unit complex numbers whose argument is the phase, $\phi$: $$\mathrm{PRTF}(\mathbf{q}) = \frac{1}{N}\left|\sum_{n=1}^N e^{i\phi_n}\right|$$ where there are $N$ independent density volumes. By convention, the azimuthal average of the PRTF is reported as a function of the radial coordinate $\left|\mathbf{q}\right|$. As described in Sec. \[sec:phasing\_steps\], the different reconstructions must be aligned in real-space before calculating the average. A shift in real space is equivalent to a phase ramp which will significantly lower the PRTF. An uncorrected central inversion will negate the phase, leading to a similar reduction [@Marchesini:2006].

One weakness of the PRTF is that it can be unjustifiably high if the support volume is chosen to be too small. As an extreme case, if the support consists of only one voxel, the PRTF (after alignment) will be unity everywhere even though the reconstruction is very poor. One should therefore have a slightly larger support mask which includes some voxels with low density. In the reconstructions performed here, the support volume (2000 voxels) is significantly larger than the nominal volume of a regular icosahedron with a size corresponding to the fringe spacing (which would be 1497 voxels).

We calculate the PRTF from 400 independent reconstructions. This number is important because it needs 