The intuitive understanding of the process of 3D reconstruction is based on a number of assumptions, which are easily made unconsciously; the most crucial is the belief that what is detected is some ...kind of projection through the structure. This ‘projection’ need not necessarily be a (weighted) sum or integral through the structure of some physical property of the latter; in principle, a monotonically varying function would be acceptable, although solving the corresponding inverse problem might not be easy. In practice, however, the usual interpretation of ‘projection’ is overwhelmingly adopted, and it was for this definition that Radon (1917) first proposed a solution. In the case of light shone through a translucent structure of varying opacity, a 3D transparency as it were, the validity of this projection assumption seems too obvious to need discussion. We know enough about the behavior of X-rays in matter to establish the conditions in which it is valid in radiography. In this chapter, we enquire whether it is valid in electron microscopy, where intuition might well lead us to suspect that it is not. Electron-specimen interactions are very different from those encountered in X-ray tomography; the specimens are themselves very different in nature, creating phase rather than amplitude contrast, and an optical system is needed to transform the information about the specimen that the electrons have acquired into a visible image.
It has been known since the early days of electron optics that the rotationally symmetric lenses employed in electron microscopes and similar instruments suffer from severe aberrations that cannot be ...eliminated by skillful lens design (Scherzer, 1936). Immense effort has been devoted to finding lenses with small aberrations and devising aberration correctors. The original demonstration that the two most important aberrations, spherical and chromatic, cannot be eliminated required that several conditions be satisfied and, by relaxing one or the other of these conditions, correctors can be designed. A nearexhaustive list was published by Scherzer (1947) and reviews charting trends in thinking about aberration correction and progress in implementing correctors are to be found in Septier (1966), Hawkes (1980), and Hawkes and Kasper (1989). These contain very full accounts of earlier attempts to correct aberrations with extensive reference lists and the material presented there is not always reproduced here. In particular, a survey of attempts to build apochromats and aplanatic lenses by H. Rose and colleagues in Darmstadt is to be found in the article by Hawkes (1980). The types of corrector that seem most promising today are examined below but first, we describe the various kinds of aberration and explain why they are important. We then look more closely at the aberration coefficients themselves, which leads naturally to a study of the correctors.
