## Corrections to linear methods for diffuse optical tomography using approximation error modelling |

Biomedical Optics Express, Vol. 1, Issue 1, pp. 209-222 (2010)

http://dx.doi.org/10.1364/BOE.1.000209

Acrobat PDF (947 KB)

### Abstract

Linear reconstruction methods in diffuse optical tomography have been found to produce reasonable good images in cases in which the variation in optical properties within the medium is relatively small and a reference measurement with known background optical properties is available. In this paper we examine the correction of errors when using a first order Born approximation with an infinite space Green’s function model as the basis for linear reconstruction in diffuse optical tomography, when real data is generated on a finite domain with possibly unknown background optical properties. We consider the relationship between conventional reference measurement correction and approximation error modelling in reconstruction. It is shown that, using the approximation error modelling, linear reconstruction method can be used to produce good quality images also in situations in which the background optical properties are not known and a reference is not available.

© 2010 Optical Society of America

## 1. Introduction

24. T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S.R. Arridge, and J.P. Kaipio, “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).

## 2. Deterministic model based reconstruction and correction methods

### 2.1. Definitions

*y*∈ ℝ

^{m}is discrete data,

*x*∈ 𝕏 is a function in some space 𝕏 representing the absorption and scattering images, and

*e*denotes the noise which is usually modelled to be Gaussian distributed

*e*~ 𝒩(0,Γ

_{e}) with zero mean and covariance Γ

_{e}∈ ℝ

^{m×m}. In the ‘true’ physical problem we assume 𝕏 =

*L*

^{∞}(Ω) but in a practical implementation the solution and forward mapping are represented in discrete vector spaces

*A*: ℝ

_{h}^{n}→ ℝ

^{m}is a linear or non-linear operator.

*D*= (3(

*μ*+

_{a}*μ*

_{s}^{′}))

^{-1}is the diffusion coefficient,

*μ*

_{s}^{′}is the reduced scattering coefficient,

*μ*is the absorption coefficient, Φ is the photon density,

_{a}*J*

^{-}is prescribed input source function, i is the imaginary unit,

*c*is the speed of light in the medium and ω is the angular modulation frequency of the input signal. Furthermore, in this paper we use the following notations

background optical parameters: | x_{0} = (μ_{a,0},μ_{s,0}^{′})^{T} |

perturbed optical parameters: | x=(μ,_{a}μ_{s}^{′})^{T}=x_{0} + δ_{x} |

estimated optical parameters: | x̂ |

measured data : | g^{obs} |

reference data : | g^{ref} |

modelled data : | y |

modelled reference data : | y_{0} |

discretized (incorrect) background optical parameters: | x_{h,0} |

discretized (incorrect) optical parameters: | x = _{h}x_{h,0} + δx_{h} |

exact model for the forward operator in domain Ω: | A |

approximate model in (incorrect) domain Ω̃: | A_{h} |

augmented model : | Ã_{h} |

Jacobian: | J |

modelling error: | ε |

### 2.2. Linear reconstruction model within inverse scattering theory

*A*in Eq. (1) as the measurement of the Green’s function

*x*) is the Green’s operator, parameterised by

*x*, that solves for the internal field Φ given the source function and the light propagation model (5) and ℳ is the measurement of Φ at the boundary of the domain Ω. In practice, measurements are needed for multiple input sources functions {

*J*

_{1}

^{-},

*J*

_{2}

^{-},…} where the functions {

*J*

_{i}

^{-}} form a basis for general functions on the boundary

*∂*Ω, leading to a vector form for the forward model

*x*

_{0}, a series solution for a perturbed state

*x*

_{0}+

*δx*is given by

*δx*) is a potential operator determined by the perturbation. Eq. (8) is a general Born series for diffuse photon density waves.

*x*

_{0}) is the discrete representation of the Fréchet derivative of the nonlinear mapping

*A*(

*x*), and seek to invert the linear expression

*g*

^{obs}are measurements and

*y*

_{0}=

*A*(

*x*

_{0}) is assumed to be an accurate prediction of the measurements that would correspond to the reference state

*x*

_{0}.

*y*

_{0}at measurement position

*r*due to source located

_{m}*r*as well as the accuracy of representation of the linearisation 𝖩(

_{s}*x*

_{0}) which both depend on the accuracy of the Green’s functions

*G*is taken to be an analytic expression such as the infinite domain Green’s function for the diffusion equation

### 2.3. Linear reconstruction model within the parameter identification method

*x*̂ denotes the estimated optical parameters and Ψ(

*x*) is a regularizing penalty functional. Taking only the linear approximation Eq. (9) leads to

### 2.4. Model correction using a reference measurement

*A*(

*x*

_{0}) represent an ‘exact’ model for the forward operator in domain Ω and state

*x*

_{0}and let

*A*(

_{h}*x*,0) represent an approximate model in the incorrect domain Ω̃ with incorrect optical parameters

_{h}*x*

_{h,0}. Consider an additive correction ε

_{0}so that the two models agree at the reference state

*x*

_{0}

*g*

^{ref}from background

*x*

_{0}and then to use the augmented model

*δx*= (

_{h}*x*-

_{h}*x*

_{h,0}). Thus, the minimisation is

## 3. Bayesian reconstruction and approximation error modelling

*A*(

*x*) and its computational approximation

*A*(

_{h}*x*). We thus define the ‘augmented model’

_{h}*first order statistics of the error model (EM-1)*is used and the linear minimization problem is of the form

*A*(

_{h}*x*

_{h,0}) is the solution of the linear forward model such as the Green’s function and 𝖩(

*x*

_{h,0}) is the Jacobian calculated with optical properties

*x*

_{h,0}.

*second order statistics of the error model (EM-2)*is considered. Thus, both the mean and the covariance of the errors between the ‘exact’ model and approximative model are taken into account. This leads to the minimisation

*ε̅*,Γ

_{ε}} are the mean and covariance of the distribution of

*ε*(

*x*) and Γ

_{h}_{e+ε}= Γ

_{e}+ Γ

_{ε}. In the linear case, this minimisation problem is written as

*ε̅*,Γ

_{ε}}, samples are drawn from an appropriate prior distribution model of the unknowns and used to determine samples from both the ‘exact’ and ‘approximate’ data distributions. In [3] this approach was used to examine the model error between fine and coarse meshes, whereas in this paper it is used to examine the error between a finite element (FE) solution of the DA and Green’s function model. The computation of the approximation error statistics is explained in more detail in Section 4.1.

**Remark 1**(Approximation error vs. reference data)

**Remark 2**(Reference method and the Rytov approximation)

## 4. Results

### 4.1. Model implementation

### 4.1.1. The target distribution

*r*= 35mm and height

*h*= 110mm. The sources and detectors were placed on two rings located 6mm above and below the central xy-plane of the cylinder. Both rings contained 16 sources and 16 detectors. The target consisted of homogeneous background with two small cubic inclusions. The edge lengths of the inclusions were

*d*= 12.2mm and the heights were

*h*= 16.4mm. The inclusions were located in such a way that the central xy-plane of one inclusion was located at the level of the upper source-detector ring and the central xy-plane of the other inclusion was located at the level of the lower source-detector ring. The simulation domain is illustrated in Fig. 1.

*ω*= 100MHz and the refractive index of the medium was 1.56. The absorption and reduced scattering coefficients of the background medium and the inclusions are given in Table 1 for the case 1 in which the true background optical properties were known and the case 2 in which the background optical properties were -30% off from the linearisation point

*x*

_{h,0}= (

*μ*

_{a,0},

*μ*

^{′}

_{s,0})

^{T}= (0.01,1)

^{T}.

*e*with standard deviation of 0.5% of corresponding amplitude and phase was added to the simulated data.

### 4.1.2. Computational forward models

*A*(

*x*) we take a FE-approximation of the DA with a dense mesh. Let {ℕ,𝕋,𝕌} represent a set of nodes ℕ = {

*N*;

_{k}*k*= 1…

*n*}, elements 𝕋 = {

_{h}*τ*;

_{j}*j*= 1…

*n*}, and shape functions 𝕌 = {

_{e}*u*(

_{k}*r*);

*k*= 1…

*n*} and 𝖦 = 𝖪

_{h}^{-1}where 𝖪 the

*n*×

_{h}*n*system matrix. We used a tetrahedra mesh with

_{h}*n*= 148276 nodes. For the representation of the optical parameters

_{h}*x*we used 8748 cubic voxels both for absorption and scattering.

_{h}*A*(

_{h}*x*) is taken to be

_{h}*G*

_{∞}(

*x*

_{h,0}) is the infinite domain Green’s function, Eq. (13). The Green’s function Jacobian 𝖩 = [𝖩

^{(μa)}

_{∞}𝖩

^{(μ′s)}

_{∞}] is constructed by sampling the forward and adjoint fields on the same mesh node points {

*N*} :

_{k}*G*

^{*}is the adjoint Green’s function and utilising the chain rule to obtain the derivatives for the scattering. In all of the approaches, the Jacobian was calculated using the homogeneous optical properties

*μ*

_{a,0}= 0.01 mm

^{-1}and

*μ*

^{′}

_{s,0}= 1mm

^{-1}.

### 4.1.3. Approximation error statistics

*π*(

*x*) and draw

*r*= 200 samples

*x*

^{(ℓ)}

_{h}from it. The correlation length for both absorption and scattering was set as 14 mm which means that this is (roughly) the prior estimate of the spatial size of the inhomogeneities in the target domain. The prior means for absorption and scattering were set as

*μ*̄

_{a}= 0.01 mm

^{-1}and

*μ*̄

^{′}

_{s}= 1mm

^{-1}, and the marginal variances for the inhomogeneity and background part were set such as 2 s.t.d. limits for the contrast of the inhomogeneities and background corresponded to 1 % of the mean values for both absorption and scattering. The approximation error was computed from the samples as

*r*multiplied by the time for the forward solutions of the accurate and approximative models [14]. The error model, however, needs to be estimated only once for a fixed measurement setup and it can be done off-line.

### 4.2. Reconstructions

*g*

^{obs}was the simulated measurement data,

*g*

^{ref}was the reference data which is the FE-solution of the DA with optical properties of the linearisation point

*x*

_{h,0}which were for the absorption and scattering

*μ*= 0.01mm

_{a}^{-1}and

*μ*

^{′}

_{s}= 1mm

^{-1}throughout the domain. Further, 𝖩(

*x*

_{h,0}) was the Jacobian which was calculated as described in Sec. 4.1, Eqs. (34) and (35),using optical parameters of the linearisation point

*x*

_{h,0}(

*μ*= 0.01mm

_{a}^{-1}and

*μ*

^{′}

_{s}= 1mm

^{-1}).

*A*(

_{h}*x*) was the infinite domain Green’s function, Eq. (33) and the Jacobian 𝖩(

_{h}*x*

_{h,0}) was calculated similarly as in the RM. The mean and the covariance of the approximation error statistics were constructed as described in Section 4.1.

### 4.2.1. Reconstructions when the background properties are known

*x*

_{h,0}=

*x*

_{0}, see case 1 of Table 1. Two horizontal and two vertical slices of the 3D reconstructions are shown in Figs. 2 and 3. The locations of the slices were chosen such that they cut through the inclusions.

### 4.2.2. Reconstructions with mismodelled background

*x*

_{0}were 10%, 20%, 30% and 40% larger or smaller than the linearisation point (

*μ*= 0.01mm

_{a}^{-1}and

*μ*

^{′}

_{s}= 1mm

^{-1}). The absorption and scattering values of the case 2, in which the background optical properties are 30% smaller than the linearisation point, are given in Table 1. The reference measurement

*g*

^{ref}, Jacobian 𝖩(

*x*

_{h,0}) and the approximation error statistics were calculated using the expected optical properties

*x*as described in Sec. 4.1.

_{h}### 4.2.3. Comparison and discussion

*x*and

*x*̂, respectively, were calculated as

## 5. Conclusions

## Acknowledgments

## References and links

1. | S. R. Arridge, “Optical tomography in medical imaging,” Inv. Probl. |

2. | S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. |

3. | 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. |

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

5. | G. Bal, “Inverse transport theory and applications,” Inv. Probl. |

6. | D. A. Boas, “A fundamental limitation of linearized algorithms for diffuse optical tomography,” Opt. Express |

7. | A. Gibson, J.C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical tomography,” Phys. Med. Biol. |

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

9. | J.M.J. Huttunen and J.P. Kaipio, “Approximation errors in nonstationary inverse problems,” Inverse Problems and Imaging |

10. | J.M.J. Huttunen and J.P. Kaipio, “Model reduction in state identification problems with an application to determination of thermal parameters,” Applied Numerical Mathematics |

11. | J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Springer, New York, 2005. |

12. | J. Kaipio and E. Somersalo, “Statistical inverse problems: Discretization, model reduction and inverse crimes,” J. Comput. Appl. Math. |

13. | A. D. Klose and A. H. Hielscher, “Optical tomography with the equation of radiative transfer,” International Journal of Numerical Methods for Heat & Fluid Flow |

14. | 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 |

15. | S.D. Konecky, G. Y. Panasyuk, K. Lee, V. Markel, A. G. Yodh, and J. C. Schotland, “Imaging complex structures with diffuse light,” Opt. Express |

16. | A. Lehikoinen, S. Finsterle, A. Voutilainen, L.M. Heikkinen, M. Vauhkonen, and J.P. Kaipio, “Approximation errors and truncation of computational domains with application to geophysical tomography,” Inverse Problems and Imaging |

17. | A. Nissinen, L.M. Heikkinen, and J.P Kaipio, “The Bayesian approximation error approach for electrical impedance tomography - experimental results,” Meas. Sci. Technol. |

18. | A. Nissinen, L.M. Heikkinen, V. Kolehmainen, and J.P Kaipio, “Compensation of errors due to discretization, domain truncation and unknown contact impedances in electrical impedance tomography,” Meas. Sci. Technol. |

19. | A. Nissinen, V. Kolehmainen, and J.P Kaipio, “Compensation of modelling errors due to unknown domain boundary in electrical impedance tomography,” IEEE Trans. Med. Imag, Submitted. |

20. | M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. |

21. | J. C. Schotland and V. Markel, “Inverse scattering with diffusing waves,” J. Opt. Soc. Am. A |

22. | J. Ripoll, V. Ntziachristos, and M. Nieto-Vesperinas, “The Kirchhoff approximation for diffusive waves,” Phys. Rev. E |

23. | N. Polydorides, “Linearization error in electical impedance tomography,” Progress In Electromagnetics Research, PIER |

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

**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: June 7, 2010

Revised Manuscript: July 9, 2010

Manuscript Accepted: July 9, 2010

Published: July 16, 2010

**Virtual Issues**

Optical Imaging and Spectroscopy (2010) *Biomedical Optics Express*

**Citation**

Tanja Tarvainen, Ville Kolehmainen, Jari P. Kaipio, and Simon R. Arridge, "Corrections to linear methods for diffuse optical tomography using approximation
error modelling," Biomed. Opt. Express **1**, 209-222 (2010)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-1-1-209

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### References

- S. R. Arridge, "Optical tomography in medical imaging," Inv. Probl. 15, R41-R93 (1999).
- S. R. Arridge, M. Cope, and D. T. Delpy, "The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis," Phys. Med. Biol. 37, 1531-1560 (1992).
- 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).
- S. R. Arridge and J. C. Schotland, "Optical tomography: forward and inverse problems," Inv. Probl. 25, 123010 (2009).
- G. Bal, "Inverse transport theory and applications," Inv. Probl. 25, 053001 (2009).
- D. A. Boas, "A fundamental limitation of linearized algorithms for diffuse optical tomography," Opt. Express 1, 404-413 (1997).
- A. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical tomography," Phys. Med. Biol. 50, R1-R43 (2005).
- J. Heino, E. Somersalo, and J. P. Kaipio, "Compensation for geometric mismodelling by anisotropies in optical tomography," Opt. Express 13, 296-308 (2005).
- J. M. J. Huttunen and J. P. Kaipio, "Approximation errors in nonstationary inverse problems," Inv. Probl. Imaging 1, 77-93 (2007).
- J. M. J. Huttunen and J. P. Kaipio, "Model reduction in state identification problems with an application to determination of thermal parameters," Appl. Numer. Math. 59, 877-890 (2009).
- J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, (Springer, New York, 2005).
- J. Kaipio and E. Somersalo, "Statistical inverse problems: Discretization, model reduction and inverse crimes," J. Comput. Appl. Math. 198, 493-504 (2007).
- A. D. Klose and A. H. Hielscher, "Optical tomography with the equation of radiative transfer," Int. J. Numer. Meth. Heat Fluid Flow 18, 443-464 (2008).
- 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).
- S. D. Konecky, G. Y. Panasyuk, K. Lee, V. Markel, A. G. Yodh, and J. C. Schotland, "Imaging complex structures with diffuse light," Opt. Express 16, 5048-5060 (2008).
- A. Lehikoinen, S. Finsterle, A. Voutilainen, L. M. Heikkinen, M. Vauhkonen, and J. P. Kaipio, "Approximation errors and truncation of computational domains with application to geophysical tomography," Inv. Probl. Imaging 1, 371-389 (2007).
- A. Nissinen, L. M. Heikkinen, and J. P Kaipio, "The Bayesian approximation error approach for electrical impedance tomography - experimental results," Meas. Sci. Technol. 19, 015501 (2008).
- A. Nissinen, L. M. Heikkinen, V. Kolehmainen, and J. P Kaipio, "Compensation of errors due to discretization, domain truncation and unknown contact impedances in electrical impedance tomography," Meas. Sci. Technol. 20, 015504 (2009).
- A. Nissinen, V. Kolehmainen, and J.P Kaipio, "Compensation of modelling errors due to unknown domain boundary in electrical impedance tomography," IEEE Trans. Med. Imag, Submitted.
- M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, "Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography," Opt. Lett. 20, 426-428, (1995).
- J. C. Schotland and V. Markel, "Inverse scattering with diffusing waves," J. Opt. Soc. Am. A 18, 2767-2777 (2001).
- J. Ripoll, V. Ntziachristos, and M. Nieto-Vesperinas, "The Kirchhoff approximation for diffusive waves," Phys. Rev. E 64, 1-8 (2001).
- N. Polydorides, "Linearization error in electical impedance tomography," Prog. Electromagn. Res., PIER 93, 323-337 (2009).
- T. Tarvainen, V. Kolehmainen, A. Pulkkinen, M. Vauhkonen, M. Schweiger, S. R. Arridge, and J. P. Kaipio, "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).

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