# Contrast transfer function

Typical contrast transfer function observed from an electron micrograph

The contrast transfer function[1][2][3] is the equivalent of the optical transfer function in light that affects images collected in a transmission electron microscope. The contrast transfer function must be corrected in the images in order to obtain high resolution structures in three-dimensional electron microscopy, especially cryo-electron microscopy.

The oscillations of contrast transfer functions have the form (not including the envelope function):

$\operatorname{CTF}(\vec{s}) \; = \sqrt{1 - A^2 \,} \cdot \sin{ \left( \gamma(\vec{s}) \right)} \, + \, A \cdot \cos{ \left( \gamma(\vec{s}) \right)}$

where A is the amplitude contrast.[4] The amplitude contrast term can be converted into a phase shift, using the linear combination trigonometry rule:

$\operatorname{CTF}(\vec{s}) \; = \sqrt{1 - A^2 \,} \cdot \sin{ \left( \gamma(\vec{s}) \right)} \, + \, A \cdot \cos{ \left( \gamma(\vec{s}) \right)} \; = \; \sin{ \left( \gamma(\vec{s}) + \varphi \right)}$

where $\varphi = \arcsin{(A)}$. The function $\gamma(\vec{s})$ is defined as:

$\gamma(\vec{s}) = \; \gamma(s, \theta) = \; -\frac{\pi}{2} \, C_s \, \lambda^3 \, s^4 \; + \; \pi \lambda \, z(\theta) \, s^2$

where r is the radius from the center of the image, Cs is the spherical aberration, λ is the wavelength of the electron beam (usually converted from the potential difference voltage) and z is the amount of defocus (using the convention that underfocus is negative and overfocus is positive)[4][5]

Furthermore, if the CTF is astigmatic, the defocus becomes a function of the angle θ where the astigmatic angle, θast given by:[6][7]

$z(\theta) \; = \; z_{\mathrm{avg}} + \frac{z_{\mathrm{diff}}}{2} \cos{\left( 2(\theta - \theta_{\mathrm{ast}}) \right)} \; = \; z_1 \!\cdot\! \cos^2{\left( \theta - \theta_{\mathrm{ast}} \right)} \; + \; z_2 \!\cdot\! \sin^2{\left( \theta - \theta_{\mathrm{ast}} \right)}$

where $z_{\mathrm{avg}} = \frac{z_1 + z_2}{2}$ is the average defocus and $z_{\mathrm{diff}} = z_1 - z_2$ is the difference between the maximal and minimal defocus in the CTF. Where the defocal difference is defined such that:

$\left| z_2 \right| > \left| z_1 \right| \;$ or $\; \frac{z_2}{z_1} > 1$

## References

1. ^ Spence, John C. H. (1988 2nd ed) Experimental high-resolution electron microscopy (Oxford U. Press, NY) ISBN 0195054059.
2. ^ Ludwig Reimer (1997 4th ed) Transmission electron microscopy: Physics of image formation and microanalysis (Springer, Berlin) preview.
3. ^ Earl J. Kirkland (1998) Advanced computing in electron microscopy (Plenum Press, NY).
4. ^ a b Malick, S.P. (2005). "ACE: Automated CTF Estimation". Ultramicroscopy 104 (1): 8–29. doi:10.1016/j.ultramic.2005.02.004.
5. ^ Maxim V. Sidorov. "What Is CTF (Contrast Transfer Function)?". ctfExplorer. Retrieved July 29, 2011.
6. ^ Mindell, J. A.; Grigorieff, N. (2003). "Accurate determination of local defocus and specimen tilt in electron microscopy". Journal of structural biology 142 (3): 334–347. PMID 12781660.
7. ^