# Spectral signal-to-noise ratio

In applied mathematics, the two-dimensional Spectral signal-to-noise ratio (SSNR) measures the normalised cross-correlation coefficient between several two-dimensional images over corresponding rings in Fourier space as a function of spatial frequency (Unser 1987). It is a multi-particle extension of the Fourier ring correlation (FRC), which is related to the Fourier shell correlation. The SSNR is a popular method for finding the resolution of a class average in cryo-electron microscopy.

## Calculation

${\displaystyle SSNR(r)={\frac {\displaystyle \sum _{r_{i}\in R}\left|\sum _{k_{i}}{F_{r_{i},k}}\right|^{2}}{\displaystyle {\frac {K}{K-1}}\sum _{r_{i}\in R}\sum _{k_{i}}{\left|{F_{r_{i},k}-{\bar {F}}_{r_{i}}}\right|^{2}}}}-1}$

where ${\displaystyle F_{r_{i},k}}$ is the complex structure Factor for image k for a pixel ${\displaystyle r_{i}}$ at radius ${\displaystyle R}$. It is possible convert the SSNR into an equivalent FRC using the following formula:

${\displaystyle FRC={\frac {SSNR}{SSNR+1}}}$