# Name

acfp - calculate the autocorrelation function of an image using padding with zeroes and multiplication in Fourier space.

# Usage

output = acfp(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
- autocorrelation function of the input image. Real. The origin of the autocorrelation 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 autocorrelation function of an image

*f*is performed first by padding with zeroes to twice the size in real space,

next by calculating Fourier transform of the image, then the modulus squared in Fourier space as ``|hat(f)|^2``, then the inverse Fourier transform, and finally the *acfp* is windowed out using the size of original images.

- In real space, this corresponds to:
``\acfp(n)=sum_(k=0)^(nx-1)f(k+n)f(k)````n = -(nx)/2, ..., (nx)/2``with the assumption that

``f(k)=0 fo\r k<0 or kgenx``Note:

*acfp*is free from "wrap around" artifacts, although coefficients with large*lag n*have large error (statistical uncertainty).

# 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.