3D reconstruction from the Fourier transform of a single superlattice image of an oblique section

An image of a thin oblique section through a 3D crystal exhibits superlattice periods much larger than the unit cell dimensions of the crystal. Within a superlattice period the contents of the unit cell of the 3D crystal are sampled at different levels, so that a 2D image of the section contains 3D information about the crystal. The 2D Fourier transform of an electron micrograph of such an oblique section thus exhibits superlattice spots, which provide an estimate of the 3D transform of the original crystal. The strengths of the observed spots are reduced from their true values by convolution with a weighting function that depends on section thickness. A method is described that uses phase relationships among symmetry-related structure factors to determine the section thickness and hence the weighting function. Wiener filter deconvolution of the section thickness is performed, in which the filter level is set by the ratio of diffraction spot intensity to background intensity. From the deconvoluted set of structure factors a 3D map of the unit cell can be computed by a standard crystallographic Fourier program. The approach is illustrated with images of oblique sections through rigor insect flight muscle.