Fourier Correlation

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FRC and FSC

Concept

Given two instances of discrete 2D data (such as an image), Fourier Ring Correlation (FRC) give the amount of consistency in frequency bands, in the same direction, between two related images/volumes.

The 3D case is often called typically called Fourier Shell Correlation (FSC).

This makes it a cross-correlation between two samples.

Conceptually, it sorts each pixel/voxel into the ring(/shell) they belong to, and does pointwise correlation between these groups's sets of pixels/voxels.

(So yes, the largest difference between frc/fsc is how you select the pixels to then treat as 1D)

The rings/shells are typically thin (1 pixel?(verify)), because that gives a more detailed (if noisier) curve. Thicker rings amount to more smoothing.

The typical form takes pixels in all directions, essentially removing direction sensitivity.

There are some variants that are more specific, such as

• Conical Fourier shell correlation, directional resolution estimation

...recognizing tomography will have always have a missing wedge

It originated in Cryo-EM to estimate the resolution of detail in a 3D reconstruction.

In this application, the more proper way to do such a check is to split the dataset in two, do separate reconstruction from each, then compare how consistent these two independent results are.

(Not using template, and not mixing any data helps validate that the structure isn't accidentally-converged fantasy in the first place)

See also

• https://en.wikipedia.org/wiki/Fourier_shell_correlation