Estimation of variance distribution in three-dimensional reconstruction. I. Theory

A theory is developed for estimating the three-dimensional (3-D) variance of a 3-D image reconstructed from projections by weighted backprojection. The theory is applicable for any data-collection schemes that produce partially redundant sampling of the angular space. The particular data collection considered here, the single-exposure random-conical scheme, is used for the reconstruction of macromolecules in electron microscopy. In this context, the purpose of the 3-D variance estimation is to detect and localize the conformational variability, to assess the significance of structural differences between two experimentally related 3-D images, and to assess the significance of local features in a 3-D image. The 3-D variance estimate of each reconstruction voxel is obtained by (i) the comparison of closest points on Fourier sections associated with difference projections, (ii) the comparison of neighbor projections in real space, or (iii) the comparison of projections with reprojections of the reconstruction.