a reconstruction with a huge amount of redundant space above and below the actual
specimen. This can be cropped from the 3D volume to save disk space. More
sophisticated software allows the user to specify the z dimensions to be reconstructed as well as any centre offset, which saves computation time, especially in the
case of more intensive algorithms.
For many years, radioastronomers, medical X-ray (CT) tomographers and, more
recently, electron tomographers have used reliable reconstruction algorithms based
on the Radon Transform. The standard reconstruction algorithm for this purpose is
the weighted back projection (syn. filtered back projection) algorithm [90]. This
algorithm is based on the theorem proved mathematically by Radon in the early
1900s [91] and which was rediscovered by radioastronomers in the 1960s [92].
Typically, each image of a single-axis tilt series corresponds to a projection of the
object onto a plane perpendicular to the tilt axis. In electron tomography, the
intensities arise from electrons that pass through the specimen, essentially along a
straight line or ‘ray’ perpendicular to the plane. The number and type of electrons
reaching the detector depends on their interactions with the specimen and the
electrostatic densities along a given path through the specimen [93]. Radon’s theorem explains that each real-space image corresponds to a 2D plane in Fourier
space that intersects the origin and is perpendicular to the viewing axis.
Back-projection algorithms make use of this theorem by repopulating the planes
according to the angular projections acquired at the microscope. In real space, this
is equivalent to an inverse projection operation, with the observed densities being
distributed equally along the ray onto all volume elements that contribute to the
projection (Fig. 1.1 (lower panel), 2, 9). Thus, back-projection algorithms are
regarded as a first-order approximation to an underdetermined system of linear
equations given by the projection images. Prior to summation, appropriate filtering
(‘weighting’) is used to reduce artefacts caused by uneven sampling in Fourier
Space (‘discretisation artefacts’), which effectively bias the distribution towards low
frequencies [94].
Post-reconstruction image processing can be applied to reduce the size of the
calculated volume or to make an arbitrary correction for X-axis tilt. Due to the
speed and robust nature of the weighted backprojection algorithm, it is common
practise to perform this calculation even if the intention is to proceed to more
time-consuming, iterative reconstruction algorithms such as the algebraic reconstruction technique (ART) or simultaneous iterative reconstruction technique
(SIRT). Both of these algorithms involve repeated back-projection steps [95]. ART
compares the differences between the reprojections of the reconstructed volume and
the measured data and corrects the volume accordingly [96, 97]. It then takes the
respective difference images and multiplies them by appropriate weighting factors
and adds them to the original back-projection model, and the projections are
recalculated until such time as a certain stopping criterion is met. ART is computationally efficient and contains all frequency information but tends to be relatively unstable with respect to noise. The SIRT algorithm is similar to ART, but
performs the update of the reconstruction volume only after all corrections have
been calculated as opposed to one at a time [90, 98]. SIRT converges more slowly
16
A. Leis
specimen. This can be cropped from the 3D volume to save disk space. More
sophisticated software allows the user to specify the z dimensions to be reconstructed as well as any centre offset, which saves computation time, especially in the
case of more intensive algorithms.
For many years, radioastronomers, medical X-ray (CT) tomographers and, more
recently, electron tomographers have used reliable reconstruction algorithms based
on the Radon Transform. The standard reconstruction algorithm for this purpose is
the weighted back projection (syn. filtered back projection) algorithm [90]. This
algorithm is based on the theorem proved mathematically by Radon in the early
1900s [91] and which was rediscovered by radioastronomers in the 1960s [92].
Typically, each image of a single-axis tilt series corresponds to a projection of the
object onto a plane perpendicular to the tilt axis. In electron tomography, the
intensities arise from electrons that pass through the specimen, essentially along a
straight line or ‘ray’ perpendicular to the plane. The number and type of electrons
reaching the detector depends on their interactions with the specimen and the
electrostatic densities along a given path through the specimen [93]. Radon’s theorem explains that each real-space image corresponds to a 2D plane in Fourier
space that intersects the origin and is perpendicular to the viewing axis.
Back-projection algorithms make use of this theorem by repopulating the planes
according to the angular projections acquired at the microscope. In real space, this
is equivalent to an inverse projection operation, with the observed densities being
distributed equally along the ray onto all volume elements that contribute to the
projection (Fig. 1.1 (lower panel), 2, 9). Thus, back-projection algorithms are
regarded as a first-order approximation to an underdetermined system of linear
equations given by the projection images. Prior to summation, appropriate filtering
(‘weighting’) is used to reduce artefacts caused by uneven sampling in Fourier
Space (‘discretisation artefacts’), which effectively bias the distribution towards low
frequencies [94].
Post-reconstruction image processing can be applied to reduce the size of the
calculated volume or to make an arbitrary correction for X-axis tilt. Due to the
speed and robust nature of the weighted backprojection algorithm, it is common
practise to perform this calculation even if the intention is to proceed to more
time-consuming, iterative reconstruction algorithms such as the algebraic reconstruction technique (ART) or simultaneous iterative reconstruction technique
(SIRT). Both of these algorithms involve repeated back-projection steps [95]. ART
compares the differences between the reprojections of the reconstructed volume and
the measured data and corrects the volume accordingly [96, 97]. It then takes the
respective difference images and multiplies them by appropriate weighting factors
and adds them to the original back-projection model, and the projections are
recalculated until such time as a certain stopping criterion is met. ART is computationally efficient and contains all frequency information but tends to be relatively unstable with respect to noise. The SIRT algorithm is similar to ART, but
performs the update of the reconstruction volume only after all corrections have
been calculated as opposed to one at a time [90, 98]. SIRT converges more slowly
16
A. Leis