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.