead of having one duck in the picture, we now had four copies of the duck in this regular lattice. And now we're going to need 20 pixels by 20 pixels to fill out this complete image with the same fidelity as the original. And so in the transform we're now going to have ten columns of 20 pixels and each one will have an amplitude and phase associated with it. So if we start to draw, and if you'll remember the pixel at the origin represents the DC component, which is just a, a plain wave across the image.

And the next pixel over here represents the wave. That pixel right there represents a wave that has one maximum within the image. Just like this. And what we can see is that there really isn't going to be a significant amplitude to such a wave, because the pattern in the image is actually a pattern that repeats twice across the box. So instead of that wave being significant, the wave that's going to be significant

is the one that has two maxima across the box like that. And it's amplitude appears right here. So let me put a check there, meaning that we're going to have a significant amplitude in that pixel.

In addition to specify all the fine details of the shape of this duck, all of the harmonics of this wave will be necessary. So for instance there's going to be a wave with four maxima across the box, and six, and eight. And so this pixel is going to be, have a significant amplitude and so will this one and so will this one.

Similarly, there's really no power in a wave that goes across the box vertically in that fashion, because the image here is a duplicate. There's two of them. And so the first wave with significant amplitude is going to be the one with two maxima across the box and all its harmonics. And so that one appears here, and then its harmonics will also be present in the image.

And there's cross harmonics, so that one and this one.

All of these pixels will have significant values. And so we start to establish a pattern where only every other pixel has a significant value. But if you count the number of pixels that actually have these important amplitudes and phases. It's the same as the number of pixels that were required to specify just one duck in the previous slide. It's that same 100 values, 50 amplitudes and 50 phases that were required to specify a duck. And so that, all that information is still here, despite the zeros in between them.

But they're spaced in a pattern. And what the pattern reveals is the positions and the pattern of where the ducks are.

And the now sparsely positioned amplitudes and phases represent information of what the duck looks like.

Now this process can be extended. What if we had four by four ducks? Now we're going to have an image with 40 pixels by 40 pixels and a transform with 20 columns of 40 pixels of amplitudes and phases. And I won't be able to draw all those pixels. So instead, if I select just the center of the transform,

and make it quite a bit bigger so that we can see more finely what the pixels will look like.

Again, the first pixel is going to be the DC component.

Now the next pixel represents a wave that has one maximum across the image and there isn't any present. So that's a zero. The next one represents a wave that has two maxima across the image. And that's not really present, either here.

The third pixel represents a wave with three oscillations across the box. And, again, there's not much of that present. However there's a substantial presence of a wave that has four maxima across the box. Because of course there are four ducks across the box now. And all of its harmonics. So the next one will be essentially zero. But then the eighth pixel will have a substantial amplitude and face. Following the same pattern, there's very little of the wave that crosses once, so that's going to be a zero. Or twice, or three times. But there will be a substantial wave that has four maxima across the box to represent the pattern of the four ducks. So this is 0, 0, and then that's going to be substantial. And then all of its harmonics off to the edge of the transform and then the cross terms will also be substantial. Okay, so, you see the pattern emerging. Now if we could see the entire transform and just not the center here, we would find that once again, there are exactly 50 pixels here with significant amplitudes and phases. The same 100 values that were needed to specify the shape of one duck. But now there pattern and the zeroes between them, their pattern reveals the pattern of the spacing of the ducks. And in one further elaboration of this, imagine that now we have. Eight ducks across the box. We now have 80 pixels by 80 pixels, and this is going to be 40 columns of 80 pixels each with their unique amplitude in phase.

And if we were to try to draw that, to analyze those pixels

we would need a very large rendition.

And as you might have guessed, there's going to be a significant amplitude in phase and the eighth pixel, both in the horizontal and vertical direction, and their cross peaks, and their harmonics out here where it pass where we can see.

The basic principle is that the transform of a crystal is a lattice of discrete spots. The information about the structure of a single unit cell is contained in the amplitudes and phases of those discrete spots.

The information about how those unit cells are patterned, how far apart they are and what is their organization in the image, is contained in the pattern of the spots.

And so, in a slightly more realistic example. If we have a crystal here, of unit cells, so each of these is a unit cell with a regular pattern inside of it, then it's Fourier transform will be a series of discrete spots.

And each spot of course has an amplitude in phase, and. The information about what is the structure of a unit cell is contained in all of the amplitudes and phases that are present here. But between these discrete spots there's a lot of pixels of just zero or very low numbers here between them. And so the pattern of where the spots are, that there's one here, that there's a spot there and there's a spot there and a spot there and a spot there. This lattice of spots contains the information of how these unit cells are arranged in real space.

And this is of course exactly what the convolution theorem would give us. Because a crystal is the convolution of some object convolved with a lattice.

And the Fourier transform of an object convolved with a lattice, let's represent lattice with the small case l, according to the convolution theorem is equal to the Fourier transform of the object times the Fourier transform of the lattice. Now, the Fourier transform of the lattice is another lattice itself. We'll represent it with a capital L. Meaning this series of discrete spots. So if you think of the original lattice as a series of delta functions. Let us just write in delta functions of each of these positions.

The Fourier transform of that lattice, big L, is going to be another series of delta functions right here at all the spots of this lattice.

And so if we multiply that series of delta functions with the Fourier transform of the object, basically what it means, it's like a screen on a filter where we only see the values of the Fourier transform of the object in the locations where we have a lattice spot. So it's as if we had the Fourier transform in the background in this image and we could only see its value in these particular spots where the Fourier transform of the lattice has a value of 1. And so you see it there.

Now, on first hearing this you might think that there must be something wrong with this, because it seems like, just the pixels that are showing up here are not enough to fully specify this complete image. This image seems to contain more information than this transform. But, remember that the image itself is just a small, unique image repeated over, and over. And so the information content of a single unit cell in fact,

is not more than just the amplitudes and phases of these discrete lattice points can reveal. And so that all applied to these hypothetical perfect crystals where every unit cell is exactly like the next cell exactly like the next unit cell. Of course in real crystals they're not exactly identical. And the information about how they're different, is actually found in the pixels in between the lattice spots. So these pixels in between the lattice spots are actually non zero for a real crystal and they contain that information about how the unit cells are actually different. The data on the lattice points contains the information of the average unit cell here. The information between the lattice spots gives you the details of how each unit cell is subtly different than its neighbor. Okay, now those were examples in two dimensions. Now it's difficult for me to draw a three dimensional crystal, but I'll give it a try here.

Imagine that we have a three dimensional crystal and then we have yet another face behind that.

So in a three dimensional crystal, its Fourier transform is going to be a series of discrete spots in all three dimensions.

So it's discreet spots in this plane. And then above that there will also be another row of discrete spots. You know, in the plane above that we'll have the same pattern of discrete spots. And like I say, it's very difficult for me to draw this in 3D.

But the principle is that the same effect that we described about having waves that need to have multiple maximum as they cross the crystal in all three directions. Means that in the transform of a 3D crystal you have discrete spots in all three directions. The lattice here is a 3D lattice. The lattice here will be the transform of that lattice which is also a 3D lattice. But what happens if, instead you have just a two dimensional crystal. And by that I mean a crystal that's a single unit cell thick.

In this case its Fourier transform has discrete spots in the xy plane, but there's no discretization of the amplitudes and phases in the vertical direction. And so each of these spots is not just a spot in the plane but it is, it's a column of amplitudes and phases in Z and that's because a, a, an object that's just a single unit cell thick. All of the waves are required to specify the kinds of structures that might exist in one single unit cell. For instance, if we have a single protein in this unit cell,

we're going to need all possible sine waves that cross that box with one maximum and two maxima and three maxima, et cetera. In order to reproduce the structure of that protein. And so, there needs to be a value here and right next to it, and right next to it, and right next to it, and right next to it. And so you have a whole column of amplitudes and phases in z at each of these discrete spots.

So now let's think about what would happen if we imaged our 2D crystal from that direction.

Well, we get an image of the crystal, where we see the pattern of the proteins.

And its Fourier transform following the projection theorem is this central section here on the xy plane that is going to have a series of discrete spots,

seeing these spots in the xy plane. And this is not surprising because this image is the image of a crystal. And that as we went through previously, the Fourier transform of such an image of a crystal should be a series of discrete spots. And so here this is what you see.

But what if we were to take a picture from this angle? In other words, we had a flat crystal, what if we tilted the crystal, and then took a picture of it?

Actually, let me draw it in this direction because I have a little bit more space on my board.

In this case, we would again see a picture of a crystal.

But the spacings in between the proteins, the spacings of this lattice, would be reduced or compressed with respect to the spacings that we're seeing previously in the untilted view. If we were to calculate its Fourier transform, what we would get is the amplitudes in phases on a tilted plane through this three dimensional transform. And it would sample the same amplitudes and phases on the xy plane there. But for other lattice lines is what these are called, we would sample values that were lower off the xy plane. Fortunately, we have a better picture of the situation here. This represents the three dimensional Fourier transform of a 2D crystal. And so the axes are spatial frequency in one direction, spatial frequency in another, and then here spatial frequency in the third direction. And in the xy plane, here labeled as a star and b star for other reasons, we see that we have a series of discrete spots just as expected, because the object is periodic in the x and y directions as a crystal. But as we move away from that xy plane, instead of having another series of discrete spots in z, as you would have for a three dimensional crystal. Instead for a two dimensional crystal, you have a continuous line of amplitudes and phases, and this we call a lattice line. So that's a lattice line, and here there's another lattice line of continuous amplitudes of phases that go both above the xy plane and down below the xy plane. And because of this, if we were to take an image of the crystal at a tilted angle,

we would get this data, shown on this plane, this central section. Let me draw the plane bigger.

In the transform of an image of the tilted crystal, we would again see a series of spots as they cross through that plane, here, here. Okay, at all of these crossing points, which would reveal the amplitude and phase of that lattice line in the position where it crosses the lattice line. The lattice line has a continuous progression of amplitudes and phases that vary as you rise above or below the xy plane. And each image of a tilted crystal measures the amplitude and phase in one position along that lattice line, as well as a position on all the other lattice lines in the 3D Fourier transform of the crystal.

To further clarify the situation, here's a depiction of the three dimensional Fourier transform of a two dimensional crystal, given in Orlova and Saibil's review here. And here plane A is the xy plane, and you can see a series of discrete spots in the transform of this two-dimensional crystal. Shown here in this plane C, which includes the y axis and this plane B, which includes the x axis.

You can see that each of these discrete spots has significant amplitudes as it rises above, and so their lattice lines complete continuous rods. And any particular image of a tilted crystal would give a cross-section through these rods. Now if we turn our attention here to this cross section through the 3D Fourier transform of a 2D crystal, we see that each image will give us data about some particular cross section through that transform. And give us information about the lattice lines, the amplitudes and phases along the lattice lines that cross through there. But because, just like in tomography, we can't tilt the sample to 90 degrees. Instead, we can only go to about 6 or 70 degrees. There will be a missing cone of information here of amplitudes and phases that we can't sample.

Here is shown a kind of a surface view of all of the cross sections that we might be able to obtain from zero to say, 60 or 70 degrees and the missing cone in 3D reciprocal space that is not measured.

And that missing cone is represented here in this figure by this

this area of amplitudes and phases that are not measured.

And just like in tomography or the random conical tilt reconstruction, because of this missing cone here of information. If we have an original object that looks like this, and we produce a three dimensional reconstruction through electron crystallography. The result of the missing cone here is that all the objects in the reconstruction will be smeared somewhat in the vertical direction. And, so the, the resolution would be isotropic, just as we've discussed in the other cases.

FT of a 2-D Crystal

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