Fourier transforms in cryo-EM | Convolution, Gaussian filters, Fourier Shell Correlation, FFT

00:59:44
https://www.youtube.com/watch?v=mRrM0cus1HE

Summary

TLDRPavel Afanasyev's lecture on Fourier transforms in cryo-EM covers the importance of these mathematical tools in analyzing and reconstructing data from electron microscopy. The lecture begins with the basics of periodic functions and Fourier transforms, explaining how complex functions can be decomposed into simpler sinusoidal components. It highlights the historical context of Fourier transforms in electron microscopy, their applications in data collection, and the significance of sampling and the Nyquist frequency for achieving high resolution. The lecture also discusses practical applications such as correlation, convolution, and filtering techniques, emphasizing the convenience of using Fourier transforms for data analysis. Afanasyev encourages the audience to utilize Fourier transforms as diagnostic tools in their work and recommends further resources for learning.

Takeaways

  • 📊 Fourier transforms are essential in cryo-EM for data analysis.
  • 🔍 Understanding periodic functions is key to grasping Fourier transforms.
  • 📈 The Nyquist frequency determines the maximum resolution achievable.
  • 🖼️ Fourier transforms allow decomposition of complex functions into simpler components.
  • 🔄 The Fast Fourier Transform (FFT) is an efficient algorithm for calculations.
  • 🧩 Convolution and correlation are important operations in image analysis.
  • 🛠️ Filtering techniques (low-pass, high-pass) help in enhancing image quality.
  • 📚 Recommended resources include Stanford lectures and classic Fourier analysis books.
  • 🔬 The Central Slice Theorem aids in reconstructing 3D objects from 2D data.
  • 🧪 Fourier transforms serve as diagnostic tools in data processing.

Timeline

  • 00:00:00 - 00:05:00

    Pavel Afanasyev introduces his lecture on the use of Fourier transforms in cryo-electron microscopy (cryo-EM), emphasizing the importance of understanding Fourier transforms for data analysis in this field. He outlines the structure of the lecture, which will cover one-dimensional periodic functions, two-dimensional images, and applications in electron microscopy.

  • 00:05:00 - 00:10:00

    The historical context of Fourier transforms in electron microscopy is discussed, highlighting the contributions of De Rosier and Klug in the mid-20th century. Their work on bacteriophage T4 tail reconstruction using Fourier analysis is presented as a foundational example of Fourier transforms' significance in cryo-EM.

  • 00:10:00 - 00:15:00

    The lecture begins with the definition of periodic functions, explaining key parameters such as amplitude, period, and phase. The relationship between frequency and period is introduced, emphasizing the relevance of these concepts in cryo-EM data analysis.

  • 00:15:00 - 00:20:00

    The concept of summing periodic functions is introduced, leading to the idea of decomposition, where any periodic function can be represented as a sum of sinusoids. The Fourier transform is defined as a mathematical tool that allows this decomposition, with a focus on its practical applications in cryo-EM.

  • 00:20:00 - 00:25:00

    The properties of Fourier transforms are discussed, including the ability to invert the transform, allowing for the reconstruction of the original function from its Fourier components. The importance of sampling and the Nyquist frequency in relation to resolution in cryo-EM is emphasized.

  • 00:25:00 - 00:30:00

    The distinction between real space and Fourier space is made, with an explanation of how functions can be decomposed into their frequency components. The concept of amplitude spectrum is introduced, highlighting the symmetry of Fourier transforms and the significance of higher frequency terms in data analysis.

  • 00:30:00 - 00:35:00

    The lecture transitions to two-dimensional Fourier transforms, using images as examples. The importance of pixel intensity and dimensionality in image representation is discussed, along with the process of decomposing images into their Fourier components.

  • 00:35:00 - 00:40:00

    The concept of scaling in both real and Fourier space is introduced, explaining how images can be binned and how this affects pixel size and resolution. The process of Fourier cropping is also explained as a method for efficient image processing.

  • 00:40:00 - 00:45:00

    Fourier filtering techniques are introduced, including low-pass, high-pass, and band-pass filters. The use of Gaussian functions for filtering in cryo-EM is emphasized, along with the importance of avoiding sharp edges in masks to prevent artifacts in the inverse Fourier transform.

  • 00:45:00 - 00:59:44

    The lecture concludes with a discussion on the application of Fourier transforms in 3D reconstruction and convolution, highlighting the efficiency of using Fourier transforms for operations like deconvolution and correlation in cryo-EM data analysis. The significance of Fourier Shell Correlation in determining resolution is also explained, along with recommendations for further reading on the topic.

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Mind Map

Video Q&A

  • What is the main topic of the lecture?

    The lecture focuses on the use of Fourier transforms in cryo-electron microscopy (cryo-EM).

  • Who is the lecturer?

    The lecturer is Pavel Afanasyev.

  • What are the key components of Fourier transforms discussed?

    The key components include amplitude, phase, and frequency.

  • What is the significance of the Nyquist frequency?

    The Nyquist frequency is crucial for determining the maximum theoretical resolution achievable in cryo-EM.

  • What are some applications of Fourier transforms in cryo-EM?

    Applications include data collection, CTF correction, validation of 3D reconstructions, and filtering techniques.

  • What is the difference between real space and Fourier space?

    Real space refers to the original data domain, while Fourier space is the domain after applying Fourier transforms.

  • What is the Fast Fourier Transform (FFT)?

    The FFT is an efficient algorithm for computing Fourier transforms.

  • What is the Central Slice Theorem?

    The Central Slice Theorem allows for the reconstruction of 3D objects from their 2D projections in Fourier space.

  • What are low-pass and high-pass filters?

    Low-pass filters allow low-frequency information to pass, while high-pass filters allow high-frequency information to pass.

  • What resources are recommended for further learning?

    Recommended resources include lectures from Stanford University and classic books on Fourier analysis.

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  • 00:00:00
    Hello my name is Pavel Afanasyev and first I would  like to apologize I couldn't make it to Brazil.
  • 00:00:06
    Nevertheless, I recorded the lecture for  you and I hope you'll find it useful.
  • 00:00:16
    The title of the lecture is "the use  of Fourier transforms in cryo-EM".
  • 00:00:20
    It's very difficult to cover the full topic  of Fourier transforms within one hour.
  • 00:00:25
    Therefore, the idea of this lecture  is to illustrate to you why Fourier
  • 00:00:32
    transforms are so important in cryo-EM  and to give you a background for an
  • 00:00:39
    intuitive use and for intuitive  understanding of the maths behind.
  • 00:00:48
    So, my talk will consist of several sections  I will start with one dimensional periodic
  • 00:00:56
    functions, I'll cover the idea of  the Fourier transforms of those.
  • 00:01:01
    Then, I'll switch to two-dimensional  periodic functions (or images).
  • 00:01:06
    We will talk about Fourier transforms of  the images and we will also talk a little
  • 00:01:12
    bit about the Fourier transforms of the  3D functions or volumes (cryo-EM maps).
  • 00:01:20
    And finally, I'll try to discuss the applications  of where it transforms in electron microscopy.
  • 00:01:29
    The first time Fourier transforms were  used in electron microscopy was in the
  • 00:01:37
    middle of the 20th century, and that  happened thanks to the crystallographers
  • 00:01:43
    who were starting doing electron microscopy. De Rosier and Klug were two scientists who studied
  • 00:01:51
    the tail of bacteriophage T4 and they used methods  from X-ray crystallography for determination of
  • 00:02:02
    3D-reconstruction from an object, and they used  these methods of the Fourier analysis in order to
  • 00:02:11
    obtain the structure of the bacteriophage T4 tail. So, in this lecture I will try to explain what
  • 00:02:21
    exactly they were doing and  what are the methods behind.
  • 00:02:26
    Today, it's very difficult to imagine the field  of cryoelectron microscopy without Fourier
  • 00:02:32
    transforms. And pretty much starting with the  data collection, we are using Fourier transforms
  • 00:02:39
    everywhere: from the CTF-correction (it's  pre-processing stages of the data analysis) up to
  • 00:02:47
    the validation of your final 3D-reconstructions  using Fourier Shell Correlation.
  • 00:02:55
    Let's start with the very basics and first I would  like to introduce (or define) periodic functions.
  • 00:03:04
    So, what is periodic? Periodic means  the function, where you can add any
  • 00:03:10
    period to the to the function and then  you'll get the same function back.
  • 00:03:14
    The easiest way to imagine that would  be if you look at the simple sinusoid.
  • 00:03:23
    This is the function sin(x) and the the function  sin(x) can be generalized in these terms: where
  • 00:03:33
    "A" would be an amplitude and we know that A=1 in  this case, so the amplitude of a function would be
  • 00:03:44
    the distance between the x-axis and  the highest crest of the function.
  • 00:03:55
    In a way this is how high  the wave of the function is.
  • 00:04:00
    Now, the second important  parameter would be the period.
  • 00:04:06
    And the period would be the distance  between two crests of the function.
  • 00:04:13
    So, in case of our function... (our sinusoid  function) the period would be equal to 2 pi.
  • 00:04:24
    Also period (you'll find sometimes), is  known as a wavelength of the function.
  • 00:04:35
    Now, the third parameter which  we'll be talking about is the phase.
  • 00:04:40
    And to talk about the phase (if we want  to talk about the phase of the function),
  • 00:04:48
    it's easier if you imagine another  function, for example sin(x - pi/4).
  • 00:04:55
    And pi/4 would be the phase, and that would  be the distance between these two functions.
  • 00:05:03
    So, this is how... (if you imagine that  this axis would be time) it means that how
  • 00:05:10
    slow or what is the delay of these two functions. And it's also sometimes denoted as a phase shift.
  • 00:05:22
    So, knowing these two functions and looking  at this general equation we can easily assign
  • 00:05:29
    values for the phase, for the period or  the wavelength, and for the amplitude.
  • 00:05:35
    Now, in cryo-EM and for us it would be  important also to talk about the frequencies,
  • 00:05:41
    and frequency is related to the  period (or the wavelength) as 1/T.
  • 00:05:49
    So, the period, sorry the frequency, would  illustrate how fast the function will be changing,
  • 00:05:59
    and it's defined as a number of  cycles per second or x axis unit.
  • 00:06:09
    In cryo-EM, as I mentioned, we would be focused  in definitions of any functions using these three
  • 00:06:17
    variables: amplitude, phase and the frequency. So, the simplest operation we could think about,
  • 00:06:25
    when we work with periodic  functions, would be summing.
  • 00:06:29
    So, what means to sum or to average two functions? Again, let's consider one function - the first one
  • 00:06:37
    which we talked about already, it would be sin(x). And the second one will have a different amplitude
  • 00:06:46
    and a different frequency  - it will be 1/3*sin(3x).
  • 00:06:50
    So, if we sum these two functions you'll see  that the second function has the minimum where
  • 00:06:57
    the first one has the maximum, and if we  average or if we sum these two functions,
  • 00:07:01
    we'll get something like this, so this  will be the sum of these two functions.
  • 00:07:06
    Let's consider the reverse operation  which is called decomposition.
  • 00:07:11
    So, the idea of the Fourier transforms actually  is in the following: any periodic function can
  • 00:07:18
    be represented as a sum of sinusoids or  as the sum of other periodic functions.
  • 00:07:25
    For example, let's consider a more complicated  function which would look like this,
  • 00:07:31
    and turns out that this function can be  represented as a sum of the following functions,
  • 00:07:39
    which you can see here on the right. So, the first one, it would be the sin(x),
  • 00:07:44
    which you can see here, and then, if  we add to sin(x) a second function
  • 00:07:48
    which would look like this, the  sum of these two would be here.
  • 00:07:53
    The blue curve will be the  sum of these two functions.
  • 00:07:58
    Now, if we consider the third function,  which will have higher frequency,
  • 00:08:05
    and if we average these three  functions, they will already
  • 00:08:12
    approximate the original function quite well. But moreover, if we take the fourth function
  • 00:08:19
    we can fully represent or we can fully reconstruct  the original functions which we started with,
  • 00:08:27
    so the four the sum of these four functions would
  • 00:08:31
    be the original function we  wanted to decompose it into.
  • 00:08:35
    And this is called decomposition - representation  of the function as a sum of other functions.
  • 00:08:42
    And you might ask: "How do we get all these  values of the amplitudes and the phases?
  • 00:08:50
    How do we know how to represent or  how to decompose the function into?"
  • 00:08:55
    And that's exactly what the  Fourier transform would be doing.
  • 00:08:59
    So, the Fourier transform allows you  to decompose any periodic function
  • 00:09:03
    into the series of sine waves. And the person who came up with
  • 00:09:09
    such an idea and who described it  was the French mathematician Fourier.
  • 00:09:14
    He came up with this formula. This formula might  look a bit complicated for you, but actually it's
  • 00:09:20
    very simple: so these two terms, they are very  much related because sine and cosine function
  • 00:09:25
    are different just by addition of of pi/2. So, we can just focus on one of them,
  • 00:09:33
    right? And then, we also have a constant  here which you can ignore for now.
  • 00:09:38
    So basically, this function would be the sum  of these sinusoids. And in order to get this
  • 00:09:47
    amplitude which, Bm would actually be, we  would have to calculate these integrals.
  • 00:09:58
    A bit complicated, but there  are ways how to calculate those,
  • 00:10:02
    and A and B would be the amplitude here. So, there is an equation which allows to
  • 00:10:09
    perform such operations of the composition of  functions as a sum of some other functions.
  • 00:10:18
    Now, the most important part about that is that  the Fourier transform can be fully inverted.
  • 00:10:24
    What does that mean? If we have our function and  we call it the function which is in "real space",
  • 00:10:32
    and if we perform the Fourier transform (if we  decompose it as a sum of some other functions),
  • 00:10:38
    we'll be working now in so-called  "reciprocal" or "Fourier space".
  • 00:10:43
    So, we perform one Fourier transform and we get
  • 00:10:48
    another function which is a sum of some  functions, and then, if we perform an
  • 00:10:54
    inverse Fourier transform, we would be able  to get our function back in real space.
  • 00:11:03
    Now, as I already mentioned,  such kind of decomposition
  • 00:11:09
    would imply that the terms which we would  be using (or the functions which would be
  • 00:11:16
    using) they would have certain properties.  And you can already see here some trends.
  • 00:11:24
    So, as we described before, we can calculate  the amplitude period and phase for each of these
  • 00:11:34
    functions and we will see that the, for example,  the wavelength of the first function would be 2 pi
  • 00:11:42
    and the frequency would would be 1/2 pi. And then, we can again determine the
  • 00:11:48
    wavelengths for the second one and  the frequency for the second one.
  • 00:11:51
    And what we can see here is that the frequency for  each further function, for each next term in this
  • 00:12:03
    decomposition will be actually higher and higher. We can see that the wavelength is getting smaller
  • 00:12:08
    and the frequency is getting higher. So, the highest frequency which we'll
  • 00:12:15
    be using for our decomposition will  be called the "Nyquist frequency".
  • 00:12:20
    And deciding on how... what would be the size  of these frequencies or would be the size of the
  • 00:12:34
    very highest frequency for the decomposition is  related to the term which is called "sampling".
  • 00:12:41
    So, it's in case of 1D case (a one  dimensional case) it would be just
  • 00:12:46
    defined by the smallest wavelength, which  we would be using for the decomposition.
  • 00:12:50
    And in case of a two-dimensional (for  two-dimensional case for the images), or for the
  • 00:12:58
    three-dimensional case that would be determined  by the pixel size or by the voxel size,
  • 00:13:04
    and in turn, these are defined just by  your magnification of the microscope.
  • 00:13:10
    Why is it important? It's important because  of the final resolution we would be aiming
  • 00:13:21
    for and and it's important to be able to  decompose any function as fine as possible.
  • 00:13:31
    Why? Because if we use frequencies which  would be too large, we would not be able to
  • 00:13:39
    reconstruct the original signal and that's what  the Nyquist-Shannon sampling theorem is about.
  • 00:13:45
    It says: "signal sample at rate  f can be fully reconstructed,
  • 00:13:49
    if it contains only frequency components  below half sampling frequency.
  • 00:13:56
    So, that means that in order to  fully reconstruct your signal
  • 00:14:01
    you need to have certain type of sampling or you  need to have certain type of your pixel size.
  • 00:14:08
    And in case of cryoelectron microscopy,  where the Nyquist frequency term will be
  • 00:14:17
    used, it is in terms of the maximum theoretical  resolution which you would be able to achieve.
  • 00:14:24
    And that would be equal to one  over two times your pixel size.
  • 00:14:28
    So now, when we defined the Fourier transforms,  when we defined the sampling, it's very important
  • 00:14:39
    again to denote two domains which we are working  with. The so-called "real space" domain or
  • 00:14:47
    "spatial domain" and "Fourier-space domain" or we  will know that it's called a "frequency domain".
  • 00:14:54
    So, we were able to decompose this  function as a sum of these four
  • 00:15:02
    and for each of these four we know its  frequencies and we know the amplitudes.
  • 00:15:09
    And we can create a plot of the frequencies  which would be the frequencies and amplitudes.
  • 00:15:19
    And Fourier transforms have properties  of being symmetric, and that's why we
  • 00:15:26
    have four amplitudes corresponding to its  frequencies as positive, and four as negative.
  • 00:15:33
    So in principle, we can, if  we know this side of the plot,
  • 00:15:38
    we would also know this side of the plot. And this is called "amplitude spectrum".
  • 00:15:43
    So any function would be... it's  possible, would be possible,
  • 00:15:48
    to think about it in terms of their amplitudes  phases and in terms of their frequencies.
  • 00:15:54
    Now, as I said before, it's very important to  have higher frequency terms in the decomposition
  • 00:16:02
    and these higher frequency terms would be  defining how good we decompose the function.
  • 00:16:11
    And, for example, if you have some kind of an  exotic function, which would look like this,
  • 00:16:16
    we would be taking more sinusoid functions  with higher and higher frequencies.
  • 00:16:23
    And they will be describing this  function better and better until
  • 00:16:28
    it converges and I have an example here  and I hope I can start it this video...
  • 00:16:39
    So, you can see here that with adding  more and more terms, the function is
  • 00:16:46
    being approximated better and better. And as I noticed, as I said before,
  • 00:16:53
    the frequency of the very last term the highest  term will be called Nyquist frequency, and it
  • 00:16:58
    reflects the sampling which we have chosen. You can already see here that for some sharp
  • 00:17:10
    features we would need to have  a lot of different functions.
  • 00:17:16
    And the truth is that actually for  the very sharp-like edges, would need
  • 00:17:23
    an infinite number of functions in order to  decompose the original function completely.
  • 00:17:33
    So, finer sampling implies using more  functions with higher frequencies.
  • 00:17:39
    And if you are familiar with complex  numbers it's very important to note that
  • 00:17:46
    the Fourier transforms can be  expressed in a slightly different form,
  • 00:17:52
    and in terms of the complex numbers we know that  the complex numbers would have their amplitudes
  • 00:17:59
    and faces just the numbers can be expressed  slightly different, and there's just another
  • 00:18:03
    representation of Fourier transform and sometimes  you'll be seeing that Fourier transforms will be
  • 00:18:08
    expressed in terms of their amplitudes... And sometimes Fourier transforms would be
  • 00:18:15
    expressed in this form - in the form of complex  numbers with their amplitudes and their phases.
  • 00:18:24
    So, I already um mentioned to you  that there are issues with the
  • 00:18:30
    sharp features of the functions, and that  would need an infinite number of functions.
  • 00:18:35
    And how... and you might be asking,  how do we deal with that in real life?
  • 00:18:41
    Because instead of a sum of  these functions we will have to
  • 00:18:46
    integrate over the infinite number of terms. And you can easily, or you can not easily,
  • 00:18:53
    but you can imagine that actually the  images, which you will be working with,
  • 00:18:59
    they are consisting of pixels and, therefore,  this decomposition of image into a pixel would
  • 00:19:11
    be somewhat its own periodic function. And when we will be talking about Fourier
  • 00:19:18
    transforms of the images we would be talking  about so-called "Discrete Fourier Transform",
  • 00:19:24
    which is implemented using "Fast Forward  Transform" algorithm, and sometimes in some
  • 00:19:30
    image processing programs you will see this term  "FFT", and that means it's just an algorithm for
  • 00:19:38
    implementation of the Fourier transform. And it's very fast - using this algorithm
  • 00:19:42
    you are able to calculate the  Fourier transforms very efficiently.
  • 00:19:49
    And these Fourier transforms would be expressed  or these coefficients, the Fourier components
  • 00:19:58
    would be expressed as a sum rather than as an  integration over the infinite number of terms.
  • 00:20:09
    So, it might have been a bit complicated for  you, and you might ask why would we bother, why
  • 00:20:15
    would we need to represent it in terms of... why  would we need to decompose any kind of function?
  • 00:20:21
    And the answer for that is because it is very  convenient, and later on you'll see (there is
  • 00:20:28
    a spoiler) that you will be able to compute it  much faster using Fourier transform, compute
  • 00:20:37
    certain things much faster than if you  would have to compute them in real space.
  • 00:20:44
    So let's now generalize our approach for  the 2D case and let's talk about the images.
  • 00:20:51
    This is an image of 20S proteasome dataset,  it's a micrograph. This was collected on the
  • 00:21:00
    Falcon 3 camera which has 4096 pixels in this  dimension and 4096 pixels in this dimension.
  • 00:21:08
    Nowadays, you might be using already the next  generation of the detector, for example Falcon
  • 00:21:13
    4 detector, but all these detectors Falcon three  two and one and four have the same dimensions:
  • 00:21:20
    4K x 4K with a size of 16.7 megapixels. Some of you might be using different cameras,
  • 00:21:28
    for example, Gatan K2 or Gatan K3, and they  have slightly different dimensions but it's
  • 00:21:36
    comparable in terms of the magnitude  they are (the size of these detectors).
  • 00:21:42
    Moreover, if you talk about just digital  cameras the camera in the latest iPhone
  • 00:21:51
    will be even bigger than these cameras (which
  • 00:21:55
    are used in cryo-EM) and it would be  composed of close to 50 megapixels.
  • 00:22:03
    Now let's talk just about..  about photos for simplicity.
  • 00:22:08
    And let's consider such a photo of Marin van Heel,
  • 00:22:12
    and he's holding here a camera. And if we start zooming in,
  • 00:22:20
    in this photograph, we will see the camera zoomed  in and then we'll see this lens, and if we zoom
  • 00:22:29
    in further, we will start seeing actually an  image with some pixels. It's a pixelated image,
  • 00:22:35
    and in this case it will be 20 by 20. So, if we zoom in at any other area of
  • 00:22:41
    these photographs of this photograph, we will see  this pixelated zoomed-in image. What does it mean?
  • 00:22:49
    So again, here I have 20 pixels in this dimension  and 20 pixels in this dimension (this is the
  • 00:22:56
    cropped area which we're talking about), like  in case of the cameras we were talking about.
  • 00:23:03
    So, each pixel will have its own intensity:  it might be whitish or it might be more
  • 00:23:12
    grayish or it might be very dark. So, that means each pixel will have its
  • 00:23:18
    own intensity, and depending on how white  or how black it would be, we can actually
  • 00:23:28
    assign different values to these intensities. And if we consider such a pellet, we can say that
  • 00:23:38
    very dark areas would be having intensity of zero,  and very white would be having intensity of 255.
  • 00:23:49
    Why 255? Or 255 plus 0 will be 256, this is  2 to the power of 8. And that describes a
  • 00:24:00
    8-bit imaging, so that means that we'll  have 256 shades of grey in our image.
  • 00:24:08
    If we consider 16-bit, we will have  much larger number of shades of gray.
  • 00:24:15
    So now, if we represent each pixel of  the image in terms of its intensities,
  • 00:24:24
    we can assign a number to each pixel,  which would be the number of them,
  • 00:24:31
    or which would be the  intensity value for each pixel.
  • 00:24:36
    And then, as you can see here, very bright areas  would have higher values, for example this white
  • 00:24:44
    one (whitish) would be 183, and then very dark  would be having something like 18 or 19 and so on.
  • 00:24:54
    And we're talking about dimensionality all the  time and then before we talked about 1D curves
  • 00:25:02
    one-dimensional curves, and we were  talking about sinusoids and so on...
  • 00:25:06
    But what 1D curve actually means, is that we have  only a single coordinate and for this coordinate
  • 00:25:15
    we can have a pair of the value which  would be another value in this plot,
  • 00:25:22
    and this is the coordinate value.
  • 00:25:24
    So, we are always plotting our curves
  • 00:25:29
    in this grid, but what we actually can do, we  can assign the intensity value for this curve
  • 00:25:41
    and, for example, everything which will have  higher intensity depending on the x-coordinate
  • 00:25:48
    will be having... will be depicted as  white here. So... and this is 1D curve.
  • 00:25:58
    So, in case of the images we will talk  again in two dimensions, but now in terms
  • 00:26:05
    of the X and Y coordinates. So, each pixel will have
  • 00:26:09
    its intensity, but also each pixel  will have coordinates X and Y.
  • 00:26:13
    And in case of 1D we have only one  coordinate. In case of 3D we'll have
  • 00:26:18
    three coordinates plus the intensities. This is about the dimensionality.
  • 00:26:24
    So now, as we noted before, since the  image can be represented in terms of
  • 00:26:32
    its intensities and we're going now in the  world of 2D, we can represent an image as
  • 00:26:40
    a linear combination of some other images. And in this case, if you want to decompose
  • 00:26:48
    an image, which would be 3x3 - a very  simple image with its own intensities
  • 00:26:54
    we can decompose it in real space. We can just decompose it as a sum
  • 00:27:00
    of such so-called "base vectors" with  a certain coefficients which would be
  • 00:27:09
    similar to the amplitudes.
  • 00:27:11
    So, any image in real space can be  decomposed in terms of again the
  • 00:27:15
    sum of some other images and this is a good  analogy to think about Fourier transforms.
  • 00:27:26
    And let's talk about the actual Fourier  transforms and what the Fourier transforms..
  • 00:27:34
    how they would look like, if we perform  Fourier transform of very simple functions.
  • 00:27:41
    For example, let's perform a Fourier  transform of this sinusoid but in 2D. So,
  • 00:27:48
    we have oscillations only in one direction here. So, the Fourier transform of such an image
  • 00:27:56
    would look like two dots in this direction. So, why two dots? It's because... two because,
  • 00:28:05
    as I mentioned before, Fourier  transforms have the symmetry,
  • 00:28:12
    and um it has it will be exactly the same in the  negative direction as in the positive direction.
  • 00:28:19
    Now, if we imagine another Fourier transform of  this... of such an image, which is similar but
  • 00:28:25
    rotated by 90 degrees, we will see two dots,  which would be again rotated by 90 degrees.
  • 00:28:32
    What's more interesting is to  consider another sinusoid which
  • 00:28:36
    would be oscillating but at higher frequencies.
  • 00:28:41
    So, the Fourier transform in Fourier  space (this image) would look like this.
  • 00:28:46
    And now comparing these two, we can see that the  frequency here is higher, and that means that the
  • 00:28:53
    distance between these two dots would be larger. Everything which is closer to zero here to the
  • 00:29:00
    centre of the image would be representing  lower frequency information, everything
  • 00:29:06
    which would be going towards the edge, will be  representing the higher frequency information.
  • 00:29:12
    And again, you can imagine easily,  that if we have something like this,
  • 00:29:16
    again this operation is  equivalent to the rotation.
  • 00:29:20
    In the perpendicular direction to  the distribution of the wave of these
  • 00:29:26
    waves we don't have any oscillations, yes? That's why in this direction we have zero.
  • 00:29:35
    And if we have a superposition of these and  the perpendicular function which would look
  • 00:29:40
    like these checkers, obviously, will  have something like four dots here.
  • 00:29:47
    So, using such decomposition we can talk about  representing our real space image from before
  • 00:29:57
    (and that represents probably  number "1" in the lens).
  • 00:30:01
    It's in real space, we can decompose the image in  2D in real space as a sum of these of these images
  • 00:30:15
    from the Fourier space. So, the same way how  we were talking about 1D Fourier transform,
  • 00:30:21
    we can create a 2D Fourier transform and to  decompose any image as a linear combination
  • 00:30:30
    of the Fourier components.
  • 00:30:35
    And the good thing about it is that the  same way as we are talking about the 1D
  • 00:30:43
    Fourier transform, you can create Fourier  transforms as many times as you want.
  • 00:30:48
    You can transform there and back, you can  create the forward in an inverse Fourier
  • 00:30:53
    transform and work with. It it's very convenient.
  • 00:30:56
    OK, the same way as we were talking about the  amplitude spectrum in 1D case, we can talk about
  • 00:31:03
    the amplitude spectrum in 2D case. So, if we take a test image,
  • 00:31:08
    if we create a Fourier transform we  will get this amplitude spectrum.
  • 00:31:13
    And when we're talking.. when we're  in the world of cryo-EM, we will be
  • 00:31:17
    talking about so-called "power spectrum". Power spectrum and amplitude spectrum basically
  • 00:31:21
    is the same thing, with the difference  that amplitude squared would be the power.
  • 00:31:30
    And it's just easier to talk in terms of the  power spectrum because it would be reflecting
  • 00:31:36
    the actual intensities which we see in the images.
  • 00:31:39
    And together with the amplitude spectrum  you should also realize that we always
  • 00:31:43
    have, when we perform a Fourier  transform, we'll always have phases.
  • 00:31:50
    And this would be represented by  so-called "phase spectrum". So when
  • 00:31:54
    we create the Fourier transform, we have  amplitudes and phases at the same time.
  • 00:31:59
    Now, as I said, as I briefly mentioned  before, everything which is closer to the
  • 00:32:05
    centre of this image, would correspond  to the lower frequency information
  • 00:32:10
    and everything which would be going farther from  the centre up to here, will be corresponding
  • 00:32:17
    to higher frequency information with the  highest frequency being a Nyquist frequency.
  • 00:32:24
    And let's consider a very simple  operation ,which we can do in both spaces:
  • 00:32:30
    real space and Fourier space. Let's talk about scaling.
  • 00:32:35
    So, if we have an image, which would be  6x6 pixels, and as I mentioned above,
  • 00:32:41
    each pixel will have its own intensity,  what we can do - we can scale our image.
  • 00:32:48
    For example, we want to bin  our image by the factor of two.
  • 00:32:52
    What would it mean? It means that  if we have an image of 6x6 pixels,
  • 00:32:58
    if we bin by the factor of two, we will get an  image of 3x3. If we had sampling (or pixel size)
  • 00:33:05
    of 1 Angstrom/pixel here, then the sampling  here would be 2 Angstrom/pixel after being,
  • 00:33:15
    and each pixel on the binned image will be  an average of four pixels from this image. S
  • 00:33:26
    o, an average of 0, 0, 1 and 3 will be 1. Now, we can move further and then we can
  • 00:33:32
    take an average of these four which will be four  and an average of these four, which will be two.
  • 00:33:38
    And this way we can calculate the values for  all the other pixels in this image which will
  • 00:33:48
    not necessarily be integers and then this way we  would be performing the beginning in real space.
  • 00:33:57
    And as I said, it's important to know  that the sampling (or the pixel size)
  • 00:34:00
    of the binned image will be twice larger  than the pixel size of the original one.
  • 00:34:07
    How would we do the scaling in Fourier space? In Fourier space we can take an image
  • 00:34:13
    and we can create a Fourier transform. And if you create a Fourier transform of
  • 00:34:17
    an image of a micrograph in cryo-EM, you'll see  this pattern which is called the "Thon rings",
  • 00:34:25
    and in later lectures you will  hear about the Thon rings.
  • 00:34:29
    So, what we can do in order  to bin this image of 4K by 4K?
  • 00:34:35
    We can take the central part of the image, and the  central part will be twice smaller than in these
  • 00:34:45
    dimensions: it will be 2048 pixels by 2048. And if here, as I said, will be the Nyquist,
  • 00:34:56
    the original Nyquist frequency, here  would be the Nyquist frequency over 2.
  • 00:35:02
    And then, what we can do, we can just crop it. We can crop the original image
  • 00:35:09
    and create an inverse Fourier transform. And if we create an inverse Fourier transform,
  • 00:35:14
    our original image will be binned to 2k x  2k, and the pixel size will be twice coarser
  • 00:35:22
    than the original pixel size. Using this  operation, we can define the cropping area,
  • 00:35:31
    which would be pretty much any, and that  means that we can bin also [by] non-integer,
  • 00:35:37
    and this type of binning allows you not to  introduce any kind of interpolations, which
  • 00:35:45
    is much better than doing binning in real space. And this operation is called "Fourier cropping"
  • 00:35:52
    so we use Fourier transforms  for binning, for example.
  • 00:35:57
    Now, another operation which you can  be doing using Fourier transforms,
  • 00:36:02
    would be Fourier filtering. Again we take an a  test image, and this image is very nice actually,
  • 00:36:08
    because it contains lots of high-frequency  information here in the clothes of the lady,
  • 00:36:12
    and also these checkers on the tablecloth.  So, [for] any image, as I said, we can
  • 00:36:20
    perform the Fourier transform of this image,  and then we will get an amplitude spectrum.
  • 00:36:26
    So, again, what we can do  with this amplitude spectrum,
  • 00:36:31
    we can consider a distribution  of different amplitudes.
  • 00:36:36
    And what if we focus now only on the  middle part of this amplitude spectrum,
  • 00:36:43
    what if we set everything else to zero? So, we'll have only the central part of
  • 00:36:49
    this amplitude spectrum, and then, if  we create an inverse Fourier transform,
  • 00:36:54
    we'll get an image like this. So, all the high-frequency information
  • 00:36:58
    all the tiny details in the image will be  gone, and this is called "low-pass filter".
  • 00:37:05
    We are allowing to pass only low-frequency  information that's why it's called low-pass.
  • 00:37:12
    Now, if we consider another case, where we cut  off everything - like 20 percent from the original
  • 00:37:25
    data (in terms of the the frequency domain), if  we cut off everything up to Nyquist over five
  • 00:37:34
    in the low-frequency information, and  we create an inverse Fourier transform,
  • 00:37:38
    we will get an image like this. And we see that all the high-frequency information
  • 00:37:43
    is preserved, however all the shades, all the  features, like large features (which correspond to
  • 00:37:54
    all the features in the image or the gradient  of Illumination, stuff like that) will be gone,
  • 00:38:02
    and using such an operation, allows us  to focus on the high-frequency details:
  • 00:38:10
    we are passing only high frequencies, and that's  why this operation is called "high-pass filter".
  • 00:38:17
    We can also consider a combination of both, and we  can use a ring over the Fourier transform over the
  • 00:38:24
    amplitude spectrum and then that will be called  a "band-pass filter" - so we are allowing to pass
  • 00:38:35
    a certain fraction of frequencies  and the high frequencies will be
  • 00:38:39
    excluded here as well as some low  frequencies will also be excluded.
  • 00:38:48
    And in cryo-EM for applying this low-pass,
  • 00:38:53
    band-pass and high-pass filters we will  be using so-called "Gaussian" functions.
  • 00:38:58
    Why Gaussian, and they would look like this,  because the Fourier transform of the Gaussian
  • 00:39:03
    function would be a Gaussian function. Why is it important to remember?
  • 00:39:07
    Let's consider four images in the real  space, and these four images, which would
  • 00:39:17
    look like something very broad and in the end  to also look as the delta-function pretty much.
  • 00:39:24
    It's a dot, so a very very narrow area -  so in 2D real space (in the real space),
  • 00:39:35
    these images would look like this. But in Fourier space everything which is broad,
  • 00:39:40
    will be condensed to a very tiny point. If  it's close to infinity (the distribution
  • 00:39:47
    of this function), that will be a delta  function in Fourier space, and inversely,
  • 00:39:55
    if there is something very broad in Fourier  space, it will be very tiny in the real space.
  • 00:40:04
    We know the relation of the Gaussian  functions and Gaussian filters actually
  • 00:40:10
    allow us to go there and back in  case of the Fourier transforms.
  • 00:40:17
    And Gaussian means that we use very smooth edges  in the masks which we are applying for the images.
  • 00:40:29
    Again, let's consider this is the Fourier  transform of one image, it's an amplitude
  • 00:40:34
    spectrum of the of the test image, and you can  see here if you're applying a band-pass filter,
  • 00:40:41
    the areas of the of the mask the edges of the  mask will be very soft and that's how it should
  • 00:40:49
    be in cryo-EM, because if edges of the mask are  very sharp, and we know that the sharp edges,
  • 00:40:57
    if it's something very small, it's a very rapid  transition - it might introduce certain artefacts,
  • 00:41:04
    if we create an inverse Fourier transform. That means, that we would we should never
  • 00:41:10
    use sharp-edged masks and in different software  packages there are parameters which correspond to
  • 00:41:20
    the soft edges of the mask, and you should always  use these masks, which would have these soft
  • 00:41:27
    edges, because if you don't, you might introduce  certain artefacts, and this is an inverse Fourier
  • 00:41:33
    transform of an image which was filtered with  a mask in Fourier space with a sharp edge.
  • 00:41:41
    OK, so we can go further in terms of the  dimensionality and we can define also the
  • 00:41:49
    Fourier transforms of 3D objects, or if you're a  mathematician, you can go up to N dimensions and
  • 00:41:57
    you can think in terms of higher dimensions. The most important is just to realize that
  • 00:42:03
    this generalization is possible and we will have  all the all the properties of Fourier transform,
  • 00:42:11
    which can be applied to the 3D case. We started with an example of this paper of
  • 00:42:21
    the reconstruction of the tale of bacteriophage  T4, and actually what was done in this work,
  • 00:42:31
    what they [did], they they took a Fourier  transforms of their images of the recorded images,
  • 00:42:38
    and using the Fourier analysis they  were able to figure out the spatial
  • 00:42:45
    orientation between these images and taking  advantage of the symmetry to reconstruct..
  • 00:42:52
    to perform an inverse Fourier  transform of these 2D images into 3D,
  • 00:42:57
    using so-called "Central Slice Theorem". It's a bit.. it might sound a bit complicated
  • 00:43:04
    "The Central Slice Theorem", but what it's  actually doing, it allows to get the 3D
  • 00:43:10
    orientation of the 2D objects in in Fourier  space, using an inverse Fourier transform.
  • 00:43:18
    In this part of the talk I will  try to illustrate a few more cases,
  • 00:43:25
    and to demonstrate what else Fourier transforms  can be useful for. Let's talk about convolution.
  • 00:43:34
    To understand the concept of convolution, you need  to imagine two functions: first a function which
  • 00:43:41
    would look like this and then a function which  would be a set of impulses, for example in this
  • 00:43:49
    case it's impulses with different amplitudes,  and one is negative and there are four of them.
  • 00:43:56
    So, what convolution would be doing is  actually.. it's blending these two functions.
  • 00:44:03
    And convolution would be expressed using  the following mathematical equation.
  • 00:44:10
    The symbol for convolution  would be this encircled cross.
  • 00:44:15
    And it might be a bit difficult to think about  this in terms of the integrals, in terms of these
  • 00:44:20
    functions and so on, but it's more important  to understand what the convolution is actually
  • 00:44:28
    doing. So, in one of them lecture notes from one  of the Stanford University professors I found the
  • 00:44:37
    following phrase, that... when he was asked and  the topic of the chapter was what is convolution,
  • 00:44:48
    and then he is noting that convolution is  what convolution does. So, don't try to
  • 00:44:53
    overthink the maths, just try to focus on their  actual properties and the the actual operation.
  • 00:45:03
    Let's consider another example, and this would  be the 2D case and convolve two functions:
  • 00:45:12
    one function would be an image or in  this case it's kind of.. it's like a
  • 00:45:19
    crystal lattice basically with ones and zeros  everywhere, and the second function will be
  • 00:45:25
    the flower. And when we're blending these two  functions, we're getting something like that.
  • 00:45:30
    The same principle and this is what would  be happening when we're talking about X-ray
  • 00:45:36
    crystallography or the same principle you'll find  you'll find in the convolutional neural networks.
  • 00:45:46
    What's important to note is again,  to compute such an operation,
  • 00:45:53
    you would be using the integral equation,  and integration is very computationally heavy
  • 00:46:05
    have a task and it takes quite a  while to to calculate integrals,
  • 00:46:10
    but a calculation of the Fourier transforms, as  I said, using the FFT algorithm is very fast.
  • 00:46:18
    What we can do, we can actually calculate the  Fourier transform of.. or we can apply the Fourier
  • 00:46:27
    transform (at least on the paper) to this part  of the equation and to this part of the equation.
  • 00:46:33
    And if we apply Fourier transform, the  Fourier transform of this double integral
  • 00:46:40
    will be the multiplication of these functions  - of the Fourier transforms of these functions.
  • 00:46:48
    So, that means that instead of  integration, we can calculate a
  • 00:46:54
    multiplication of the Fourier transforms. That means that we can perform an inverse
  • 00:47:01
    Fourier transform of these two  terms which are just multiplied.
  • 00:47:07
    And this is much faster and computationally  much cheaper to produce it in Fourier space.
  • 00:47:15
    So, we're talking about multiplication versus  integration so it's very efficient and later
  • 00:47:24
    on you'll figure out that you can perform  deconvolution for your light microscopy data.
  • 00:47:32
    So, what is deconvolution you can again imagine  the following situation: so you have a test
  • 00:47:39
    image and this test image is multiplied with a  Point Spread Function, and Point Spread Function
  • 00:47:47
    is the function which is describing  imperfection of your optical system.
  • 00:47:52
    And then, if you zoom in here, you'll  see so-called Airy pattern, and the Point
  • 00:47:57
    Spread Function in your light microscope  might act as a Gaussian low-pass filter.
  • 00:48:05
    And in the end, if the original object  of your interest looks like this,
  • 00:48:11
    with all the high frequency information, the data  which we're getting in the microscope will be
  • 00:48:17
    somewhat distorted, so in order to reconstruct the  original image, we can apply deconvolution, and
  • 00:48:25
    we can take advantage of the Fourier transforms  and this is how this operation is being performed
  • 00:48:31
    in the light microscopy software packages. Somewhat similar operation is performed also
  • 00:48:40
    in cryo-EM, and this is called CTF-correction,  and again we're performing two Fourier transforms.
  • 00:48:48
    Now, the second operation which is very much  similar to convolution, would be correlation,
  • 00:48:58
    and correlation sometimes they  call "similarity measurement",
  • 00:49:01
    and for illustration of the correlation  we need to consider again two functions.
  • 00:49:07
    One function will be a cropped area of this  image - so something which belongs to this image.
  • 00:49:20
    But if we crop an area like this,  it will be very hard to figure out
  • 00:49:24
    where it actually belongs to, because  there are many similar areas like that.
  • 00:49:28
    So, to calculate the correlation  between these two functions,
  • 00:49:36
    or to compare these two functions in order to find  where it matches, what we would have to do in real
  • 00:49:44
    space (would be let's say, for simplicity, it  will be like an image or just of four pixels),
  • 00:49:52
    and it will have to go one-by-one pixel  in this direction, and to compare whether
  • 00:50:00
    this image belongs to this area or not. And if we are lucky enough, we will compare
  • 00:50:06
    different areas, and then some point we will  figure out that the image, the first function
  • 00:50:16
    will be exactly the same as the second function. And we can actually measure measure the value of
  • 00:50:23
    the similarity, and in order to measure the  value of such a similarity again, we'd have
  • 00:50:28
    to calculate such a complicated integral. And as you might already guess, if we apply
  • 00:50:35
    for a transform to both sides, instead of  integration, we will get multiplication.
  • 00:50:43
    And that means, that in order to  calculate the similarity measurement,
  • 00:50:47
    we will be able to take an inverse inverse  Fourier transform of the multiplied Fourier
  • 00:50:54
    transforms of each of these functions. And using this approach, we can actually
  • 00:50:59
    calculate the correlation map, so again calculate  the correlation map between these two images,
  • 00:51:05
    and we will see something like that, where  the bright areas would correspond to high
  • 00:51:12
    correlation and dark areas to low correlation. And the highest value here will correspond
  • 00:51:18
    to the solution of the problem and  where the image actually belongs to,
  • 00:51:24
    the cropped area of the image belongs to. So, in cryo-EM, we will apply Fourier transforms
  • 00:51:31
    and the calculations of the correlation,  and these values you'll find being
  • 00:51:41
    called cross-correlation coefficients. You'll find these operations in particle
  • 00:51:46
    picking, in image alignments, projection  matching and in Fourier Shell Correlation.
  • 00:51:52
    So speaking about Fourier Shell Correlation.. it's  the last concept I would like to explain to you.
  • 00:52:02
    To simplify explanation of the Fourier  Shell Correlation, which is correlation
  • 00:52:06
    in 3D of the Fourier components, we  will consider Fourier Ring Correlation.
  • 00:52:12
    And Fourier Ring Correlation is a correlation  of the Fourier components of two images.
  • 00:52:23
    So, what we can do: there are two images,  which are not really related - it's not
  • 00:52:29
    the same image. It might have a similar  object, but these two images are different.
  • 00:52:33
    So, what we can create, we can create Fourier  transforms of each of these images, and then
  • 00:52:39
    if we calculate the Fourier transforms,  we will see different amplitude spectra.
  • 00:52:44
    So now, what we can do, we can create or we  can draw certain rings of certain diameters,
  • 00:52:52
    and we can we can define these  rings in the size, that they would
  • 00:53:00
    cover the whole area of our amplitude spectra. So, they will be a larger and larger with larger
  • 00:53:10
    and larger diameter. And then what we can do, we  can pairwise create cross-correlations between
  • 00:53:20
    the Fourier components within these rings. And  using such a comparison, which will be dependent
  • 00:53:32
    on the frequency - here is the frequency and  here is the correlation value, we will build
  • 00:53:38
    a plot which is called Fourier Ring Correlation. If two images are independent from each other,
  • 00:53:43
    their Fourier components will not be correlated,  and this curve will oscillate around zero.
  • 00:53:54
    This are the cross correlation values. In this plot you also see the curve which
  • 00:53:59
    is called "half-bit criterion", which actually  reflects the amount of information in all these
  • 00:54:06
    Fourier shells [rings], but it's a bit advanced  topic for now. Fourier Ring Correlation of two
  • 00:54:14
    similar objects can be Illustrated in the  following example: I took this truck image,
  • 00:54:21
    and then I added noise to this image. The noise will suppress high-frequency details.
  • 00:54:26
    You can see still some low-frequency  details, some shape of the car.
  • 00:54:32
    If we calculate the Fourier transform for  each of the images, you can see here in
  • 00:54:38
    the amplitude spectrum, that higher  frequency components are suppressed.
  • 00:54:44
    If we calculate the Fourier  Shell [Ring] Correlation
  • 00:54:48
    between these two images we will see that there  is certain values for the cross correlation of
  • 00:54:54
    the lower frequency components and it will slowly  be going down to towards the Nyquist frequency.
  • 00:55:04
    And here you can see actually also again the  half-bit criterion curve, and the crossing will
  • 00:55:11
    define the information content of this image. So what's important to understand is this cross
  • 00:55:19
    correlation curve, which would reflect  the similarity between these images.
  • 00:55:26
    If we generalize 2D case into 3D, so  we can easily imagine that during the
  • 00:55:36
    refinement procedure of our single-particle  pipeline approach, we can split our data set
  • 00:55:44
    in two halves, and we can determine the 3D  reconstruction for each of these halves.
  • 00:55:50
    Then, what we can do, we can create a Fourier  transform, 3D-Fourier transform of each of these
  • 00:55:57
    half-maps, and then the same way we would compare  shells (this time shells, not the rings) from the
  • 00:56:08
    origin of the image towards the Nyquist frequency. And then we will see, that the lower components of
  • 00:56:15
    the lower Fourier components will correlate higher  than the higher frequency components, and this
  • 00:56:23
    plot is called "Fourier Shell Correlation",  and where it crosses the criteria for
  • 00:56:30
    determination of the resolution will determine  actually the resolution of our reconstruction.
  • 00:56:37
    And it's different compared to X-ray  crystallography, where you would have
  • 00:56:43
    resolution as the distance between the origin  and the peak of the highest reflection detected.
  • 00:56:51
    In cryo-EM, together with amplitudes we are  having also phases and this is the difference.
  • 00:56:58
    So, finally we'll find the natural Fourier  transforms which are happening inside your
  • 00:57:06
    microscope, and the objective lenses will  be serving as the Fourier transforms of
  • 00:57:14
    your object and in the back focal plane you'll  see a diffraction pattern - this would be the
  • 00:57:22
    amplitudes which will correspond to the projection  image of your object and the second lens can serve
  • 00:57:29
    as a inverse Fourier transform, and then the  image you will be having in the microscope,
  • 00:57:34
    would be an image, which would be actually  affected by imperfections of your optical system.
  • 00:57:41
    So, the objective lens in the electron  microscope will work as a Fourier Transform.
  • 00:57:48
    So, I've shown you many examples in this lecture,  but what I would like you to take with you home
  • 00:57:56
    is understanding that Fourier transform is  a mathematical tool which allows you just
  • 00:58:06
    to decompose a function it's a one-dimensional  function or an image or three-dimensional function
  • 00:58:12
    as a sum of some other periodic functions and  it's just a different representation of your data.
  • 00:58:21
    A different representation, but it's an important  representation because it allows you to perform
  • 00:58:28
    certain convenient computational operations. So therefore, Fourier transforms allow you
  • 00:58:36
    to facilitate number of your operations with your  data analysis which would be impossible to perform
  • 00:58:42
    in real space, such as correlation, alignments,  filtering, resolution measurements and some other.
  • 00:58:49
    It's also important to start using  Fourier transform as a diagnostic
  • 00:58:54
    tool of your data analysis as soon as  you can and just to get used to that.
  • 00:58:59
    Because each time you're processing your data  you can calculate the Fourier transform look
  • 00:59:05
    at the amplitude spectrum and check whether  you have some issues with your data or not.
  • 00:59:11
    if you're interested in this topic, I would  highly recommend you the lectures from this
  • 00:59:18
    professor from Stanford University - these  are the lecturers for electrical engineering
  • 00:59:24
    students and this course is free on YouTube,  it's like 30 lectures which are very exciting.
  • 00:59:29
    And two books, which are classic books  on the Fourier analysis: from Goodman,
  • 00:59:35
    "Introduction to Fourier Optics" and Bracewell,  "The Fourier transform and its applications".
  • 00:59:40
    And with that I would like to  thank you for your attention!
Tags
  • Fourier transforms
  • cryo-EM
  • data analysis
  • sampling
  • Nyquist frequency
  • image processing
  • correlation
  • convolution
  • filtering
  • mathematics