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fundamentals.cpp  fundamentals.py 
Name
cnvnp  calculate the normalized convolution function between two images using padding with zeroes and multiplication in Fourier space.
Usage
output = cnvnp(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 convolution function between image and ref. Real. The origin of the convolution 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 normalized convolution 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, both images are padded with zeroes to twice the size in real space, Fourier transforms of both images are calculated, their product in Fourier space calculated as `hat(f)hat(g)`, then the inverse Fourier transform, and finally the cnvnp is windowed out using the size of original images.
 In real space, this corresponds to:
`c\nvnp(n)=(1/(nx)(sum_(k=0)^(nx1)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: cnvnp 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.