Direct 3D Reconstruction from Projections with Initially Unknown Angles

Abstract

The problem of reconstructing an object from its projections arises in many diverse fields, such as radio astronomy, medical imaging, and electron microscopy (1). Normally, the reconstruction geometry, i.e., the angular relations between the various projections, is known a priori. In medical computerised tomography, for example, one knows the positions of the X-ray source and detectors exactly for each of the measured line projections. This knowledge, however, is not always available. In particular, we consider as an example the application of macromolecular structure determination by electron microscopy (2). In an electron micrograph made of a preparation of biological macromolecules, one may find hundreds of macromolecular images, each representing a projection through the 3D electron density distribution of the macromolecule. Since the macromolecules are lying in random orientation on the support, each image represents a projection in a different direction of the structure of an “average” macromolecule. These projection directions must somehow be initially determined and later refined. Moreover, three Eulerian angles are needed to describe the orientational relations of any two molecular projections, making the problem a “true” or direct 3D reconstruction problem (3,4).