Ville Kolehmainen, Martin Schweiger, Ilkka Nissilä, Tanja Tarvainen, Simon R. Arridge, and Jari P. Kaipio, "Approximation errors and model reduction in three-dimensional diffuse optical tomography," J. Opt. Soc. Am. A 26, 2257-2268 (2009)
Model reduction is often required in diffuse optical tomography (DOT), typically because of limited available computation time or computer memory. In practice, this means that one is bound to use coarse mesh and truncated computation domain in the model for the forward problem. We apply the (Bayesian) approximation error model for the compensation of modeling errors caused by domain truncation and a coarse computation mesh in DOT. The approach is tested with a three-dimensional example using experimental data. The results show that when the approximation error model is employed, it is possible to use mesh densities and computation domains that would be unacceptable with a conventional measurement error model.
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is the number of nodes, is the number of tetrahedra elements in the mesh, and is the number of voxels in the representation of and in Eqs. (23, 24). t is the wall clock time for one complete forward solution (all sources).
is the number of nodes, is the number of tetrahedra elements in the mesh, and is the number of voxels in the representation of and in Eqs. (23, 24). t is the wall clock time for one complete forward solution (all sources). The model domain is a truncated approximation of the true domain Ω. The height of the domain is 14.5 mm. The truncated domain is denoted by the top and bottom of the three circles at the center of the cylinder in Fig. 1.
Table 3
Results of Initial Estimation (29) for Test Case 1 (Figs. 4, 5)
Error Model
Forward Model
η
ϕ
CEM
0.0079
1.086
−0.487
0.011
CEM
0.0066
0.8318
−2.246
−0.025
EEM
0.0080
1.0637
−0.454
0.033
Table 4
Reconstruction Times for Test Case 1 (Figs. 4, 5)a
Error Model
Forward Model
(s)
(s)
(s)
CEM
126 min 20 s
173 min 22 s
299 min 44 s
CEM
1 min 11 s
7 min 18 s
8 min 29 s
EEM
28 s
7 min 34 s
8 min 2 s
is the (wall clock) time for initial estimation (29), for the MAP estimation, and the total reconstruction time (initial + MAP).
Table 5
Results of Initial Estimation (29) for Test Case 2 (Figs. 6, 7)
Error Model
Forward Model
η
ϕ
CEM
0.0004
1.148
−1.0095
0.035
CEM
1.068
−0.845
0.039
EEM
0.0082
1.125
−0.471
0.004
EEM
0.0094
1.0795
−0.349
0.006
Table 6
Reconstruction Times for Test Case 2 (Figs. 6, 7)a
Error Model
Forward Model
(s)
(s)
(s)
CEM
18 min 58 s
20 min 14 s
39 min 12 s
CEM
14 s
1 min 31 s
1 min 45 s
EEM
14 min 56 s
17 min 35 s
32 min 31 s
EEM
17 s
1 min 34 s
1 min 51 s
is the (wall clock) time for initial estimation, (29), for the MAP estimation, and the total reconstruction time (initial + MAP).
Tables (7)
Table 1
Algorithm 1: Construction of the enhanced error model
Draw the set of random samples from the
prior .
fordo
Compute the solution of the accurate model .
Compute the solution of the coarse target model
where the model reduction operator P is given by Eq. (9).
Store the realization of the
modeling error.
end for
Using the set of realizations of the modeling error
compute the mean and covariance of the modeling error as
is the number of nodes, is the number of tetrahedra elements in the mesh, and is the number of voxels in the representation of and in Eqs. (23, 24). t is the wall clock time for one complete forward solution (all sources).
is the number of nodes, is the number of tetrahedra elements in the mesh, and is the number of voxels in the representation of and in Eqs. (23, 24). t is the wall clock time for one complete forward solution (all sources). The model domain is a truncated approximation of the true domain Ω. The height of the domain is 14.5 mm. The truncated domain is denoted by the top and bottom of the three circles at the center of the cylinder in Fig. 1.
Table 3
Results of Initial Estimation (29) for Test Case 1 (Figs. 4, 5)
Error Model
Forward Model
η
ϕ
CEM
0.0079
1.086
−0.487
0.011
CEM
0.0066
0.8318
−2.246
−0.025
EEM
0.0080
1.0637
−0.454
0.033
Table 4
Reconstruction Times for Test Case 1 (Figs. 4, 5)a
Error Model
Forward Model
(s)
(s)
(s)
CEM
126 min 20 s
173 min 22 s
299 min 44 s
CEM
1 min 11 s
7 min 18 s
8 min 29 s
EEM
28 s
7 min 34 s
8 min 2 s
is the (wall clock) time for initial estimation (29), for the MAP estimation, and the total reconstruction time (initial + MAP).
Table 5
Results of Initial Estimation (29) for Test Case 2 (Figs. 6, 7)
Error Model
Forward Model
η
ϕ
CEM
0.0004
1.148
−1.0095
0.035
CEM
1.068
−0.845
0.039
EEM
0.0082
1.125
−0.471
0.004
EEM
0.0094
1.0795
−0.349
0.006
Table 6
Reconstruction Times for Test Case 2 (Figs. 6, 7)a
Error Model
Forward Model
(s)
(s)
(s)
CEM
18 min 58 s
20 min 14 s
39 min 12 s
CEM
14 s
1 min 31 s
1 min 45 s
EEM
14 min 56 s
17 min 35 s
32 min 31 s
EEM
17 s
1 min 34 s
1 min 51 s
is the (wall clock) time for initial estimation, (29), for the MAP estimation, and the total reconstruction time (initial + MAP).