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Traditionally, in computed tomography practiced in radiology, the resolution of the reconstruction is expressed in terms of the number of evenly spaced projections required for the faithful reconstruction of an object that has a given diameter (see equation (10) below).The tacit assumption is that projection data have a sufficient spectral signal-to-noise ratio (SSNR) in the whole frequency range in order to reproduce the object faithfully. In electron microscopy, the situation is dramatically different, as the electron dose limitations result in very low SSNR in the individual projections. The suppression of signal is particularly severe in high spatial frequencies, where the signal is affected by the envelope function of the microscope and the high amount of ambient noise, as well as in some low spatial frequency regions (due to the influence of the contrast transfer function (CTF) of the electron microscope). In single-particle reconstruction, a satisfactory level of the SSNR in the 3D reconstruction is achieved by including a large number of 2D projections (tens to hundreds of thousands) that are averaged during the reconstruction process. Except for rare cases (), the angular distribution of projections is not an issue, as the large number of molecules and the randomness of their orientations on the support grid all but guarantee uniform coverage of angular space. The concern is whether the number of projections per angular direction is sufficient to yield the desired SSNR or whether the angular distribution of projections is such that the oversampling of the 3D Fourier space achieved during the reconstruction process will yield the desired SSNR.

The crucial problem inherent to electron tomography is radiation damage or, related to this, the choice of the correct electron dose: an excessive dose destroys the specimen, especially biological ones, while an insufficient dose results in images that are noisy and lack information. Sophisticated and highly automated techniques have been developed both for data acquisition with the aim of keeping the electron dose as low as possible, and for image processing, in order to extract reliable information from the recorded data. However, the tolerable dose is very small, especially for unstained, frozen-hydrated specimens. As a rule of thumb, 5000e/nm are tolerable for such specimens. According to the dose fractionation theorem (Hegerl and Hoppe, 1978), the total tolerable dose has to be divided by the number of projection views in order to find the dose allowed for each image of a tilt series. In addition, the low scattering power of biological material results in low-contrast images. For instance, assuming a tilt series of 50 images, a pixel size of 1nm, phase contrast imaging with a contrast of 10%, and considering only the shot noise of the electrons, the signal-to-noise ratio (SNR defined as energy of signal over energy of noice) in the projection images is in the order of 1. An increase in the number of projection images, a decrease of the pixel size and additional noise arising from the image recording system push the SNR below 1.

The intuitive understanding of the process of segmentation is that of a compartmentalization of the image into coherent regions and the extraction of independent objects. Perhaps the most sophisticated segmentation mechanism is human vision, which is capable of interpreting a large variety of groups, associating them into classes and compartments, as well as finding relationships among them. Computer-based image segmentation algorithms typically perform only a single task, which is coupled to a specific application. Humans use a large variety of different criteria to segment images, e.g. similarity, proximity, continuity and symmetry. In electron tomography, the observer usually searches for a known shape or multiply occurring shapes to guide his segmentation. The separation criteria used are the gray value and the contrast between the feature and the environment. In a general sense, the aim is to group pixels or voxels into subsets which correspond to meaningful regions or objects.When regarding pictures by eye, one has an intuitive sense for the boundaries of meaningful objects and regions. When using the computer, however, it is difficult to find quantitative criteria which define meaningful areas on the basis of pixel properties such as contours, brightness, color, texture, etc.

Electron tomography is a method for determining 3D structure by electron microscopy, using multiple tilt views of the specimen (; ; ). Since electron tomography does not employ averaging or require the presence of symmetry, it can be used in biological applications to image single copies of subcellular components . When specimen preparation is optimized by use of rapid freezing, and imaged either directly in the frozen-hydrated state, or after freeze substitution and plastic embedding, electron tomography provides a relatively high-resolution view of biological structure in a native, or near-native, cellular context.

Cryoelectron tomography aims to act as an interface between two levels of 3D imaging: cell imaging and techniques achieving atomic resolution (e.g., X-ray crystallography). This most likely will happen through a computational motif search by mapping structures with atomic resolution into lower-resolution tomograms of cells and organelles. There exist a large variety of pattern recognition techniques in engineering, which can perform different types of motif search. This chapter will focus on cross-correlation techniques, which aim to identify a motif within a noisy 3D image (the tomogram or the 3D reconstruction). Generally, the success of the crosscorrelation approach depends on the resolution of the tomograms, the degree of corruption of the motif by noise as well as the fidelity with which the template matches the motif. For maximal detection signal, the template should have the same impulse response as the motif, which in this case is the macromolecule sought. Since the noise in the tomogram cannot be significantly decreased after data recording, the task of designing an accurate template reduces to the determination of the precise parameters of the image recording conditions, so that the searched motifs may be modeled as accurately as possible.

Electron tomography offers opportunities to study structures that are not amenable to 3D imaging by any of the classical methods, such as singleparticle reconstruction (), helical reconstruction (; ) or electron crystallography () that require either a repetitive structure, or multiple copies of identical structures. Since electron tomography can produce a 3D image of a single copy of a structure, it is finding wide application in cell biology and material science. Paracrystalline specimens constitute another class of structure for which electron tomography can be particularly useful for obtaining detailed 3D images (). Paracrystals (para—Greek prefix meaning faulty) are arrays with various kinds of intrinsic disorder. Spatial averaging of such specimens usually blurs or even erases the disordered component, which may eliminate the functionally interesting feature. For this chapter, we define a paracrystalline specimen as one with partial ordering such that one component of the specimen may be highly regular while another may be irregular due to either low occupancy, lattice irregularity or both.