Electron tomography is a type of axial or computational tomography. The latter term is well known from medical applications under the acronym CT. The need to calculate a three-dimensional (3D) reconstruction from a set of initially acquired data is common to both x-ray based CT and electron tomography. The acquired data contains a series of projections obtained from the object of interest viewed along different directions. In order to obtain a projection, the specimen has to be penetrated by electrons. The transmitted electrons form an image detected on the specimen side opposite to the illumination source (Figure 1). In contrast to x-rays, electrons have a smaller range in matter. Hence, specimens have to be prepared with an adequate size and geometry, that is, they ideally exhibit a needle shape. Depending on the material under investigation, the needle diameter must be less than 1 µm or even less than 100 nm. In that way, the specimen is transparent to electrons during a full tilt series.

formation of projection
Figure 1 - Transmission of electrons through a specimen and formation of a projection. The needle-shape of the specimen allows to transmit electrons through the specimen in all directions perpendicular to the tilt axis. Here, the needle predominantly consists of GaSb and the diameter is approximately 150 nm.

Prior to the calculation of the 3D reconstruction, the tilt series of images has to be aligned. The following videos demonstrate the raw and the aligned series. In the former one, the specimen seems to jump on a scale of several 10 nm which arises due to mechanical inaccuracies of the specimen stage. In the latter one, the images are shifted in order to approximately align them with respect to a significant object.

Figure 2 - The videos show a tilt series of images as acquired (left) and after rough alignment (right). The series shows projections of the specimen taken in steps of 2° over a tilt range of 180°.

Different algorithms are available for the calculation of the 3D reconstruction from the aligned tilt series. The selection of an appropriate algorithm depends, for instance, on the sampling parameters (tilt range and angle step), the signal-to-noise ratio in the data set and the intended evaluation of 3D data. In general, the contemporary performance of a desktop computer with a dedicated graphics card satisfies the computation requirements of common algorithms.
An illustrative explanation of the reconstruction (Figure 3) is given according to the "back-projection" algorithm. The set of aligned images is located around the volume which is to be reconstructed. The grey value of each image pixel is traced as ray into the volume. The superposition of rays results in a approximated reconstruction of the original object. The reconstructed volume consists of "voxels" (analogue to pixels in a 2D image). They posses a grey scale value.

schematic illustration of the reconstruction
Figure 3 - The schematic illustrates the reconstruction of a 3D object from a set of 2D images assembled in a tilt series.

The evaluation of the tomographic data requires a certain representation of the reconstructed volume. A typical representation - as common in medical analyses - is the sequence of slices through the volume (original meaning of "tomogram"). These slices are 2D images as shown in the left video of Figure 4. The pixel grey values are equivalent to the respective voxel values. In order to keep the 3D impression, the objects of interest have to be rendered. The video in the middle of Figure 4 presents an opaque isosurface of the needle shaped specimen. The isosurface corresponds to voxels with only one selected grey value. If necessary, the surface is smoothed and/or shown semitransparent. Alternatively, the 3D data is "volume rendered" as shown in the right video of Figure 4. Here, a color scale is allocated to the grey values and the voxels are shown semitransparent. Eventually, it might be necessary to manually render 3D objects. That is, contours have to be drawn into the slices in order to render the objects of interest. The set of contours from different slices making up one object are subsequently joined and represented as a 3D (opaque or semitransparent) object.
The representation as video is a compromise between a 2D image and the capability to actively regard the object in a 3D viewer. In the former case, shadows and specular surfaces have to evoke the impression of spatiality. In the latter case, the observer intentionally rotates the object on a computer with the available software controls for object movements.
In the first instance, the choice of the representation has to answer the demands of the analysis, as for instance the revelation of morphology, size and distribution of objects or their spatial location and orientation, if necessary, with respect to other features within the specimen. On the other hand, aspects of visibility, comprehensibility and aesthetic have to be concerned.

Figure 4 - The videos show different representations of a 3D reconstruction: slices through the volume (left), an opaque and, subsequently, a semitransparent isosurface (middle) and a "volume rendering" (right).