order to increase the sensitivity of the correlation, the mean is subtracted in each resolution shell before calculating the cross-correlation i.e. a Pearson correlation coefficient is calculated in each shell independently. $$\mathrm{CC}_{1/2}(q) = \frac%
  {\sum\limits_{|\mathbf{q}_i| = q} \left(\mathrm{I}_1 - \overline{\mathrm{I}_1}\right) \left(\mathrm{I}_2 - \overline{\mathrm{I}_2}\right)}%
  {\sqrt{\sum\limits_{|\mathbf{q}_i| = q} \left(\mathrm{I}_1 - \overline{\mathrm{I}_1}\right)^2}%
   \sqrt{\sum\limits_{|\mathbf{q}_i| = q} \left(\mathrm{I}_2 - \overline{\mathrm{I}_2}\right)^2}}$$ where $\mathrm{I}_k$ is shorthand for $\mathrm{I}_k(\mathbf{q}_i)$ and $\overline{\mathrm{I}_k}$ is the mean intensity in the resolution shell $\overline{\mathrm{I}_k(q)}$. The increased sensitivity due to subtracting the mean is most apparent when there is spherically symmetric background in the intensity reconstruction, as is the case here.

\[sec:prtf\]Phase retrieval transfer function (PRTF)
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The other metric is the phase retrieval transfer function (PRTF) [@Shapiro:2005]. This metric measures the reliability of iterative phasing by (in effect) averaging complex values over may instances of the phasing process.

The first step in the calculation of this metric is to reconstruct a large number of independent density volumes from different random starting guesses. At any given reciprocal-space voxel, $\mathbf{q}$, the argument of the complex Fourier transform of the density (the phase) can be slightly different in each random start. The value of the PRTF at that voxel is the complex sum of the 