# Fourier Shell Correlation

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

- for each shell index i=1:N/2
- compute a discretized shell n(i) as a volume with entries 0 or 1
- a fourier pixel with a distance
*k*from the center is in shell 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 vector with as many entries as non-zero elements in 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) as a volume with entries 0 or 1

`
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: the zero-th component of the Fourier transform. In the other extreme of resolution, FSC(N/2) would be measuring the similarity of those Fourier modes in A and B that are *strictly worse* that the resolution 1/(N/2), but *better* than the resolution 1/(N/2-1).

Don't get confused about the notation. FSC(i)|_{i=1} corresponding to the Fourier component of order zero (and not one) just means that we start counting the components of the FSC vector with index i=1 (up to i=N/2). The first fourier component A(0) has fixed definition, and it corresponds to
`
`

N

A(k) = sum a(n)*exp(-j*2*pi*(k-1)*(n-1)/N), 1 <= k <= N.

n=1

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for k = 0 (i.e, the total intensity of particle `*a* in Fourier space).