# 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

$SSNR (r) = \frac{\displaystyle\sum_{r_i \in R}\left|\sum_{k_i}{ F_{r_i,k} }\right|^2} {\displaystyle \frac{K}{K-1} \sum_{r_i \in R}\sum_{k_i}{ \left|{ F_{r_i,k} - \bar{F}_{r_i}}\right|^2}} -1$

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

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