Now that we understand that a two dimensional Fourier transform is a decomposition of an image into its component sine waves, let's look at how that transform could be represented in another image. So let's being here with an imaginary image of just some simple object that I drew. And let's suppose that in this image we have ten pixels in x, and ten pixels in y. And it's a digital image. It's a grey scale. So the pixels have values from zero up to nine where the image is very dark and zero where it's, it's light. And as you can see in some pixels where the object only cuts through part of a pixel, it gets an intermediate value of say six. Here in this case the object only touches a little bit of the pixel so we give it a value of one. These are just numbers that I obviously made up. But the point is that you could have an image of an object that was digitized into a series of numbers. Now what would it look like if we sent that digital image to a Fourier transform routine in the computer? What would it reply with. It turns out that the output of the Fourier transform routine will be another two dimensional array of numbers and the patterns of numbers is readily understood. So here the Fourier transform of this two dimensional image will be a series of numbers, a two dimensional array of numbers. And in this case, instead of being organized as x and y values now the numbers have h and k indices. These are the same Miller indices that we've talked about previously. And so the values of h are going to be zero, one, two, three, four and five. In an image that has ten pixels across. And the values of k are from zero to positive five and to negative five. And, for instance this first set of values this would be the amplitude of the h equals zero k equals zero wave. And the phase of the h equals zero, k equals zero wave. Well what is the zero, zero wave? This is a wave that has zero oscillations across the box in x and zero oscillations across the box in y. In other words, it's flat. It's just a simple, constant value. Just like the the first term in a one-dimensional Fourier transform represented the average value of the function. So here, the zero zero wave represents the average value of all the pixels in this whole image. And so it's, it's it just represents a constant number across that image. The next position. Let's look at this one. These two values that the computer would respond with, are amplitude of the h equals zero, k equals one. Wave and the phase of that same zero one wave, and so this is the wave that has one oscillation across the box in x and none in y, that very first wave that we've represented a couple times now. And here is its amplitude, and here is its face. Let's look in this position, the computer will respond with. The amplitude and the phase of the one zero wave. So you see these indices of the h and k values of the waves that is how the computer will arrange the amplitudes and phases of all the Fourier components of the image. As it responds back to us as the output of the Fourier transform routine. In similar fashion, this is where we'll find the amplitude and the phase of the zero five wave. This is the amplitude and phase of the zero minus five wave. This is the amplitude in phase of the two zero wave. The five zero wave, the five, five wave. The five negative five wave, and the pixels in between. For instance here, add amplitudes and phases of all the other possible waves present in the image. So for instance here this is the amplitude and phase of the h equals two, k equals three wave. A wave that has two oscillations across the box in x, and three oscillations in y. Now one of the important principles here is that the original digital image with all its pixel values. And its Fourier transforms, namely the set of amplitudes and phases of all of these different waves, have exactly the same information content. In other words if in the original image we have n squared numbers, n squared pixel values,. In the Fourier transform, so two we have approximately n squared numbers. Notice that here, h goes from zero to five, and k goes from minus five to positive five, approximately ten values. And so 10 times 5 sine waves is 50 sine waves. But each sine wave has both an amplitude and a phase, and so to describe each sine wave we have two numbers. Amplitude and phase. And so these are all of the sine waves that you need to represent this digital image. There aren't any others. So for instance, remember we talked about the Nyquist frequency. That, in a one dimensional function, the Nyquist frequency was the wave that oscillated in every other box. Every other pixel. In two dimensional images, there's also a Nyquist frequency. It's here. This is a wave that oscillates, at least across x, every other pixel. If there's ten pixels in the image then Nyquist frequency is the frequency with five wavelengths across the box. So it's going up and down every other pixel. And the Nyquist frequency in the vertical direction in k is also five, meaning that represents a wave that's going up and down every other pixel in y. Now in a two dimensional image, the highest frequency wave present is this one in a corner, which actually has five oscillations in x and five oscillations in y. And so it grows across the box diagonally a little over seven times. So that's now the highest spacial frequency present. But the point is there is no component in this image of a sine wave that has six oscillations. Across the box because that would start to be oscillations up and down all within this same pixel. And so that wave is simply not present in a digital image that's been sampled with just ten pixels across. These waves are all that you need to perfectly represent the original image. They're called Nyquist. Now, the reason that the Fourier transform has a few extra than N squared values that are returned back is because there is an extra row here of h equals 0 sine waves,. And some redundancies. For instance, the 0 minus 5 wave would looks just like the 0 5 wave. You can't tell whether its going up or whether it's going down across the box. So, there are few extra values that come through in a Fourier transform. Beyond the N squared numbers that defined the original image. But to a first approximation, if you have an image with N squared numbers, the Fourier transform comes back with N squared numbers. They're different numbers. They represent amplitudes, and phases of sine waves. It's like saying the same thing in two different languages. This is the real space language, which are actual pixel values. This is the Fourier transform language, where the values represent amplitudes, and phases of component sine waves. Now, it's difficult to appreciate the information content, if we look at the actual numerical values in the 2-dimensional Fourier transform. Instead, what we like to do is plot them in 2-dimensions. And let me introduce you to that now. So, this is a simple 2-dimensional image, and its transform. We'll learn later, that the transform of an image is, is the same as its diffraction pattern, and so that's why I've written that here. So, let's look at a very simple 2-D image. Here it is, and you'll recognize now that it's simply 1 sine wave that's oscillating back and forth, as it crosses the box, and we call this, and, and the coordinates are x and y, and this is real space that's the actual image. Now, if we record its Fourier transform, that's what appears here. The axes in a Fourier transform is for spatial frequency in the x direction, which is indexed with that Miller indis, index h, and the vertical axis is spatial frequency in the y direction, indexed with the Miller index, index k. And so, the transform of this image is very simple. It only has one sign component in it. And so, it's transform its two dots, one right here, and another right here. So, in this image this is a sign wave with ten oscillations across the box. So, it's the h equals ten, k equals zero sine wave. And we can plot the amplitude of the h equals ten, ken, k equals zero wave, we can plot it's amplitude right here. And typically, instead of plotting amplitudes we plot power, or the intensity, which is just the amplitude squared. And so, we get a, a strong intensity here, and a strong intensity here. This represents the minus ten, zero wave, which from a computer's point of view, it's the same wave. We can't tell whether the wave is going to the left, or to the right. It looks the same. And so, the ten, zero wave, and the minus ten, zero wave are just mirrors of each other, they're the same thing. And so the computer responds that there are two waves present in this image. Here, and here, and if we plot their amplitudes, we've see values here, and here. And the rest of the plot is zero, because there are no other waves present in this image. And so, we call this the reciprocal space, and the frequencies here, we call them spacial frequencies. Okay. Let's do another simple 2-D image, and its transform. So, here is the real space image in x and y. And you might notice, well, this is a sine wave that's oscillating vertically, and it has ten oscillations across the box. So, here is our reciprocal space with. Spatial frequencies in the x and the y direction, and what would the Fourier transform of this image look like? Can you guess? Well, the Fourier transform of this image just has two non-zero values, one right here, and one right here, representing the zero, ten wave, and the zero minus ten wave. These are waves that have ten oscillations across the box vertically. No oscillations in x, and again the ten, zero, or, or the zero, ten. And the zero minus ten waves are indistinguishable, so we see those two values appearing in the Fourier transform. Now, let's look at some more complex 2-D images, and their transforms. Let's start with this 2-D image. This is a diagonal wave with one, two, three, four, four oscillations across the box in x, and one, two, three, four in y, what would its transform look like? Well, the answer is two spots, very near the origin. So, in this case the origin of this Fourier transform is about there, and this is spacial derec, spacial frequency in the x direction, and this is spacial frequency in the y direction. From zero spatial frequency, or which represents the sine wave that's flat all the way across the box, to the edge of this image is the Nyquist frequency. Which represents a sine wave that's going up, and down every other pixel. Now, this is not going up, and down every other pixel, we, we can't tell how finely this image is sampled. But the sine wave that's present in this image is just one sine wave. And it's the h equals four, k equals four wave, so we have a single non-zero value right there, and then it's mirror image across the origin representing the minus four, minus four wave. Which are in, indistinguishable. So, the Fourier transform of [SOUND] this image is just two dots. One here, and one there, and the rest is all zero. Well, how about this image? In this image, we still just have one sine wave. But there's a couple differences. First of all, it has very many oscillations in x. And very many oscillations in y. And the cress are headed towards the lower right corner. So, what would be, what would the foray transform look like? Well, in this case, the Fourier transform is again two dots equal distant, equi distant from the origin. And it's an h equals, perhaps 20. And k equals minus 20 wave. So, it has approximately 20 oscillations in this direction, and 20 oscillations in this direction across the box. And it's a minus k wave. And that is what tells us that it's headed towards the lower right corner. So, if you'll notice, if you have sine waves the position of the dot in the reciprocal space that represents that sine wave, is in a direction that's perpendicular to the cress of the wave. In this case, perpendicular with this direction, and so the spots appeared in that direction in the Fourier transform. In this case the perpendicular to the cress is in this direction, and so we move in that direction to find the position of that wave in reciprocal space. And then it has it's mirror image across the origin. Okay, let's look at the next wave. So, in this case, it's still a single sine wave, but now the frequency is much lower. The oscillation is very, there's far fewer oscillations across the box. And it has a little bit different direction. What will its Fourier Transform look like? Well the answer is it's too very closely spaced out on either side of the origin. And the reason they're so close to the origin is because it represents the h equals one wave. because there's just one oscillation across the box in x, and the k equals 2 wave. Because there's just two oscillations across the box vertil, vertically. So the very close to the origin. It represents a very slowly varying wave, and it is positioned one unit over in x and two units down from the origin. And so, if you look at the perpendicular to the crest, that's the direction from the origin, where that spot is found. Okay, now we're ready for this last, most complicated, image. Suppose this is an image that you recorded. What would its Fourier Transform be? Now I'll give you a hint. This image is the sum of this one and this one and this one. Do you see how this pattern is the sum of these three images? So what would the Fourier Transform of this be? Did you guess it right? It is all of the spots present in these three Fourier transforms. In other words an image that is, the sum of three different sine waves. In the Fourier transform, you'll see spots representing all three sine waves present. So the origin is approximately there, in this image. This spot represents the sine wave over here, that's oscillating across the box towards the upper right corner. This wave represents this fastly varying high frequency wave that's going from the upper left to the lower right. And this dot represents the slowest varying wave that is headed in this direction. And these three spots are their mirror images across the origin. So in other words, what we see in the Fourier Transform of the image is each com point separated as a single spot in the image. Now the way that this image is related to the array that I showed you that the computer would return as the output of the Fourier Transform routine. That array had all of the different pixel values from h in case I showed you h equals zero to five, k from negative five to positive five, and there was all those different pixels. Well, if we simply plot the square of the amplitude of each pixel in a space like this. Then, where a sign wave is present and it has a finite amplitude it will appear we can plot its intensity by how dark we make that spot. So in this case, all of these three waves had essentially the same amplitude and so they all look like three similarly intense spots. And there's nothing around them because all the other waves in between have an amplitude of zero. Meaning they're not present in the image. And that's why this power spectrum looks so simple with just six dots representing three different waves. And they all have approximately the same amplitude. But real images are, of course, far more complex. But these principles will help you understand their 4A transforms. So, we're already to a point where we can understand one of the figures of a recent paper from my group. And in this project we were imaging the chemoreceptor arrays of bacterial cells. So here is a slice of a Three-dimensional tomogram of a cell. And all that you need to know at this point is that it's a Two-dimensional image of a section through a cell. And this cell has a series of receptors that bind together in a hexagonal lattice. So you can begin to see these receptors bound together in a hexagonal lattice. It's an EM image, and we these receptors, we're interested in how their packed together. And with images like this we could see that there's some kind of pattern. So to discover the pattern we calculated for a transforms of the images. Now in this particular paper, we we're comparing how the receptors packed in about 11 different organisms. So this panel is a slice of a tomagram, where we see the pattern in one particular species and this is its 4A transform. And you see a very nice, clean pattern of one dot in the middle and then six, dots surrounding it. And that's because this image is almost entirely dominated by three sine waves. This one, this one, and this one. That travel in this direction, this direction, and this direction. And together they build up this hexagonal lattice. This is the characteristic for a transform of a hexagonal pattern and so we, we could see that in this species, the, the receptors were arranged in a hexagonal lattice. Now in the next species, we saw that the receptors were again arranged in a nice hexagonal lattice, as evidenced by the Fourier Transform of that image. And so in all the species that we imaged and here are all of the examples, the same Fourier Transform evidenced the same basic hexagonal lattice. And so in this case, the Fourier Transforms of all these images were all that we needed to argue that the receptors were packed similarly in all these different species. Now the images of many objects are even more complex than hexagonal pattern. They could be any kind of an image. So lets look at a, a classic example in structural biology. Which is the image of a duck. So here's the image of a duck. It's black and white and here, is its Fourier transform. So, you need to remember that in a Fourier transform this is plotted as a function of two different variables, spatial frequency in the x direction and spatial frequency in the y direction. And so this image is a series of pixels, I'm drawing them extra big here, but it's a series of pixels. Pixels, you know, fill this image, and, [SOUND] each pixel represents one of the possible sine waves that could be present or not, in this image. Now, more precisely, all of these sine waves are present in this image, just some are at very low amplitude and some are at very high amplitude. The ones at very low amplitude here look black. because we're plotting their intensity, or their amplitude squared. So, these are very low numbers and here near the origin these sine waves are present with high amplitude in the image and so they appear bright. Now note that we don't plot the phase anywhere. These two-dimensional plots of the Fourier transform only give you information about the amplitudes of each. Fourier component. That's another term that we use to mean each sine wave in the image, each sine wave can be called a Fourier component. And we simply plot their amplitude squared, or their intensity. We, we don't represent their phase here in these images. These images are also called power spectrum and you'll hear that word a lot. This is a power spectrum plural is power spectra. And the power spectra show you the power of each Fourier component or sound wave in this image. And so we use power as a synonym for the intensity, or the amount, of each sine wave present. Now we can introduce the meaning of the word, of the term resolution. We have before considered that the pixels near the origin in these power spectra represents sine waves that oscillate slowly. Maybe once across the box, that's the first pixel. Maybe three or five times across the box, that's the third pixel, and the fifth pixel. But these pixels near the origin represent very slowly varying features in the image. And so we call those low resolution. On the other hand, pixels near the edge of the power spectra, these represent waves that are oscillating very rapidly in the image. Perhaps, every other pixel, they're up and down. That would be the Nyquist frequency. And so we talk about those as high resolution, high frequency components of the image. So let's consider this picture of a person in front of a black board. And you can clearly see the features of his face, you can even see what he's written on the black board. Now, if we calculate the Fourier transform of that image, we get this complicated pattern. And remember, that this is spatial frequency in the x direction, spatial frequency in the y direction. So these pixels near the origin, they represent slowly varying waves that are present in this image. Pixels at the extreme edges of the Fourier transform represents rapidly oscillating waves. For instance, waves that even may oscillate up and down in every other pixel. These are oscillations across the x direction. These represent waves that are oscillating in the vertical direction. These are waves, for instance, that are oscillating diagonally across the image. And everything in between. Every pixel in this foray transform represents one two-dimensional sine wave that could be present in this image. And the, and the brightness of the pixel represents its intensity. Well, if we then took the Fourier transform of this Fourier transform, we took another Fourier transform. What we'll, we would get back is the original image of the person with his writing on the chalkboard. In other words, if you take the f, a Fourier transform moves you from real space to reciprocal space. Then if you take another Fourier transform, you might call it an inverse Fourier transform, but it works the same way. You would recover the original image in real space. And so each time you take the Fourier transform, it just switches the representation from real to reciprocal to real to reciprocal, vice versa. Fourier transforms just change which space the image is being represented in. And so you would get an identical image here to the original by taking two Fourier transforms. However, if instead of using all the pixels in this Fourier transform, if we just use a set of them, so here this is the same Fourier transform of this image. But this time, we're only going to include the pixels in the middle of the image, near the origin. And we're going to 0, just multiple by 0 all of these other values. So we just 0 out the Fourier transform all around here. If we then take the Fourier transform of that, we get an image that resembles the original but the details are gone. So here, you see the outlines of the man's face, but we've lost details, for instance particular hair. We've lost the hair, we've lost the details of the writing on the chalkboard many of the features of the eye, and this is called a low pass filter, because we've only transmitted. We, we have filtered the Fourier transform and blocked off all of the high frequency components and passed only the low frequency components. So in a low pass filter, you pass only those sine waves near the origin which are slowly varying, and you get an image that's lost the details. Now as you can imagine, you can do the inverse. Instead you can take this original Fourier transform, with all the information you need, and instead you can zero out the pixel values near the origin. So here these are blocked out, but all of the higher frequency components are allowed to pass through. And if we take the Fourier transform of this, now we get an image that again resembles the original image. In this case, it has all the high frequency details. All of the little details like you can read the writing on a chalkboard just fine. You can see individual locks of his hair, some details of the shirt. But now what's missing is the low frequency components, because the wave that contains that information, namely that this side of the image is dark, and this side of the image is generally bright. That wave would be one that oscillated, it would have a minimum here and a maximum over here. And that is a slowly varying wave, and so that wave would appear very close to the origin here. And it is zeroed out. And so in the filtered image you don't see, you lose this information that this side was generally dark and that was generally bright, and they look rather comparable. But you still have the high frequency details, because they were contained in these pixels. This is a called a high pass filter, because you're passing the high frequency components, the details. Well you can imagine any number of filters. This filter would contain neither of the lowest frequency components, nor the highest frequency components. But it would instead, it would only pass a few of the sine waves present in this middle range. And this is called a band pass filter because we're passing through the Fourier transform just a single band of spatial frequencies. And the image, these images can be hard to interpret, you can se, still see the general outline of the man's head. But neither the high frequency details, like the writing on the chalkboard nor the low frequency information, like, that this is generally dark and that's supposed to be generally white. All of that is gone, and you only see the middle-range detail present in the image.
