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One of the key steps in Electron Microscopy is the tomographic reconstruction of a three-dimensional (3D) map of the specimen being studied from a set of two-dimensional (2D) projections acquired at the microscope. This tomographic reconstruction may be performed with different reconstruction algorithms that can be grouped into several large families: direct Fourier inversion methods, back-projection methods, Radon methods, or iterative algorithms. In this review, we focus on the latter family of algorithms, explaining the mathematical rationale behind the different algorithms in this family as they have been introduced in the field of Electron Microscopy. We cover their use in Single Particle Analysis (SPA) as well as in Electron Tomography (ET).

Electron Microscopy has been established as one of the key players in Structural Biology with the goal of elucidating the three-dimensional structure of macromolecular complexes in order to better understand their function and molecular mechanisms [

The situation in Electron Tomography (ET) is, in general, specially involved, due to the combined effect of a smaller number of projection images, the existence of a missing wedge, and the fact that images are very large, posing an important challenge to the computational resources (especially in terms of memory; for a comparison between Single Particle Analysis and Electron Tomography, see Jonic et al. [

In this review, we focus on the family of iterative reconstruction algorithms, also known as series expansion methods. The classical algorithms (ART, Block ART, and SIRT) have been followed by a number of more modern algorithms like Conjugate Gradient, Subgradient Descent, Projected Subgradient, Superiorization, ADMM, and so forth, allowing all kinds of regularizations with a special emphasis on sparsity promoting regularizers. We start by showing how the 3D reconstruction problem can be posed as a problem of solving a linear system of equations. Then, we put the more classical methods in a common algebraic framework. Finally, we introduce the rationale behind the more modern methods.

Images collected by the electron microscope can be understood as the parallel projection of the Coulomb potential of the macromolecule being imaged. The relationship between the 3D model,

In practice, this ideal image is never observed but it is corrupted by random noise (whose nature is related to the structure of the ice surrounding the macromolecule, the random arrival of electrons, etc.) and the projected image is not known at any position

Let us assume now that we express the volume as a linear combination of functions,

The goal of the tomographic problem is to determine the basis coefficients,

We have presented the linear equation system only in connection to the data collection geometry. However, the Contrast Transfer Function (CTF) of the microscope (i.e., how the microscope blurs the ideal images) can easily be incorporated. The convolution can be represented by a matrix multiplication using a Toeplitz matrix, such that instead of the

Interestingly, posing the tomographic problem as a linear equation system can also be done in Fourier space. Thanks to the Central Slice Theorem, the relationship between the 3D Fourier transform of the macromolecule model and the 2D Fourier transform of the projection can be expressed as [

Given the inconsistent equation system

The most straightforward approach to solving this equation system is the use of the Moore-Penrose pseudoinverse:

Note that all equation systems posed so far (see (

A very simple, but general, approach to produce an iterative algorithm solving the equation system in (

The reasons why these numerical schemes succeed in solving the equation system

SIRT is one of the most popular reconstruction algorithms used in Electron Tomography. As we show below, SIRT is the result of the Jacobi algorithm applied to the normal equations with a particular weighting scheme. Let us consider the normal equations of the Weighted Least Squares problem in real space:

In EM, we are used to formulate SIRT as

In the following, let us show that both formulations (see (

Let us now work on (

The case where

Given a volume

In this way, we may rewrite the SIRT iteration with

That is, at every iteration, we update the volume with a weighted sum of the orthogonal projections of the current solution onto the set of hyperplanes defined by the experimental measurements. The relaxation factor

Actually, we may update our estimate of the current solution after a single hyperplane projection; that is, we do not have to wait to “see” all measurements at the same time but we may update the volume just after seeing each pixel value:

The index

Additionally, we do not need to be restricted to orthogonal projections, and oblique projections can be undertaken. Given a symmetric, positive definite matrix

All SIRT algorithms can be written in a compact matrix form:

Stated in this matrix form, we may easily find another SIRT algorithm that is very popular in image processing: the Landweber iteration. A possible solution to the Weighted Least Squares problem,

Another interesting variant of this family of traditional algorithms is the possibility to use unmatched projectors, for example, a relatively complicated forward operator,

In recent years, one of the most popular algorithms for solving a linear equation system is Conjugate Gradient [

In the context of EM, the Conjugate Gradient was used by Chen and Förster [

In practice, none of these iterative algorithms are run to convergence. Instead, the algorithms are typically run for a fixed number of iterations (typically

The preconditioning

The feasibility problem of (

Landweber iterations can also be set in a constrained setup. Let us assume that we have the a priori knowledge that

As seen in the two examples above, one of the most interesting ideas of this constrained optimization is the possibility to alternate between the standard tomographic update (

These ideas of constrained optimization can be further extended to nondifferentiable, convex functions (the

At a differentiable point of

The problem at a nondifferentiable point (which occurs normally at the frontiers of the intersection of convex sets) is that the gradient is not well defined. We may define instead the subgradient. A vector

Superiorization has been proposed as an alternative to Projected Subgradients [

A different approach to constrained optimization which has been actively explored in EM is by defining new equations that must be simultaneously solved along with the equation system coming from the measurements [

Iteratively steering algorithms tend to promote reconstructions with certain characteristics. For instance, [

Sparse representations has been one of the most active research fields in the area of image and signal processing in the last 10–15 years [

As stated above, sparse representations are mostly interesting in

The first class of algorithms we will review involve an

Let

The second class of algorithms substitute the

In the above iteration, we have assumed an orthonormal dictionary or transformation, but equivalent formulas can be found for nonorthonormal dictionaries.

The use of Lagrangian augmented objective functions of the form

We might have gone one step further to the generalized Tikhonov regularization,

If we assume independence between the preprocessed coefficients, most maximum a posteriori (MAP) algorithms could be written using

If we take

The approach of Albarqouni et al. [

We can see that the above regularized problems include either some a priori knowledge on the volume (

Related to these sparse reconstruction problems is the one of compressed sensing. The idea is to perform a 2D or 3D reconstruction problem starting, not from a full image, but from a “few” incoherent points from the projection images [

The field of iterative reconstruction algorithms has been very much studied, particularly in its application to Electron Microscopy data, as we have shown in this review. The 3D reconstruction problem is no longer seen as a bottleneck in Single Particle Analysis (this is a technique in which many single particles, assumed to come from an homogeneous population but at different angular orientations, are combined into a single 3D map; Jonic et al. [

The authors declare that they have no conflicts of interest.

The authors would like to acknowledge economical support from Comunidad de Madrid through Grant CAM (S2010/BMD-2305), the NSF through Grant 1114901, the Spanish Ministry of Economy and Competitiveness through Grants AIC-A-2011-0638, BIO2010-16566, and BIO2013-44647-R, Instituto de Salud Carlos III (PT13/0001/0009), and Fundación General CSIC (Programa ComFuturo). This work was funded by Instruct, part of the European Strategy Forum on Research Infrastructures (ESFRI), and supported by national member subscriptions.