The types of digital electron microscope image processing of greatest interest are not the same in the three major types of electron microscope. In the TEM, electron tomography and crystallography, particularly for 3-D reconstruction of biological specimens, and procedures that enable the physicist to establish the atomic structure, and especially defects in crystalline material, from one or more of a host of different records - bright- and dark-field images, traditional or convergent-beam diffraction patterns, energy-loss spectra, holograms - have attracted the greatest attention. In the SEM, image enhancement and image analysis (counting and measurement of geometrical or compositional features of the image) have always been important and the earliest attempts to process SEM images were in these fields. In the STEM, it is the possibility of using all the information in the far-field diffraction pattern from every element of the object that has provoked the most exciting and original ideas.
Electron images began to be processed by computer or by analog means just 30 years ago and during those years, images of every kind have likewise been enhanced or restored or analysed: in astronomy and medicine, in forensic science and in agriculture, in textile studies and the geosciences, similar algorithms have been developed, often with little awareness of related work. In order to harmonize all this material, an attempt to find a common language for representing it has been sought and the result is 'image algebra'. This is a simple mathematical structure in which the basic element is not the pixel value but the image, which is a general notion: an array, the elements of which may be real or complex numbers or one- or two-dimensional arrays of such numbers. An image whose pixel values are themselves images plays a very important part and is referred to as a 'template'.
We show how the various image processing algorithms that have been found useful in electron microscopy can be expressed in terms of this algebra. Moreover, image algebra is not merely a convenient way of representing these algorithms; its simplicity and conciseness make it easy to recognize generalizations and variants of the original procedures. We draw attention to some of these and point out promising directions for future work, including the use of a new transform, the 'slope transform', for interpreting scanning tunnelling microscope images.