A memory efficient method for fully three-dimensional object reconstruction with HAADF STEM

• The full 3D approach to atomic resolution object retrieval has high memory load.
• For incoherent imaging the projection process is a matrix–vector product.
• Carrying out this product implicitly as Fourier transforms reduces memory load.
• Reconstructions are demonstrated from HAADF STEM and depth sectioning simulations.

The conventional approach to object reconstruction through electron tomography is to reduce the three-dimensional problem to a series of independent two-dimensional slice-by-slice reconstructions. However, at atomic resolution the image of a single atom extends over many such slices and incorporating this image as prior knowledge in tomography or depth sectioning therefore requires a fully three-dimensional treatment. Unfortunately, the size of the three-dimensional projection operator scales highly unfavorably with object size and readily exceeds the available computer memory. In this paper, it is shown that for incoherent image formation the memory requirement can be reduced to the fundamental lower limit of the object size, both for tomography and depth sectioning. Furthermore, it is shown through multislice calculations that high angle annular dark field scanning transmission electron microscopy can be sufficiently incoherent for the reconstruction of single element nanocrystals, but that dynamical diffraction effects can cause classification problems if more than one element is present.