### Introduction

In the context of biology and medicine, electron tomography is a means of estimating the internal structure of an object from measurements of the intensity of a high-voltage electron beam impinging upon it. The object is assumed to possess a degree of opacity with respect to individual electrons that results in an attenuation of the beam via effects such as scattering and absorption. The device that generates the electron beam, positions and orients the object in the beam, and measures the beam's intensity is called a transmission electron microscope (TEM), with the label “transmission” referring to the passage of the electron beam through the object being investigated. A measurement of the intensity of the electron beam usually takes the form of a one- or two-dimensional grey-scale image referred to by various names, with “tilt”, “micrograph”, “electron micrograph”, “transmission electron micrograph”, “TEM micrograph”, and “projection” being among the most common. In the present document, we will refer to all such measurements using the umbrella term **projection**.

### Light microscopy versus electron microscopy

Theoretically speaking, the resolution limit of an imaging system depends directly on the range of energy wavelengths it is capable of detecting. Since the wavelength of visible light is between 390 to 750 nanometres (10^{-9}m), standard light microscopes are incapable of resolving features smaller than 1/10th of a micrometre (10^{-7}m). Because of the inverse relationship between the relativistic momentum of an electron and its wavelength, electrons that are accelerated to an appreciable fraction (e.g., 70%) of the speed of light possess wavelengths on the order of picometres (10^{-12}m). In terms of the electromagnetic spectrum, this is the wavelength of gamma rays. Modern TEMs can therefore achieve magnifications 1,000X that of standard light microscopes, easily resolving features in the micro- to nanometre range (10^{-6}m--10^{-9}m), with the highest achieved resolutions (ca. 2005) in the 1/100ths of nanometres. Biological structures falling within this resolution range include mammalian cell nuclei (~6μm in diameter), human red blood cells (~6--8μm in diameter), *Caulobacter crescentus* bacteria (4--6μm long), mitochrondria (0.5--10μm), Polio virus capsids (30nm in diameter), and typical cell membranes (6--10nm thick).

### Projection acquisition

When producing projections intended to be used for tomographic reconstruction (a term defined below), the object being imaged is usually required to be a thin section of tissue between 150--500nm thick. The tissue is often stained to increase its electron opacity, most commonly with compounds containing heavy metals such as osmium, lead, or uranium (e.g., osmium tetroxide and uranyl acetate). The high voltages used to accelerate the electrons in the TEM column require that the column be evacuated to prevent electrical arcing and undesirable interactions between the electrons and atmospheric gas molecules. To withstand the vacuum of the TEM column, biological material must also be fixated prior to imaging. Both osmium tetroxide and uranyl acetate, in addition to their staining properties, behave as biological fixatives; plastic embedding provides alternative means of fixation. At some point during sample preparation, the object is marked with small particles, such as ~100nm diameter colloidal gold, which act as **fiducial markers** (or simply **fiducials**). If this marking occurs after fixation and any subsequent shaving down of the section block, only the exterior surfaces of the block will be marked. Following the sample preparation, the object is placed on a specimen stage, which is the part of the TEM that positions and orients the object in the electron beam. Projections of the object are taken at a series of orientations, typically 1- or 2-degree incremental rotations about a single axis (“single tilt”) or two orthogonal axes (“double tilt”) -- in either case, the current rotation axis is often referred to as the **tilt axis**. Note that these two projection geometries are the most common only because they are the most mechanically convenient.

Figure 1: This diagram serves as an illustration of the initial projection geometry inside an electron microscope. In this and subsequent diagrams, the electron source is located at the origin, but here the beam is turned off. The object being imaged is the opaque grey block in the center of the diagram; note the black spheres embedded in the surface of the block -- these represent fiducial markers. There are also fiducial markers embedded on the far side of the object. The line parallel to the |

Figure 2: In this diagram the electron beam has been turned on; the dashed line running along the negative |

Figure 3: In this diagram, the opaque material of the object has been rendered transparent to give the viewer a better idea of what is going on. There are actually four structures inside the object: a rectangular solid, a small sphere, an elliptical solid, and a bowl-shaped structure. Note that the projection of the rectangular solid at this orientation more-or-less completely shadows the small sphere. A mathematical description of this orientation would be something like ω = ( |

Figure 4: In this animated diagram, the object is rotated about the |

Figure 5: In this animated diagram, the specimen stage holding the object has been rotated 90.0 degrees about the |

A careful reader may have noticed a number of additional peculiarities in the preceding diagrams; for further discussion, click here. (A link to a page containing further discussion of the figures.)

In the animated diagrams above, the rotation angles about the tilt axis range in magnitude from 0.0 to 60.0 degrees. This matches what is usually found in practice: assuming that the object at orientation ω = (tilt-axis ≠ *z*-axis, 0.0) has a thickness along the *z*-axis that is much less than its extents along the *x*- and *y*-axes, as the object's rotation about the tilt axis approaches +/-90.0 degrees, the thickness of the object along the *z*-axis becomes very large. For most biological samples oriented at high degree, the object's thickness along the *z*-axis becomes too great for the electron beam to effectively penetrate.

### Tomographic reconstruction

A single projection of an object gives some indication of its internal structure, but because of shadowing, particularly in the case of perfectly electron-opaque structures, much of the detail is obscured. Furthermore -- especially in the case of perfectly electron-opaque structures -- a single projection leaves spatial arrangement along the *z*-axis completely ambiguous. Multiple projections of an object remove some of this spatial ambiguity, but the shadowing problem remains. (It would be instructive at this point to consider why the geometries of only three of the four structures in the object imaged in the diagrams above can be completely determined from their projections.) What is desired is the calculation of a 3D scalar function, a **tomogram**, representing the density of the object with respect to the electron beam. The word “tomogram” is a modern combination of two words from Attic Greek: “tómos”, a cut or section, and “grámma”, something written or drawn. Indeed, once calculated, a tomogram can be viewed as a stack of 2D slices, but there exist other techniques for viewing a tomogram as a whole volume. The technique of reconstructing an object's tomogram from its projections is called **tomographic reconstruction** or, simply, **tomography**.

At this point is necessary to touch briefly upon the mathematical theory underpinning tomographic reconstruction. Tomographic reconstruction is what physicists and mathematicians refer to as an **inverse problem**. An inverse problem is one that attempts to determine the values of a set of model parameters from a set of observations. Assume that the synthetic TEM in the diagrams above is perfectly stable, and that its physical geometry is completely known. Furthermore, assume that we have a sensible model for the physics of electrons, the relationship between the attenuation of an electron beam passing through matter and that matter's density, and the electron detector comprising the projection screen. The projections and their corresponding object orientations thus stand as a set of observations, and the model parameters we seek are therefore associations between the points in the 3D space occupied by the object and scalar densities. Mathematical models of image formation in a TEM are usually couched in the language of line integrals: the intensity associated with a particular point on a given projection is the integral along a line passing through that point and the electron source. Tomographic reconstruction can be understood as the inverting of these line integrals. In theory, *completely* reconstructing an object's tomogram with *perfect fidelity* requires that the object be of finite extent, that *all* line integrals not passing through the object equal zero, and that *all* line integrals passing through the object be computed. If a set of line integrals are missing, they are assumed to equal some value (usually zero), which will degrade the accuracy of the tomogram. As mentioned in the previous section, in the case of typical biological specimens there is an entire range of orientations at which projections cannot be taken of the object. These missing projections and their associated orientations are known as the **missing wedge** and are a source of error in tomographic reconstructions.

Given a set of projections of an object, there are three common approaches to reconstructing its tomogram, only one of which concerns us at present: **filtered back-projection**. The filtered back-projection algorithm is comprised of three steps. The first step, **alignment**, calculates a geometrical association -- a set of parameters referred to as a **projection map** -- between the various orientations of the object and the detector at a fixed reference orientation (alternatively, the geometrical associations can be understood to be between projections oriented with respect to the object at a fixed reference orientation). This is usually achieved via approximations that depend on tracked 2D projections of a set of fiducial markers, although, theoretically, anything that can be tracked with tolerable accuracy throughout the set of projections can be used. The second step, **filtration**, is too mathematically abstract to be discussed much here, though it can accurately be construed as a sharpening of the projections. It is worth noting, however, that while in theory filtration depends directly on alignment, often in practice this dependence is assumed to take a very simplified form, which amounts to a sharpening along the horizontal pixel rows in the projection. The third step, **reconstruction**, is typically implemented as follows: for each 3D point *x* in the tomogram, for each filtered projection indexed by orientation ω, find the 2D point *x*_{p} in the current projection that *x* projects to according to projection map indexed by ω and add the filtered intensity at *x*_{p} to the intensity of the tomogram at *x*. The mapping from *x* to *x*_{p} is called **forward-projection**, while the carrying of the filtered intensity at *x*_{p} back to *x* is called **back-projection**. Given some means of calculating the projection trajectory that passes through point *x*_{p} in the filtered projection indexed by ω (alternatively, given a simplified projection geometry), forward-projection is not a theoretical necessity, and the filtered intensity at *x*_{p} can be back-projected at constant value along its trajectory through the tomogram. In practice, however, particularly when the projection trajectories are curvilinear, this leads to sampling errors in the tomogram and foward-projection must be used.