## Compensation of optode sensitivity and position errors in diffuse optical tomography using the approximation error approach |

Biomedical Optics Express, Vol. 4, Issue 10, pp. 2015-2031 (2013)

http://dx.doi.org/10.1364/BOE.4.002015

Acrobat PDF (1323 KB)

### Abstract

Diffuse optical tomography is highly sensitive to measurement and modeling errors. Errors in the source and detector coupling and positions can cause significant artifacts in the reconstructed images. Recently the approximation error theory has been proposed to handle modeling errors. In this article, we investigate the feasibility of the approximation error approach to compensate for modeling errors due to inaccurately known optode locations and coupling coefficients. The approach is evaluated with simulations. The results show that the approximation error method can be used to recover from artifacts in reconstructed images due to optode coupling and position errors.

© 2013 OSA

## 1. Introduction

1. S. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inv. Probl. **25**, 123010 (2009). Topical Review. [CrossRef]

3. A. Gibson, J. Hebden, and S. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. **50**, R1–R43 (2005). Topical Review. [CrossRef] [PubMed]

*ill-posed*. The ill-posedness means that even small errors in measurements or modeling can cause large errors in the reconstructions. In contact based DOT measurement setup, lasers are coupled to fiber optodes and these are used to direct light into the tissue. The transmitted light is collected using another set of optodes coupled to photo-multipliers. In practical experiments, the positions of the source and detector optodes are not always known accurately. There are also other uncertain model parameters such as coupling coefficients that include source strength, coupling losses of the optodes and detector efficiency or gain. It has previously been observed that small errors in source and detector positions and coupling coefficients can cause large artifacts in the reconstructed images [4

4. J. J. Stott, J. P. Culver, S. R. Arridge, and D. A. Boas, “Optode positional calibration in diffuse optical tomography.” Appl. Opt. **42**, 3154–62 (2003). [CrossRef] [PubMed]

5. T. Tarvainen, V. Kolehmainen, M. Vauhkonen, A. Vanne, A. P. Gibson, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “Computational calibration method for optical tomography.” Appl. Opt. **44**, 1879–88 (2005). [CrossRef] [PubMed]

6. E. M. C. Hillman, J. C. Hebden, F. E. W. Schmidt, S. R. Arridge, M. Schweiger, H. Dehghani, and D. T. Delpy, “Calibration techniques and datatype extraction for time-resolved optical tomography,” Rev. Sci. Instrum. **71**, 3415 (2000). [CrossRef]

7. V. Ntziachristos, B. Chance, and A. G. Yodh, “Differential diffuse optical tomography,” Opt. Express **5**, 565–570 (1999). [CrossRef]

8. H. Xu, B. W. Pogue, R. Springett, and H. Dehghani, “Spectral derivative based image reconstruction provides inherent insensitivity to coupling and geometric errors,” Opt. Lett. **30**, 2912–2914 (2005). [CrossRef] [PubMed]

6. E. M. C. Hillman, J. C. Hebden, F. E. W. Schmidt, S. R. Arridge, M. Schweiger, H. Dehghani, and D. T. Delpy, “Calibration techniques and datatype extraction for time-resolved optical tomography,” Rev. Sci. Instrum. **71**, 3415 (2000). [CrossRef]

5. T. Tarvainen, V. Kolehmainen, M. Vauhkonen, A. Vanne, A. P. Gibson, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “Computational calibration method for optical tomography.” Appl. Opt. **44**, 1879–88 (2005). [CrossRef] [PubMed]

9. I. Nissila, T. Noponen, K. Kotilahti, T. Katila, L. Lipiainen, T. Tarvainen, M. Schweiger, and S. Arridge, “Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography,” Rev. Sci. Instrum. **76**, 044302 (2005). [CrossRef]

10. C. H. Schmitz, H. L. Graber, H. Luo, I. Arif, J. Hira, Y. Pei, A. Bluestone, S. Zhong, R. Andronica, I. Soller, N. Ramirez, S. L. Barbour, and R. L. Barbour, “Instrumentation and calibration protocol for imaging dynamic features in dense-scattering media by optical tomography.” Appl. Opt. **39**, 6466–86 (2000). [CrossRef]

11. B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and K. D. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. **38**, 2950–2961 (1999). [CrossRef]

12. D. Boas, T. Gaudette, and S. Arridge, “Simultaneous imaging and optode calibration with diffuse optical tomography.” Opt. Express **8**, 263–70 (2001). [CrossRef] [PubMed]

13. M. Schweiger, I. Nissilä, D. A. Boas, and S. R. Arridge, “Image reconstruction in optical tomography in the presence of coupling errors,” Appl. Opt. **46**, 2743–2756 (2007). [CrossRef] [PubMed]

14. R. Fukuzawa, S. Okawa, S. Matsuhashi, T. Kusaka, Y. Tanikawa, Y. Hoshi, F. Gao, and Y. Yamada, “Reduction of image artifacts induced by change in the optode coupling in time-resolved diffuse optical tomography.” J. Biomed. Opt. **16**, 116022 (2011). [CrossRef] [PubMed]

15. S. Oh, A. B. Milstein, R. P. Millane, C. A. Bouman, and K. J. Webb, “Source-detector calibration in three-dimensional Bayesian optical diffusion tomography.” J. Opt. Soc. Am. A **19**, 1983–1993 (2002). [CrossRef]

18. Kaipio and E. Somersalo, “Discretization model reduction and inverse crimes,” J. Comput. Appl. Math. **198**, 493–504 (2007). [CrossRef]

19. V. Kolehmainen, T. Tarvainen, S. R. Arridge, and J. P. Kaipio, “Marginalization of uninteresting distributed parameters in inverse problems - application to diffuse optical tomography,” Int. J. Uncertainty Quantification **1**, 1–17 (2011). [CrossRef]

20. S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, “Approximation errors and model reduction with an application in optical diffusion tomography,” Inv. Probl. **22**, 175–195 (2006). [CrossRef]

21. J. Heino, E. Somersalo, and J. Kaipio, “Compensation for geometric mismodelling by anisotropies in optical tomography,” Opt. Express **13**, 296–308 (2005). [CrossRef] [PubMed]

23. V. Kolehmainen, M. Schweiger, I. Nissilä, T. Tarvainen, S. R. Arridge, and J. P. Kaipio, “Approximation errors and model reduction in three-dimensional diffuse optical tomography,” J. Opt. Soc. Am. A **26**, 2257–2268 (2009). [CrossRef]

24. T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “An approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography,” Inv. Probl. **26**, 015005 (2010). [CrossRef]

25. T. Tarvainen, V. Kolehmainen, J. P. Kaipio, and S. R. Arridge, “Corrections to linear methods for diffuse optical tomography using approximation error modelling.” Biomed. Opt. Express **1**, 209–222 (2010). [CrossRef]

19. V. Kolehmainen, T. Tarvainen, S. R. Arridge, and J. P. Kaipio, “Marginalization of uninteresting distributed parameters in inverse problems - application to diffuse optical tomography,” Int. J. Uncertainty Quantification **1**, 1–17 (2011). [CrossRef]

## 2. Diffuse optical tomography

### 2.1. Diffusion approximation model

*, where*

^{n}*n*is the dimension of the medium (

*n*= 2, 3), model this object domain. In a diffusive medium like soft tissue, the commonly used light transport model for DOT is the diffusion approximation (DA) for the radiative transport equation (RTE) [26]. In this paper the frequency domain version of the diffusion approximation model is used as the model for light propagation in tissues [27

27. S. Arridge, M. Schweiger, M. Hiraoka, and D. Delpy, “A finite element approach to modelling photon transport in tissue,” Med. Phys. **20**, 299–309 (1993). [CrossRef] [PubMed]

*(*

_{i}*r*) := Φ

*is the photon density for the*

_{i}*i*:th source,

*μ*(

_{a}*r*) :=

*μ*is the absorption coefficient, i is the imaginary unit,

_{a}*ω*is the angular modulation frequency of the input signal,

*c*is the speed of light in the medium and

*q*

_{0,i}(

*r*) :=

*q*

_{0}is the source within object domain Ω operating at frequency

*ω. κ*(

*r*) :=

*κ*is the diffusion coefficient. The diffusion coefficient

*κ*is given by

*κ*(

*r*) = 1/(

*n*(

*μ*(

_{a}*r*) +

*μ*(

_{s}*r*))), where

*μ*(

_{s}*r*) :=

*μ*is the reduced scattering coefficient.

_{s}*m*⊂

_{i}*∂*Ω,

*i*= 1...

*N*

_{s}and the location of detector optodes by

*n*⊂

_{j}*∂*Ω,

*j*= 1...

*N*

_{d}. When the source is modeled as diffuse boundary source,

*q*

_{0,i}(

*r*) = 0 and the source term can be written in the boundary condition where

*n̂*is the outward normal to the boundary at point

*r*,

*q*is the strength of the boundary source at location

_{i}*m*,

_{i}*γ*is dimension dependent constant (

*γ*= 1/

*π*when Ω ⊂ ℝ

^{2},

*γ*= 1/2 when Ω ⊂ ℝ

^{3}) and

*α*is a parameter governing the internal reflection at the boundary

*∂*Ω. The measurable quantity, the exitance, Γ

*at detector*

_{i,j}*j*under illumination from source

*i*is defined by

^{NsNd}denote vector of complex valued measurement data corresponding to measurement between all source-detector pairs

*i*,

*j*with single indexation A typical frequency domain DOT measurement setup collects the amplitude and phase as measurement data where y ∈ ℝ

^{2NsNd}is data vector that contains the measured log amplitude and phase for all source-detector pairs.

*m*and

_{i}*n*are surface patches of known length in 2D and area in 3D. We parameterize the locations by the center point of the source and detector optodes and use notation for the vector of source and detector location parameters. Using this notation, the observation model is where

_{j}*e*∈ ℝ

^{2NsNd}models the random noise in measurements, x = (

*μ*,

_{a}*μ*)

_{s}^{T}∈ ℝ

^{2Nn}is discretized optical coefficients and the mapping

*A*is typically based on the finite element method (FEM) solution of Eq. (1–3).

### 2.2. Source and detector coupling coefficients

13. M. Schweiger, I. Nissilä, D. A. Boas, and S. R. Arridge, “Image reconstruction in optical tomography in the presence of coupling errors,” Appl. Opt. **46**, 2743–2756 (2007). [CrossRef] [PubMed]

*q*in (2) by a complex valued multiplicative coupling coefficient

_{i}*ŝ*∈ ℂ, leading to photon density where is the source coupling coefficient with amplitude factor

_{i}*s*and phase factor

_{i}*δ*. Similarly the coupling losses in measurement optodes are modeled with multiplicative coupling coefficients leading to exitance [13

_{i}13. M. Schweiger, I. Nissilä, D. A. Boas, and S. R. Arridge, “Image reconstruction in optical tomography in the presence of coupling errors,” Appl. Opt. **46**, 2743–2756 (2007). [CrossRef] [PubMed]

*g*(

*ζ*) ∈ ℂ

^{NdNs}such that Using these notations and taking the log transform of vector of data of the form (7) leads to the separation of coupling coefficients into additive components leading to observation model where

*ε*

_{1}(

*ζ*) ∈ ℝ

^{2NsNd}is the discrepancy in the log transformed measurement compared to the ideal (no coupling losses) model (5) for given realization

*ζ*. Notice that when ideal sources and detectors are assumed (no losses), we have

*s*= 1,

_{i}*δ*= 0 for all

_{i}*i*,

*d*= 1,

_{j}*η*= 0 for all

_{j}*j*and

*ε*

_{1}≡ 0, i.e., model (10) becomes equal to (5).

### 2.3. Statistical inversion in DOT

#### 2.3.1. Posterior model

*π*(

*x*,

*ζ*,

*ξ*|

*y*) is a probability density on a high-dimensional space. In computation of point estimate(s) from the posterior model, the most common choice is the

*maximum a posteriori*(MAP) estimate. In principle, one could attempt to compute MAP estimate for all the unknown model parameters However, this would lead to computationally extensive and complicated problem, and to our knowledge the simultaneous estimation of (

*x*,

*ξ*,

*ζ*) has not been performed (for estimation of

*x*and coupling coefficients

*ζ*, see [12

12. D. Boas, T. Gaudette, and S. Arridge, “Simultaneous imaging and optode calibration with diffuse optical tomography.” Opt. Express **8**, 263–70 (2001). [CrossRef] [PubMed]

**46**, 2743–2756 (2007). [CrossRef] [PubMed]

15. S. Oh, A. B. Milstein, R. P. Millane, C. A. Bouman, and K. J. Webb, “Source-detector calibration in three-dimensional Bayesian optical diffusion tomography.” J. Opt. Soc. Am. A **19**, 1983–1993 (2002). [CrossRef]

*ξ*,

*ζ*) in a similar manner than treating the uncertainty in

*e*, that is, by marginalizing the posterior model as and then compute estimate for the primary unknowns from the posterior

*π*(

*x*|

*y*). However, the solution of the integration (14) has closed form solution only in case of purely linear and Gaussian models. In case of non-linear problems such as DOT, the solution of (14) would require Markov chain Monte Carlo integration which would in most cases be infeasible for practical applications.

*π̃*(

*x*|

*y*) for the posterior model (14) such that the marginalization over the uncertainty in the values of (

*ξ*,

*ζ*) is carried out

*approximately*but in a computationally feasible way.

*ξ*,

*ζ*), we first review the standard DOT reconstruction approach where

*ξ*=

*ξ*

_{0}and

*ζ*=

*ζ*

_{0}are treated as known and fixed conditioning variables.

#### 2.3.2. Conventional measurement error model (*ζ* and *ξ* treated as known fixed parameters)

*ζ*and

*ξ*are treated as known deterministic parameters and independent of the optical coefficients. Within the Bayesian framework, any

*known*parameter (whether measured or otherwise known) is interpreted as a conditioning variable. In the case at hand, if we take

*ζ*and

*ξ*to be known, we would have

*x*and

*e*as the only unknowns, and would obtain the conditional density

*ξ*=

*ξ*

_{0}and

*ζ*=

*ζ*

_{0}the observation model becomes

*y*=

*A*(

*x*,

*ξ*

_{0}) +

*ε*

_{1}(

*ζ*

_{0}) +

*e*. Using Gaussian prior models for the unknown optical parameters

*x*and the random measurement noise

*e*where

*x̄*∈ ℝ

^{2Nn}and

*ē*∈ ℝ

^{2NsNd}are the means, and Γ

*∈ ℝ*

_{x}^{2Nn×2Nn}and Γ

*∈ ℝ*

_{e}^{2NsNd×2NsNd}are the covariance matrices, and marginalizing over the unknown measurement errors

*e*as the posterior model becomes [19

19. V. Kolehmainen, T. Tarvainen, S. R. Arridge, and J. P. Kaipio, “Marginalization of uninteresting distributed parameters in inverse problems - application to diffuse optical tomography,” Int. J. Uncertainty Quantification **1**, 1–17 (2011). [CrossRef]

### 2.4. Approximation error model

*ξ*

_{0}and

*ζ*

_{0}be the fixed realizations of the optode locations and coupling parameters that are to be used in estimation of the optical properties

*x*. Evidently, if these fixed values are erroneous, these errors will lead to artefacts in the estimate of

*x*.

*ε*

_{1}and

*ε*

_{2}are

*approximation errors*that describe the discrepancy between the accurate model and the target model in which the optode parameters have the fixed values

*ζ*

_{0}and

*ξ*

_{0}. The measurement model (20) is called the approximation error model.

*x*can be efficiently computed by using the existing optimization codes for conventional measurement error model or Tikhonov regularized least squares.

### 2.5. Estimation of approximation error statistics

*ε̄*

_{1}and

*ε̄*

_{2}and the covariances Γ

_{ε1}and Γ

_{ε2}in equation (22). Closed form solutions for these are only available in purely linear and Gaussian case. In case of non-linear models, the means and covariances of

*ε*

_{1}and

*ε*

_{2}have to be estimated numerically by a simple Monte Carlo integration procedure. The following gives the outline of this estimation procedure.

#### 2.5.1. Estimation of *ε̄*_{1} and Γ_{ε1}

*ε*

_{1}(

*ζ*), we specify prior models

*π*(

*s*) and

*π*(

*δ*) for the vectors of amplitude and phase coupling coefficients of the sources and prior models

*π*(

*d*) and

*π*(

*η*) for the amplitude and phase coupling coefficients of the detectors, respectively. The prior models are used for drawing sets of

*N*random samples of each of the coefficient vectors {

*s*

^{(ℓ)},

*ℓ*= 1...,

*N*}, {

*δ*

^{(ℓ)},

*ℓ*= 1...,

*N*} and and {

*d*

^{(ℓ)},

*ℓ*= 1...,

*N*}, {

*η*

^{(ℓ)},

*ℓ*= 1...,

*N*}. These sets are used to construct a set of

*N*samples of

*ζ*as Given the samples, we compute

*N*samples of the detector and source coupling error

*ε*

_{1}as

#### 2.5.2. Estimation of *ε̄*_{2} and Γ_{ε2}

*ε*

_{2}(

*x*,

*ξ*), we draw

*M*random samples from prior models

*π*(

*x*) and

*π*(

*ξ*) =

*π*(

*m*)

*π*(

*n*). The samples are used to generate samples of

*ε*

_{2}as The mean and covariances are estimated as Notice that, whereas the estimation of mean and covariance of

*ε*

_{1}is computationally fast since it requires only evaluations of the mapping

*g*(

*ζ*) (equations (8–9)), the estimation of the mean and covariance of

*ε*

_{2}is computationally somewhat intensive as 2

*M*forward solutions need to be evaluated. However, these computations need to be done only once for a fixed measurement setup and this estimation can be done off-line.

## 3. Results

### 3.1. Simulation of the measurement data

^{2}was a disk with radius

*r*= 25mm. The measurement setup consisted of

*N*= 16 sources and

_{s}*N*= 16 detectors. The source and detector optodes were modeled as 1mm wide surface patches located at equi-spaced angular intervals on the boundary

_{d}*∂*Ω. With this setup, the vector of DOT measurements (4) was

*y*∈ ℝ

^{512}. A target with background optical properties

*μ*

_{a}= 0.01mm

^{−1},

*μ*

_{s}= 1mm

^{−1}containing an absorption inclusion with

*μ*

_{a}= 0.02mm

^{−1}and scatter inclusion with

*μ*

_{s}= 2mm

^{−1}was constructed. The simulated measurement data was generated using FE approximation of the DA in a mesh with 33806 nodes and 67098 triangular elements. Random measurement noise

*e*, that was drawn from a zero-mean Gaussian distribution where the standard deviations

*σ*were specified as 1% of the absolute value of simulated noise free measurement data, was added to the simulated measurement data.

_{e,k}28. M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss-Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. **50**, 2365–2386 (2005). [CrossRef] [PubMed]

*ē*= 0 and covariance Γ

*were assumed known.*

_{e}### 3.2. Prior models and computation of approximation error statistics

*x*= (

*μ*

_{a},

*μ*

_{s})

^{T}. The same prior model was used both in the construction of the enhanced error model for

*ε*

_{2}and all the MAP estimates based on the conventional measurement error model (19) and the approximation error model (24).

*μ*

_{a}and

*μ*

_{s}were modeled as mutually independent Gaussian random fields with a joint prior model where In the construction of the mean vectors

*μ̄*

_{a},

*μ̄*

_{s}and covariances Γ

_{μa}and Γ

_{μs}, the random field, say

*f*(i.e., either

*μ*

_{a}or

*μ*

_{s}), is considered in the form where

*f*

_{in}is a spatially inhomogeneous parameter with zero mean, and

*f*

_{bg}is a spatially constant (background) parameter with non-zero mean. For the latter, we can write

*f*

_{bg}=

*q*

^{Nn}is a vector of ones and

*q*is a scalar random variable with distribution

_{in,f}, the approximate correlation length can be adjusted to match the size of the expected inhomogeneities and the marginal variances of

*f*:s are tuned based on the expected contrast of the inclusions. We model the distributions

_{k}*f*

_{in}and

*f*

_{bg}as mutually independent, that is, the background is mutually independent with the inhomogeneities. Thus, we have See [17, 19

**1**, 1–17 (2011). [CrossRef]

20. S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, “Approximation errors and model reduction with an application in optical diffusion tomography,” Inv. Probl. **22**, 175–195 (2006). [CrossRef]

29. C. Lieberman, K. Willcox, and O. Ghattas, “Parameter and state model reduction for large-scale statistical inverse problems,” SIAM J. Sci. Comput. **32**, 2523–2542 (2010). [CrossRef]

*π*(

*x*) were selected as follows. The mean for background absorption and scatter were set as

*μ*

_{a,*}= 0.01mm

^{−1}and

*μ*

_{s,*}= 1mm

^{−1}and the standard deviations

*σ*

_{bg,μa}and

*σ*

_{bg,μs}of the background values were chosen such that 2 s.t.d. limits equaled 25% of the mean values

*μ*

_{a,*}and

*μ*

_{s,*}. In the construction of Γ

_{in,}

*the correlation length for both*

_{f}*μ*

_{a}and

*μ*

_{s}was set as 16mm. The marginal standard deviations were set to equal values in each pixel and

*σ*

_{in,μa}and

*σ*

_{in,μs}were chosen such that 2 s.t.d. limits equaled 50% of the mean values

*μ*

_{a,*}and

*μ*

_{s,*}. Thus, the overall marginal standard deviations (i.e., square root of diagonal elements of Γ

_{μa}and Γ

_{μs}) were such that 2

*σ*

_{μa}= 0.0056mm

^{−1}and 2

*σ*

_{μs}= 0.56mm

^{−1}. This gives overall 2 s.t.d. intervals

*μ*

_{a}∈ [0.0044, 0.0156]mm

^{−1}and

*μ*

_{s}∈ [0.44, 1.56]mm

^{−1}, i.e., the values of absorption and scatter are expected to lie within theses intervals with

*prior*probability of 95%.

*ζ*= (

*s*,

*δ*,

*d*,

*η*)

^{T}, all the optode parameters were considered as mutually independent where We modeled the amplitude parameters by uniform prior distributions between a selected minimum value and the ideal value 1, and for the phase parameters we used uniform prior models between ideal value of zero and a selected maximum phase shift In the construction of the prior model

*π*(

*ξ*) for the optode location vector

*ξ*= (

*m*,

*n*)

^{T}, we modeled the locations of the optodes mutually independent and used parameterization where

*ξ*

_{0,k}is the angular location of the center point of optode

*k*in the fixed parameterization

*ξ*

_{0}that is to be used in the inverse problems and

*δθ*∼ U(−

_{k}*δθ*

_{max},

*δθ*

_{max}) models an error in the angular location.

*ε*

_{1}(

*ζ*) were estimated using four different values of (

*s*

_{min},

*d*

_{min},

*δ*

_{max},

*η*

_{max}) in the prior models. Similarly, the mean and covariance of

*ε*

_{2}(

*x*,

*ξ*) was estimated using four different settings for

*δθ*

_{max}. The different values that were used are tabulated in Table 3.2. In the estimation of the statistics of

*ε*

_{1}, equations (25–26),

*N*= 10000 random samples of the optode coupling parameters were used. In the construction of the statistics of

*ε*

_{2}, equations (29–30),

*M*= 2000 random samples of

*x*and the optode location vectors

*ξ*were used. The correlation structure of the covariance matrices Γ

_{ε1}and Γ

_{ε2}corresponding to prior parameters in the first row of Table 3.2 are displayed in Fig. 1.

### 3.3. Case 1: Separated and combined approximation errors

*μ*

_{a}top,

*μ*

_{s}bottom). The second column in Fig. 2 shows the MAP estimate of

*μ*

_{a}and

*μ*

_{s}with conventional measurement error model (CEM), equation (19), when there are no optode coupling or location errors present. This estimate gives the reference estimate with the conventional model in the ideal case that there are no optode errors present.

*ζ*that was used for simulating the measurement data with coupling losses was drawn from the prior model

*π*(

*ζ*) using the parameters in the first row of Table 3.2. The images on the first and second row of column three in Fig. 2 show the conventional MAP estimate (19) when the optode coupling

*ζ*

_{0}is modeled (incorrectly) as ideal (measurement model

*y*=

*A*(

*x*,

*ξ*) +

*e*). The images on the first and second row of column 4 in Fig. 2 show the MAP estimate (24) with the approximation error model

*y*=

*A*(

*x*,

*ξ*) +

*ε*

_{1}+

*e*where the approximation error

*ε*

_{1}due to unknown optode coupling coefficients has been accounted for.

*ξ*of optode locations was drawn from the prior

*π*(

*ξ*) with the parameters given in the first row of Table 3.2, and measurement data was simulated as

*y*=

*A*(

*x*,

*ξ*)+

*e*. The third column shows the conventional MAP estimate (19) using the incorrect fixed realization

*ξ*

_{0}that corresponds to the equispaced locations (i.e., measurement model

*y*=

*A*(

*x*,

*ξ*

_{0}) +

*e*). The fourth column shows the MAP estimate (24) using the approximation error model

*y*=

*A*(

*x*,

*ξ*

_{0}) +

*ε*

_{2}(

*ξ*) +

*e*where the approximation error

*ε*

_{2}due to poorly known locations is taken into account.

*ξ*and

*ζ*from the their prior models using the parameters in the first row of Table 3.2 and then the data was computed as

*y*=

*A*(

*x*,

*ξ*) +

*ε*

_{1}(

*ζ*) +

*e*. The third column shows the conventional MAP estimate (19) using the inexact fixed realizations (

*ξ*

_{0},

*ζ*

_{0}), where

*ξ*

_{0}correspond to equispaced locations and

*ζ*

_{0}to ideal coupling (i.e., measurement model

*y*=

*A*(

*x*,

*ξ*

_{0}) +

*e*). The fourth column shows the MAP estimate (24) using the approximation error model

*y*=

*A*(

*x*,

*ξ*

_{0}) +

*ε*

_{1}+

*ε*

_{2}+

*e*where both approximation errors are taken into account.

*y*=

*A*(

*x*,

*ξ*) +

*ε*

_{1}+

*e*, the second column shows the reconstruction using the optode location error model

*y*=

*A*(

*x*,

*ξ*

_{0})+

*ε*

_{2}(

*ξ*)+

*e*and the third column show the estimate with the combined coupling and location error model

*y*=

*A*(

*x*,

*ξ*

_{0}) +

*ε*

_{1}+

*ε*

_{2}+

*e*. As can be seen, the approximation error model estimates are similar to the reference estimate in the second column of Fig. 2, which corresponds to correct noise model in this case. The difference in the estimates against the reference estimate is slightly larger in the cases of optode coupling error model and combined model than in the pure optode location error model. This discrepancy arises from the selection of the prior models for the optode coupling parameters; with the prior models used in the present case, the mean of optode coupling error

*ε*

_{1}is non-zero and consequently the noise realization

*n*=

*e*+

*ε*

_{1}with

*ε*

_{1}= 0 has relatively low probability density with respect the actual noise model. The results indicate that the approximation error model performs also robustly in the ideal case when there are no optode coupling or location errors present in the measurement data.

### 3.4. Case 2: Magnitude of errors in (ζ, ξ) and sensitivity with respect the prior model

*π*(

*ζ*) and

*π*(

*ξ*) for estimation of the approximation error statistics and drawing the realizations of

*ζ*and

*ξ*that were used in simulating the measurement data with optode coupling or/and location errors. Basically, this case corresponds to a situation in which we know the actual prior probability distribution of the nuisance parameters. To investigate the impact of incorrect prior models of (

*ζ*,

*ξ*) and how large errors the approximation error approach can tolerate in these parameters, we performed a test case where we simulated measurement data and constructed the approximation error statistics for

*ε*

_{1}and

*ε*

_{2}using the four different uniform prior distributions that are listed in Table 3.2. The results for the case of optode coupling errors are shown in Fig. 4 and for the case of optode location errors in Fig. 5. Both of these figures show a 4 × 4 table of MAP estimates (24) with the approximation error model such that in each image pair the left image shows

*μ*

_{a}and the right image shows

*μ*

_{s}. The support of the uniform prior models that were used for estimation of the approximation error statistics grows wider column wise from left to right and the support of the uniform prior that were used in the simulation of the measurement data increases from top to bottom. In the estimates indicated with small arrows the approximation error was trained with the same prior distribution that was used in the simulation of the measurement data, i.e., the image pairs on the diagonal of the 4 × 4 table correspond to the case that the actual prior distribution of

*ζ*or

*ξ*is known. We can see that using a prior that has a too restricted (too narrow) support compared to the actual distribution of the optode parameters leads to less efficient recovery from the modeling errors. However, training the approximation error statistics with a wider prior model than the actual distribution of the uncertainly known parameter does not seem to lead to deterioration in the recovery from the modeling error. Thus, we can say that it is safe to overestimate the uncertainty (up to a limit).

*s*,

_{i}*d*∼ U(0, 1.5) for the amplitude (zero corresponds to complete coupling failure!), and phase errors drawn from

_{j}*δ*,

_{i}*η*∼ U(0, 3

_{j}*π*), and location errors drawn from

*δθ*∼ U(−20°, +20°).

## 4. Conclusion

## Acknowledgments

## References and links

1. | S. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inv. Probl. |

2. | D. Leff, O. Warren, L. Enfield, A. Gibson, T. Athanasiou, D. Patten, J. Hebden, G. Yang, and A. Darzi, “Diffuse optical imaging of the healthy and diseased breast: A systematic review,” Breast Cancer Res. Tr. |

3. | A. Gibson, J. Hebden, and S. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. |

4. | J. J. Stott, J. P. Culver, S. R. Arridge, and D. A. Boas, “Optode positional calibration in diffuse optical tomography.” Appl. Opt. |

5. | T. Tarvainen, V. Kolehmainen, M. Vauhkonen, A. Vanne, A. P. Gibson, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “Computational calibration method for optical tomography.” Appl. Opt. |

6. | E. M. C. Hillman, J. C. Hebden, F. E. W. Schmidt, S. R. Arridge, M. Schweiger, H. Dehghani, and D. T. Delpy, “Calibration techniques and datatype extraction for time-resolved optical tomography,” Rev. Sci. Instrum. |

7. | V. Ntziachristos, B. Chance, and A. G. Yodh, “Differential diffuse optical tomography,” Opt. Express |

8. | H. Xu, B. W. Pogue, R. Springett, and H. Dehghani, “Spectral derivative based image reconstruction provides inherent insensitivity to coupling and geometric errors,” Opt. Lett. |

9. | I. Nissila, T. Noponen, K. Kotilahti, T. Katila, L. Lipiainen, T. Tarvainen, M. Schweiger, and S. Arridge, “Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography,” Rev. Sci. Instrum. |

10. | C. H. Schmitz, H. L. Graber, H. Luo, I. Arif, J. Hira, Y. Pei, A. Bluestone, S. Zhong, R. Andronica, I. Soller, N. Ramirez, S. L. Barbour, and R. L. Barbour, “Instrumentation and calibration protocol for imaging dynamic features in dense-scattering media by optical tomography.” Appl. Opt. |

11. | B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and K. D. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. |

12. | D. Boas, T. Gaudette, and S. Arridge, “Simultaneous imaging and optode calibration with diffuse optical tomography.” Opt. Express |

13. | M. Schweiger, I. Nissilä, D. A. Boas, and S. R. Arridge, “Image reconstruction in optical tomography in the presence of coupling errors,” Appl. Opt. |

14. | R. Fukuzawa, S. Okawa, S. Matsuhashi, T. Kusaka, Y. Tanikawa, Y. Hoshi, F. Gao, and Y. Yamada, “Reduction of image artifacts induced by change in the optode coupling in time-resolved diffuse optical tomography.” J. Biomed. Opt. |

15. | S. Oh, A. B. Milstein, R. P. Millane, C. A. Bouman, and K. J. Webb, “Source-detector calibration in three-dimensional Bayesian optical diffusion tomography.” J. Opt. Soc. Am. A |

16. | J. P. Culver, A. M. Siegel, J. J. Stott, and D. A. Boas, “Volumetric diffuse optical tomography of brain activity,” Opt. Lett. |

17. | J. Kaipio and E. Somersalo, |

18. | Kaipio and E. Somersalo, “Discretization model reduction and inverse crimes,” J. Comput. Appl. Math. |

19. | V. Kolehmainen, T. Tarvainen, S. R. Arridge, and J. P. Kaipio, “Marginalization of uninteresting distributed parameters in inverse problems - application to diffuse optical tomography,” Int. J. Uncertainty Quantification |

20. | S. R. Arridge, J. P. Kaipio, V. Kolehmainen, M. Schweiger, E. Somersalo, T. Tarvainen, and M. Vauhkonen, “Approximation errors and model reduction with an application in optical diffusion tomography,” Inv. Probl. |

21. | J. Heino, E. Somersalo, and J. Kaipio, “Compensation for geometric mismodelling by anisotropies in optical tomography,” Opt. Express |

22. | J. Heiskala, V. Kolehmainen, T. Tarvainen, J. P. Kaipio, and S. R. Arridge, “Approximation error method can reduce artifacts due to scalp blood flow in optical brain activation imaging,” J. Biomed. Opt. |

23. | V. Kolehmainen, M. Schweiger, I. Nissilä, T. Tarvainen, S. R. Arridge, and J. P. Kaipio, “Approximation errors and model reduction in three-dimensional diffuse optical tomography,” J. Opt. Soc. Am. A |

24. | T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, “An approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography,” Inv. Probl. |

25. | T. Tarvainen, V. Kolehmainen, J. P. Kaipio, and S. R. Arridge, “Corrections to linear methods for diffuse optical tomography using approximation error modelling.” Biomed. Opt. Express |

26. | A. Ishimaru, |

27. | S. Arridge, M. Schweiger, M. Hiraoka, and D. Delpy, “A finite element approach to modelling photon transport in tissue,” Med. Phys. |

28. | M. Schweiger, S. R. Arridge, and I. Nissilä, “Gauss-Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. |

29. | C. Lieberman, K. Willcox, and O. Ghattas, “Parameter and state model reduction for large-scale statistical inverse problems,” SIAM J. Sci. Comput. |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

(170.6960) Medical optics and biotechnology : Tomography

(290.7050) Scattering : Turbid media

**ToC Category:**

Image Reconstruction and Inverse Problems

**History**

Original Manuscript: May 2, 2013

Revised Manuscript: July 25, 2013

Manuscript Accepted: August 29, 2013

Published: September 6, 2013

**Citation**

Meghdoot Mozumder, Tanja Tarvainen, Simon R. Arridge, Jari Kaipio, and Ville Kolehmainen, "Compensation of optode sensitivity and position errors in diffuse optical tomography using the approximation error approach," Biomed. Opt. Express **4**, 2015-2031 (2013)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-4-10-2015

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