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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Algorithm 1: Construction of the enhanced error model

Draw the set of random samples $S=\{{x}_{\delta}^{\left(1\right)},{x}_{\delta}^{\left(2\right)},\dots ,{x}_{\delta}^{\left(K\right)}\}$ from the

prior $\pi \left({x}_{\delta}\right)$.

for$l=1\dots K$do

Compute the solution of the accurate model ${A}_{\Omega ,\delta}\left({x}_{\delta}^{\left(l\right)}\right)$.

Compute the solution of the coarse target model ${A}_{\stackrel{\u0303}{\Omega},h}\left(P{x}_{\delta}^{\left(l\right)}\right)$

where the model reduction operator P is given by Eq. (9).

Store the realization ${\epsilon}^{\left(l\right)}=[{A}_{\Omega ,\delta}\left({x}_{\delta}^{\left(l\right)}\right)-{A}_{\stackrel{\u0303}{\Omega},h}\left(P{x}_{\delta}^{\left(l\right)}\right)]$ of the

modeling error.

end for

Using the set $\{{\epsilon}^{\left(1\right)},\dots ,{\epsilon}^{\left(K\right)}\}$ of realizations of the modeling error

compute the mean and covariance of the modeling error as

${N}_{n}$ is the number of nodes, ${N}_{e}$ is the number of tetrahedra elements in the mesh, and ${n}_{p}$ is the number of voxels in the representation of ${\mu}_{\mathrm{a}}$ and ${\mu}_{\mathrm{s}}$ in Eqs. (23, 24). t is the wall clock time for one complete forward solution (all sources).

Table 2

Mesh Details for Test Case 2^{
a
}

Model

${N}_{n}$

${N}_{e}$

${n}_{p}$

t (s)

${A}_{\Omega ,\delta}$

148 276

843 750

7 668

178

${A}_{\stackrel{\u0303}{\Omega},\delta}$

23 413

124 416

1 408

15.3

${A}_{\stackrel{\u0303}{\Omega},h}$

1 085

4 608

1 408

0.2

${N}_{n}$ is the number of nodes, ${N}_{e}$ is the number of tetrahedra elements in the mesh, and ${n}_{p}$ is the number of voxels in the representation of ${\mu}_{\mathrm{a}}$ and ${\mu}_{\mathrm{s}}$ in Eqs. (23, 24). t is the wall clock time for one complete forward solution (all sources). The model domain $\stackrel{\u0303}{\Omega}$ is a truncated approximation of the true domain Ω. The height of the domain $\stackrel{\u0303}{\Omega}$ 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)

Reconstruction Times for Test Case 1 (Figs. 4, 5)^{
a
}

Error Model

Forward Model

${t}_{\mathrm{init}}$ (s)

${t}_{\mathrm{MAP}}$ (s)

${t}_{\mathrm{tot}}$ (s)

CEM

${A}_{\Omega ,\delta}$

126 min 20 s

173 min 22 s

299 min 44 s

CEM

${A}_{\Omega ,h}$

1 min 11 s

7 min 18 s

8 min 29 s

EEM

${A}_{\Omega ,h}$

28 s

7 min 34 s

8 min 2 s

${t}_{\mathrm{init}}$ is the (wall clock) time for initial estimation (29), ${t}_{\mathrm{MAP}}$ for the MAP estimation, and ${t}_{\mathrm{tot}}$ the total reconstruction time (initial + MAP).

Table 5

Results of Initial Estimation (29) for Test Case 2 (Figs. 6, 7)

Reconstruction Times for Test Case 2 (Figs. 6, 7)^{
a
}

Error Model

Forward Model

${t}_{\mathrm{init}}$ (s)

${t}_{\mathrm{MAP}}$ (s)

${t}_{\mathrm{tot}}$ (s)

CEM

${A}_{\stackrel{\u0303}{\Omega},\delta}$

18 min 58 s

20 min 14 s

39 min 12 s

CEM

${A}_{\stackrel{\u0303}{\Omega},h}$

14 s

1 min 31 s

1 min 45 s

EEM

${A}_{\stackrel{\u0303}{\Omega},\delta}$

14 min 56 s

17 min 35 s

32 min 31 s

EEM

${A}_{\stackrel{\u0303}{\Omega},h}$

17 s

1 min 34 s

1 min 51 s

${t}_{\mathrm{init}}$ is the (wall clock) time for initial estimation, (29), ${t}_{\mathrm{MAP}}$ for the MAP estimation, and ${t}_{\mathrm{tot}}$ the total reconstruction time (initial + MAP).

Tables (7)

Table 1

Algorithm 1: Construction of the enhanced error model

Draw the set of random samples $S=\{{x}_{\delta}^{\left(1\right)},{x}_{\delta}^{\left(2\right)},\dots ,{x}_{\delta}^{\left(K\right)}\}$ from the

prior $\pi \left({x}_{\delta}\right)$.

for$l=1\dots K$do

Compute the solution of the accurate model ${A}_{\Omega ,\delta}\left({x}_{\delta}^{\left(l\right)}\right)$.

Compute the solution of the coarse target model ${A}_{\stackrel{\u0303}{\Omega},h}\left(P{x}_{\delta}^{\left(l\right)}\right)$

where the model reduction operator P is given by Eq. (9).

Store the realization ${\epsilon}^{\left(l\right)}=[{A}_{\Omega ,\delta}\left({x}_{\delta}^{\left(l\right)}\right)-{A}_{\stackrel{\u0303}{\Omega},h}\left(P{x}_{\delta}^{\left(l\right)}\right)]$ of the

modeling error.

end for

Using the set $\{{\epsilon}^{\left(1\right)},\dots ,{\epsilon}^{\left(K\right)}\}$ of realizations of the modeling error

compute the mean and covariance of the modeling error as

${N}_{n}$ is the number of nodes, ${N}_{e}$ is the number of tetrahedra elements in the mesh, and ${n}_{p}$ is the number of voxels in the representation of ${\mu}_{\mathrm{a}}$ and ${\mu}_{\mathrm{s}}$ in Eqs. (23, 24). t is the wall clock time for one complete forward solution (all sources).

Table 2

Mesh Details for Test Case 2^{
a
}

Model

${N}_{n}$

${N}_{e}$

${n}_{p}$

t (s)

${A}_{\Omega ,\delta}$

148 276

843 750

7 668

178

${A}_{\stackrel{\u0303}{\Omega},\delta}$

23 413

124 416

1 408

15.3

${A}_{\stackrel{\u0303}{\Omega},h}$

1 085

4 608

1 408

0.2

${N}_{n}$ is the number of nodes, ${N}_{e}$ is the number of tetrahedra elements in the mesh, and ${n}_{p}$ is the number of voxels in the representation of ${\mu}_{\mathrm{a}}$ and ${\mu}_{\mathrm{s}}$ in Eqs. (23, 24). t is the wall clock time for one complete forward solution (all sources). The model domain $\stackrel{\u0303}{\Omega}$ is a truncated approximation of the true domain Ω. The height of the domain $\stackrel{\u0303}{\Omega}$ 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)

Reconstruction Times for Test Case 1 (Figs. 4, 5)^{
a
}

Error Model

Forward Model

${t}_{\mathrm{init}}$ (s)

${t}_{\mathrm{MAP}}$ (s)

${t}_{\mathrm{tot}}$ (s)

CEM

${A}_{\Omega ,\delta}$

126 min 20 s

173 min 22 s

299 min 44 s

CEM

${A}_{\Omega ,h}$

1 min 11 s

7 min 18 s

8 min 29 s

EEM

${A}_{\Omega ,h}$

28 s

7 min 34 s

8 min 2 s

${t}_{\mathrm{init}}$ is the (wall clock) time for initial estimation (29), ${t}_{\mathrm{MAP}}$ for the MAP estimation, and ${t}_{\mathrm{tot}}$ the total reconstruction time (initial + MAP).

Table 5

Results of Initial Estimation (29) for Test Case 2 (Figs. 6, 7)

Reconstruction Times for Test Case 2 (Figs. 6, 7)^{
a
}

Error Model

Forward Model

${t}_{\mathrm{init}}$ (s)

${t}_{\mathrm{MAP}}$ (s)

${t}_{\mathrm{tot}}$ (s)

CEM

${A}_{\stackrel{\u0303}{\Omega},\delta}$

18 min 58 s

20 min 14 s

39 min 12 s

CEM

${A}_{\stackrel{\u0303}{\Omega},h}$

14 s

1 min 31 s

1 min 45 s

EEM

${A}_{\stackrel{\u0303}{\Omega},\delta}$

14 min 56 s

17 min 35 s

32 min 31 s

EEM

${A}_{\stackrel{\u0303}{\Omega},h}$

17 s

1 min 34 s

1 min 51 s

${t}_{\mathrm{init}}$ is the (wall clock) time for initial estimation, (29), ${t}_{\mathrm{MAP}}$ for the MAP estimation, and ${t}_{\mathrm{tot}}$ the total reconstruction time (initial + MAP).