Angular Reconstitution: A Posteriori Assignment of Projection Directions for 3D Reconstruction

Ultramicroscopy. 1987;21(2):111-23. doi: 10.1016/0304-3991(87)90078-7.


In computerized tomography as well as in most problems of three-dimensional reconstruction from projections, one knows from the experimental set-up the angular relationships between the projections from which the reconstruction is to be calculated. A serious difficulty is encountered when the angles are not known. In this paper, a method of "angular reconstitution" is described, which allows the a posteriori determination of the relative angular orientations of the projections and thus enables the three-dimensional reconstruction of the object to be calculated. For asymmetric objects, a minimum of three projections is required, which should not be related by a tilt around a single rotation axis. The method can be applied to determine the three-dimensional structure of biological macromolecules based on electron micrographs of randomly oriented individual molecules. Angular reconstitution, in combination with multivariate statistical techniques to classify and average the characteristic views of a molecule forms a complete, self-contained methodology for molecular structure analysis by electron microscopy.

MeSH terms

  • Macromolecular Substances*
  • Mathematics
  • Microscopy, Electron / methods*
  • Multivariate Analysis


  • Macromolecular Substances