So now, I've talked to you about one-dimensional Fourier transforms. Then we talked about images and two-dimensional Fourier transforms. But in lots of fields, three-dimensional Fourier transforms are involved. In structural biology, these include X-ray crystallography, 3-D microscopy, 3-D NMR. And all these situations were dealing with 3-D representations of an object. You might call it an image, 3-D volumes, and their transforms. So for instance, in X-ray crystallography, we might be interested in the structure of this protein. So I've represented a protein with an alpha helix and two beta strands here in a box. And if we're studying the structure of that protein, we might like to record its Fourier transform, or its diffraction pattern. And so, what would the three-dimensional Fourier transform look like? This is a 3-D object, so its Fourier transform is going to be a three-dimensional Fourier transform. Well, following what we've already discussed, its Fourier transform is going to be a series of three-dimensional sine waves. So now, we have to think about for a second what is a three-dimensional sine wave? What does that look like? And I'm just going to, they're difficult to draw, but I'm going to go ahead and sketch some example three-dimensional sine waves. [NOISE] So here is a real space cube. A simple 3-D sine wave would look like this. If I were to color in the position of its maximum. And then if I were to color in the position of its minimum in this different colored green, so this is a minimum, imagine a wave that starts here down on the plane of z equals 0. So it's on this plane. And as it passes through the volume, through the cube vertically, it starts at a value of 0. And then the values rise to a maximum after it's a quarter of the way up through the box. So it reaches a maximum of 1. Then as you look into pixels that are approaching the mid-plane of this cube, the value of the sine wave again approaches 0. And then as you move forward higher in z, now the values of the function reach a minimum, negative 1. And as we go higher eventually they reach back to 0. So, a 3-D sine wave, all I'm drawing here is the crest, the maximum and the minimum. Each voxel inside this cube has a value. The value is the sine function at that position. And all of the voxels on this plane have a value of 1. All the boxes here are negative 1, and and the oscillation is between those. Now, you can imagine another three-dimensional sine wave [NOISE] which would be travelling across the box in the other direction. This would be a sine wave that starts at zero along the x equals 0 phase of the cube. This being x direction, this is the y direction, and this is the z direction. So the x equals 0 phase here, this phase, the wave could travel from here across the box this way. And it could start at 0 along this phase and, as it moves across, it reaches maximum of 1. Then halfway across the box, here in the midsection, it would be back to 0. And then at three quarters position. And then at three quarters position, it would reach its minimum of negative 1. And as it moves further across again at the x equals 1. If we call that the edge of the box, on that phase, the value of the function would return again to 0. So this would be another three-dimensional sine wave. Finally, you can imagine a third three-dimensional sine wave that was parallel to the, the phases of the cube. And in, [NOISE] in this case. The sine wave would start out at 0 at, on the z equals 0 phase of the cube. And then a quarter of the way across that box, here, it would reach its maximum. And, as you might guess, three-quarters of the way across the box, in z, it would reach a minimum. And so this is a way that travels through the box in z. Now, in addition to these simple 3-D waves whose crests were parallel to one of the phases of the cube, there could be other waves that cross it in a more diagonal fashion. For instance, there could be a wave whose crest of maximum values crosses the box in that kind of a direction. And its minimum might be seen over here. [NOISE] And we could draw any number of waves that cross the box in different directions. You could have a wave that had a maximum a crest, a maximum crest like that. And its minimum might [NOISE] be a phase looking like that as it crosses the box diagonally. And so it's difficult to draw a 3-D sine wave but you can at least draw the crests of maxima and minima. And so this is what 3-D sine waves look like. In a 3-D Fourier transform of a three-dimensional object, it's going to be the amplitudes and phases of all the possible three-dimensional sine waves that cross this cube. And so, if we were to digitize this image by sampling it in discrete voxels, giving each voxel a density value. And then send that array of densities through a Fourier transform routine on a computer, it would respond with another three-dimensional array of values. And their pattern is, is very specific. There would be three coordinates. The first we call h. The Miller inde, index h, which is spacial frequency in their x direction. The next would be k, which represents spacial frequency in the y direction. And now, in a three-dimensional Fourier transform, we introduce a third Miller index, l, which is spatial frequency in a z direction. Now, the values of these two pixels are the amplitude and the phase of the 0, 0, 0 wave. Now, this is a wave that oscillates 0 times in x, y, and z, and z. In other words, it's the constant, it's a constant value throughout the entire cube. And in fact, you would find the average density of all the voxels in this cube would be returned here as the amplitude of that wave. On the other hand, these pixels would give the amplitude and the phase of a wave that had 0 oscillations across the box in x. But 5 oscillations across the box in y. And 0 oscillations across the box in z. So it'd be one of those flat waves that we drew that were, whose crests were parallel to the in this case, the y equals 0 phase of the cube. All of these pixels represent specific three-dimensional sine waves. This is the amplitude and phase of the sine wave with 5 oscillations, 5 oscillations, and 0 oscillations across the box. In the case of a 3-D Fourier transform, there's another dimension of values. So, for instance now this, these 2 pixels represent the amplitude and phase of a 0, 5, 0 wave that has 5 oscillations across the box in y and 0 in x and z. Meanwhile, this, these 2 pixels represent the amplitude and phase of the wave that oscillates across the box five times in x, y and z. So it passes diagonally across from the origin to the upper right corner of the cube. And this this is the most rapidly oscillating Fourier component present in this image. Now again, if this image has N cubed voxels, in the Fourier transform, there are approximately N cubed values. These N cubed values represent the amplitude and the phase of N cubed over 2 sine waves, 3-D sine waves. N over 2 Fourier components. Now, notice that in this case, while the Miller index h goes from 0 to 5, both k and l go from 0 to, from positive 5 to minus 5. This again is for a real space image with 10 pixels across. But k and l both have positive and negative values. Meanwhile, h would only have positive values because the negative h, k, l values would be the same as the positive h, k, l values. Because you can't tell a 3-D sine wave, whether it's going forwards, or backwards. And so, the non-redundant set of terms only two of these indices has to go both positive and negative. The other can be simply positive. In summary, the Fourier transform of any function is the list of sine waves that are needed to reproduce that function when they're added up. In one dimension, [COUGH] these were one-dimensional sine waves. For a two-dimensional object, an image, these sine waves were two-dimensional sine waves, characterized by the indexes h and k. In 3-D, the sine waves were three-dimensional sine waves represented by three indices, h, k and l. And each of these sine waves has an amplitude and a phase. And if you were to add up all of these sine waves with their respective amplitudes and phases, it would exactly reproduce the object of interest. Now that you understand Fourier transforms, we're prepared to go back to the electron microscope, and think about image formation.