# Name

ccfnpl - calculate the normalized cross-correlation function between two images using padding with zeroes, multiplication in Fourier space, and normalization of the result by the actual number of pixels used for calculating the ccf coefficients.

# Usage

output = ccfnpl(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
- 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

In order to calculate the normalized and lag-normalized cross-correlation function between image

*f*and reference image*g*, first both images are normalized by subtracting their respective averages and by dividing by their respective standard deviations. Next, Fourier transforms of both images are calculated, then their product in Fourier space as``hat(f)hat(g)^**``, then the inverse Fourier transform, the*ccfnpl*is windowed out using the size of original images, and the resulting cross-correlation function is normalized by the lag, i.e., the actual number of pixels in image that entered the calculation.- In real space, this corresponds to:
``c\cfnpl(n)=(1/(nx-|n|)sum_(k=0)^(nx-1)f((k+n)-Ave_f)(g(k)-Ave_g))/(sigma_f sigma_g)````n = -(nx)/2, ..., (nx)/2``with the assumption that

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

*ccfnpl*is free from "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.cpp

# Maturity

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

# Bugs

None. It is perfect.