# Name

Fourier Ring/Shell Correlation

# Usage

frsc = fsc(img1, img2 ,[w, filename])

## Input

- Input images can be 2-D or 3-D, they can have either real or Fourier format, but have to have the same size

- img1
- first image
- img2
- second image
- w
- ring/shell width in Fourier space. Default: w=1.
- filename
- result of this function is stored in the filename provided.
- Note
- w and filename are optional.

## Output

- frsc
- a list with three columns:

- frsc[0] - the absolute Fourier frequency
- frsc[1] - FSC
- frsc[2] - number of Fourier coefficients n within given ring/shell.

The length of each column (i.e., frsc[0][0] ... frsc[0][len(frsc[0])-1]) depends on the image size and ring width

*w*.

Note:: This is the output when the result is not written to the file but just displayed on the screen when the script is executed.

- filename
- the name of the output text file where the results are stored in three columns:
- the absolute Fourier frequency
- FSC
- number of Fourier coefficients n within given ring/shell.

Note:: This is the output when the result is written to a file and when the contents of the file are displayed.

- Note that error of the FSC is proportional to 1.0/sqrt(n). In particular, 3.0/sqrt(n) yields the so-called "3 sigma" criterion
according to which resolution is equal to the spatial frequency at which fsc <= 3sigma. Other criteria can be based on Signal-To-Noise Ratio in the data, which can be calculated using the relation:

``SNR = 2 (FSC)/(1-FSC)``

A reasonable cutoff value is

``SNR = 1``(power of noise equal to the power of signal in the results), which corresponds to``FSC = 1/3 = 0.33``.

# Description

The calculation is done in Fourier space as

``FSC(f,g;r)=(sum_(||bbb{y}_n|-r| le w)hat(f)(bbb(y)_n)hat(g)^**(bbb(y)_n))/([(sum_(||bbb{y}_n|-r| le w)|hat(f)(bbb(y)_n)|^2)(sum_(||bbb{y}_n|-r| le w)|hat(g)(bbb(y)_n)|^2)]^(1/2)``

where ``hat(f)`` and ``hat(g)`` are Fourier transforms of two input images and summation is performed over rings/shells in Fourier space at spatial frequency ``|bbb{y}_n|`` and within ring/shell width *w*.

# Method

For real input images the FFTs are calculated (without padding) and FSC is calculated for each ring. The format of input images is not changed.

# Reference

W. O. Saxton and W. Baumeister, ‘‘The correlation averaging of a regularly arranged bacterial envelope protein,’’ J. Microsc. (Oxford) 127, 127–138 (1982).

P. A. Penczek, ‘‘Three-dimensional spectral signal-to-noise ratio for a class of reconstruction algorithms,’’ J. Struct. Biol. 138, 34–46 (2002).

# Author / Maintainer

Pawel Penczek

# Keywords

- category 1
- STATISTICS.
- category 2
- FOURIER.

# Files

statistics.py

# Maturity

- stable
- works.

# Bugs

None. It is perfect.