# Name

ccfn - calculate the normalized circulant cross-correlation function between two images using multiplication in Fourier space.

# Usage

output = ccfn(image, ref, center=True)

## Input

- image
- input image (real)
- ref
- second input image (real) (in the alignment problems, it should be the reference image).
- 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 cross-correlation function between image and ref. Real. The origin of the cross-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 cross-correlation function between image

*f*and reference image*g*is performed by

first normalization of both images by subtracting their respective averages and by dividing them by their respective

standard deviations. Next, Fourier transforms are calculated, multiplied as

``hat(f)_"normalized"hat(g)_"normalized"^**``, and an inverse Fourier

transform is calculated to yield *ccfn*.

- In real space, this corresponds to:
``\c\c\f\n(n)=(1/(nx)(sum_(k=0)^(nx-1)(f((k+n+nx)mod\nx)-Ave_f)(g(k)-Ave_g)))/(sigma_f sigma_g````n = -(nx)/2, ..., (nx)/2``Note: for image size

*nx*and object size*m*, the circulant*ccfn*is valid only within``+//- (nx-m/2)``pixels

from the origin. More distant *ccfn* values are corrupted by the "wrap around" artifacts.

# Reference

Pratt, W. K., 1992. Digital image processing. Wiley, New York.

# 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.