# Difference between revisions of "Fourier Shell Correlation"

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#:: A<sub>i</sub> = A|n(i) ; a vector with as many entries as non-zero elements in n(i) | #:: A<sub>i</sub> = A|n(i) ; a vector with as many entries as non-zero elements in n(i) | ||

#:: B<sub>i</sub> = B|n(i) ; | #:: B<sub>i</sub> = B|n(i) ; | ||

− | #: compute cross correlation of | + | #: compute cross correlation of vectors A<sub>i</sub>, B<sub>i</sub>: |

− | #:: FSC(i) = A<sub>i</sub> * | + | #:: FSC(i) = A<sub>i</sub> * B<sub>i</sub> / |A<sub>i</sub>| |B<sub>i</sub>| |

+ | #:: ''*'' represents pixelwise multiplication, | . | is the vector norm. | ||

</tt> | </tt> | ||

− | With this definition, the first element in a FSC would measure the similarity of the | + | With this definition, the first element in a FSC would measure the similarity of A and B restricted to the first shell A<sub>1</sub> and B<sub>1</sub>. This shell is just the single central Fourier pixel in both cases. In the other extreme of resolution, FSC(N/2) would be measuring the similarity of Fouier modes that are strictly worse that the resolution 1/(N/2), but better than the resoultion 1/(N/2-1). |

## Revision as of 10:11, 25 April 2016

The Fourier Shell Correlation (FSC) between two volumes is a measure of their similarity. It assigns a coefficient between -1 and 1 to each chosen frequency range.

FSC computations are used in different areas of *Dynamo*

- During adaptive bandpassing, the FSC of the two half averages is used to measure the
*resolution*attained at each iteration. - During a regular project, the FSC between the freshly computed iteration average and the one generated in the previous iteration is also computed. This information is used to measure the
*convergence*.

The command line that computes the FSC is `dfsc`. In its default settings, the FSC for two volumes with a sidelength of *N* pixels would be a vector with a length of "N"/2 pixels.
`
`

- compute A as fourier transform of real space volume a
- compute B as fourier transform of real space volume a

- for each i=1:N/2
- compute a discretized shell n(i)
- a fourier pixel with a distance
*k*from the center is in n(i) if i-1<=|k|<i

- a fourier pixel with a distance
- locate elements of A and B inside shell n(i)
- A
_{i}= A|n(i) ; a vector with as many entries as non-zero elements in n(i) - B
_{i}= B|n(i) ;

- A
- compute cross correlation of vectors A
_{i}, B_{i}:- FSC(i) = A
_{i}* B_{i}/ |A_{i}| |B_{i}| -
***represents pixelwise multiplication, | . | is the vector norm.

- FSC(i) = A

- compute a discretized shell n(i)

`
With this definition, the first element in a FSC would measure the similarity of A and B restricted to the first shell A`_{1} and B_{1}. This shell is just the single central Fourier pixel in both cases. In the other extreme of resolution, FSC(N/2) would be measuring the similarity of Fouier modes that are strictly worse that the resolution 1/(N/2), but better than the resoultion 1/(N/2-1).