Differences between revisions 1 and 3 (spanning 2 versions)
 ⇤ ← Revision 1 as of 2006-08-04 18:54:00 → Size: 1946 Editor: penczek Comment: ← Revision 3 as of 2007-06-28 21:48:54 → ⇥ Size: 1931 Editor: cpe-24-167-47-215 Comment: Deletions are marked like this. Additions are marked like this. Line 8: Line 8: par1:: image - input image (real)    par2:: ref - second input image (real) (in the alignment problems, it should be the reference image). image::   input image (real)    ref:: second input image (real) (in the alignment problems, it should be the reference image). Line 13: Line 13: out1:: 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. 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. Line 20: Line 20: . {{{`c\cfnp(n)=(1/(nx)(sum_(k=0)^(nx-1)f((-k+n)-Ave_f)(g(k)-Ave_g)))/(sigma_f sigma_g`}}} . {{{`c\nvnp(n)=(1/(nx)(sum_(k=0)^(nx-1)f((-k+n)-Ave_f)(g(k)-Ave_g)))/(sigma_f sigma_g`}}}

# Name

cnvnp - calculate the normalized convolution function between two images using padding with zeroes and multiplication in Fourier space.

# Usage

output = cnvnp(image, ref)

## Input

image
input image (real)
ref
second input image (real) (in the alignment problems, it should be the reference image).

## 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)^(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: 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.

Pawel A. Penczek

category 1
FUNDAMENTALS
category 2
FOURIER

fundamentals.cpp

# Maturity

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

# Bugs

None. It is perfect.

cnvnp (last edited 2013-07-01 13:12:24 by localhost)