## Preliminaries

Given a data collection scheme that produces a set of projections over an angular range of sufficient size, with angle assignments for each projection, there is still a choice among different techniques for obtaining the reconstruction. Under "technique" we understand the mathematical algorithm and—closely linked to it—its numerical and computational realization. The value of a reconstruction technique can be judged by its mathematical tractability, speed, computational efficiency, stability in the presence of noise, and various other criteria (see, for instance, Sorzano et al., 2001). Competing techniques can be roughly grouped into weighted back-projection, real-space iterative techniques, and techniques based on Fourier interpolation. Weighted back-projection (section 4.2) is a real-space technique which has gained wide popularity on account of the fact that it is very fast compared to all iterative techniques (section 4.4). However, iterative techniques can be superior to weighted back-projection according to several criteria, such as sensitivity to angular gaps and smoothness (e.g., Sorzano et al., 2001). The quality of Fourier interpolation techniques (including so-called gridding techniques) (section 4.3), which match weighted back-projection in speed, depends critically on the interpolation method used.

Apart from computational efficiency, two mutually contradictory criteria that are considered important in the reconstruction of single macromolecules from electron micrographs are the linearity of a technique and its ability to allow incorporation of constraints. Here, what is meant by "linearity" is that the reconstruction technique can be considered as a black box with "input" and "output" channels, and that the output signal (the reconstruction) can be derived by linear superimposition of elementary output signals, each of which is the response of the box to a delta-shaped input signal (i.e., the projection of a single point). In analogy with the point-spread function, defined as the point response of an optical system, we could speak of the "point-spread function'' of the combined system formed by the data collection and the subsequent reconstruction. This analogy is well developed in Radermacher's (1988) treatment of general weighting functions for random-conical reconstruction.

Strictly speaking, linearity holds for weighted back-projection only in the case of even angular spacing, where the weighting is by |k|. For the more complicated geometries with randomly distributed projection angles, linearity is violated by the necessary practice of limiting the constructed weighting functions to prevent noise amplification (see section 4.2). Still, the property of linearity is maintained in an approximate sense, and it has been important in the development and practical implementation of the random conical reconstruction method because of is mathematical tractability. [Some iterative techniques such as the algebraic reconstruction technique (ART) and simultaneous iterative reconstruction technique (SIRT), which also share the property of linearity, have not been used for random conical reconstruction until more recently because of their relatively low speed.] The practical importance of linearity also lies in the fact that it allows the 3D variance distribution to be readily estimated from projection noise estimates (see section 2.2 in chapter 6).

On the other hand, the second criterion—the ability of a technique to allow incorporation of constraints—is important in connection with efforts to fill the angular gap. Weighted back-projection as well as Fourier reconstruction techniques fall into the class of linear reconstruction schemes, which make use only of the projection data and do not consider the noise explicitly. In contrast, the various iterative algebraic techniques lend themselves readily to the incorporation of constraints and to techniques that take the noise statistics explicitly into account. However, these techniques are not necessarily linear. For example, the modified SIRT technique described by Penczek et al. (1992) incorporates nonlinear constraints.

In comparing the two different approaches, one must bear in mind that one of the disadvantages of weighted back-projection—its failure to fill the missing gap—can be mitigated by subsequent application of restoration, which is, however, again a nonlinear operation. Thus, when one compares the two approaches in their entirety—weighted back-projection plus restoration versus any of the nonlinear iterative reconstruction techniques—the importance of the linearity stipulation is somewhat weakened by its eventual compromise.