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:

1. the absolute Fourier frequency
2. FSC
3. 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)`

  where the factor two is due to the fact that the FSC was calculated from averages of half datasets.

  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.