3. The signal model

Assume that the particle has a standard orientation and that a rigid orthogonal reference frame is attached to the particle (Figure 3). This frame is the particle reference frame or the attached frame.

The term "structure of a particle" refers to the 3D electrostatic potential experienced by the electron as it passes near the particle. The structure can be described by a real-valued function S defined in the 3D space of the attached frame. Embedding the particle in ice at a random orientation and exposing it to a vertical electron beam are equivalent to taking the particle in its standard orientation and exposing it to the beam from a random direction in the attached frame. Let the beam direction be given by a unit vector n in the attached frame (Figure 3). The tomographic projection of S along n is the image f defined on a 2D plane perpendicular to n (see Figure 3). The value of f at a point u in the plane is given by f uð Þ¼ <sup>Ð</sup> <sup>∞</sup> �∞S uð Þ <sup>þ</sup> <sup>ρ</sup><sup>n</sup> <sup>d</sup>ρ.

We can write this relation concisely using the operator notation by saying that the tomographic projection operator P<sup>n</sup> gives the projection f of S via f ¼ Pnð ÞS . Continuing to use the operator notation, the effect of CTF on the projection is the action of the CTF operator C and is given by CPnð ÞS . This action is, of course, the filtering of Pnð ÞS with the CTF. The filtered image appears embedded in a micrograph with an additional 2D rotation and translation (Figure 3). Let Rθ,<sup>t</sup> be an operator that rotates a planar image by angle θ and translates it by the vector t. Then, the image as observed in the micrograph is I ¼ Rθ,<sup>t</sup> CPnð Þþ S n, where n is the noise. The signal flow diagram corresponding to this equation is shown in Figure 4.

Tomographic projection in a particle reference frame and its embedding in a micrograph.

A Gentle Introduction to Cryo-EM Single-Particle Reconstruction Algorithms DOI: http://dx.doi.org/10.5772/intechopen.90099

Figure 4. Signal flow for cryo-EM.

A final fact to consider is that cryo-EM images are noisy. Noise is introduced into the image by the camera and potentially also by the beam. A simple model for noise—one that is used in most reconstruction algorithms—is the Gaussian.

turn limit the amount of "signal" in the images.

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In engineering terms, it can be less than �10 db.

3. The signal model

by f uð Þ¼ <sup>Ð</sup> <sup>∞</sup>

Figure 3.

46

reference frame or the attached frame.

�∞S uð Þ <sup>þ</sup> <sup>ρ</sup><sup>n</sup> <sup>d</sup>ρ.

The effect of noise is compounded by the fact that long exposures to the electron beam damage the particles (they are bombarded by high-energy electrons), thereby altering their structure. To limit this damage, exposures are typically short, which in

The result is that the signal-to-noise ratio (SNR) in cryo-EM images is quite low.

Assume that the particle has a standard orientation and that a rigid orthogonal reference frame is attached to the particle (Figure 3). This frame is the particle

We can write this relation concisely using the operator notation by saying that the tomographic projection operator P<sup>n</sup> gives the projection f of S via f ¼ Pnð ÞS . Continuing to use the operator notation, the effect of CTF on the projection is the action of the CTF operator C and is given by CPnð ÞS . This action is, of course, the filtering of Pnð ÞS with the CTF. The filtered image appears embedded in a micrograph with an additional 2D rotation and translation (Figure 3). Let Rθ,<sup>t</sup> be an operator that rotates a planar image by angle θ and translates it by the vector t. Then, the image as observed in the micrograph is I ¼ Rθ,<sup>t</sup> CPnð Þþ S n, where n is the noise. The signal flow diagram corresponding to this equation is shown in Figure 4.

Tomographic projection in a particle reference frame and its embedding in a micrograph.

The term "structure of a particle" refers to the 3D electrostatic potential experienced by the electron as it passes near the particle. The structure can be described by a real-valued function S defined in the 3D space of the attached frame. Embedding the particle in ice at a random orientation and exposing it to a vertical electron beam are equivalent to taking the particle in its standard orientation and exposing it to the beam from a random direction in the attached frame. Let the beam direction be given by a unit vector n in the attached frame (Figure 3). The tomographic projection of S along n is the image f defined on a 2D plane perpendicular to n (see Figure 3). The value of f at a point u in the plane is given

At this point, it is useful to introduce the adjoints of the projection operator P<sup>n</sup> and the CTF operator C. The adjoint of the projection operator is the back-projection operator denoted by P<sup>∗</sup> <sup>n</sup> . The back-projection operator takes an image f and produces a 3D function S as follows: S xð Þ¼ fð Þ Πnð Þ x , where x is a point in threedimensional space and Πnð Þ x is the orthogonal projection of x onto the image plane (which is perpendicular to n). Note that P<sup>∗</sup> <sup>n</sup> is the adjoint (loosely speaking, the "transpose") of Pn, and not its inverse. The CTF operator C is self-adjoint, so its adjoint is itself.

Every image picked from a micrograph has its own projection direction n, CTF, and in-plane rotation and translation. Suppose N images Ii, i ¼ 1, … , N are picked, then

$$I\_i = R\_{\theta\_i, t\_i} C\_i P\_{n\_i}(\mathcal{S}) + \mathfrak{n}\_i,\tag{1}$$

where n<sup>i</sup> is the projection direction of Ii, Ci is the CTF operator of image Ii, θ<sup>i</sup> and ti are the in-plane rotations and translations of image Ii, and n<sup>i</sup> is the noise in image Ii.

Of all the terms that appear in the right-hand side of Eq. (1), only the CTF Ci is known (because the defocus at which the image micrograph was obtained is known). The challenge in cryo-EM reconstruction is to estimate S given that ni, θi, ti, and noise are also unknown. Note that the set of all possible values of n<sup>i</sup> is identical to the set of points on the surface of the unit sphere in 3D. The set of all possible values of θ<sup>i</sup> is the interval [0; 2π). And the set of all possible values of ti is identical to the set of points in some square in the plane (it is not the entire plane because the particle is usually located somewhere close to the center of the image).

Although Eq. (1) is commonly used, there are two variations of the equation that are worth noting. The first simply applies the in-plane rotations and translations to the image rather than to the projected structure:

$$R\_{\theta\_i, t\_i} I\_i = C\_i P\_{\mathfrak{n}\_i}(\mathbb{S}) + \mathfrak{n}\_i. \tag{2}$$

The second introduces a positive scalar ρi, which models contrast change due to the variable ice thickness:

$$I\_i = \rho\_i R\_{\theta\_i t\_i} \mathbf{C}\_i \mathbf{P}\_{\mathfrak{n}\_i}(\mathbf{S}) + \mathfrak{n}\_i,\\ \text{or } R\_{\theta\_i t\_i} I\_i = \rho\_i \mathbf{C}\_i \mathbf{P}\_{\mathfrak{n}\_i}(\mathbf{S}) + \mathfrak{n}\_i. \tag{3}$$

In this version, scalar ρ<sup>i</sup> is also unknown.

The unknown variables in the cryo-EM reconstruction problem are the structure S, the set of projection directions N ¼ f g n1, … , n<sup>N</sup> , the set of 2D rotations and translations T ¼ fðθ1,t1Þ, … ,ðθN,tNÞg, and the set of scalars ρ ¼ ρ<sup>1</sup> f g , … , ρ<sup>N</sup> (if using the model of Eq. (3)). Of these variables, we are only interested in S; the rest are nuisance variables.

Cryo-EM reconstruction algorithms can be classified according to how they treat the nuisance variables. On the one hand, there are algorithms that simultaneously

estimate the structure as well as the nuisance variables. I will call these algorithms (for reasons that will become clear below) best-alignment algorithms. On the other hand, there are algorithms that only determine the structure. These are based on the expectation-maximization algorithm.
