## Scattering and absorption transport sensitivity functions for optical tomography

Optics Express, Vol. 7, Issue 13, pp. 492-506 (2000)

http://dx.doi.org/10.1364/OE.7.000492

Acrobat PDF (303 KB)

### Abstract

Optical tomography is modelled as an inverse problem for the time-dependent linear transport equation. We decompose the linearized residual operator of the problem into absorption and scattering transport sensitivity functions. We show that the adjoint linearized residual operator has a similar physical meaning in optical tomography as the ‘backprojection’ operator in x-ray tomography. In this interpretation, the geometric patterns onto which the residuals are backprojected are given by the same absorption and scattering transport sensitivity functions which decompose the forward residual operator. Moreover, the ‘backtransport’ procedure, which has been introduced in an earlier paper by the author, can then be interpreted as an efficient scheme for ‘backprojecting’ all (filtered) residuals corresponding to one source position *simultaneously* into the parameter space by just solving one adjoint transport problem. Numerical examples of absorption and scattering transport sensitivity functions for various situations (including applications with voids) are presented.

© Optical Society of America

## 1 The transport equation in optical tomography

*(*

^{n}*n*=2,3) with smooth boundary ∂Ω.

*S*

^{n-1}denotes the set of direction vectors

*θ*which have unit length in ℝ

*.*

^{n}*v*(

*x*) denotes the outward unit normal to

*∂*Ω at the point

*x*∊

*∂*Ω. The solution of (1)–(3),

*u*(

*x*,

*θ*,

*t*), describes the density of particles (photons) which travel in Ω at time

*t*through the point

*x*in the direction

*θ*. The velocity

*c*of the particles is assumed to be normalized to

*c*=1cms

^{-1}, such that we will drop it in the notation from now on.

_{-}(resp. Γ

_{+}) denotes the set of points (

*x*,

*θ*,

*t*) which correspond to incoming (resp. outgoing) radiation (particles) at the boundary

*∂*Ω:

_{±}:={(

*x*,

*θ*,

*t*)∊

*∂*Ω×

*S*

^{n}^{-1}×[0,

*T*], ±

*ν*(

*x*)·

*θ*>0}.

*a*(

*x*) is the absorption cross section,

*b*(

*x*) the scattering cross section, and

*µ*(

*x*)=

*a*(

*x*)+

*b*(

*x*) the total cross section or attenuation. These parameters are assumed to be real and strictly positive, and they depend only on the position

*x*. The absorption mean free path

*l*is defined by

_{a}*l*(

_{a}*x*):=

*a*

^{-1}(

*x*), the scattering mean free path

*l*by

_{s}*l*(

_{s}*x*):=

*b*

^{-1}(

*x*), and the transport mean free path

*l*by

*l*(

*x*):=

*µ*

^{-1}(

*x*). Typical values in optical tomography are

*l*≈1.0–10.0 cm,

_{a}*l*≈

_{b}*l*≈0.005-0.01 cm [3

3. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems **15** (2), R41–R93 (1999) [CrossRef]

*η*(

*θ*·

*θ′*) describes the probability for a particle which enters a scattering event with the direction of propagation

*θ′*, to leave this event with the direction

*θ*. It is normalized to

*η*depends only on the cosine of the scattering angle

*θ*·

*θ′*=:cos

*ϑ*and, in particular, is independent of the location of the scattering event

*x*. A frequently used scattering function is the following Henyey-Greenstein scattering function

*ϑ*=

*θ*·

*θ′*, where

*g*∊]-1,1[is some parameter. A value of

*g*=0 means isotropic scattering, whereas values between

*g*=0.9 and

*g*=0.95 are more typical for situations in optical tomography and describe scattering events which are primarily forward directed. We will use this scattering function in our numerical experiments. The right hand side of (1) models the source distribution

*t*=0 at the position

*x*into the direction

_{s}*θ*. The symbols

_{s}*δ*(

*x*-

*x*),

_{s}*δ*(

*t*) and

*δ*(

*θ*-

*θ*) denote Dirac delta functions in the corresponding variables. If the source is positioned at the boundary,

_{s}*x*∊

_{s}*∂*Ω, we assume that

*v*(

*x*)·

_{s}*θ*<0. In our numerical experiments in section 5 we will use

_{s}*v*(

*x*)·

_{s}*θ*=-1 (normally incident radiation).

_{s}*q*which, however, might be situated at the boundary

*∂*Ω. We want to mention that an alternative way of describing laser sources at the boundary is to use an inhomogeneous boundary condition instead of (3) and a zero source

*q*=0. Both choices are equivalent, although care has to be taken when transforming one into the other. For more details see for example [5].

*u*is the solution of (1)–(3) with source

*q*and parameters

*a*,

*b*and

*x*and

_{r}*t*denote the receiver location and receiving time, respectively. The symbol

_{r}*S*

^{n}^{-1}

_{+}denotes the subset of direction vectors

*θ*∊

*S*

^{n}^{-1}for which

*v*(

*x*)·

*θ*>0. The inverse transport problem in optical tomography can now be formulated as follows.

*Inverse transport problem of optical tomography*. Assume that we measure for

*p*≥1 different sources of the form (5) the corresponding data which are given by (6), where

*u*solves the transport problem (1)–(3). Provided with this information, and knowing the scattering function

*η*, we want to reconstruct both coefficient functions

*a*(

*x*) and

*b*(

*x*) inside of Ω

*simultaneously*.

## 2 The linearized residual operator and its adjoint

*a*or

*b*by

*P*, and the space of data

*M*by

_{a,b}*D*. The residuals

*R*(

*a*,

*b*) are defined as the difference between the physically measured data

*G̃*(

*x*,

*t*) and the data which correspond to the parameter distribution (

*a*,

*b*). More formally we have with (6)

*a*,

*b*), since the solution

*u*in (6) depends in a nonlinear way on (

*a*,

*b*). When solving the inverse problem, we want to find a set of parameters (

*â*,

*b̂*) from the (usually noisy) data

*G̃*such that the residuals (7) are minimized in some given norm.

*R*is nonlinear, many reconstruction approaches involve the calculation of its ‘derivative’ or ‘linearized operator’

*R′*which has to be calculated at the most recent best guess (

_{a b}*a, b*) for the parameters. This operator is often called ‘Fréchet-derivative’, ‘Jacobian’, or ‘sensitivity matrix’. It is a mapping from the parameter space

*P*×

*P*into the data space

*D*.

*a*(

*x*)→

*a*(

*x*)+

*δ*

*a*(

*x*) and

*b*(

*x*)→

*b*(

*x*)+

*δb*(

*x*) and plug this into (1)–(3). Assuming that the corresponding solution

*u*(

*x*) of (1)–(3) responds to this perturbation according to

*u*(

*x*)→

*u*(

*x*)+

*w*(

*x*), and neglecting all terms in (1) which are of higher than linear order in

*δa*,

*δb*and

*w*, we arrive at the following result [9, 19].

*The linearized residual operator*

*R′*

_{a,b}*is given by*

*where*

*w*

*solves the linearized equation*

*S*

^{n}^{-1}×[0,

*T*]

*with the homogeneous initial condition*

*and an absorbing boundary condition*

*Here, the*‘

*scattering sources*’

*Q*

_{δa}*and*

*Q*

_{δb}*are defined as*

*where u is the solution of*(1)–(3)

*with parameters a and b*.

*δa*(

*x*),

_{sc}*δb*(

*x*)), where the index

_{sc}*sc*stands for ‘scatter’, the value of

*R′*(

_{a,b}*δa*,

*δb*) is a function in the variables

*x*and

_{r}*t*, where

_{r}*x*is the receiver location and

_{r}*t*the receiver time. This explains the somewhat complicated notation in (8). The physical interpretation of this result is that the perturbations

_{r}*δa*and

*δb*create scattering sources

*Q*and

_{δa}*Q*, which give rise to a distribution

_{δb}*w*(

*x*,

*θ*,

*t*) (which can be positive or negative) of virtual ‘secondary particles’ propagating in the unperturbed medium to the receivers, where they are detected as the (linearized) residuals in the data.

*adjoint linearized residual operator*

*R′**

*is formally defined by the identity*

_{a,b}*D*, and where the brackets denote the inner products in the underlying function spaces

*P*×

*P*and

*D*. The adjoint linearized residual operator is a mapping from the data space

*D*into the parameter space

*P*×

*P*.

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

*Let z denote the solution of the following adjoint linear transport equation*

*with the*‘

*final value condition*’

*and the inhomogeneous outgoing boundary condition*

*and let u be the solution of the forward problem*(1)–(3).

*Then we have*

*with*

*Solving*(15)–(17)

*is called*‘

*backtransport*’.

*D*, i.e., it is a function of detector position

*x*and detection time

_{r}*t*. The values of ζ, which in our applications will be the (filtered) residuals, are attached as time-dependent adjoint sources at these receiver positions in (17), and transported backward in direction and in time into the medium Ω by solving (15)–(17). Notice the sign change in front of the time derivative

_{r}*∂*and the space derivative

_{t}*θ*·∇ in (15) compared to (1), which indicates a reversed time and direction. This is also the reason why we have used a final value condition (16) and an ‘outgoing flux condition’ (17) for uniquely specifying the solution of (15). Notice also that the (filtered) residuals ζ are applied uniformly in all directions in (15), which compensates for the averaging

*θ*-integration of the measurement process (6).

*δa, δb*) and which have caused the mismatch in the data. The correction to our best guess (which is just

*R′**

*ζ as it is described in section 4) is then calculated by combining these backpropagated densities with the actual densities*

_{a,b}*u*of our forward problem (1)–(3). For example, if no particles of the forward solution reach a given point

*x*during the experiment, i.e.

_{sc}*u*(

*x*,

_{sc}*θ*,

*t*)=0 for all

*θ*∊

*S*

^{n}^{-1}and all

*t*∊[0,

*T*], then the update (

*R′**

*ζ)(*

_{a,b}*x*) in (19), (20) will be zero at this location

*x*. This makes sense physically since no secondary particles could have been generated in (12), (13) at this point in the medium by a parameter change (

_{sc}*δa*(

*x*),

_{sc}*δb*(

*x*)) and reach the detector via (9). This means also that the ‘sensitivity’ of our source-receiver pair to parameter changes at this location will be zero. This observation motivates the introduction of (linearized) ‘sensitivity functions’, which quantify the sensitivity of a given source-receiver pair to parameter perturbations at each point

_{sc}*x*in the medium. We will discuss these sensitivity functions for the linear transport equation in the following section.

_{sc}## 3 Transport sensitivity functions

*backtransport*, as it is derived above, and the general idea of

*backprojection*, which is well-known in x-ray tomography [17]. A short derivation of this result is given in the appendix.

*For a given source (5), we can find functions*ψ

*a*(

*x*,

_{r}*t*;

_{r}*x*) and ψ

_{sc}*(*

_{b}*x*,

_{r}*t*;

_{r}*x*)

_{sc}*with the following properties*.

*Projection*’

*Step*:

*The action of the linearized residual operator*

*R′*

_{a,b}*on the perturbation*((

*δa*,

*δb*)

*in parameter space can be described as follows*

*Backprojection*’

*Step*:

*The action of the adjoint linearized residual operator*

*R*′*

_{a,b}*on the vector*ζ

*in data space can be described as follows*

*The functions*ψ

*(*

_{a}*x*,

_{r}*t*;

_{r}*x*) and ψ

_{sc}*(*

_{b}*x*,

_{r}*t*;

_{r}*x*)

_{sc}*are called*‘

*transport sensitivity functions*’.

*δa*(

*x*),

_{sc}*δb*(

*x*)) at a position

_{sc}*x*with a large sensitivity value ψ

_{sc}_{a,b}(

*x*,

_{r}*t*;

_{r}*x*) will have a relatively strong influence on the data measured at the location

_{sc}*x*and at the time

_{r}*t*. Formula (22), on the other hand, says that a residual at a receiver with position

_{r}*x*, which is detected at time

_{r}*t*, will produce relatively large updates of the parameters (

_{r}*a*,

*b*) in the medium at positions

*x*where the corresponding sensitivity functions ψ

_{sc}*(*

_{a,b}*x*,

_{r}*t*;

_{r}*x*) have large values.

_{sc}*nonlinear*inverse problem. The adjoint linearized residual operator

*R′**

*changes here (as part of an iterative reconstruction scheme) with the latest best guess (*

_{a,b}*a, b*). Moreover, it is not applied to the data but to the differences between the data and the calculated data. Moreover, the geometric patterns onto which the data are ‘backprojected’ are different here. Instead of backprojecting over lines, as it is done in x-ray tomography, we have to backproject onto more complicated shapes which are given by the sensitivity functions ψ

*(*

_{a,b}*x*,

_{r}*t*;

_{r}*x*). These sensitivity functions depend on the source and receiver locations, the most recent parameter distribution in the medium, and the detection time of the given data.

_{sc}## 4 The Transport-BackTransport (TBT) algorithm

*a, b*),

*simultaneously*by just solving one adjoint transport problem (15)–(17) on a computer. In many applications, however, we have data given which correspond to many source positions. A possible way of addressing these problems and making use of the adjoint scheme is to consider only the data for one source at a time. We want to outline this procedure in the following.

*δa*,

*δb*) to our best guess (

*a, b*) by looking for a solution of the linearized problem

*C*to the unknown operator (

*R′**

_{a,b}R′*)*

_{a,b}^{-1}, which maps from the data space to the data space and which can be considered as a ‘filtering operator’. We have shown that all we have to do in order to find (

*δa*,

*δb*)

^{†}is to solve one forward transport problem (1)–(3) with our given source

*q*, and one adjoint transport problem (15)–(17) with the (filtered) residual values ζ=

*CR*(

*a, b*) attached as ‘adjoint sources’ to the receiver positions. The correction (

*δa*,

*δb*)

^{†}is then calculated from these two solutions

*u*and

*z*by the formulas (19) and (20). We mention that additional regularization criteria can be incorporated in (24), but these refinements of the inversion scheme will not be discussed in the present paper.

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

18. F. Natterer and F. Wübbeling, “A propagation-backpropagation method for ultrasound tomography,” Inverse Problems **11**, 1225–1232 (1995) [CrossRef]

*δa*,

*δb*) is applied immediately to the parameter distribution (

*a, b*) after being calculated from the data of one source position, which yields an updated best guess (

*a*+

*δ*+

_{a, b}*δ*). Then, the data corresponding to the next source position are used to find a new correction (

_{b}*δa*,

*δb*) to these parameters by running one forward and one adjoint problem on the corrected guess. This is done iteratively marching from source to source in some cyclic order, until the method converges. Since with each update the reference parameter distribution (

*a, b*) changes, the described inversion method is

*nonlinear*. For a more detailed description of this single-step adjoint field inversion scheme in the application of optical tomography we refer to [8, 9, 10

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

## 5 Numerical examples of transport sensitivity functions

**14**, 1107–1130 (1998) [CrossRef]

^{2}which is discretized into 120×120 elementary cells or pixels. One individual time step lasts 0.2 s. Assuming a (normalized) velocity of

*c*=1 cm/s, a photon needs for example 6 s or 30 time steps in order to travel from one boundary of the domain parallel to the axes to the opposite boundary without being scattered. The directional variable

*θ*is discretized into 12 individual direction vectors equidistantly distributed over the unit circle

*S*

^{1}, four of them pointing in the directions of the positive and negative

*x*and

*y*coordinate axes. For a more detailed description of this discretization scheme for (1)–(3) we refer to [9, 10

**14**, 1107–1130 (1998) [CrossRef]

*b*=100 cm

^{-1}and

*a*=0.1 cm

^{-1}everywhere. In the other three experiments, we have embedded into this background distribution so-called ‘clear regions’ or ‘voids’ of various shapes. These clear regions can be found in many physical situations where near-infrared light is used for medical imaging [4

4. S. R. Arridge, H. Dehgani, M. Schweiger, and E. Okada, “The Finite Element Model for the Propagation of Light in Scattering Media: A Direct Method for Domains with Non-Scattering Regions,” Medical Physics , **27** (1), 252–264 (2000) [CrossRef] [PubMed]

15. V. Kolehmainen, S. R. Arridge, W. R. B. Lionheart, M. Vauhkonen, and J. P. Kaipio, “Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data” Inverse Problems **15**, 1375–1391 (1999) [CrossRef]

20. E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope, and D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt. **36** (1), 21–31 (1997) [CrossRef] [PubMed]

*b*=1.0 cm

^{-1}and

*a*=0.01 cm

^{-1}. Their shapes can be seen in Figure 1. We mention that in all of our numerical examples a Henyey-Greenstein scattering law (4) is used with parameter

*g*=0.9.

*M*(

_{a,b}*x*,

_{r}*t*), defined in (6), for these four experiments. In the first image, a delta-like laser source (5) is positioned at the boundary point

_{r}*P*1(which is marked in Figure 1 in the upper left image) with coordinates (40, 0). The receiver is situated at the boundary point

*P*3 (see again Figure 1) and has the coordinates (60, 120). In the second image, the source is positioned at the boundary point

*P*2 with coordinates (120,40), and the receiver is again situated at the boundary point

*P*3. The data curves correspond in both images, from bottom to top, to the experiments 1 (blue, dash-dotted), 4 (red, solid), 2 (green, stars), and 3 (magenta, dashed). We see that the presence of clear regions has a significant influence on the first arrival times as well as on the amplitudes and shapes of the data.

4. S. R. Arridge, H. Dehgani, M. Schweiger, and E. Okada, “The Finite Element Model for the Propagation of Light in Scattering Media: A Direct Method for Domains with Non-Scattering Regions,” Medical Physics , **27** (1), 252–264 (2000) [CrossRef] [PubMed]

*(*

_{a}*x*,

_{r}*t*;

_{r}*x*) where the source is situated at position

_{sc}*P*1 and the receiver location

*x*is at

_{r}*P*3. The receiving time is

*t*=10 s, which corresponds to the time step

_{r}*T*1=50. Therefore, looking at the data curves on the left hand side of Figure 2, this figure describes the sensitivity of the early photons (which are sometimes called ‘snake’ photons) to parameter perturbations in the imaging domain. We can clearly see the ‘wave-guiding effect’ which occurs in the experiments 2 and 3 due to the near-boundary clear layer.

*t*=24 s, which corresponds to time step

_{r}*T*2=120. See again the image on the left hand side of Figure 2 for locating these receiver times in the general form of the data curves. We see that the shapes of the sensitivity functions at these later times are much broader than those of the snake photons. In addition, the wave-guiding effect of the clear boundary layers has disappeared. Figures 5 and 6 show the sensitivity functions which correspond to Figures 3 and 4 but where the source is now located at position

*P*2 instead of

*P*1. The discussion of these two figures is completely analogous to the discussion of Figures 3 and 4 and is therefore omitted here.

*b*(

*x*,

_{r}*t*;

_{r}*x*) for the same situations as in Figures 3 and 4. We see that the data corresponding to early times as well as to later times are equally sensitive to parameter changes inside the clear layers. Notice also that the norms

_{sc}*N*of these sensitivity functions are much smaller than those for the absorption parameter. This is due to the strong scattering in the domain. Directional information corresponding to local perturbations in the scattering parameter is quickly lost due to the many succeeding scattering events of the photons.

23. J. M. Schmitt, A. H. Gandjbakhche, and R. F. Bonner, “Use of polarized light to discriminate short-path photons in a multiply scattering medium,” Appl. Opt. **31**, 6535–6546 (1992) [CrossRef] [PubMed]

22. L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion **24**, 327–370 (1996) [CrossRef]

## 6 Appendix

*transport Green functions*

*G*[

*x*,

_{sc}*θ*,

_{sc}*t*](

_{sc}*x*,

*θ*,

*t*) as solutions of the problem

*S*

^{n}^{-1}×[0,

*T*] with the homogeneous initial condition

*adjoint transport Green functions*

*G*̂[

*x*,

_{r}*θ*,

_{r}*t*](

_{r}*x*,

*θ*,

*t*) are defined accordingly by

*S*

^{n}^{-1}×[0,

*T*] with the ‘final value condition’

*[*

_{a}*x*,

_{r}*θ*,

_{r}*t*] are defined by

_{r}*δb*=0. Then the linearized residual operator corresponding to our source (5) can be written in the explicit form

*r*stands for ‘receiver’. Using the following

*generalized reciprocity relation*

*(*

_{a}*x*,

_{r}*t*;

_{r}*x*) by

_{sc}*δb*) ∊

*P*×

*P*and define the functions Φ

*[*

_{b}*x*,

_{r}*θ*,

_{r}*t*] by

_{r}*(*

_{b}*x*,

_{r}*t*;

_{r}*x*) as

_{sc}*R′**

*are derived in a similar way, such that we will not present them here explicitly. In the derivation we make use of the fact that a boundary condition*

_{a,b}*z*(

*x*,

*θ*,

*t*)=ζ(

*x*,

*t*) on Γ

_{+}can be replaced by an equivalent ‘surface source distribution’

*q*∂Ω=

*v*(

*x*)·θζ(

*x*,

*t*) along the boundary. For more details see for example [5].

*x*,

*t*)=δ(

*x*-

*x*)

_{r}*δ*(

*t*-

*t*) in order to calculate the sensitivity functions ψ

_{r}*(*

_{a}*x*,

_{r}*t*;

_{r}*x*) and ψ

_{sc}*(*

_{b}*x*,

_{r}*t*;

_{r}*x*) corresponding to a given receiver position

_{sc}*x*and a receiver time

_{r}*t*. The sensitivity functions are then automatically given by ψ

_{r}*(*

_{a}*x*,

_{r}*t*;

_{r}*x*)=Δ

_{sc}*(*

_{a}*x*) and ψ

_{sc}*(*

_{b}*x*,

_{r}*t*;

_{r}*x*)=Δ

_{sc}*(*

_{b}*x*) with Δ

_{sc}*and Δ*

_{a}*defined in (19) and (20).*

_{b}## 7 Acknowledgment

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19. | F. Natterer, “Numerical Solution of Bilinear Inverse Problems,” Preprints “Angewandte Mathematik und Informatik” 19/96-N, Münster (1996) |

20. | E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope, and D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt. |

21. | J. Riley, H. Dehghani, M. Schweiger, S. R. Arridge, J. Ripoll, and M. Nieto-Vesperinas, “3D Optical Tomography in the Presence of Void Regions,” to appear in Optics Express, (this issue) |

22. | L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion |

23. | J. M. Schmitt, A. H. Gandjbakhche, and R. F. Bonner, “Use of polarized light to discriminate short-path photons in a multiply scattering medium,” Appl. Opt. |

24. | J. C. Schotland, J. C. Haselgrove, and J. S. Leigh, “Photon hitting density,” Appl. Opt. |

**OCIS Codes**

(170.3880) Medical optics and biotechnology : Medical and biological imaging

(170.5280) Medical optics and biotechnology : Photon migration

(170.6960) Medical optics and biotechnology : Tomography

(170.7050) Medical optics and biotechnology : Turbid media

**ToC Category:**

Focus Issue: Diffuse optical tomography

**History**

Original Manuscript: October 27, 2000

Published: December 18, 2000

**Citation**

Oliver Dorn, "Scattering and absorption transport sensitivity functions for optical tomography," Opt. Express **7**, 492-506 (2000)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-7-13-492

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### References

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- E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope and D. T. Delpy, "Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head," Appl. Opt. 36 (1), 21-31 (1997) [CrossRef] [PubMed]
- J. Riley, H. Dehghani, M. Schweiger, S. R. Arridge, J. Ripoll and M. Nieto-Vesperinas, "3D Optical Tomography in the Presence of Void Regions," Opt. Express 7, 462-467 (2000), http://www.opticsexpress.org/oearchive/source/26894.htm
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- J. C. Schotland, J. C. Haselgrove and J. S. Leigh, "Photon hitting density," Appl. Opt. 32, 448-453 (1993) [CrossRef] [PubMed]

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