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11.1.1.5 Discrete Algebraic Reconstruction Technique (DART)
DART algorithm belongs to the field of discrete tomography (DT), dealing with
volume reconstruction from a low number of projections [7, 8]. It is driven by the
idea of a priori object segmentation (i.e. component separation assigning a single
grey level for each material present in the object). It operates iteratively in two steps:
(1) reconstruction step, (2) segmentation step. For simplicity, let us consider a binary
case (i.e. only one material species and the background) for a 2D reconstruction
from 1D projections, that can be easily generalized to 3D volume reconstructions.
The problem (reconstruction) can be represented by a linear system
p = W
x, where p
are the projection values, x the image/object and W represents the projection process.
The reconstruction step is a continuous tomographic reconstruction method: ART
[13], SIRT [14] or any other iterative algorithm can be used. From now on, we refer to
this step as the algebraic reconstruction method (ARM). The segmentation is carried
out separating the material and the background (thresholding the grey levels), and
identifying the set boundary B
(t) and ‘fixed’ F
(t) pixels (where (t) accounts for the
iteration step).
The set B
(t) will be updated by the ARM algorithm in each iteration, while F
(t)
will remain fixed. This way, the number of variables in the linear system are reduced
and the computational power required is greatly decreased. Since the segmentation
is applied upon the ARM-reconstructed images (possibly suffering from missing
wedge), the separation of grey levels can be defective at first. To allow the creation
of boundary regions inside the material, a random set of pixels from F
(t) is introduced
in the ARM as well with a p probability (adjustable from 0 to 1, depending on each
problem). Also, a smoothing filter is applied after each iteration, to counter the strong
local oscillations in the grey levels of B
(t) after the ARM step.
It has been proven that DART converges faster than any ARM available, when
the number of grey levels does not exceed 5 [7]. The accuracy of the reconstruction (measured comparing phantom images and the results of the reconstruction for
different algorithms) is usually higher, independently of the number of projections
or the angular range. Tests on the effect of the missing wedge reveal a much-reduced
impact in final solutions achieved by DART. Since electron dose is reduced when a
lower number of projections are required, DART appears as a viable option in cases
of severe sample damage due to beam sensitivity.
Despite the advantages described for DART, it remains a heuristic algorithm,
meaning that convergence cannot always be assured. This circumstance makes the
selection of a unique termination criteria difficult. It also relies on previous knowledge of the sample composition. Although it is robust with respect to the chosen
segmentation criteria (threshold), if the number of components (different grey levels)
is not correctly identified, the reconstruction may present serious image artefacts
(Fig. 11.2).
P. Torruella et al.
11.1.1.5 Discrete Algebraic Reconstruction Technique (DART)
DART algorithm belongs to the field of discrete tomography (DT), dealing with
volume reconstruction from a low number of projections [7, 8]. It is driven by the
idea of a priori object segmentation (i.e. component separation assigning a single
grey level for each material present in the object). It operates iteratively in two steps:
(1) reconstruction step, (2) segmentation step. For simplicity, let us consider a binary
case (i.e. only one material species and the background) for a 2D reconstruction
from 1D projections, that can be easily generalized to 3D volume reconstructions.
The problem (reconstruction) can be represented by a linear system
p = W
x, where p
are the projection values, x the image/object and W represents the projection process.
The reconstruction step is a continuous tomographic reconstruction method: ART
[13], SIRT [14] or any other iterative algorithm can be used. From now on, we refer to
this step as the algebraic reconstruction method (ARM). The segmentation is carried
out separating the material and the background (thresholding the grey levels), and
identifying the set boundary B
(t) and ‘fixed’ F
(t) pixels (where (t) accounts for the
iteration step).
The set B
(t) will be updated by the ARM algorithm in each iteration, while F
(t)
will remain fixed. This way, the number of variables in the linear system are reduced
and the computational power required is greatly decreased. Since the segmentation
is applied upon the ARM-reconstructed images (possibly suffering from missing
wedge), the separation of grey levels can be defective at first. To allow the creation
of boundary regions inside the material, a random set of pixels from F
(t) is introduced
in the ARM as well with a p probability (adjustable from 0 to 1, depending on each
problem). Also, a smoothing filter is applied after each iteration, to counter the strong
local oscillations in the grey levels of B
(t) after the ARM step.
It has been proven that DART converges faster than any ARM available, when
the number of grey levels does not exceed 5 [7]. The accuracy of the reconstruction (measured comparing phantom images and the results of the reconstruction for
different algorithms) is usually higher, independently of the number of projections
or the angular range. Tests on the effect of the missing wedge reveal a much-reduced
impact in final solutions achieved by DART. Since electron dose is reduced when a
lower number of projections are required, DART appears as a viable option in cases
of severe sample damage due to beam sensitivity.
Despite the advantages described for DART, it remains a heuristic algorithm,
meaning that convergence cannot always be assured. This circumstance makes the
selection of a unique termination criteria difficult. It also relies on previous knowledge of the sample composition. Although it is robust with respect to the chosen
segmentation criteria (threshold), if the number of components (different grey levels)
is not correctly identified, the reconstruction may present serious image artefacts
(Fig. 11.2).