CTF

(Modified 2003 Aug 21)

CTF correction

How do different correction schemes affect the overall transfer function?

At perfect focus, biological specimens produce little contrast in vitreous ice. So, we take pictures somewhat underfocus, which produces phase contrast but which leads to a systematic alteration of the image data. This alteration is described as the contrast transfer function (CTF). A theoretical description can be found in Erickson & Klug (1971).

The CTF can be thought of as a scalar function multiplying the Fourier transform of the specimen:

An example CTF profile generated using the program ICE.

The horizontal axis is spatial frequency: reciprocal Angstroms at the bottom (unreadable), Angstroms at the top.

For a perfect image, the transfer function would be +1 everywhere. Here, instead the CTF suppresses spatial information (where the function goes to zero) and inverts the contrast in zones where the function changes sign (equivalent to flipping the phase by 180 degrees).

Faced with the absence of information at the CTF nodes (or zeroes), the typical solution is to collect images at a series of defocus values. The goal is, where some images carry no information at a certain frequency, other images will supply complementary information. For example:

In red is a CTF profile from above, corresponding to 2.3 microns underfocus.

In green the CTF profile corresponding to 1.7 microns underfocus

The green profile above goes to zero at around 20 Angstroms (or strictly speaking, 1/20 Angstroms^-1), but the red profile is still strong there. Similarly, where the red profile goes to zero at 17 Angstroms, the green profile is strong. Both profiles may have coincident zeroes (e.g., at ~11.5 Angstroms), so images at other defocus values can be collected.

There are multiple methods for dealing with the summation of image data collected at different defocus values:

1. do nothing
2. correct the phases but not the amplitudes
3. correct the amplitudes and phases

We will describe the result of each of these three methods here.

Let the structure S be expressed in Fourier space in polar coordinates. For the sake of simplicity, we will describe the CTF as a function of only radius, which is closely related to the scattering angle and the resolution. (With astigmatism, the CTF is also a function of the angle in the plane.) The CTF may vary with each image, but we assume the underlying structure is the same in each image. The observed image I is the multiplication of the structure with the CTF. When we average images, we are assuming that the underlying structure is the same, though obscured differently by noise and by the CTF.

Summing the images without regard for the CTF leads to the following situation:

The net effect of the straight sum is to multiply the structure by the sum of the CTF's. The appearance of such a CTF-sum based on defocus values from a real data set (defocus parameters shown here) is shown below. The same defocus parameters are used for subsequent examples. The defocus ranged from 1.3 to 2.7 microns with images collected at 200 keV.

A trap has been implemented to set the CTF at low resolution to -1 where normally it would start near zero and go to -1.

No envelope function has been implemented, i.e., the CTF for any single image will oscillate between -1 and +1 without decay.

At low resolution, the profile of the CTF does not depend so strongly on the defocus. In fact, even up to the second zero, the CTF's are well synchronized. At higher resolution, the CTF's get out of sync and begin to interfere destructively. The expectation value of an uncorrelated sum is zero, which is approached at high resolution.

The simplest solution beyond doing nothing is to do a phase-only correction. Here, where the CTF switches sign, we multiply it by -1. Let's compare this to the uncorrected case:

A phase only correction is thus equivalent to multiplying the structure by the sum of the absolute values of the CTF. The sum of |CTF| is shown here:

The overall transfer function is not bad, reaching a minimum around 0.4 where the first zeroes are somewhat synchronized. Where, in the uncorrected case above, the CTF's could vary from -1 to +1, the |CTF|'s will vary from 0 to +1, so the agreement won't be too bad. If we approximate the CTF as a sine function, the asymptote of of the sum of |CTF|'s would approach 2/Pi (~0.634).

One drawback of the phase-correction scheme, as pointed out to me by Michael Radermacher, is that one essentially multiplies the Fourier data by a function which varies from +1 to -1 with sharp edges. The sharp edges will cause some rippling in real space. Below with a Wiener filter, we will multiply by a function which varies from +1 to -1 with a gentle falloff: namely, the CTF itself.

A more sophisticated correction involves the use of a Wiener filter. The Wiener filter corrects for both phases and amplitudes. One perspective is to think of the CTF as a convolution, i.e., a multiplication in Fourier space. If so, one could correct this convolution by dividing by it. However, the CTF goes to zero in places, so division is forbidden. Instead, with a Wiener filter, we first multiply the image by the CTF, which serves to correct for the sign-reversals. (The image is already under the effect of the CTF, so multiplying by the CTF again will be equivalent to the "perfect" structure multiplied by the always-positive CTF-squared.) Then, we divide by the sum of the squares of the CTF plus a noise-to-signal term. The n/s term effectively serves as a kind of trap to avoid possibly dividing by some tiny number, which would have the effect of blowing up weak Fourier data.

In a sense, instead of dividing by the CTF, we're almost doing the equivalent by multiplying by the CTF and then dividing by the CTF-squared (the n/s trap notwithstanding). The overall transfer function is shown here:

For simplicity the n/s ratio was set to 1.

Note the vertical axis ranges from 0.95 to 1.00.

If the CTF as we have modeled it here were the only aberration, the Wiener filter would be nearly a perfect correction. As things stand though, there are other factors which will dominate at higher resolutions, like the envelope function which suppresses high-resolution data.

To summarize we will show the different schemes and transfer functions side by side. Note the changes in vertical scale.

No correction

Phase only

Wiener filter

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