9.1. Tomogram Resolution and the Need for Filtering
The noise or graininess in a reconstruction from well-aligned data arises from two main sources: noise in the projection images because they are not taken at very high dose; and artifacts in the back-projection because the tilt increment was too large for the thickness of volume being reconstructed. The angular increment is one of the major factors governing the resolution of a tomogram. The classic resolution formula of Crowther, DeRosier, and Klug (1970) can provide a rough guide for how to adjust the radial filter, even though it is strictly applicable only to the case of a full 180º range of angles. The formula is:
d = D * Δβ
where:
d is the resolution in real-space units (nm or pixels),
D is the diameter of volume reconstructed,
Δβ is the angular increment in radians.
Alternatively, if Δβ is in degrees, the resolution f in
reciprocal-space (frequency) units is
f = 57.3 / (D * Δβ)
Or, the equation can be expressed in terms of n, the number of views, and the maximum tilt angle βmax:
f = (28.5 * n) / (D * βmax)
which for the case of a tilt range of ±60º, reduces to:
f = 0.48 * n / D
One of the uncertainties in these formulas is the meaning of the diameter D. In rough terms it corresponds to the thickness of the section; but if electron-dense material in the section is relatively sparse, it may correspond more closely to the size of clusters of material within the section, implying a higher resolution. It has been argued that in extended slabs of material, D corresponds to the maximum thickness of the section when tilted, but a formula based on this assumption gives resolutions much too low to be a guide for filtering the reconstruction. The last version of the formula implies that the angular increment does not limit the resolution when the number of views is comparable to the section thickness in pixels. This makes intuitive sense if one thinks of the projections as providing information needed to solve for densities in the volume, since then the number of measurements will match the number of unknowns.