# Name

scfn - calculate the normalized circulant self-correlation function of an image using multiplication in Fourier space.

# Usage

output = scfn(image, center=True)

## Input

- image
- input image (real)
- center
- if set to True (default), the origin of the result is at the center; if set to False, the origin is at (0,0), the option is much faster, but the result is difficult to use

## Output

- output
- normalized circulant self-correlation function of an input image. Real. The origin of the self-correlation function (term ccf(0,0,0)) is located at (int[n/2], int[n/2], int[n/2]) in 3D, (int[n/2], int[n/2]) in 2D, and at int[n/2] in 1D.

# Method

Calculation of the circulant self-correlation function of an image

*f*is performed by first normalization of the input image by subtracting its average and by dividing it by its standard deviation. Next, Fourier transform is calculated, modulus calculated as``|hat(f)_"normalized"|``, and an inverse Fourier transform is calculated to yield*scfn*.- This expression does not have any corresponding expression in real space - it can be considered to be adaptive filtration.
Note: for image size

*nx*and object size*m*, the circulant*scfn*is valid only within``+//- (nx-m/2)``pixels from the origin. More distant*scfn*values are corrupted by the "wrap around" artifacts.

# Reference

van Heel, M., Schatz, M., Orlova, E., 1992. Correlation functions revisited. Ultramicroscopy 46, 307-316.

# Author / Maintainer

Pawel A. Penczek

# Keywords

- category 1
- FUNDAMENTALS
- category 2
- FOURIER

# Files

fundamentals.py

# Maturity

- stable
- works for most people, has been tested; test cases/examples available.

# Bugs

None. It is perfect.