8.3.3 Updating the 3D Map Volume
The original updating for ART was done per ray [19]:
f
ðk þ 1Þ
j
¼ f
ðkÞ
j
þ ke
ðkÞART
i
w ij
where k is a damping or relaxation parameter (<1) to avoid instability. However,
the updating did not account for the contributions from different micrographs, and
tended to become chaotic, generating “salt and pepper” type artifacts. Gilbert [20]
suggested that a better technique is to calculate a combined adjustment from all
micrographs during each iteration, i.e., SIRT. The update now becomes:
f
ðk þ 1Þ
j
¼ f
ðkÞ
j
þ k
P
i e
ðkÞSIRT
i
w ij
P
i w ij
where the summation is over all difference images. This results in increased
computation and very slow convergence. Anderson and Kak proposed an algorithm
intermediate between ART and SIRT, where the volume is updated with every
difference image, i.e., SART [21]. Instead of difference images, Wolf et al. [32]
updates the map at each iteration with appropriately weighted images (i.e., a WBP
reconstruction).
A variant of these methods is DART (discrete algebraic reconstruction technique) [33], where the volume is segmented during each iteration into discrete
levels. The application is appropriate where it is known that the different density
values correspond to different elements or homogenous regions.
In MEM, various schemes have been devised to update the reconstruction [34].
A common technique is to update the volume based on the gradients of the entropy
and the chi square metric of the difference between the calculated projection and
micrograph [24, 26].
For PSRT, the addition of each Gaussian sphere is evaluated as to whether it
improves the comparison with the micrographs or not. If the difference decreases,
the sphere is accepted [22].
8.3.4 The Number of Iterations
It was noted in the original development of SIRT in synthetic test cases that an
optimal correspondence is reach with the original map, after which it diverges [20].
It was reported that in ART the divergence is faster than SIRT. As an example,
Fig. 8.2c shows part of a SIRT reconstruction after 10 iterations, and in Fig. 8.2d,
after 50 iterations. Evidently the latter is dominated by noise. In this implementation
of IMOD, the iterative process is started from a WBP reconstruction. The comparisons are not initially weighted, so that the low frequencies are amplified.
218
J. Bernard Heymann
The original updating for ART was done per ray [19]:
f
ðk þ 1Þ
j
¼ f
ðkÞ
j
þ ke
ðkÞART
i
w ij
where k is a damping or relaxation parameter (<1) to avoid instability. However,
the updating did not account for the contributions from different micrographs, and
tended to become chaotic, generating “salt and pepper” type artifacts. Gilbert [20]
suggested that a better technique is to calculate a combined adjustment from all
micrographs during each iteration, i.e., SIRT. The update now becomes:
f
ðk þ 1Þ
j
¼ f
ðkÞ
j
þ k
P
i e
ðkÞSIRT
i
w ij
P
i w ij
where the summation is over all difference images. This results in increased
computation and very slow convergence. Anderson and Kak proposed an algorithm
intermediate between ART and SIRT, where the volume is updated with every
difference image, i.e., SART [21]. Instead of difference images, Wolf et al. [32]
updates the map at each iteration with appropriately weighted images (i.e., a WBP
reconstruction).
A variant of these methods is DART (discrete algebraic reconstruction technique) [33], where the volume is segmented during each iteration into discrete
levels. The application is appropriate where it is known that the different density
values correspond to different elements or homogenous regions.
In MEM, various schemes have been devised to update the reconstruction [34].
A common technique is to update the volume based on the gradients of the entropy
and the chi square metric of the difference between the calculated projection and
micrograph [24, 26].
For PSRT, the addition of each Gaussian sphere is evaluated as to whether it
improves the comparison with the micrographs or not. If the difference decreases,
the sphere is accepted [22].
8.3.4 The Number of Iterations
It was noted in the original development of SIRT in synthetic test cases that an
optimal correspondence is reach with the original map, after which it diverges [20].
It was reported that in ART the divergence is faster than SIRT. As an example,
Fig. 8.2c shows part of a SIRT reconstruction after 10 iterations, and in Fig. 8.2d,
after 50 iterations. Evidently the latter is dominated by noise. In this implementation
of IMOD, the iterative process is started from a WBP reconstruction. The comparisons are not initially weighted, so that the low frequencies are amplified.
218
J. Bernard Heymann