International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 369371

The algebraic methods have been derived based on the observation that when the projection equation (2.5.6.5) is discretized, it forms a set of linear equations. Thus, pixels from all available N projections are placed (in an arbitrary order) in a vector and the voxels of the 3D object in a vector (in an order derived from the order of f by algebraic relations). Note we left the exact sizes of f and g undetermined, as the major advantage of algebraic methods is that we can include in the reconstruction only pixels located within an arbitrary support in two dimensions and this support can be different for each projection; similarly, the support of the object in three dimensions can be arbitrary. Thus, the number of elements in f and g are at most K^{2}N and K^{3}, respectively, with K^{2} being the number of pixels within chosen support. In the algebraic formulation the operation of projection is defined by the projection matrix P whose elements are the interpolation weights. Their values are determined by the interpolation scheme used, but for the bi and trilinear interpolations . The algebraic version of (2.5.6.5) isMatrix P is rectangular and since in singleparticle reconstruction the number of projections exceeds the linear size of the object () the system of equations is overdetermined. It can be solved in a leastsquares sense:which yields a unique structure that corresponds to the minimum of . As in the case of direct inversion in Fourier space (2.5.6.17), the approach (2.5.6.20) is impractical because of the very large size of the projection matrix. Indeed, the size of P is K^{2}N × K^{3}, which for a modest number of projections N = 10 000 and image size K^{2} = 64^{2} = 4096 yields a matrix of size ~(4 × 10^{7}) × (2 × 10^{6})! Nevertheless, in some cases the projections of the structure are orthoaxial, the full 3D reconstruction reduces to a series of independent 2D reconstructions, and it becomes possible to solve (2.5.6.19) by using the singular value decomposition (SVD) of the matrix :where and contain eigenvectors of matrices and , respectively, and is a diagonal matrix containing the first r nonnegative square roots of the eigenvalues of (Golub & Van Loan, 1996). The solution is given bywhere is the inverse of the matrix containing the largest elements of with the remaining values set to zero. By selecting l appropriately we achieve a measure of regularization. The advantage of the approach is that for a given geometry the decomposition has to be calculated once; thus, the method becomes very efficient if the reconstruction has to be performed repeatedly for the same distribution of projections or if additional symmetries, such as helical, are taken into account (Holmes et al., 2003).
In the general 3D case, a leastsquares solution can be found using one of the iterative approaches that take advantage of the fact that the projection matrix is sparse. Indeed, if bilinear interpolation is used, a row of will contain only about 4N nonzero elements (out of the total ~K^{2}N). The main idea is that the matrix is not explicitly calculated or stored; instead, its elements are calculated repeatedly during iterations as needed. In the simultaneous iterative reconstruction technique (SIRT) we find the minimum ofby selecting the initial 3D structure (usually set to zero) and by iteratively updating its current approximation using the gradient of the objective function (2.5.6.23) ,Setting the relaxation parameter yields Richardson's algorithm (Gilbert, 1972). In singleparticle reconstruction applications convergence (i.e., a solution that is not dominated by the noise) is usually reached in ~100 iterations.
SIRT is extensively used in singleparticle reconstruction (Frank et al., 1996) because it yields superior results over a wide range of experimental conditions (Penczek et al., 1992) and, when presented with angular gaps in the distribution of projections, produces the least disturbing artifacts. Note that strictly speaking, filtered backprojection and direct Fourier inversion algorithms are not applicable to data that do not cover the Fourier space fully. In addition, SIRT offers considerable flexibility in EM applications. First, it is possible to accelerate the convergence by adjusting relaxation parameters: setting results in a steepest decent algorithm. Secondly, even faster convergence (in ~10 iterations) is achieved by solving (2.5.6.23) using the conjugategradient method, but this requires addition of a regularizing term to (2.5.6.23) in order to prevent excessive enhancement of noise. Such a term has the form , where matrix B is a discrete approximation of a Laplacian or higherorder derivatives. Thirdly, it is possible to take into account the underfocus settings of the microscope by including the contrast transfer function (CTF) of the microscope in (2.5.6.23) and by solving the problem for the structure g that in effect will be corrected for the CTF,Here we assumed that the projection data were grouped according to the defocus settings indexed by and we introduced a Lagrange multiplier whose value determines the smoothness of the solution g (Zhu et al., 1997). represents the spaceinvariant pointspread function of the microscope for the κth defocus group; thus has a blockToeplitz structure (Biemond et al., 1990) and . If we assume that there is no astigmatism (which means that the pointspread function is rotationally invariant), can be either applied to the projection of the structure or to the 3D structure itself. With these assumptions, (2.5.6.25) can be solved using the iterative scheme (Penczek et al., 1997)The second sum can be precalculated and stored as a 3D volume in the computer memory. Thus, the input projections have to be read only once and are never accessed again during the course of the iterations, which eliminates the need to store them in the memory [as is also the case in (2.5.6.24)]. In addition, the product B^{T}B is the 3D Laplacian, which can be applied efficiently without actually creating the matrix B^{T}B. Finally, because of the large size of the matrix it is more convenient to apply it in Fourier space and to modify the Fourier transform of the volume instead of using matrix multiplication or realspace convolution. Algorithms (2.5.6.24)–(2.5.6.26) are implemented in the SPIDER package (Frank et al., 1996) and (2.5.6.24) in the SPARX package (Hohn et al., 2007).
The algebraic reconstruction technique (ART) predates SIRT; in the context of tomographic reconstructions it was proposed by Gordon and coworkers (Gordon et al., 1970) and later it was recognized as a version of Kaczmarz's method for iteratively solving (2.5.6.23) (Kaczmarz, 1993). We write (2.5.6.19) as a set of systems of equations, each relating single pixels f_{n}, in projections with the 3D structure,Note (2.5.6.27) and (2.5.6.19) are equivalent, as and . With this notation and a relaxation parameter , ART comprises the following steps:
Although the mathematical forms of update equations in SIRT (2.5.6.24) and in ART (2.5.6.28) are very similar, there are profound differences between them. In SIRT, all voxels in the structure are corrected simultaneously after projections and backprojections of the current update of the structure are calculated. In ART, the projection/backprojection step (2.5.6.28) involves only correction with respect to an individual pixel in a single projection immediately followed by the update of the structure. This results in a much faster convergence of ART as compared to SIRT. Further acceleration can be achieved by selecting the order in which pixels enter the correction step (2) in (2.5.6.28). It was observed that if a pixel is selected such that its projection direction is perpendicular to the projection direction of the previous pixel, the convergence is achieved faster (Hamaker & Solmon, 1978; Herman & Meyer, 1993). Interestingly, a random order works almost equally well (Natterer & Wübbeling, 2001).
In singleparticle reconstruction, ART has been introduced in the form of `ART with blobs' (Marabini et al., 1998) and is available in the Xmipp package (Sorzano et al., 2004). In this implementation, the reconstruction structure is represented by a linear combination of spherically symmetric, smooth, spatially limited basis functions, such as Kaiser–Bessel window functions (Lewitt, 1990, 1992; Matej & Lewitt, 1996). Introduction of blobs significantly reduces the number of iterations necessary to reach an acceptable solution (Marabini et al., 1998).
The major advantage of iterative reconstruction methods is the ability to take advantage of a priori knowledge, i.e., any information about the protein structure that was not initially included in the data processing, and introduce it into the reconstruction process in the form of constraints. Examples of such constraints include similarity to the experimental (measured) data, positivity of the protein mass density (only valid in conjunction with the CTF correction), bounded spatial support etc. Formally, the process of enforcing selected constraints is best described in the framework of the projections onto convex sets (POCS) theory (Sezan & Stark, 1982; Youla & Webb, 1982; Sezan, 1992) introduced into EM by Carazo and coworkers (Carazo & Carrascosa, 1986, 1987; Carazo, 1992).
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