This time we're going to talk about Fourier transform and Fourier space, which is a very important concept in understanding microscopy. Well, Fourier transform is all about math, like here. So, I'm going to make it simple. Let's start with a simple sine wave. As simple as possible, you see sine oscillation, very smooth oscillations. To describe this sine wave, well, you have to have some mathematical formula, like here, as simple as we can go: I, which is the intensity at a certain point, equals sine of the coordinate x. And there are more properties of this wave. For example, you can have different spacing or different frequencies, and in this case you can see 3 white stripes in the entire view, so we can say this has a frequency of 3. And in the case when you have a frequency of 4, k=4, you have 4 white stripes. And higher frequency, k=6, and even higher frequencies, where you have k=30, then there's a lot of little white stripes showing here... very dense, high frequency waves. So, that's one aspect of a pure sine wave. Now, in addition to frequency, there's also another thing, which is the amplitude. Now you have a coefficient, A, in front of the sine function. And in this case you have amplitude 1, and when you have amplitude 0.5 it's dimmer or lower amplitude, smaller oscillations. And an amplitude of 0.25, it's going to be even dimmer in the image, or even lower contrast. That's now two of the properties of the sine wave: the frequency and the amplitude. Getting a little bit more complicated, we have an additional phase factor in this sine function. What we can see here is one additional ɸ in the bracket of the sine function. And in this case, this phase is zero, that means the sine oscillation, here, starts from zero, and continues oscillating. If we have an additional phase shift of 45°, and you can see the entire wave gets translated and it starts somewhere mid-way. And we can have more phase shifts, like 90°, 135, 180, 225, 270, 315, until we get a phase shift of 360°, that's one entire circle, so it's come back to the original sine wave. This is the phase factor. And a fourth one to consider is that we're describing a two-dimensional image. In this case, the sine wave function is basically oscillating along the x direction -- you can oscillate it along an axis that's slightly off from the x direction. And in this case we have to have a little bit more complicated math to describe it. And you can see in the sine function here, we've changed both the k and x into vectors, and the k essentially describes, as a vector... and the k, as a vector, now, essentially describes the propagation function of this sine wave. So, we have our x and y axis, and we have a vector k. In this case, the k has a 30° angle to the x axis and that is the propagation angle. And in a different case, this vector has been rotated to have a 60° angle to the x axis and you can see the entire waveform gets rotated. So those are the four parameters to describe a two-dimensional sine wave. And we have our frequency, and we have our orientation. Both of these are included in this k vector. And then we have our amplitude in a, and we have our phase in ɸ0. We need to find a graphical way of describing these sine functions, because looking at the mathematical formula and looking at pure numbers, that's very boring. Then we can have our coordinate system, and in this case to avoid confusion I called it kx and ky, instead of x and y, but it's basically the same thing. And then we have the k vector. Just now, I said the orientation or the direction of the k vector tells the propagation direction of the sine wave, and in fact the length of this k vector, the wave vector, tells the frequency. And so the k describes the frequency and orientation as a vector. For each point on this two-dimensional kx-ky plane, we can have two values. One value is A as a function of the k vector, that tells the amplitude of the sine wave, and another value is ɸ, that tells the phase of the sine wave. So a total of four numbers describes this sine wave and then, because we're looking at a 2-D plane, we can draw them into a picture, and this is basically a full picture of describing a lot of different sine waves. Each point on this picture describes one sine wave with a certain frequency and orientation and the value of this point describes the amplitude and the phase. And that is how we can graphically represent a collection of different sine waves. So, so far we are talking about simple sine waves, but whatever cell structure we're looking at, they are not sine waves. They have fairly complicated structures, and how can we use sine waves to describe the structure. Well, again, we start with an extremely simple example. Now, instead of sine waves, we have black and white stripes. That's simple enough. But even that differs significantly from a pure sine wave. Just in cross-section, you can see a sine wave is a fairly smooth oscillation, but here, now, we have very sharp steps. We can try to compensate for that, though, for example, here, at the edge, this sine wave has a smooth drop, whereas this black and white pattern has a very sharp transition. So we can imagine if we have yet another peak, here. And that will somewhat compensate for this. And at the same time, if we have a negative peak in the middle, that will also compensate for this peaky behavior of the sine wave to make it more flat. So this is what we do, we have yet another sine wave which is 3 times the frequency and, summed up, you can see, because its 3 times the frequency and it has a negative in the middle and two positive peaks at the edge, and that makes this overall shape more step-like. But this is still not perfect yet. We can consider more, further compensating for it. This time, we have another one, 5 times the frequency. And when you have an infinite amount of these adding up, then we will approximate our black and white streak pattern. So this is the process of how we can use a collection of sine waves, or a collection of different spatial frequencies, to describe a real image. And we can write it in mathematical form -- and this is important -- as, well, our original image, F(x) equals a sum of different sine functions, and then G(x) actually describes the amplitude and phase of these functions, it's how they are summed. So now we have two functions: one function is F(x), that describes the real image in the coordinate x; the other function is G(k), that describes our collection of sine waves that can sum up into this F(x) function, and that's in the coordinate of k(x). If you look into image, you can see discrete spots. The position of these spots describes the spatial frequency, k equals 1, 3, 5, and, very dim, 7, and there can be an infinite number of them. And the intensity of these spots essentially describes the amplitude. In this case, we're not considering the phase shift yet. So, in this case, you can see that we have some relationship between F(x) and G(k), and to derive G(k) from F(x) is called the Fourier transform. In this case, we established the connection between the original image, which is in so-called real space, and the Fourier image, which is in frequency space. This is a very, very simple example. Now, let's give you a slightly more complicated image to show you how you can read the Fourier space information. We're looking at the Fourier transform of an image of Fourier himself. That's the Fourier transform of Fourier, and in this case the original image has an x and a y coordinate. The Fourier space image has a kx and ky coordinate. And one very simple thing to think about this... the things close to the origin will have very short k vectors and that's describing the low frequency, low spatial frequency information. And things further away from the original will describe the highest spatial frequency information. Just to illustrate that, if we take this image and we mask out the highest spatial frequency information and we inverse transform it back -- so, this is call the inverse Fourier transform -- well, we still see the image of Fourier, but at a much lower resolution. For example, here, the details of his coat are completely lost, and this is because is on Fourier space we masked out all the high spatial frequency information. We can do it the other way, we can mask out the low spatial frequency information and do the inverse Fourier transform -- this is what we get. Well, we lose the overall image shape, we lose the overall shape of Fourier, but we do preserve these fine details in the coat, these very sharp edges, so that's what we see if we only keep the high spatial frequency information out of the image. And that's one simple example of how you can read an image in Fourier space. Okay, that's enough math and that's enough very abstract concepts. Why are we talking about Fourier transform? How is that related to microscopy? Okay, let's come to the very basics of image formation of a microscope, and the key element in the microscope is the objective. And you can think of it as a lens, so it has the back focal plane, which is behind the objective, and the sample is placed at the front focal plane of this objective. And what our lens does, for any parallel light from the back, it's going to focus it down to one point in the front focal plane or the sample plane. And if the angle of this parallel light has an angle of α to the optical axis, it's going to focus down to a point at x = f sinα, f is the focal length. This is basically the design of an objective, so this is how the objectives are designed. Now, making this simple, we only consider two light rays. And the question we ask is that, at the back focal plane, because this is where light comes in, we have some light intensity distribution, say, at a distance of k from the center of the objective, we're going to have some light intensity of A as a function of k. Now, what's the light intensity in the sample plane when it's focused down, with all these different light rays? We can calculate that. Now, two light rays are going to propagate from the back focal plane all the way to the sample, and they go through two different paths and there are two different lengths to these paths. To calculate the distance difference of these two light rays to get to the sample, we simply draw a perpendicular line through the center of the back focal plane, and in this case this distance, d, will tell you how much extra length this light ray has traveled to get to the sample. And to... this distance, d, we can calculate it very easily assuming the refractive index is 1, that's in air. Using very simple triangular mathematics, d equals k times the sine of this angle α. And we know, here, x = f sinα, we can plug that back in, we get d=kx/f, which is the focal length of the objective. So, what does this extra distance mean? Light propagates as waves from this plane all the way to the sample and then these two waves, or many waves, can then add together, one wave, here, another wave has traveled an extra distance, so there's some phase delay. And light is also a sine wave, so we can write that in, so the intensity, here, at the sample, I, is going to be the sum of the intensity over the back focal plane times the effect of different phase delays, and that is a sine function. So, what we get, here, is the sum of A(k) times sine of d, that's the delay, [times 2π over λ], λ is the [wavelength] of the light. And then we can plug in what we get from d. Here is... inside the sin function, we we have kx times... kx times 2π over λf. If you look at this, this appears similar. If you remember the mathematical formula I showed before, which I said is very important, we can see this is exactly a Fourier transform. That's what a lens does. It takes the image at its front focal plane and produces a Fourier transform of the image at the back focal plane, and that's why Fourier transform is so tightly associated to microscopy. Indeed, if we take an image of Fourier, put it at the sample plane of the microscope, at the back focal plane we will observe exactly the same thing as we have mathematically calculated here -- the Fourier transform of Fourier. And this means a lot to microscope image formation. For example, as well we have discussed, in the Fourier space, towards the center, that's where the low spatial frequency information is, towards the edge, that's the high spatial frequency. And the high spatial frequency means very fine features. On the other hand, well, a microscope objective cannot be infinitely large. Here, we're talking about some real space in the Fourier... in the back focal plane, and so it has only some finite size of the back aperture. This means that low spatial frequency information, within this back aperture, will be able to go through the objective and reach out camera and get detected. But high spatial frequency information, outside of this back aperture, will get cut off by the objective. And this is essentially the fundamental reason why microscopes have a finite spatial resolution. It cannot resolve anything infinitely sharp. Let's be more quantitative. The size of the back aperture determines some maximum k value that the objective will allow to go through, and this k value is related to the spatial frequency in the sample domain in this formula, k... spatial frequency = k times 2π over λf, where f is the focal length, λ is the wavelength of light. Then, what is the maximum value of k? Well, from the previous slides, we know that given a light ray that has an angle of α to the optical axis, the k will be f times sinα, and the maximum value of k will be determined by the maximum value of α, or how high of an angle this light ray can be from the optical axis. And that is essentially described by the numerical aperture of this objective. So, kmax = f times sin, or maximum value, of α, and sin of maximum value of α is exactly the numerical aperture of the objective, so kmax = f times numerical aperture. And then... and what does this mean to resolution? Okay, now we're talking about resolving two sine waves, and one sine wave like this, another sine wave like this, and if we're not considering this periodicity, we're just singly looking at two peaks, and you can think, or an easy criteria is, that to be able to resolve these two peaks, the sum of these two peaks needs to have a dip in the middle, and in this case the maximum value of the second peak is going to fall at exactly the same point as the minimum value of the first peak, and then the two sine waves will have a phase difference of π. We can plug that in, here. The resolution is related to the maximum spatial frequency as π over maximum spatial frequency. And then, putting all these numbers together, we get that the resolution equals λ/2NA. This is exactly what we have seen a lot of the time in the microscopy literature, how the numerical aperture of the objective is connected to the resolution of the objective. We can also understand it in this way. Now, we think about the microscope objective focusing the light, then we have our circular numerical aperture is completely filled by light, and that is going to converge down to one single point, and, by definition, this point, or this focal point, is the point spread function of the microscope objective. And now we also know that from the left side and the right side, the connection here is Fourier transform, so it means that the Fourier transform of the light intensity, and of course the phase, at the back aperture, is the point spread function of the microscope. So, that's the lecture. We've talked about real space and that's connected to frequency space through the Fourier transform, and in microscopy this Fourier transform is done by the objective itself, connecting the sample plane and the back focal plane.
