Fourier shell correlation

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 space, i.e., as a function of spatial frequency, a three-dimensional extension of the Fourier ring correlation (FRC).

Calculation

$FSC(r)={\frac {\sum _{i}{F_{1}(r_{i})\cdot F_{2}(r_{i})^{\ast }}}{\sum _{i}{\left|F_{1}(r_{i})\right|^{2}}\cdot \sum _{i}{\left|F_{2}(r_{i})\right|^{2}}}}$

where $F_{1}$ is the complex structure Factor for volume 1, $F_{2}^{\ast }$ is the complex conjugate of the structure Factor for volume 2, and $r_{i}$ is the individual voxel element at radius $r$. In this form, the FSC takes two three-dimensional data sets and converts them into a one-dimensional array.

Application in Cryo-electron microscopy

In cryo-electron microscopy, the resolution of a structure is typically measured by the Fourier shell correlation (FSC). To measure the FSC, the data needs to be separated into two groups. Typically, the even particles form the first group and odd particles the second based on their order. This is commonly referred to as the even-odd test. Most publications quote the FSC 0.5 cutoff, which refers to the when the correlation coefficient of the Fourier shells is equal to 0.5. Determining the resolution remains a controversial topic. 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.

See also

• Resolution_(electron_density)

References

• Harauz, G. (1986). "Exact filters for general geometry three dimensional reconstruction". Optik. 73: 146–156. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• van Heel, M. (1982). "Detection of objects in quantum-noise limited images". Ultramicroscopy. 8: 331–342.
• Saxton, W.O. (1982). "The correlation averaging of a regularly arranged bacterial cell envelope protein". J Microscopy. 127: 127–138. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Böttcher, B. (1997). "Determination of the fold of the core protein of hepatitis B virus by electron microscopy". Nature. 386: 88–91. doi:10.1038/386088a0. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Frank, Joachim (2006). Three-Dimnsional Electron Microscopy of Macromolecular Assemblies. New York: Oxford University Press. ISBN 0-19-518218-9.
• van Heel, M. (2005). "Fourier shell correlation threshold criteria". Journal of Structural Biology. 151: 250–262. doi:10.1016/j.jsb.2005.05.009. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)