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.