Estimation of missing cone data in three-dimensional electron microscopy.

The range of tilt angles for which projected images of two-dimensionally periodic specimens can be obtained in electron microscopy is limited both by technical aspects, such as goniometer design, and by the more fundamental limitation of object thickness. The lack of a full set of projections causes a missing cone in the reciprocal space data for the object, which will give an anisotropic resolution in a three-dimensional reconstruction and may cause the quality to be impaired by spurious features. The problem is governed by a linear operator which maps the three-dimensional object onto the set of projections. The eigenvalue spectrum of this operator is determined by the range of tilt angles and the spatial extent of the object. If the object is spatially restricted, the eigenvalues are all positive, and it is in principle possible to retrieve experimentally unavailable structure data from those that are measured. However, with restricted angle data, some of the eigenvalues are extremely small, so the problem is 'ill-conditioned' or sensitive to small perturbations in the data, such as noise, and it is necessary to regularize the solution. We applied two methods of band-limited extrapolation and inference on electron microscope data. Alternating projections onto convex sets regularized by a regularization parameter and a least squares estimation regularized by the Shannon entropy functional yield similar results if a close object extent constraint is available. The criterion of maximum entropy, however, allows a relaxation of this constraint.