# Theoretical Resolution of the 3D Reconstruction

We recall that resolution is a boundary in reciprocal space defining the 3D domain within which Fourier components contribute significantly to the density map. Since a number of different effects (such as electron-microscopic contrast transfer function, recording medium, reconstruction algorithm) produce independent limitations, we can visualize the resolution limitation of the whole system as the effect of passing the Fourier transform of the structure through a series of 3D masks centered on the origin. It is always the most restrictive one that determines the final resolution. (Since each of the boundaries may be nonspherical, a more complicated situation can be imagined where the final boundary is a vignette, with restrictions contributed by different masks in different directions.)

In that context, "theoretical resolution" is only the resolution limitation imposed by the choice of reconstruction algorithm and data collection geometry. It will determine the outcome only if all other resolution limits are better (i.e., represented by a wider mask in 3D Fourier space).

The theoretical resolution of a reconstruction is determined by the formula given by Crowther et al. (1970) (section 2.1). It represents the maximum radius of a spherical Fourier domain that contains no information gaps. Corresponding formulas have been given by Radermacher (1991) for the case of a conical projection geometry. Here, we mention the result for an even number of projections:

where d = 1/R is the resolution distance, i.e., the inverse of the resolution, D is the object diameter, and 90 is the tilt of the specimen grid. Furthermore, for such data collection geometry, the resolution is direction dependent, and the above formula gives only the resolution in the directions perpendicular to the direction of the electron beam. In directions that form an angle oblique to those planar directions, the resolution is degraded. In the beam direction, the effect of the missing cone is strongest, and the resolution falls off by a factor of 1.58 (for 90 = 45°) or 1.23 (for 90 = 60°).