8.6.6 Beltrami Flow: Isotropic Diffusion
Beltrami flow is an isotropic diffusion algorithm where the rate of diffusion is
dependent on the presence of edges. Fernandez [57] implemented it with an edge
enhancement option (Fig. 8.4c).
8.6.7 Non-linear Anistropic Diffusion (NAD)
NAD is intended to smooth the data (i.e., remove noise) in such a way that contiguous structures are preserved (Fig. 8.4d–f) [58]. The parameters can be tuned to
favor needle-like or disc-like structures, depending on the specimen [59]. The
diffusion process is essentially a simulation with a very small time step, so that
many iterations are required to achieve a meaningful level of denoising. The kernels
over which the structure and diffusion tensors are calculated are small (3 Â 3 Â 3),
but the computational cost is very high. While considered one of the best denoising
algorithms, other methods that are computationally more efficient may be more
appropriate.
NAD is typically done on the 3D map, but it can also be implemented ahead of
reconstruction [37, 41]. Here the aim is to decrease the appearance of streaking
artifacts in the reconstruction, resulting in easier interpretation.
8.7 Information Recovery
8.7.1 Missing Information
In the current tilt geometry of the microscopes, the grid cannot be tilted beyond 70°,
leaving a significant gap in the sampled views. This is reflected in the reconstruction as missing information in frequency space: a missing wedge for single a
single tilt series. Methods to reduce the missing wedge include doing double tilt
[60–62] or conical tilt [63] series reconstructions. The effort involved in acquiring
the data and dealing with dose and distortion issues make these approaches less
attractive.
Missing information in the reconstruction means that the resolution will always
be lower in the z direction. In addition, as the tilt angle increases, the planar
specimen effectively becomes thicker. This decreases contrast and diminishes the
information content further in high tilt images. At 60°, the effective thickness is
twice that of the specimen thickness, and at 70°, it is almost three times!
The effects of missing information includes artifacts such streaks emanating
from high density regions (e.g., gold particles) and the apparent elongation of
structures in the z direction. Filtering to smooth the transitions between known and
226
J. Bernard Heymann
Beltrami flow is an isotropic diffusion algorithm where the rate of diffusion is
dependent on the presence of edges. Fernandez [57] implemented it with an edge
enhancement option (Fig. 8.4c).
8.6.7 Non-linear Anistropic Diffusion (NAD)
NAD is intended to smooth the data (i.e., remove noise) in such a way that contiguous structures are preserved (Fig. 8.4d–f) [58]. The parameters can be tuned to
favor needle-like or disc-like structures, depending on the specimen [59]. The
diffusion process is essentially a simulation with a very small time step, so that
many iterations are required to achieve a meaningful level of denoising. The kernels
over which the structure and diffusion tensors are calculated are small (3 Â 3 Â 3),
but the computational cost is very high. While considered one of the best denoising
algorithms, other methods that are computationally more efficient may be more
appropriate.
NAD is typically done on the 3D map, but it can also be implemented ahead of
reconstruction [37, 41]. Here the aim is to decrease the appearance of streaking
artifacts in the reconstruction, resulting in easier interpretation.
8.7 Information Recovery
8.7.1 Missing Information
In the current tilt geometry of the microscopes, the grid cannot be tilted beyond 70°,
leaving a significant gap in the sampled views. This is reflected in the reconstruction as missing information in frequency space: a missing wedge for single a
single tilt series. Methods to reduce the missing wedge include doing double tilt
[60–62] or conical tilt [63] series reconstructions. The effort involved in acquiring
the data and dealing with dose and distortion issues make these approaches less
attractive.
Missing information in the reconstruction means that the resolution will always
be lower in the z direction. In addition, as the tilt angle increases, the planar
specimen effectively becomes thicker. This decreases contrast and diminishes the
information content further in high tilt images. At 60°, the effective thickness is
twice that of the specimen thickness, and at 70°, it is almost three times!
The effects of missing information includes artifacts such streaks emanating
from high density regions (e.g., gold particles) and the apparent elongation of
structures in the z direction. Filtering to smooth the transitions between known and
226
J. Bernard Heymann