This chapter introduces mathematical principles underlying the relationship between a three-dimensional object and its projections, the projection theorem and Radon's theorem. The shape transform is introduced as an important concept, and it is explained that recovery of the object from a finite set of projections is feasible when the object is of finite extent. It also explains the basis for Crowther's formula, which relates the number of projections to the object's size and the resolution to which it is to be recovered. The next topic is the determination of orientations, either ab initio, or by reference to an existing 3D template (three-dimensional projection alignment). Ab initio methods widely used are introduced: random-conical data collection and common lines (or angular reconstitution). Several important reconstruction techniques are covered: weighted back-projection, Fourier interpolation, and iterative methods such as the algebraic reconstruction technique, or ART. Subsequent topics of the chapter include resolution assessment, contrast transfer correction, and reconstructions from heterogeneous datasets by supervised classification.
Keywords: algebraic reconstruction technique, angular reconstitution, common lines, Crowther's formula, Fourier interpolation, projection theorem, Radon's theorem, Random-conical reconstruction, supervised classification
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