If the data coverage in angular space is isotropic, then the resolution boundary in Fourier space roughly follows the surface of a sphere (of course this is irrespective of the criterion used). Often, however, there exists a situation where the resolution is direction dependent. In that case, the resolution boundary in Fourier space is better approximated by an ellipsoid or a more complicated shape, having "good" and ''bad'' directions. For example, a random-conical reconstruction has data missing in a cone defined by the experimental tilt angle, so that the resolution boundary resembles the surface of a yo-yo.
There is as yet no convention on how to describe the direction dependence of the experimental resolution parametrically. Thus, the resolution sometimes relates to an average over the part of the shell within the measured region of 3D Fourier space; while in other instances (e.g., Boisset et al., 1993a, 1995; Penczek et al., 1994) it might relate to the entire Fourier space, without exclusion of the missing cone. It is clear that, as a rule, the latter figure must give a more pessimistic estimate than the former.
A tool for measuring the direction dependence of resolution in three dimensions is now available in the 3D SSNR (Grigorieff, 1998; Penczek, 2002a). In particular, Penczek (2002a) used this quantity to characterize and visualize the anisotropy of resolution in a single-particle reconstruction, of a molecule from some 3700 particle images, as the surface of the resolution domain in Fourier space. The resolution domain (whose boundary was defined by a threshold value for the 3D SSNR) had a shape vaguely resembling a potato, in its irregularity and unequal axes.
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