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[[Image:Contrast transfer function.jpg|thumb|Typical contrast transfer function observed from an electron micrograph]]
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The '''contrast transfer function'''<ref name="Spence1982">Spence, John C. H. (1988 2nd ed) ''Experimental high-resolution electron microscopy'' (Oxford U. Press, NY) ISBN 0195054059.</ref><ref name="Reimer97">Ludwig Reimer (1997 4th ed) ''Transmission electron microscopy: Physics of image formation and microanalysis'' (Springer, Berlin) [http://books.google.com/books?id=3_84SkJXnYkC preview].</ref><ref name="Kirkland1998">Earl J. Kirkland (1998) ''Advanced computing in electron microscopy'' (Plenum Press, NY).</ref> 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 (mathematics)|envelope]] function):
 
:<math>
\operatorname{CTF}(\vec{s}) \; = \sqrt{1 - A^2 \,} \cdot \sin{ \left( \gamma(\vec{s}) \right)} \, + \, A \cdot \cos{ \left( \gamma(\vec{s}) \right)}
</math>
 
where ''A'' is the amplitude contrast.<ref name=malick05>{{cite journal |last1=Malick |first1=S.P. |year=2005 |title=ACE: Automated CTF Estimation |journal=Ultramicroscopy |volume=104  |pages=8–29 |issue=1 |doi=10.1016/j.ultramic.2005.02.004 }}</ref> The amplitude contrast term can be converted into a phase shift, using the [[List of trigonometric identities#Linear combinations|linear combination trigonometry rule]]:
 
:<math>
\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)}
</math>
 
where <math>\varphi = \arcsin{(A)}</math>. The function <math>\gamma(\vec{s})</math> is defined as:
 
:<math>
\gamma(\vec{s}) = \; \gamma(s, \theta) = \; -\frac{\pi}{2} \, C_s \, \lambda^3 \, s^4  \; + \; \pi \lambda \, z(\theta) \, s^2
</math>
 
where ''r'' is the radius from the center of the image, ''C<sub>s</sub>'' is the [[spherical aberration]], &lambda; 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)<ref name=malick05/><ref>{{cite web |url=http://www.maxsidorov.com/ctfexplorer/webhelp/background.htm |title=What Is CTF (Contrast Transfer Function)? |accessdate=July 29, 2011 |author=Maxim V. Sidorov |work=ctfExplorer }}</ref>
 
Furthermore, if the CTF is [[astigmatic]], the defocus becomes a function of the angle ''&theta;'' where the astigmatic angle, ''&theta;<sub>ast</sub>'' given by:<ref>{{Cite pmid|12781660}}</ref><ref>{{Cite pmid|8867604}}</ref>
 
:<math>
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)}
</math>
 
where <math>z_{\mathrm{avg}} = \frac{z_1 + z_2}{2}</math> is the average defocus and <math>z_{\mathrm{diff}} = z_1 - z_2</math> is the difference between the maximal and minimal defocus in the CTF. Where the defocal difference is defined such that:
 
<math>\left| z_2 \right| > \left| z_1 \right| \;</math>  or <math>\; \frac{z_2}{z_1} > 1</math>
 
== See also ==
 
* [[Optical transfer function]]
* [[Point spread function]]
* [http://www.wadsworth.org/spider_doc/spider/docs/techs/ctf/ctf.html Contrast transfer function (CTF) correction]
* [[Airy disk]], different but similar phenomena in light
* [http://www.youtube.com/watch?v=I3_4HF1ZeIQ Talk on the CTF by Henning Stahlberg]
* [http://em-outreach.ucsd.edu/web-course/ref2.html CTF reading list]
 
== References ==
<references/>
 
[[Category:Microscopes]]
[[Category:Protein structure]]

Latest revision as of 16:47, 18 June 2014

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