The problem of image reconstruction from indirect measurements is considered. Reconstruction methods for the following types of measurement are presented in detail: (a) Radiant intensity at the image plane of the transmission electron microscope. (b) Radiant intensity of X-ray diffraction from paracrystalline and fibrous macromolecular specimens. (c) Projections which may be incomplete in linear extent, as well as sampled and finite in number. Image reconstruction from measurements (a) and (b) is affected by the phase problem. It is shown how: (a) Off-set holography might be achieved in the electron microscope when examining a crystalline specimen or an aperiodic specimen deposited on to a crystalline substrate. A diffraction plane mask selects one of the diffracted beams from the crystal to act as the holographic reference. Microscope aberrations may be compensated after reconstruction: from the hologram. An optical simulation of the process is reported. (b) Under certain conditions it is possible to deduce the continuous diffraction pattern of a molecule from measured crystal structure factor intensities. Theoretical results are derived for the diffraction from generalised helix-like structures. A recently proposed model of the DNA molecule is shown to be consistent with much of the available X-ray data. (c) Preprocessing of projection data may be effected efficiently when the data are measured with a fan beam of radiation. (d) Useful images may be reconstructed from projections which are incomplete in the sense that they are "hollow" or "truncated".

The modified back-projection method of image reconstruction is analysed in detail. “Hollow" and "truncated" projections are defined to be projections which have their inner and outer parts missing, respectively. Theoretical considerations show that unambiguous reconstruction is possible from hollow but not from truncated projections. Practical methods are presented which preprocess the incomplete projections so that reconstructions can be obtained from them using the modified back-projection method. Examples showing reconstructions of a test object from computer-generated incomplete projections are used to demonstrate the effectiveness of the preprocessing methods.