## Optical Tomography in weakly scattering media in the presence of highly scattering inclusions |

Biomedical Optics Express, Vol. 2, Issue 3, pp. 440-451 (2011)

http://dx.doi.org/10.1364/BOE.2.000440

Acrobat PDF (1274 KB)

### Abstract

We consider the problem of optical tomographic imaging in a weakly scattering medium in the presence of highly scattering inclusions. The approach is based on the assumption that the transport coefficient of the scattering media differs by an order of magnitude for weakly and highly scattering regions. This situation is common for optical imaging of live objects such an embryo. We present an approximation to the radiative transfer equation, which can be applied to this type of scattering case. Our approach was verified by reconstruction of two optical parameters from numerically simulated datasets.

© 2011 OSA

## 1. Introduction

1. P. J. Shaw, D. A. Agard, Y. Hiraoka, and J. W. Sedat, “Tilted view reconstruction in optical microscopy. Three-dimensional reconstruction of Drosophila melanogaster embryo nuclei,” Biophys. J. **55**(1), 101–110 (1989). [CrossRef] [PubMed]

2. C. S. Brown, D. H. Burns, F. A. Spelman, and A. C. Nelson, “Computed tomography from optical projections for three-dimensional reconstruction of thick objects,” Appl. Opt. **31**(29), 6247–6254 (1992). [CrossRef] [PubMed]

3. J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science **296**(5567), 541–545 (2002). [CrossRef] [PubMed]

7. A. Bassi, D. Brida, C. D’Andrea, G. Valentini, R. Cubeddu, S. De Silvestri, and G. Cerullo, “Time-gated optical projection tomography,” Opt. Lett. **35**(16), 2732–2734 (2010). [CrossRef] [PubMed]

8. L. Wang, S. L. Jacques, and L. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. **47**(2), 131–146 (1995). [CrossRef] [PubMed]

11. O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. **14**(5), 1107–1130 (1998). [CrossRef]

18. L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **79**(3), 036607 (2009). [CrossRef] [PubMed]

19. L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography: simultaneous reconstruction of scattering and absorption,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **81**(1), 016602 (2010). [CrossRef] [PubMed]

## 2. Methodology

### 2.1. Direct problem

*B*in the source term reads

*I*denotes the intensity of light; (ii)

*μ̃*=

*μ*+

*iω*/

*c*is the complex extinction coefficient, where

*μ*is the transport coefficient, which is a sum of the scattering,

*μ*, and absorption,

_{s}*μ*, coefficients,

_{a}*ω*is the Fourier parameter, and

*c*is the speed of light; (iii)

*λ*denotes the albedo of a single scattering event such that

*λ*∈ [0,1],

*μ*=

_{s}*λμ*and

*μ*= (1−

_{a}*λ*)

*μ*; (iv)

*p*(

**s**·

**s**′) is the phase function. The simplest form of an anisotropic phase function is assumed in this study

*ε*∈ [−1,1]. The last term in Eq. (2) is scattered once direct radiation

*I*

_{0}(

**r**,

**s**

_{0}). Note that the transport coefficient,

*μ*, and albedo,

*λ*, are chosen instead of conventional scattering and absorption coefficients. The albedo is a photon's probability to survive a single scattering event and, therefore, it controls the true absorption. The transport coefficient, which is reciprocal to photon's mean path length between successive scattering events, describes scattering properties of the medium. When the scattering medium is a mixture of two types of particles with different scattering properties then the resulting transport coefficient and the albedo are found as

*μ*=

*μ*

_{1}+

*μ*

_{2}and

*λ*= 2

*λ*

_{1}

*λ*

_{2}(

*λ*

_{1}+

*λ*

_{2})

^{−1}, where

*μ*and

_{j}*λ*(

_{j}*j*= 1, 2) are corresponding parameters for each type of particle. Parameters of a mixture of more than two types of scattering particles are computed recursively.

**s**

_{0}. The intensity of each ray (the Green function) is found by solving the equation

*Q*

_{0}is the source amplitude and

**r**

_{0}belongs to a source plane

**r**

_{0}·

**s**

_{0}=

*const.*The solution of Eq. (4) is given by

**r**rather than from

**r**

_{0}. The source and observation points are connected by a line

**r**=

**r**

_{0}+

**s**

_{0}

*l*, and, therefore, the integration in exponent in Eq. (5) can be rewritten in the form

^{l}_{0}

*μ̃*(

**r**

_{0}+

**s**

_{0}

*l*′)

*dl*′ = ∫

^{l}_{0}

*μ̃*(

**r**−

**s**

_{0}

*l*′)

*dl*′.

**r**is given by

*I*

_{0}, Eq. (5), must be added to

*I*when

**s**=

**s**

_{0}. To correspond to physical reality we add the direct radiation to the scattered intensity when

**s**·

**s**

_{0}≥ 1−

*δ*, where

*δ*is finite but small number.

*B*is known. Exact knowledge of the function

*B*is equivalent to computing the solution of the RTE. Here we suggest an approximation to the function

*B*according to the assumption that the medium consists of weakly and highly scattering regions, whose transport coefficients differ by an order of magnitude. We further assume that recorded photons coming from weakly scattering regions are scattered only once, i.e. 1/

*μ*is a length scale on the order of physical dimensions of the scattering domain. This assumption implies a presence of photons scattered more than once. However, they do not reach the CCD array. Therefore, the method of successive approximations applies [20] when only the first approximation is retained. Thus, as an approximation for

*B*we take the scattered once direct radiation

*p*(

**s**·

**s**

_{0})

*I*

_{0}(

**r**,

**s**

_{0}), which is the last term in Eq. (2). On the other hand, in highly scattering regions the intensity in Eq. (2) is approximated by

*u*denotes the average intensity defined by

*κ*is the diffusion coefficient

**s**and integration over the whole solid angle leads to a system of two first order differential equations for

*u*and the flux

**q**= −

*κ*∇

*u*. This system is closed when the phase function is assumed in the form of Eq. (3). Elimination of the flux in this system results in the Helmholtz equation for the integrated intensity [21

21. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. **15**(2), R41 (1999). [CrossRef]

*ρ*, represents direct intensity averaged over the whole solid angle [21

21. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. **15**(2), R41 (1999). [CrossRef]

*B*to

*B*is used for computing the intensity

*I*, Eq. (6), where in weakly scattering regions the integrated intensity,

*u*, is neglected.

### 2.2. Implementation details

23. R. L. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. **12**(2), 252–255 (1985). [CrossRef] [PubMed]

*x*,

*y*, and

*z*-axes. The final three-dimensional ray's path is found by applying a merge sort to these three sets of distances.

*ελμκ*

**s**·∇

*u*along a ray in accordance with Eq. (12). For the sake of simplicity a constant value of the parameter

*ε*is assumed everywhere in the domain. Then, we are looking for an inexpensive way of numerical evaluation of the following term

*u*satisfies

*u*across a cell's interface at

*l*=

*l*along the direction

_{i}**s**. Then, the line integral (Eq. (13)) is approximated by a sum

*l*=

*l*. The diffusion coefficient at the cell's interface,

_{i}*λ*and

*μ*are substituted. The distance Δ

*l*is the length of the ray's path within a cell provided by Siddon's algorithm. Here, the index

_{j}*j*enumerates cells on the ray path and

*μ̃*is the extinction coefficient of

*j*-th cell. The cell's interface values of parameters

*λ*} = (1/2) (

*λ*

^{+}+

*λ*

^{−}),

*μ*} = (1/2) (

*μ*

^{+}+

*μ*

^{−}),

*mm*and, therefore, physical dimensions of the computational domain are: (i) 10

*mm*in

*x*- and

*y*-axis, and (ii) 20

*mm*in

*z*-axis. The parameter ε was set to 1/2.

*z*-axis. It has the background transport coefficient

*μ*= 0.1

*mm*

^{−1}and the albedo

*λ*= 0.999. The value of the transport coefficient for each ball is set to

*μ*= 0.75

*mm*

^{−1}. Two spirals have the background value of the albedo and one absorbing spiral has the value of albedo

*λ*= 0.25. In regions where balls overlap, the density of scattering particles is increased and, therefore, values of parameters are computed accordingly as described in section 2.1. The direct radiation

*I*

_{0}enters the domain along the direction

**s**

_{0}=2

^{−1/2}(1,0,−1)

*. The amplitude of the direct radiation is set to 1. A camera rotates around the*

^{T}*z*-axis. It is convenient to describe a position of the camera by a normal vector

**n**to its CCD array. We set the initial position of the camera as

**n**= (1,0,0)

*. In Fig. 1a the camera was rotated by 153° with respect to the initial position. In the figure on the right (Fig. 1b) two highly scattering cylinders are embedded in a weakly scattering cylinder. The weakly scattering cylinder has the same optical properties as above. Both highly scattering cylinders have*

^{T}*μ*= 0.75

*mm*

^{−1}, one of them has a low value of the albedo,

*λ*= 0.25. The direct light enters the domain along the same direction as above. The camera was rotated by 117° from its initial position around the

*z*-axis in the positive direction. In addition to these figures, two multimedia files show animated camera's views over 360° of the triple helix and cylinders (Media 1, Media 2). The camera was rotated around the

*z*-axis with 3° angular step. These two cases will be used below for reconstruction experiments.

### 2.3. Inverse problem

*μ*from projection datasets due to domination of the direct radiation in the transmitted light. However, this approach has limited applicability because it does not take into account forward scattered intensity, and does not allow reconstruction of two parameters. Inversion formulae for the attenuated Radon transform ( [25–28

25. O. Tretiak and C. Metz, “The exponential Radon transform,” SIAM J. Appl. Math. **39**(2), 341–354 (1980). [CrossRef]

*B*in Eq. (6) depends on the direct radiation, whose direction s0 varies. Moreover, the diffusive nature of light transport in highly scattering regions makes the inverse problem to be three-dimensional. Therefore, we use a variational framework [29].

*I*and

_{E}*I*are experimentally recorded and computed intensities in the direction

**s**, respectively. The function

*ξ*(

**s**) is introduced for convenience and represent sampling of the camera's positions

**s**

*can be replaced with the normal vector to the CCD array*

_{n}**n**, and

*N*is the number of positions of the camera. Similarly, the functions

*χ*and

_{θ}*ς*represent sampling of measurements in space and frequency

*M*is the number of the camera's pixels;

*L*denotes the number of samples in the Fourier domain (

*ω*); the vector

**r**

*denotes the surface points visible by the CCD camera. Factors*

_{m}*σ*are surface areas supporting rm such that ∫

_{m}*χ*(

**r**)

*d*

^{3}

**r**gives the total visible area. This form of 𝓔 is chosen in order to simplify a variational procedure. Thus, the function

*χ*allows to replace a sum over surface points visible by the CCD camera with a volume integral. Analogously, the function

*ς*replaces a sum over samples in the Fourier domain with an integral.

*J*is the adjoint intensity. The dynamic form of the penalty term is chosen, which depends on (

*k*+ 1)-th and

*k*-th iterations as

*μ*=

*μ*

_{k+1}−

*μ*, Δ

_{k}*λ*=

*λ*

_{k+1}−

*λ*,

_{k}*α*and

_{μ}*α*are Tikhonov regularization parameters.

_{λ}*δℱ*(

*I,u, J,μ,λ*) vanishes. Variation of

*J*recovers Eq. (1) while the variation of

*I*results in the adjoint transport equation

**s**= −

**s**

*. Therefore, the adjoint intensity*

_{n}*J** propagates from the CCD array in direction of its normal

**n**. Variations of optical parameters

*μ*and

*λ*results in two equations

*f*and

_{μ}*f*are computed according to

_{λ}*ω*/

*μc*≪1 and, therefore, terms containing this parameter in Eqs. (24) and (25) can be neglected. It is also interesting to notice that the second term in Eq. (24) disappears for the time-independent case. Formally, Eqs. (23) represent an iterative backprojection algorithm, when backprojected functions

*f*and

_{μ}*f*are products of

_{λ}*J** with various combinations of

*I*

_{0},

*I*,

*u*and

**s**·∇

*u*.

*α*first. From Eqs. (23) we find that

_{μ}*μ*∥ must decay together with 𝓔

^{1/2}and, therefore

*C*

^{(k)}

_{μ}in general depends on the iteration number

*k*and decreases with

*k*. Analogously, we find

*C*

^{(k)}

_{λ}is analogous to

*C*

^{(k)}

*. Iterations are terminated when the functional 𝓔 + ϒ attains its minimal value.*

_{μ}## 3. Numerical experiments

**s**

_{0}, was rotated by 5° around the

*z*-axis over 360° starting from its initial direction

**s**

_{0}= (1,0,0)

*. The source plane*

^{T}**r**

_{0}·

**s**

_{0}=

*const*was parallel to the

*z*-axis (

**e**

*·*

_{z}**s**

_{0}= 0) and set at some distance from the scattering domain. Free space was assumed to be non-absorbing and non-scattering. For each direction of

**s**

_{0}images were acquired by rotating the CCD camera. The camera's viewing direction is controlled by its normal

**n**, which was rotated independently of

**s**

_{0}around the

*z*-axis with angular step of 90° starting from the its initial direction

**n**= (1,0,0)

*. The first approximation to the transport coefficient was computed from the projection dataset when*

^{T}**s**

_{0}and

**n**differ by 180°. The albedo was reconstructed from angularly selective intensity measurements, when

**s**

_{0}and

**n**differ by 90° and 270°. The first approximation to the transport coefficient was further corrected from angularly selective intensity measurements.

*z*= 10

*mm*in Fig. 2. The left slice (Fig. 2a) demonstrates reconstruction of the transport coefficient

*μ*, while the right slice (Fig. 2b) displays reconstructed albedo.

*z*-axis. These kinks are caused by obstructions of one cylinder by another for projection acquisition and by the cylinders' shadows for angularly selective measurements. It is worthwhile to mention that obstructions and shadows result in opposite effects on reconstruction.

*z*= 9

*mm*, while the second row show parameters at

*z*=11

*mm*. Left column displays the transport coefficient

*μ*and the right column shows the albedo

*λ*. Quantitatively our reconstruction approach demonstrates accurate results. Even increase of the transport coefficient in a regions where scattering balls overlap with each other is clearly seen. However, there is some noise present on the weakly scattering background.

## Acknowledgments

## References and links

1. | P. J. Shaw, D. A. Agard, Y. Hiraoka, and J. W. Sedat, “Tilted view reconstruction in optical microscopy. Three-dimensional reconstruction of Drosophila melanogaster embryo nuclei,” Biophys. J. |

2. | C. S. Brown, D. H. Burns, F. A. Spelman, and A. C. Nelson, “Computed tomography from optical projections for three-dimensional reconstruction of thick objects,” Appl. Opt. |

3. | J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science |

4. | C. Vinegoni, C. Pitsouli, D. Razansky, N. Perrimon, and V. Ntziachristos, “In vivo imaging of Drosophila melanogaster pupae with mesoscopic fluorescence tomography,” Nat. Methods |

5. | M. Fauver, E. J. Seibel, J. R. Rahn, M. G. Meyer, F. W. Patten, T. Neumann, and A. C. Nelson, “Three-dimensional imaging of single isolated cell nuclei using optical projection tomography,” Opt. Express |

6. | J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J Biophotonics |

7. | A. Bassi, D. Brida, C. D’Andrea, G. Valentini, R. Cubeddu, S. De Silvestri, and G. Cerullo, “Time-gated optical projection tomography,” Opt. Lett. |

8. | L. Wang, S. L. Jacques, and L. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. |

9. | Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express |

10. | E. Alerstam, W. C. Yip Lo, T. D. Han, J. Rose, S. Andersson-Engels, and L. Lilge, “Next-generation acceleration and code optimization for light transport in turbid media using GPUs,” Biomed. Opt. Express |

11. | O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. |

12. | T. Tarvainen, M. Vauhkonen, and S. R. Arridge, “Gauss-Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. |

13. | T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Numer. Methods Eng. |

14. | K. Ren, G. S. Abdoulaev, G. Bal, and A. H. Hielscher, “Algorithm for solving the equation of radiative transfer in the frequency domain,” Opt. Lett. |

15. | A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer - Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transf. |

16. | A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. |

17. | G. Bal, “Radiative transfer equations with varying refractive index: a mathematical perspective,” J. Opt. Soc. Am. A |

18. | L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

19. | L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography: simultaneous reconstruction of scattering and absorption,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

20. | V. V. Sobolev, A Treatise on Radiative Transfer (D. Van Nostrand Company, Inc., 1963). |

21. | S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. |

22. | S. R. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. |

23. | R. L. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. |

24. | S. R. Deans, The Radon Transform and some of its applications, (Dover Publications, 2007). |

25. | O. Tretiak and C. Metz, “The exponential Radon transform,” SIAM J. Appl. Math. |

26. | F. Natterer, “On the inversion of the attenuated Radon transform,” Numer. Math. |

27. | F. Natterer, “Inversion of the attenuated Radon transform,” Inverse Probl. |

28. | D. V. Finch, “The attenuated X-Ray transforms: recent developments,” Inverse Probl. |

29. | J. Nocedal, and S. J. Wright, Numerical Optimization, (Springer-Verlag Inc., 1999). |

**OCIS Codes**

(170.0170) Medical optics and biotechnology : Medical optics and biotechnology

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

(290.0290) Scattering : Scattering

(290.7050) Scattering : Turbid media

**ToC Category:**

Image Reconstruction and Inverse Problems

**History**

Original Manuscript: November 8, 2010

Manuscript Accepted: January 18, 2011

Published: January 31, 2011

**Citation**

Vadim Y. Soloviev and Simon R. Arridge, "Optical Tomography in weakly scattering media in the presence of highly scattering inclusions," Biomed. Opt. Express **2**, 440-451 (2011)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-2-3-440

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### References

- P. J. Shaw, D. A. Agard, Y. Hiraoka, and J. W. Sedat, “Tilted view reconstruction in optical microscopy. Three-dimensional reconstruction of Drosophila melanogaster embryo nuclei,” Biophys. J. 55(1), 101–110 (1989). [CrossRef] [PubMed]
- C. S. Brown, D. H. Burns, F. A. Spelman, and A. C. Nelson, “Computed tomography from optical projections for three-dimensional reconstruction of thick objects,” Appl. Opt. 31(29), 6247–6254 (1992). [CrossRef] [PubMed]
- J. Sharpe, U. Ahlgren, P. Perry, B. Hill, A. Ross, J. Hecksher-Sørensen, R. Baldock, and D. Davidson, “Optical projection tomography as a tool for 3D microscopy and gene expression studies,” Science 296(5567), 541–545 (2002). [CrossRef] [PubMed]
- C. Vinegoni, C. Pitsouli, D. Razansky, N. Perrimon, and V. Ntziachristos, “In vivo imaging of Drosophila melanogaster pupae with mesoscopic fluorescence tomography,” Nat. Methods 5(1), 45–47 (2008). [CrossRef] [PubMed]
- M. Fauver, E. J. Seibel, J. R. Rahn, M. G. Meyer, F. W. Patten, T. Neumann, and A. C. Nelson, “Three-dimensional imaging of single isolated cell nuclei using optical projection tomography,” Opt. Express 13(11), 4210–4223 (2005). [CrossRef] [PubMed]
- J. McGinty, K. B. Tahir, R. Laine, C. B. Talbot, C. Dunsby, M. A. A. Neil, L. Quintana, J. Swoger, J. Sharpe, and P. M. W. French, “Fluorescence lifetime optical projection tomography,” J Biophotonics 1(5), 390–394 (2008). [CrossRef] [PubMed]
- A. Bassi, D. Brida, C. D’Andrea, G. Valentini, R. Cubeddu, S. De Silvestri, and G. Cerullo, “Time-gated optical projection tomography,” Opt. Lett. 35(16), 2732–2734 (2010). [CrossRef] [PubMed]
- L. Wang, S. L. Jacques, and L. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995). [CrossRef] [PubMed]
- Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express 17(22), 20178–20190 (2009). [CrossRef] [PubMed]
- E. Alerstam, W. C. Yip Lo, T. D. Han, J. Rose, S. Andersson-Engels, and L. Lilge, “Next-generation acceleration and code optimization for light transport in turbid media using GPUs,” Biomed. Opt. Express 1(2), 658–675 (2010). [CrossRef] [PubMed]
- O. Dorn, “A transport-backtransport method for optical tomography,” Inverse Probl. 14(5), 1107–1130 (1998). [CrossRef]
- T. Tarvainen, M. Vauhkonen, and S. R. Arridge, “Gauss-Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 109(17-18), 2767–2778 (2008). [CrossRef]
- T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Finite element model for the coupled radiative transfer equation and diffusion approximation,” Int. J. Numer. Methods Eng. 65(3), 383–405 (2006). [CrossRef]
- K. Ren, G. S. Abdoulaev, G. Bal, and A. H. Hielscher, “Algorithm for solving the equation of radiative transfer in the frequency domain,” Opt. Lett. 29(6), 578–580 (2004). [CrossRef] [PubMed]
- A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, “Optical tomography using the time-independent equation of radiative transfer - Part 1: forward model,” J. Quant. Spectrosc. Radiat. Transf. 72(5), 691–713 (2002). [CrossRef]
- A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202(1), 323–345 (2005). [CrossRef]
- G. Bal, “Radiative transfer equations with varying refractive index: a mathematical perspective,” J. Opt. Soc. Am. A 23(7), 1639–1644 (2006). [CrossRef] [PubMed]
- L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(3), 036607 (2009). [CrossRef] [PubMed]
- L. Florescu, J. C. Schotland, and V. A. Markel, “Single-scattering optical tomography: simultaneous reconstruction of scattering and absorption,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 81(1), 016602 (2010). [CrossRef] [PubMed]
- V. V. Sobolev, A Treatise on Radiative Transfer (D. Van Nostrand Company, Inc., 1963).
- S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41 (1999). [CrossRef]
- S. R. Arridge and J. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl. 25(12), 123010 (2009). [CrossRef]
- R. L. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. 12(2), 252–255 (1985). [CrossRef] [PubMed]
- S. R. Deans, The Radon Transform and some of its applications, (Dover Publications, 2007).
- O. Tretiak and C. Metz, “The exponential Radon transform,” SIAM J. Appl. Math. 39(2), 341–354 (1980). [CrossRef]
- F. Natterer, “On the inversion of the attenuated Radon transform,” Numer. Math. 32(4), 431–438 (1979). [CrossRef]
- F. Natterer, “Inversion of the attenuated Radon transform,” Inverse Probl. 17(1), 113–119 (2001). [CrossRef]
- D. V. Finch, “The attenuated X-Ray transforms: recent developments,” Inverse Probl. 47, 47–66 (2003).
- J. Nocedal, and S. J. Wright, Numerical Optimization, (Springer-Verlag Inc., 1999).

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