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[[Image:Contrast transfer function.jpg|thumb|Typical contrast transfer function observed from an electron micrograph]]
 
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]]

Revision as of 18:07, 23 January 2014

File:Contrast transfer function.jpg
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):

CTF⁡(s→)=1−A2⋅sin⁡(γ(s→))+A⋅cos⁡(γ(s→))

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

CTF⁡(s→)=1−A2⋅sin⁡(γ(s→))+A⋅cos⁡(γ(s→))=sin⁡(γ(s→)+φ)

where φ=arcsin⁡(A). The function γ(s→) is defined as:

γ(s→)=γ(s,θ)=−π2Csλ3s4+πλz(θ)s2

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(θ)=zavg+zdiff2cos⁡(2(θ−θast))=z1⋅cos⁡2(θ−θast)+z2⋅sin⁡2(θ−θast)

where zavg=z1+z22 is the average defocus and zdiff=z1−z2 is the difference between the maximal and minimal defocus in the CTF. Where the defocal difference is defined such that:

|z2|>|z1| or z2z1>1

See also

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).
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  5. ↑ Template:Cite web
  6. ↑ Template:Cite pmid
  7. ↑ Template:Cite pmid
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