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Tomography is a method for reconstructing the interior of an object from its projections. The word literally means the visualization of slices, and is applicable, in the strict sense of the word, only in the narrow context of the single-axis tilt geometry: for instance, in medical computerized axial tomography (CAT-scan imaging), the detector-source arrangement is tilted relative to the patient around a single axis (Fig. 1a). In electron microscopy, where the beam direction is fixed, the specimen holder is tilted around a single axis (Fig. 1b). However, the usage of this term has recently become more liberal, encompassing arbitrary geometries, . In line with this relaxed convention, we will use the term for any technique that employs the transmission electron microscope to collect projections of an object that is tilted in multiple directions and uses these projections to reconstruct the object in its entirety. Excluded from this definition are ‘single-particle’ techniques that make use of multiple occurrences of the object in different orientations, with or without the additional aid of symmetry (Fig. 1c). These techniques are covered elsewhere (non-symmetric: , ; symmetric: ).

Just as fossil insects embalmed in amber are extraordinarily preserved, so are biological samples that have been embedded in plastic for electron microscopy. The success of embedding samples in plastic lies in the astounding resilience of the sections in the electron microscope, albeit after initial changes. The electron microscope image results from projection of the sample density in the direction of the beam, i.e. through the depth of the section, and therefore is independent of physical changes in this direction. In contrast, the basis of electron tomography is the constancy of the physical state of the whole section during the time that different views at incremental tilt angle steps are recorded.

The technique of cryoelectron tomography of frozen-hydrated biological specimens is opening a new window on cellular structure and organization. This imaging method provides full 3D structural information at much higher resolution (typically 5–10 nm) than is attainable by light microscopy, and can be applied to cells and organelles that are maintained in a state that is as close to native as can be achieved currently in electron microscopy. Not only can cryoelectron tomography be used to visualize directly extended cellular structures, such as membranes and cytoskeleton, but it can also provide 3D maps of the location, orientation and, perhaps, the conformation of large macromolecular complexes, the cell’s ‘molecular machinery’. This information complements that coming from single-particle cryoelectron microscopy (, ) and X-ray crystallography, about the subnanometer structure of the same molecular assemblies after isolation. As with studies using single-particle cryoelectron microscopy, specimens smaller than 1 µ in size can be prepared for cryoelectron tomography by plunge-freezing (). Cells or organelles can be rapidly frozen directly on an electron microscope grid in thin layers of glass-like, amorphous ice, without the formation of ice crystals that would otherwise disrupt fine structure (). Specimens are imaged directly, without chemical fixation, dehydration or staining with heavy metals.

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.

Electron tomography is an imaging technique that provides 3D images of a specimen with nanometer scale resolution. The range of specimens that can be investigated with this technique is particularly wide, from large (500–1000 nm) unique variable structures such as whole cells to suspensions of thousands of small identical macromolecules (>200 kDa).When applied to cryofixed frozen-hydrated biological material, the technique is often referred to as cryotomography. In combination with automated low-dose data collection and advanced computational methods, such as molecular identification based on pattern recognition, cryotomography can be used to visualize the architecture of small cells and organelles and/or to map macromolecular structures in their cellular environment. The resolution that can be obtained with cryotomography depends on several fundamental and technical issues related to specimen preparation, microscopy and subsequent image processing steps, but will typically be in the range of 5–10 nm.

Accurate alignment of projection images is an important step in obtaining a high-quality tomographic reconstruction. Ideally, all images should be aligned so that each represents a projection of the same 3D object at a known projection angle. Inadequate image alignment will result in blurring or smearing of features in the reconstruction. The problem is made more difficult because exposure to the electron beam during the acquisition of a tilt series induces geometric changes in many samples (primarily plastic-embedded material). For any sample, but particularly for ones that are not rigid during imaging, a powerful method of image alignment uses the measured coordinates of fiducial markers through the series of images; these can be fit to equations that describe the image projection. Alternative methods are based on cross-correlation of images. The fiducial marker method has the advantage that it guarantees a consistent alignment among the images from the full range of tilt angles. It is also more easily adapted to correct for changes in the sample that occur during imaging.

In computing high-accuracy reconstructions from transmission electron microscope (TEM) tilt series, image alignment currently has an important role. Though most are automated devices today, the imaging systems have certain non-idealities which give rise to abrupt shifts, rotations and magnification changes in the images. Thus, the geometric relationships between the object and the obtained projections are not precisely known initially. In this chapter, refers to the computation of the projection geometry of the tilt series so that most of the above deviations from the assumed ideal projection geometry could be rectified by using simple 2D geometric transformations for the images before computing a tomographic reconstruction.

Since the 1970s, it has become increasingly evident that transmission electron microscopy (TEM) images of typical thin biological specimens carry a large amount of information on 3D macromolecular structure. It has been shown many times how the information contained in a set of TEM images (2D signals) can determine a useful estimate of the 3D structure under study.

Traditionally, 3D reconstruction methods have been classified into two major groups, and (e.g. ; ). Fourier methods are defined as algorithms that restore the Fourier transform of the object from the Fourier transforms of their projections and then obtain the real-space distribution of the object by inverse Fourier transformation. Included in this group are also equivalent reconstruction schemes that use expansions of object and projections into orthogonal function systems (e.g. , ; ; Chapter 9 of this volume). In contrast, direct methods are defined as those that carry out all calculations in real space. These include the convolution back-projection algorithms (; ; ) and iterative algorithms (; ). Weighted back-projection methods are difficult to classify in this scheme, since they are equivalent to convolution back-projection algorithms, but work on the real-space data as well as the Fourier transform data of either the object or the projections. Both convolution back-projection and weighted back-projection algorithms are based on the same theory as Fourier reconstruction methods, whereas iterative methods normally do not take into account the Fourier relationships between object transform and projection transforms.

In 1917, Johann Radon posed the question of whether the integral over a function with two variables along an arbitrary line can uniquely define that function such that this functional transformation can be inverted. He also solved this problem as a purely mathematical one, although he mentioned some relationships to the physical potential theory in the plane. Forty-six years later, A. M. Cormack published a paper with a title very similar to that by Radon yet still not very informative to the general reader, namely ‘Representation of a function by its line integrals’—but now comes the point: ‘with some radiological applications’. Another point is that the paper appeared in a journal devoted to applied physics. Says Cormack, ‘A method is given of finding a real function in a finite region of a plane given by its line integrals along all lines intersecting the region. The solution found is applicable to three problems of interest for precise radiology and radiotherapy’. Today we know that the method is useful and applicable to the solution of many more problems, including that which won a Nobel prize in medicine, awarded to A. M. Cormack and G. N. Hounsfield in 1979. Radon’s pioneering paper (1917) initiated an entire mathematical field of integral geometry. Yet it remained unknown to the physicists (also to Cormack, whose paper shared the very same fate for a long time). However, the problem of projection and reconstruction, the problem of tomography as we call it today, is so general and ubiquitous that scientists from all kinds of fields stumbled on it and looked for a solution-without, however, looking back or looking to other fields.