8.2.1 Weighted Back-Projection (WBP)
This is by far the most common reconstruction algorithm used due to its simplicity
(notice several implementations in Table 8.1). The back-projection algorithm (also
called a direct method or direct summation method) is an effective reversal of the
projection operation in real space [5, 9]. Each image is propagated (“smeared”) into
the direction of projection through the reconstruction volume. The result of
back-projecting many images from different angles is that where these propagation
lines agree, the signal is enhanced, and where they disagree, the signal is suppressed. The more the angular range is covered, the better the reconstruction gets.
The propagation algorithm requires an interpolation to calculate values at 3D
grid points from 2D pixels. This is typically done by linear interpolation, assigning
weights to neighboring voxels based on the distance between source pixels and
destination voxels [10, 11].
One consequence of this process is that the large features (low frequencies) are
emphasized compared to the detail (high frequencies), giving rise to a reconstruction with a decidedly low-pass filtered appearance. The cause is best understood as the overlap of central sections in frequency space. The reconstruction must
be appropriately weighted, thus the term “weighted back-projection” [10]. In a
single tilt tomographic series, the central sections all overlap along the tilt axis, but
diverge perpendicular to the tilt axis. The appropriate filter is therefore a ramp
function perpendicular to the tilt axis and proportional to the frequency (see
Fig. 8.2a for an example). For more complicated view distributions (such as double
or conical tilt series), the weighting function must be suitable to account for the
distribution of views. If the view distribution has good coverage (usually not
applicable to tomography), an isotropic ramp filter can be used (generally referred
to as filtered back-projection [11].
8.2.2 Frequency Space Reconstruction (FSR)
The central section theorem [8] provides a basis for reconstruction in frequency
space. The 2D Fourier transform of each micrograph is added to the 3D reconstruction volume as a central section in the correct orientation. The weight at each
frequency space voxel is just the sum of the contributions from the micrographs,
accounting for overlaps of the central sections. Once the summation is done, the
complex value at each voxel is divided by the weight. This avoids the weighting
complications encountered for WBP and produces fewer artifacts [12, 13].
The integration requires some form of interpolation, and several schemes have
been devised [9]. The simplest is the nearest neighbor interpolation, where every pixel
in the 2D image transform is assigned to the closest 3D frequency space voxel. This
seems crude, but works reasonably well (see Fig. 8.2b for an example). It can be made
more sophisticated by weighing each contribution by the distance between the source
8 Tomographic Reconstruction from Electron Micrographs
213
This is by far the most common reconstruction algorithm used due to its simplicity
(notice several implementations in Table 8.1). The back-projection algorithm (also
called a direct method or direct summation method) is an effective reversal of the
projection operation in real space [5, 9]. Each image is propagated (“smeared”) into
the direction of projection through the reconstruction volume. The result of
back-projecting many images from different angles is that where these propagation
lines agree, the signal is enhanced, and where they disagree, the signal is suppressed. The more the angular range is covered, the better the reconstruction gets.
The propagation algorithm requires an interpolation to calculate values at 3D
grid points from 2D pixels. This is typically done by linear interpolation, assigning
weights to neighboring voxels based on the distance between source pixels and
destination voxels [10, 11].
One consequence of this process is that the large features (low frequencies) are
emphasized compared to the detail (high frequencies), giving rise to a reconstruction with a decidedly low-pass filtered appearance. The cause is best understood as the overlap of central sections in frequency space. The reconstruction must
be appropriately weighted, thus the term “weighted back-projection” [10]. In a
single tilt tomographic series, the central sections all overlap along the tilt axis, but
diverge perpendicular to the tilt axis. The appropriate filter is therefore a ramp
function perpendicular to the tilt axis and proportional to the frequency (see
Fig. 8.2a for an example). For more complicated view distributions (such as double
or conical tilt series), the weighting function must be suitable to account for the
distribution of views. If the view distribution has good coverage (usually not
applicable to tomography), an isotropic ramp filter can be used (generally referred
to as filtered back-projection [11].
8.2.2 Frequency Space Reconstruction (FSR)
The central section theorem [8] provides a basis for reconstruction in frequency
space. The 2D Fourier transform of each micrograph is added to the 3D reconstruction volume as a central section in the correct orientation. The weight at each
frequency space voxel is just the sum of the contributions from the micrographs,
accounting for overlaps of the central sections. Once the summation is done, the
complex value at each voxel is divided by the weight. This avoids the weighting
complications encountered for WBP and produces fewer artifacts [12, 13].
The integration requires some form of interpolation, and several schemes have
been devised [9]. The simplest is the nearest neighbor interpolation, where every pixel
in the 2D image transform is assigned to the closest 3D frequency space voxel. This
seems crude, but works reasonably well (see Fig. 8.2b for an example). It can be made
more sophisticated by weighing each contribution by the distance between the source
8 Tomographic Reconstruction from Electron Micrographs
213