11 Electron Tomography
259
us consider a single tilt axis experiment. In the tilt axis direction, the resolution is
that of the projected images, but in the other two orthogonal axes, it is limited by
the number of projections acquired and the diameter of the reconstruction volume,
where lower step improves the resolution, especially at higher angles. However, this
leads to an increment in the acquisition time, possibly jeopardizing sample stability
and increasing sample damage. A compromise between angle step and acquisition
time, avoiding a significant loss of resolution, is thus necessary.
Furthermore, since the space in the pole piece inside the TEM column is limited, it
is practically impossible to cover the whole range of angles ±90
◦ with conventional
sample holders. This causes an undersampling in the illumination direction and is
responsible of the most important artefact in TEM tomography: the missing wedge
[2]. It degrades the resolution causing an elongation in the illumination direction, and
it must be corrected by a factor of: =
√ (α + sin α cos α)/(α − sin α cos α). There
are several approaches to deal with the effect of the missing wedge. Experimentally,
the preparation of samples in a needle shape [4, 5] is one possible solution (to cover
the ±90
◦ tilting range), but in principle excludes the study of nanoparticles. Double
tilt experiments [6] are a viable option, reducing the missing wedge to a missing
cone, at the cost of increasing the difficulty during image acquisition. DART [7, 8]
algorithms appear as an alternative option when the composition of the sample is
well known.
11.1.1 Mathematical Principles and Reconstruction Methods
11.1.1.1 Radon Transform and Fourier Methods
The first attempts to solve problems of reconstruction of 3D objects by tomography
were based on the Radon transform [9] formulation and the Fourier space properties,
through the application of the so-called central section theorem: the projection of an
object at a given angle is a central section through the Fourier transform of that object
(i.e. calculating the Fourier transform of the acquired projected images at different
angles is equivalent to sampling the Fourier space of the projected object, that can
be recovered performing an inverse Fourier transform).
The definition of the missing wedge is straightforward in terms of the Fourier
space. Applying the central section theorem, we can relate the undersampling in the
projection tilt series with non-sampled spatial frequencies in the Fourier space and,
thus, to the resolution degradation and elongation of the reconstruction [2].
Although theoretically and historically relevant, the discretization imposed by
the actual experiment inevitably forces a certain degree of interpolation in the reconstruction and, thus, an increment in the computational time. This triggered the development of the so-called direct methods. In them, the reconstruction is carried out in
direct space instead of Fourier space.
259
us consider a single tilt axis experiment. In the tilt axis direction, the resolution is
that of the projected images, but in the other two orthogonal axes, it is limited by
the number of projections acquired and the diameter of the reconstruction volume,
where lower step improves the resolution, especially at higher angles. However, this
leads to an increment in the acquisition time, possibly jeopardizing sample stability
and increasing sample damage. A compromise between angle step and acquisition
time, avoiding a significant loss of resolution, is thus necessary.
Furthermore, since the space in the pole piece inside the TEM column is limited, it
is practically impossible to cover the whole range of angles ±90
◦ with conventional
sample holders. This causes an undersampling in the illumination direction and is
responsible of the most important artefact in TEM tomography: the missing wedge
[2]. It degrades the resolution causing an elongation in the illumination direction, and
it must be corrected by a factor of: =
√ (α + sin α cos α)/(α − sin α cos α). There
are several approaches to deal with the effect of the missing wedge. Experimentally,
the preparation of samples in a needle shape [4, 5] is one possible solution (to cover
the ±90
◦ tilting range), but in principle excludes the study of nanoparticles. Double
tilt experiments [6] are a viable option, reducing the missing wedge to a missing
cone, at the cost of increasing the difficulty during image acquisition. DART [7, 8]
algorithms appear as an alternative option when the composition of the sample is
well known.
11.1.1 Mathematical Principles and Reconstruction Methods
11.1.1.1 Radon Transform and Fourier Methods
The first attempts to solve problems of reconstruction of 3D objects by tomography
were based on the Radon transform [9] formulation and the Fourier space properties,
through the application of the so-called central section theorem: the projection of an
object at a given angle is a central section through the Fourier transform of that object
(i.e. calculating the Fourier transform of the acquired projected images at different
angles is equivalent to sampling the Fourier space of the projected object, that can
be recovered performing an inverse Fourier transform).
The definition of the missing wedge is straightforward in terms of the Fourier
space. Applying the central section theorem, we can relate the undersampling in the
projection tilt series with non-sampled spatial frequencies in the Fourier space and,
thus, to the resolution degradation and elongation of the reconstruction [2].
Although theoretically and historically relevant, the discretization imposed by
the actual experiment inevitably forces a certain degree of interpolation in the reconstruction and, thus, an increment in the computational time. This triggered the development of the so-called direct methods. In them, the reconstruction is carried out in
direct space instead of Fourier space.