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

1. During adaptive bandpassing, the FSC of the two half averages is used to measure the resolution attained at each iteration.
2. 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.

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

1. 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

   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)

   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.

With this definition, the first element in a FSC would measure the similarity of A and B restricted to the first shell A₁ and B₁. 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) is traditionally defined as corresponding to

            N
A(k) =  sum  a(n)*exp(-j*2*pi*(k)*(n-1)/N), 0 <= k <= N-1.
           n=1

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