Recall that defocus is required to generate contrast in phase contrast imaging, and that phase contrast is the dominant source of contrast in cryo-TEM images. The power spectra of cryo-TEM images collected using a higher level of defocus often exhibits a pattern of concentric circles, called Thon rings. The intensity of the Thon rings varies depending on the nature of the material being imaged; e.g. continuous carbon will give stronger Thon rings than amorphous ice, and provided the specimen is sufficiently stable, the Thon rings will become clearer with longer exposures. The position of the Thon rings also depends on the optical parameters of the microscope including defocus and astigmatism. Recall also that in the power spectrum, information towards the centre of the image is low-resolution information, and towards the edge of the information is the high-resolution information. Therefore, in a well aligned microscope (i.e. where the Thon rings are perfectly circular), the effect of the contrast transfer function (CTF) is uniform for a given spatial frequency (or resolution).

An example of a power spectrum from an image taken using defocus showing the obvious Thon rings. Image courtesy of Lou Brillault, University of Queensland.

You will remember from the How Images are Formed section that in phase-contrast imaging, waves scattered by the sample and then focussed by the objective, will travel different distances as they arrive at the detector. They will therefore undergo differing amounts of phase shift relative to each other. When waves are out of phase they can enhance or cancel each other out to some extent. This means that some of the information about the sample is strongly represented in the image and some is reduced or absent altogether. This interference is particularly pronounced when defocus is used to generate contrast.

The Contrast Transfer Function (CTF) is a measure of how much the phase shift and microscope aberrations have corrupted the image. This function describes how much contrast is recorded in the image for a given feature in the sample. Describing the CTF allows us to correct for this corruption and get back the true information.

Because the amount of defocus and aberrations in the microscope are known, their effect can be calculated and compensated for to obtain the maximum amount of true high-resolution information about the original sample.

If we consider a true image of the sample – one that is not corrupted by the CTF or other aberrations – we would not expect to see Thon rings in the power spectrum. Rather, there would be a continuous distribution of information throughout the power spectrum, with fluctuations in the intensity of the data points that reflect only the underlying structural features of the sample. In perfectly focussed samples the power spectrum is closer to this ideal, although there is still some fall off in the average intensity of the data points at high spatial frequencies. This fall off is separately described as representing the envelope function of the imaging system. Combining the theoretical CTF with the envelope function gives us a picture of the effective CTF.

Images and their corresponding power spectra for images taken at 5mm (A and D), 2mm (B and E) and in focus (C and F). The particles are more readily obvious at the higher defocus and almost invisible in the in focus image. In the power spectrum for the 5mm defocus the Thon rings start closer to the centre and do not extend as far out into the higher spatial frequencies. The opposite is true for the 2mm defocus power spectrum. In the in focus power spectrum no Thon rings are visible. Images courtesy of Juanfang Ruan, University of New South Wales.

This CTF can be represented by a plot that corresponds to a line through the power spectrum from the centre to the edge. The shape of the plot shows how much the microscope aberrations and defocus have degraded the signal reaching the detector and therefore reveals the quality of the information (peak height) you can get from the image at increasing spatial frequencies (increasing resolution). For the perfect microscope operated at focus the CTF would be 1.

A CTF plot.

However, for the real microscope using defocus, the CTF fluctuates between alternating positive and negative maxima, corresponding to positive and negative contrast transferred to the image. The zero-crossings indicate frequencies where no information has been captured. The negative values indicate useful information but where the signal has become inverted. These negative values can be made positive by phase flipping. This is done by simply multiplying the Fourier transform by -1 over the appropriate frequency ranges. This serves to correct aspects of the phase contrast effects in the image.

The information missing from the CTF, as indicated by the zero-crossings, cannot be retrieved from the single image represented in a given power spectrum. However, as you will learn later on cryo-EM reconstructions are obtained by averaging together many thousands or even millions of particle images. By recording images at different defocus settings, the information loss will be spread out. So that the different spatial frequencies that are absent from different images are recovered by combining information from many different images.

Limitations of the microscope also contribute to the CTF. These factors are called envelope functions and dampen the CTF at high spatial frequencies (high resolution). This is visualised by the decreasing height of the peaks at higher resolution. The limit to which the high spatial frequencies (outer rings) can be seen in the power spectrum indicates the maximum resolution that can be obtained from a given image. This maximum resolution is called the Nyquist limit.

You can calculate the theoretical optimal CTF of the microscope for the conditions you used. You can then compare it to the CTF for the actual data you collected. Those images that show poor correlation to the shape and resolution limit of the theoretical CTF can be discarded.

Representative image from a dataset taken of E.coli ATP synthase particles. The corresponding CTF showing a comparison of a theoretical CTF with the real CTF for the image. Scalebar is 50nm. Images courtesy of Alastair Stewart, Victor Chang Cardiac Research Institute.

As the CTF is directly related to defocus, typical plots show the following:

With high defocus (more phase contrast in the images)

• There are frequent zero crossings indicating substantial loss of information
• The initial maxima are at low spatial frequency indicating that you get better information at lower spatial frequencies (low resolution)
• A better signal at low frequency means high contrast images
• More high frequency information is lost at high defocus

With low defocus (less phase contrast in images)

• There are fewer zero crossings indicating that high resolution information is better preserved
• Initial maxima are at higher frequencies
• This causes the features in the images to show low contrast, which may not be visible to the naked eye

Comparison of a CTF for an image with a A. high defocus and a B. low defocus. With high defocus there are many zero crossings, the initial maxima is at low frequency. In the low defocus CTF There are fewer zero crossings, the initial maxima is at higher frequency and more high frequency information is preserved.

The CTF can be empirically determined using the formula below:

where Cs is a spherical aberration of microscope; λ is the electron wavelength from the voltage; f is spatial frequency, d is the applied defocus (negative for defocus), and E(f) is the envelope function. Thus, it can be seen that the value of the CTF can be theoretically approximated if all of the other components of the equation are known. For a given microscope operated at a fixed voltage, the Cs and wavelength (λ) are known. The nominal defocus is usually known for a given image but is usually fine-tuned based on the position of the zero crossings (this is actually how the autofocusing function of the data acquisition software on your microscope works). The envelope function is approximated from the damping of the maximum peak heights in the CTF

If we assume that the image recorded by an electron microscope is a CTF-corrupted image, we can recover the true image by modelling the CTF and then removing it from the image by inverse multiplication. This generates the corrected Fourier transform, which can then be inverted to generate the corrected image.

Figure showing the process of CTF correction. A Fourier transform is performed then an estimation of the CTF is calculated. Then a CTF correction can be applied to restore the correct information to the original image. The CTF here is shown as its power spectrum. It’s compared against a theoretical CTF in the top left-hand quadrant to confirm its accuracy. Images courtesy of Lou Brillault, University of Queensland.