## Contrast transfer function (CTF) correction

### What is the contrast transfer function?

The image formation in bright field electron microscopy can be described by the action of the contrast transfer function (CTF) H(k). Accordingly, the relationship between the object o(r) and the image contrast i(r) can be written as i(r) = o(r)* h(r), where * stands for the convolution operation, and h(r) is the point spread function, which is the Fourier transform of H(k). Thus, following the convolution theorem, I(k) = O(k)H(k).

defocus [A] - which describes the deviation in the focus of the objective lens from the "Gaussian focus."
spherical aberration coefficient [mm] - which describes the (third order) spherical aberration of the wave front in the objective lens.
source size [1/A] - which describes the illumination divergence, expressed as a size in the back focal plane (hence a quantity in reciprocal space).
defocus spread - which describes the spread of defocus due to the spread of electron energies or to the fluctuation of lens current.

The only parameter being varied in the experiment is the defocus. Depending on the defocus setting, different features of the object appear enhanced or suppressed in the image. This is because the CTF oscillates between -1 (negative contrast transfer) and +1 (positive contrast transfer) as we go from low to high spatial frequencies. The exact locations of the zero crossings (where no contrast is transferred, and information is lost) depends on the defocus.

## ### CTF correction

In CTF correction, we attempt to retrieve the undistorted object from the image. This attempt is compromised by the presence of noise; i.e., recovery of the object is never ideal. Since the CTF always has zero crossings, part of the information about the object is lost. This is why we make use of several images obtained at different defocus settings, hoping that the resulting CTFs Hn(k) jointly (after appropriate weighting) cover the whole Fourier space without gap.

The Wiener filter is the least square solution to the problem of signal recovery in the presence of noise. Let's assume we have N images in (r) (with Fourier transforms In(k)) whose CTFs are Hn(k). In that case, the best estimation of the object transform O(k) is where and SNR is the signal-to-noise ratio, defined as the ratio of signal to noise variances.

### Example of CTF correction

In the SPIDER procedure file, ctfexample.spi, we first simulate the action of the electron microscope, by applying a CTF to an "object", which is the projection of the 3D density map of the ribosome, and adding noise to the result, and then use the 2D Wiener filtering operation described above to retrieve the original. The gallery of images describes the progress of these operations. Note that the choice SNR=100 makes the Wiener filter quite aggressive, and this benefits the recovery of low spatial frequencies responsible for defining the particle's boundary and overall shape. ### CTF parameters in SPIDER and single particle reconstruction

Various CTF parameters are used in SPIDER's 'TF' operations. See the glossary for definitions. These parameters, along with some others, are listed in a parameter document file. CTF parameter estimation is done on micrographs using 'CTF FIND' to estimate defocus and astigmatism.

• In single particle reconstruction using projection matching without defocus groups:
CTF correction is applied to the windowed particle images before 3D reconstruction using the 'TF CT' & 'TF COR' operations.

• In legacy methods for single particle reconstruction using projection matching with defocus groups:
A separate volume is created for each defocus group. Then 3-dimensional CTF correction is carried out for each defocus group volume and these volumes are merged to form a single, CTF-corrected volume.
This uses the 'TF CT3' & 'TF COR' operations.
For details, see Penczek et al.,1997.

### CTF graphical tools

A number of python graphical tools for analyzing the CTF are included with SPIDER:

• ctfdemo : A graphical interface that lets you experiment with the various CTF parameters used in SPIDER.
• ctfmatch : A tool for analyzing the output from SPIDER's 'TF ED' operation.
• ctfgroup : A legacy tool to graphically assign micrographs to defocus groups.

### References

Frank, J. (2006) Three-Dimensional Electron Microscopy of Macromolecular Assemblies. Oxford University Press, New York.

P.A. Penczek, J. Zhu, R. Schröder, J. Frank (1997) Three Dimensional Reconstruction with Contrast Transfer Compensation from Defocus Series Special Issue on Signal and Image Processing, Scanning Microscopy Volume 11, 1997, page 147.

G. T. Herman and J. Frank, Editors (2014) Computational Methods for Three-Dimensional Microscopy Reconstruction Birkhauser, Basel 260pg. DOI: 0.1007/978-1-4614-9521-5

Updated 3 Nov. 2015