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In [[applied mathematics]], the three-dimensional '''Fourier shell correlation (FSC)''' measures the normalised [[cross-correlation]] coefficient between two 3-dimensional volumes over corresponding shells in [[Fourier Transform|Fourier]] space (i.e., as a function of spatial frequency<ref name=Harauz1986>Harauz & van Heel, 1986</ref>). The FSC is the three-dimensional extension of the two-dimensional Fourier ring correlation (FRC);<ref name=vanHeel1982>van Heel, 1982</ref> also known as: spatial frequency correlation function.<ref name=Saxton1982>Saxton & Baumeister, 1982</ref>


== Calculation ==
:<math>
FSC(r) =
\frac{\displaystyle\sum_{r_i \in r}{ F_1(r_i) \cdot F_2(r_i)^{\ast} }}
{\displaystyle\sqrt[2]{\sum_{r_i \in r}{ \left|F_1(r_i)\right|^2} \cdot \sum_{r_i \in r}{\left|F_2(r_i)\right|^2}}}
</math>


where <math>F_1</math> is the complex structure Factor for volume 1, <math>F_2^{\ast}</math> is the complex conjugate of the structure Factor for volume 2, and <math>r_i</math> is the individual voxel element at radius <math>r</math>.<ref>{{cite web | title=Image Science's FSC: Program to calculate the Fourier Shell Correlation (FSC) of two 3D volumes | url=http://www.imagescience.de/fsc/ | work=fsc | publisher=Image Science | accessdate=2009-04-09}}</ref><ref>{{cite web | title=RF 3 - Phase Residual & Fourier shell correlation | url=http://www.wadsworth.org/spider_doc/spider/docs/man/rf3.html | work=SPIDER | publisher=Wadsworth Center | accessdate=2009-04-09}}</ref><ref name=vanHeel2005>van Heel & Schatz, 2005</ref> In this form, the FSC takes two three-dimensional data sets and converts them into a one-dimensional array.
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The FSC originated in [[cryo-electron microscopy]] and gradually proliferated to other fields. To measure the FSC, two independently determined 3D volumes are required. In [[cryo-electron microscopy]], the two volumes are the result of two three-dimensional reconstructions, each based on half of the available data set. Typically, the even particle images form one half and the odd particles the other half of the data set. Most publications quote the FSC 0.5 resolution cutoff, which ''ad hoc'' criterion refers to when the correlation coefficient of the Fourier shells is equal to 0.5.<ref name=Bottcher1997>Böttcher et al., 1997</ref><ref name=Frank2006>Frank, 2006, p250-251</ref> However, determining the resolution threshold remains a controversial issue: fixed-value thresholds were argued to be based on incorrect statistical assumptions.<ref name=vanHeel2005/> Many other criteria using the FSC curve exist, including 3-σ criterion, 5-σ criterion, and the 0.143 cutoff. The half-bit criterion indicates at which resolution we have collected enough information to reliably interpret the 3-dimensional volume, and the (modified) 3-[[sigma]] criterion indicates where the FSC systematically emerges above the expected random correlations of the background noise.<ref name=vanHeel2005/>
 
== See also ==
* [[Resolution (electron density)]]
 
== Notes ==
{{reflist|2}}
 
== References ==
<div class="references-small">
*{{cite journal
  | last = Harauz
  | first = G.
  | coauthors = M. van Heel
  | title = Exact filters for general geometry three dimensional reconstruction
  | journal = Optik
  | volume = 73
  | pages = 146&ndash;156
  | year = 1986}}
*{{cite book
  | last = van Heel
  | first = M.
  | coauthors = Keegstra, W.; Schutter, W.; van Bruggen E.F.J.
  | title = Arthropod hemocyanin studies by image analysis, in: Structure and Function of Invertebrate Respiratory Proteins,EMBO Workshop 1982, E.J. Wood 
  | journal = Life Chemistry Reports
  | volume = Suppl. 1
  | pages = 69&ndash;73
  | year = 1982
  | isbn = 9783718601554}}
*{{cite journal
  | last = Saxton
  | first = W.O.
  | coauthors = W. Baumeister
  | title = The correlation averaging of a regularly arranged bacterial cell envelope protein
  | journal = J Microscopy
  | volume = 127
  | pages = 127&ndash;138
  | year = 1982
  | doi = 10.1111/j.1365-2818.1982.tb00405.x
  | issue = 2}}
*{{cite journal
  | last = Böttcher
  | first = B.
  | coauthors = Wynne, S.A.; Crowther, R.A.
  | title = Determination of the fold of the core protein of hepatitis B virus by electron microscopy
  | journal = Nature
  | volume = 386
  | pages = 88&ndash;91
  | doi=10.1038/386088a0
  | year = 1997
  | pmid = 9052786
  | issue = 6620}}
*{{cite book
  | last = Frank
  | first = Joachim
  | authorlink = Joachim Frank
  | title = Three-Dimensional Electron Microscopy of Macromolecular Assemblies
  | publisher = [[Oxford University Press]]
  | location = New York
  | year = 2006
  | pages =
  | isbn = 0-19-518218-9 }}
*{{cite journal
  | title=Fourier shell correlation threshold criteria
  | journal=Journal of Structural Biology
  | year= 2005
  | last= van Heel
  | first= M.
  | coauthors = Schatz, M.
  | volume=151
  | pages=250&ndash;262
  | doi=10.1016/j.jsb.2005.05.009
  | pmid=16125414
  | issue=3  }}
</div>
 
==External links==
* [http://pdbe.org/emstats EMstats] Trends and distributions of maps in EM Data Bank (EMDB), e.g. resolution trends
 
{{DEFAULTSORT:Fourier Shell Correlation}}
[[Category:Applied mathematics]]

Latest revision as of 22:59, 6 January 2015


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