## Imaging complex structures with diffuse light

Optics Express, Vol. 16, Issue 7, pp. 5048-5060 (2008)

http://dx.doi.org/10.1364/OE.16.005048

Acrobat PDF (387 KB)

### Abstract

We use diffuse optical tomography to quantitatively reconstruct images of complex phantoms with millimeter sized features located centimeters deep within a highly-scattering medium. A non-contact instrument was employed to collect large data sets consisting of greater than 10^{7} source-detector pairs. Images were reconstructed using a fast image reconstruction algorithm based on an analytic solution to the inverse scattering problem for diffuse light.

© 2008 Optical Society of America

## 1. Introduction

1. M. C.W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy and diffusion,” Rev. Mod. Phys. **71**, 313–371 (1999). [CrossRef]

2. A. Yodh and B. Chance, “Spectroscopy and imaging with diffuse light,” Phys. Today pp. 34–40 (1995). [CrossRef]

5. S. B. Colak, M. B. van der Mark, G. W. Hooft, J. H. Hoogenraad, E. S. van der Linden, and F. A. Kuijpers, “Clinical optical tomography and NIR spectroscopy for breast cancer detection,” IEEE J. Sel. Top. Quantum Electron. **5**, 1143–1158 (1999). [CrossRef]

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

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

22. J. C. Schotland, “Continuous wave diffusion imaging,” J. Opt. Soc. Am. A **14**, 275–279 (1997). [CrossRef]

^{8}measurements may be readily acquired with non-contact DOT systems [27–29

27. G. Turner, G. Zacharakis, A. Soubret, J. Ripoll, and V. Ntziachristos, “Complete-angle projection diffuse optical tomography by use of early photons,” Opt. Lett. **30**, 409–411 (2005). [CrossRef] [PubMed]

*optically-thin*sample by employing time gating of early-arriving photons [27

27. G. Turner, G. Zacharakis, A. Soubret, J. Ripoll, and V. Ntziachristos, “Complete-angle projection diffuse optical tomography by use of early photons,” Opt. Lett. **30**, 409–411 (2005). [CrossRef] [PubMed]

23. V. A. Markel and J. C. Schotland, “Inverse scattering for the diffusion equation with general boundary conditions,” Phys. Rev. E **64**, R035,601 (2001). [CrossRef]

30. V. A. Markel, V. Mital, and J. C. Schotland, “The inverse problem in optical diffusion tomography. III. Inversion formulas and singular value decomposition,” J. Opt. Soc. Am. A **20**, 890–902 (2003). [CrossRef]

31. V. A. Markel and J. C. Schotland, “Symmetries, inversion formulas and image reconstruction for optical tomography,” Phys. Rev. E **70**, 056,616 (2004). [CrossRef]

29. Z.-M. Wang, G. Y. Panasyuk, V. A. Markel, and J. C. Schotland, “Experimental demonstration of an analytic method for image reconstruction in optical tomography with large data sets,” Opt. Lett. **30**, 3338–3340 (2005). [CrossRef]

29. Z.-M. Wang, G. Y. Panasyuk, V. A. Markel, and J. C. Schotland, “Experimental demonstration of an analytic method for image reconstruction in optical tomography with large data sets,” Opt. Lett. **30**, 3338–3340 (2005). [CrossRef]

## 2. Methods

### 2.1. Instrumentation

*µ*m multi-mode fiber to a pair of galvanometer-controlled mirrors (Innovations in Optics, Woburn, MA). The mirrors raster scanned the beam (on a 35×35 rectangular grid with 4 mm step size) over a 13.6×13.6 cm

^{2}square on one side of an imaging tank containing a target. For each laser beam position, the light transmitted through the tank was collected by a

*f*=25 mm F/0.95 lens and focused on a front-illuminated thermoelectric-cooled 16-bit CCD array (DV887ECS-UV, Andor Technology, Belfast, Ireland). A 20×20 cm

^{2}square area of the opposite surface was mapped onto a square grid of 512×512 CCD pixels; this corresponds to a grid of detectors with 0.4 mm step. The signal recorded by each CCD pixel for a given source beam position defined an independent measurement. The total size of the data set recorded in a single experiment was (35×512)

^{2}≈3×10

^{8}. Somewhat smaller subsets of the data were used for image reconstruction.

*L*=6 cm, and was filled with a mixture of water, a highly scattering fat emulsion (Liposyn III, 30%, Abbott Laboratories, Chicago, IL), and India ink (Black India 4415, Sanford, Bellwood, IL). The absorption and the reduced scattering coefficients of the mixture at 786 nm were

*µ*=0.05 cm

_{a}^{-1}and

*µ*′

*=7.5 cm*

_{s}^{-1}, respectively. The transport mean free path was

*ℓ**=1/(

*µ*+

_{a}*µ*′

*) ≈ 1.3 mm. The diffuse wave number (i.e. the inverse characteristic length over which the diffuse waves decay exponentially) was*

_{s}*ℓ**, and thus the experiments were carried out in the diffusion regime. In addition

*k*≈6.6, and thus the transmitted light was substantially attenuated. These parameters are typical of human breast tissues in the NIR spectral region.

_{d}L### 2.2. Targets

*µ*was four times larger. Targets were made in the shape of letters (3 cm tall, 2 cm wide, 5 mm thick with individual components 3 mm in width), and bars (6 cm tall, 5 mm thick, with widths of 7–9 mm).

_{a}### 2.3. Theoretical background

1. M. C.W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy and diffusion,” Rev. Mod. Phys. **71**, 313–371 (1999). [CrossRef]

*µ*≪

_{a}*µ*′

*. This condition was satisfied everywhere in the experimental medium. In the diffusion approximation, the electromagnetic energy density*

_{s}*u*(

**r**) inside the medium obeys the diffusion equation

*D*=

*cℓ**/3=

*c*/3(

*µ*+

_{a}*µ*′

*) the diffusion coefficient,*

_{s}*c*is the average speed of light in the medium and

*S*is the power density of the source. The diffusion equation (1) is supplemented by the boundary condition

*ℓ*is the extrapolation distance [33, 34

34. R. Aronson, “Boundary conditions for diffuse light,” J. Opt. Soc. Am. A **12**, 2532–2539 (1995). [CrossRef]

*D*≈

*c*/3

*µ*′

*. Thus, if the reduced scattering coefficient*

_{s}*µ*′

*is the same inside the target and the surrounding medium, we have*

_{s}*D*≈

*D*

^{(0)}=const. In this case, the goal of DOT is to reconstruct

*µ*(

_{a}**r**) from a set of boundary measurements of

*u*(

**r**), assuming

*D*

^{(0)}is known. We, however, do not imply that reconstruction of

*µ*′

*(*

_{s}**r**) is not possible or not important in general. As was discussed in references [30

30. V. A. Markel, V. Mital, and J. C. Schotland, “The inverse problem in optical diffusion tomography. III. Inversion formulas and singular value decomposition,” J. Opt. Soc. Am. A **20**, 890–902 (2003). [CrossRef]

31. V. A. Markel and J. C. Schotland, “Symmetries, inversion formulas and image reconstruction for optical tomography,” Phys. Rev. E **70**, 056,616 (2004). [CrossRef]

*µ*and

_{a}*µ*′

*is possible with the use of either time-resolved or frequency resolved measurements. The fast image reconstruction method described in Section 2.4 can easily be generalized to the case of heterogeneous*

_{s}*µ*′

*[30*

_{s}30. V. A. Markel, V. Mital, and J. C. Schotland, “The inverse problem in optical diffusion tomography. III. Inversion formulas and singular value decomposition,” J. Opt. Soc. Am. A **20**, 890–902 (2003). [CrossRef]

31. V. A. Markel and J. C. Schotland, “Symmetries, inversion formulas and image reconstruction for optical tomography,” Phys. Rev. E **70**, 056,616 (2004). [CrossRef]

*µ*(

_{a}**r**) as

*µ*(

_{a}**r**)=

*µ*

^{(0)}

*+*

_{a}*δµ*(

_{a}**r**). where

*µ*a is the constant value of the absorption coefficient in the fluid while

^{(0)}*δµ*(

_{a}**r**) represents a spatial fluctuation due to the target. Then the Green’s function satisfies the Dyson equation

*µ*=

_{a}*µ*

^{(0)}

*.*

_{a}*G*

_{0}satisfies Eq. (3) with the substitution

*µ*(

_{a}**r**)→

*µ*

^{(0)}

*and can be found analytically in an infinite slab [30*

_{a}**20**, 890–902 (2003). [CrossRef]

*G*(

**r**,

**r**

*), which appears in the right-hand side, can not be measured since the point*

_{s}**r**lies inside the medium. Since

*G*depends on

*δµ*, the right-hand side of (4) is, generally, a nonlinear functional of the latter. Several approximate linearization methods can be used to transform Eq. (4) to an equation which is linear in

_{a}*δµ*. We have used the first Rytov approximation, according to which, the right-hand side in (4) is replaced by the expression

_{a}*δµ*. We then define the data function as

_{a}*T*(

**r**

*,*

_{d}**r**

*)≡*

_{s}*I*(

**r**

*,*

_{d}**r**

*)/*

_{s}*I*

_{0}(

**r**

*,*

_{d}**r**

*) is the experimentally measurable transmission function. Here*

_{s}*I*

_{0}(

**r**

*,*

_{d}**r**

*) is the result of a measurement in a reference homogeneous medium with*

_{s}*δµ*=0. We then obtain the following linear integral equation:

_{a}**r**

*,*

_{d}**r**

*;*

_{s}**r**)=

*G*

_{0}(

**r**

*,*

_{d}**r**)

*G*

_{0}(

**r**,

**r**

*) is known analytically and the right-hand side*

_{s}*ϕ*(

**r**

*,*

_{d}**r**

*) is measured experimentally. Note that we have used the fact that*

_{s}*I*(

**r**

*,*

_{d}**r**

*)/*

_{s}*I*

_{0}(

**r**

*,*

_{d}**r**

*)=*

_{s}*G*(

**r**

*,*

_{d}*r*)/

_{s}*G*

_{0}(

**r**

*,*

_{d}**r**

*) to eliminate the coupling constants*

_{s}*C*and

_{d}*C*. Equations (5) and (6) are the main equations of linearized DOT. The method we have used for inverting (6) is briefly described below.

_{s}### 2.4. Inversion formula

*G*

_{0}as

*g*(

**q**;

*z*,

*z*′) are given in Ref. [30

**20**, 890–902 (2003). [CrossRef]

*g*(

**q**;

*z*,

*z*′) and

*G*

_{0}(

*,*

**ρ***z*;

*′,*

**ρ***z*′) are real for continuous-wave measurements. We substitute this plane-wave decomposition of

*G*

_{0}into (6) and take a four-dimensional Fourier transform with respect to the source and detector transverse coordinates to obtain

*ψ*(

**q**,

**p**)=

*(*ϕ ˜

**q**+

**p**,-

**p**) and

*κ*(

**q**,

**p**;

*z*)=

*cg*(

**q**+

**p**;

*z*,

_{s}*z*)

*g*(-

^{p};

*z*,

*z*).

_{d}*κ*(

**q**,

**p**;

*z*). We do so for a discrete set of samples of the Fourier variable q and then use the inverse Fourier transform to obtain

*δµ*(

_{a}**ρ**,

*z*). Note that Eq. (11) is inverted for each sample of

**q**by utilizing multiple discrete values of

**p**. To this end, we construct analytically the pseudoinverse to the linear integral operator of the left-hand side of (11). The mathematical details of this construction are given in Ref. [30

**20**, 890–902 (2003). [CrossRef]

*δµ*(

_{a}**ρ**,

*z*) by the inverse Fourier transform accounting for sampling and truncation of real-space data is discussed in Refs. [25

25. V. A. Markel and J. C. Schotland, “Effects of sampling and limited data in optical tomography,” Appl. Phys. Lett. **81**, 1180–1182 (2002). [CrossRef]

**70**, 056,616 (2004). [CrossRef]

**70**, 056,616 (2004). [CrossRef]

*O*(

*N*

_{q}N^{3}

*), where*

_{p}*N*is the number of discrete values of the vector

_{q}**q**, while

*N*is the number of discrete values of

_{p}**p**used to invert each one-dimensional equation (11). This is the computational cost of inverting the integral operator only. To this one should add the cost of Fourier-transforming the real-space data function and computing

^{ψ}(

**q**,

**p**). With the use of the fast Fourier transform, the latter scales as

*O*(

*N*

^{2}

*ln*

_{q}*N*). If the same amount of data and the same grid of voxels is used with a purely algebraic image reconstruction method, the computational cost of matrix inversion scales as

_{q}*O*((

*N*)

_{q}N_{p}^{3}). In the above estimate, we have assumed that the number of measurements is equal to the number of voxels. We thus see that the fast image reconstruction methods exploit the block structure of the linear operator that couples the data to the unknown function. Instead of inverting a large matrix of size

*N*, these methods require inversion of

_{q}N_{p}*N*matrices of the size

_{q}*N*each, thus gaining a factor of

_{p}*N*

^{2}

*in computation time. We note that reconstructions which utilized data sets of up to 10*

_{p}^{7}source-detector pairs required less than one minute of CPU time on a 1.3GHz workstation.

### 2.5. Simulated data

*δµ*(

_{a}**r**) as a delta-function. Data for infinite lattices of sources and detectors was computed directly in the Fourier domain using (8). For finite lattices we computed

*G*

_{0}numerically using (7) and substituted the result into (9); in the latter formula, summation over real space variables was truncated.

*I*

_{0}to have the same maximum values as were experimentally measured. The resulting data function with shot noise was then

*R*is a random variable with a Gaussian distribution and a variance of one. We note that since

*I*

_{0}≫

*ϕ*, the shot noise depends primarily on the reference intensity, not on the data function itself. In order to simulate the effect of read noise and dark current, we determined the stardard deviation σ

_{bg}of a background image taken with the light source blocked. This error was propagated as

## 3. Experimental results

*I*

_{0}when the target is not present (scattering fluid only), the intensity

*I*when the target is present, and the Rytov data,

*ϕ*=-log(

*I*/

*I*

_{0}), which is used in the reconstruction algorithm (see

*Methods*). Note that the letters cannot be identified by simply inspecting images of the transmitted light. Structure is visible in the Rytov data when the letters “PENN” are close to the detector plane (Fig. 2(a)). However, when the letters “DOT” are in the center of the tank (Fig. 2(b)) their shape is completely blurred.

^{7}measurements. The effect of changing the size of the data set was investigated by sampling the detectors on a grid with a step size of 2 mm, five times larger than the minimum experimentally available detector spacing. We found that increasing the number of detectors up to the experimentally available maximum did not improve image quality. That is, reconstructions performed using 2mmsource separations or with a denser sampling of CCD pixels were visually indistinguishable from those in Fig. 3. However,

*decreasing*the number of data points does result in poorer image quality, as is illustrated in Fig. 4. From left to right, three separate reconstructions of the 8 mm bar target are shown. The data for these reconstructions were taken from a single experiment with the target positioned in the center of the tank. All reconstruction parameters are kept constant, except for the number of measurements used. The reconstruction on the left uses 8×10

^{6}measurements with sources and detectors sampled with 4 mm and 2 mm steps, respectively. In the center reconstruction, we use 4 mm spacing for both the sources and detectors, which corresponds to 2×10

^{6}data points. In the reconstruction on the right, the sampling is 8 mm for sources and 4 mm for detectors, or, approximately, 5×10

^{5}data points. It can be seen that as the number of measurements decreases, image artifacts become more prominent and resolution is lost. We note that the optimal number measurements for a given experiment will vary depending on factors such as experimental geometry and noise level. For example, smaller/larger data sets would be optimal if the reconstruction field of view was smaller/larger.

*µ*/

_{a}*µ*

^{(0)}

*is plotted against the expected contrast. It can be seen that the absorption is quantitatively reconstructed with a linear dependence on ink concentration over nearly a decade in absorption contrast. Deviation from linearity occurs at higher concentrations, as expected.*

_{a}## 4. Discussion

### 4.1. Transverse resolution and Fourier analysis of data

*z*

_{0}-

*h*/2<

*z*<

*z*

_{0}+

*h*/2, where

*h*is small. We seek to reconstruct the deviation of the absorption coefficient in this slice,

*δµ*(

_{a}*,*

**ρ***z*

_{0}), from the respective value in the surrounding fluid, as a function of the transverse variable

*=(*

**ρ***x*,

*y*). The formula (11) derived in the

*Methods*Section becomes in this case

*ψ*(

**q**,

**p**) is the measurable data function,

*κ*(

**q**,

**p**) is known analytically (see

*Methods*for precise definitions of

*ψ*and

*κ*), and

*δµ*(

_{a}**q**,

*z*

_{0}) is the transverse Fourier transform of

*δµ*(

_{a}*,*

**ρ***z*

_{0}) with respect to

*. The real-space function*

**ρ***δµ*(

_{a}*,*

**ρ***z*

_{0}) is obtained by inverse Fourier transformation, namely,

**p**in the above formula is arbitrary since the problem is overdetermined (we use four-dimensional data to reconstruct a two-dimensional function); it is sufficient to choose

**p**=0. We then use the data function

*ψ*(

**q**,0) for reconstruction. The latter is taken from experiment and contains noise. It is important to note that the “ideal” data function is rapidly (exponentially) decreasing with |

**q**|. However, the experimental noise is approximately white, i.e., the amplitude of its Fourier transform is approximately constant in the range of |

**q**| which is of interest. In image reconstruction, integration in the above formula is over a disc |

**q**| <

*q*

_{max}such that outside of this disc, the signal-to-noise ratio in

*ψ*(

**q**,0) becomes smaller than unity. It then follows that the minimum spatial feature that can be resolved has a characteristic transverse dimension of Δ

_{x}~

*π*/

*q*

_{max}.

*π*/

*q*

_{max}. Second, one can attempt to use additional degrees of freedom (i.e.,

*N*distinct values of

**p**) to improve the resolution. This approach, however, is not expected to yield a significant improvement since the noise amplitude decreases no faster than 1/√

*N*, while the function

*ψ*(

**q**,

**p**) decreases exponentially with |

**q**|.

*ψ*(

**q**,0) is in this case cylindrically symmetric and we can write

*ψ*(

**q**,0)=Ψ(

*q*), where

*q*=|

**q**|. The five curves shown in Fig. 6 contain (i) simulated ideal data corresponding to infinitely large and dense grids of noiseless sources and detectors, (ii) simulated data (without noise) for the finite grids of sources and detectors that were used in the experiments, (iii) simulated data for the finite grids with background noise added (i.e. Gaussian distributed noise with a variance equal to the variance of the detected signal when the laser is off), (iv) simulated data for the finite grid with shot noise added (i.e. Gaussian distributed noise with a variance equal to the number of photoelectrons detected experimentally), and (v) data from an experiment with a small (4 mm) highly absorbing target in the tank (see

*Methods*for details of the simulations). The early onset of noise in the simulated data with shot noise, suggests the latter is the limiting factor in our experiments. We thus conclude that the other errors discussed above (e.g. the finite numbers of sources and detectors, CCD noise, errors in the diffusion model, non-ideal optics, and non-linearity) do not play a significant role in limiting image resolution. We see that noise begins to dominate the data at

*q*

_{max}/

*π*≈0.1 mm

^{-1}. This corresponds to a spatial resolution of ℓ1 cm. For objects closer to the surface, the power spectrum decays more slowly. For example, for a point absorber located 1 cm from the surface, shot noise begins to dominate the data at

*q*

_{max}/

*π*≈0.25 mm

^{-1}, which corresponds to a spatial resolution of ℓ4 mm.

### 4.2. Titration experiments

## 5. Conclusion

## Acknowledgments

## References and links

1. | M. C.W. van Rossum and T. M. Nieuwenhuizen, “Multiple scattering of classical waves: microscopy, mesoscopy and diffusion,” Rev. Mod. Phys. |

2. | A. Yodh and B. Chance, “Spectroscopy and imaging with diffuse light,” Phys. Today pp. 34–40 (1995). [CrossRef] |

3. | S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems |

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

5. | S. B. Colak, M. B. van der Mark, G. W. Hooft, J. H. Hoogenraad, E. S. van der Linden, and F. A. Kuijpers, “Clinical optical tomography and NIR spectroscopy for breast cancer detection,” IEEE J. Sel. Top. Quantum Electron. |

6. | D. J. Hawrysz and E. M. Sevick-Muraca, “Developments toward diagnostic breast cancer imaging using near-infrared optical measurements and fluorescent contrast agants,” Neoplasia |

7. | Y. Xu, X. J. Gu, L. L. Fajardo, and H. B. Jiang, “In vivo breast imaging with diffuse optical tomography based on higher-order diffusion equations,” Appl. Opt. |

8. | J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. |

9. | Q. I. Zhu, M. M. Huang, N. G. Chen, K. Zarfos, B. Jagjivan, M. Kane, P. Hedge, and S. H. Kurtzman, “Ultrasound-guided optical tomographic imaging of malignant and benign breast lesions: Initial clinical results of 19 cases,” Neoplasia |

10. | X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. |

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12. | S. D. Jiang, B. W. Pogue, T. O. McBride, M. M. Doyley, S. P. Poplack, and K. D. Paulsen, “Near-infrared breast tomography calibration with optoelastic tissue simulating phantoms,” J. Electron. Imaging |

13. | D. A. Benaron, J. van Houten, D. C. Ho, S. D. Spilman, and D. K. Stevenson, “Imaging neonatal brain injury using light-based optical tomography,” Pediatr. Res. |

14. | J. P. van Houten, W. F. Cheong, E. L. Kermit, T. R. Machold, D. K. Stevenson, and D. A. Benaron, “Clinical measurement of brain oxygenation and function using light-based optical tomography,” Pediatr. Res. |

15. | A. Y. Bluestone, G. Abdoulaev, C. H. Schmitz, R. L. Barbour, and A. H. Hielscher, “Three-dimensional optical tomography of hemodynamics in the human head,” Opt. Express |

16. | G. Strangman, D. A. Boas, and J. P. Sutton, “Non-invasive neuroimaging using near-infrared light,” Biol. Psychiatry |

17. | J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, “Three-dimensional optical tomography of the premature infant brain,” Phys. Med. Biol. |

18. | J. C. Hebden, “Advances in optical imaging of the newborn infant brain,” Psychophysiology |

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

20. | A. M. Siegel, J. P. Culver, J. B. Mandeville, and D. A. Boas, “Temporal comparison of functional brain imaging with diffuse optical tomography and fMRI during rat forepaw stimulation,” Phys. Med. Biol. |

21. | G. Q. Yu, T. Durduran, D. Furuya, J. H. Greenberg, and A. G. Yodh, “Frequency-domain multiplexing system for in vivo diffuse light measurements of rapid cerebral hemodynamics,” Appl. Opt. |

22. | J. C. Schotland, “Continuous wave diffusion imaging,” J. Opt. Soc. Am. A |

23. | V. A. Markel and J. C. Schotland, “Inverse scattering for the diffusion equation with general boundary conditions,” Phys. Rev. E |

24. | V. A. Markel and J. C. Schotland, “The inverse problem in optical diffusion tomography. II. Inversion with boundary conditions,” J. Opt. Soc. Am. A |

25. | V. A. Markel and J. C. Schotland, “Effects of sampling and limited data in optical tomography,” Appl. Phys. Lett. |

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

27. | G. Turner, G. Zacharakis, A. Soubret, J. Ripoll, and V. Ntziachristos, “Complete-angle projection diffuse optical tomography by use of early photons,” Opt. Lett. |

28. | R. Schulz, J. Ripoll, and V. Ntziachristos, “Noncontact optical tomography of turbid media,” Opt. Lett. |

29. | Z.-M. Wang, G. Y. Panasyuk, V. A. Markel, and J. C. Schotland, “Experimental demonstration of an analytic method for image reconstruction in optical tomography with large data sets,” Opt. Lett. |

30. | V. A. Markel, V. Mital, and J. C. Schotland, “The inverse problem in optical diffusion tomography. III. Inversion formulas and singular value decomposition,” J. Opt. Soc. Am. A |

31. | V. A. Markel and J. C. Schotland, “Symmetries, inversion formulas and image reconstruction for optical tomography,” Phys. Rev. E |

32. | R. Choe, “Diffuse Optical Tomography and Spectroscopy of Breast Cancer and Fetal Brain,” Ph.D. thesis, University of Pennsylvania (2005). |

33. | A. Ishimaru, |

34. | R. Aronson, “Boundary conditions for diffuse light,” J. Opt. Soc. Am. A |

**OCIS Codes**

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

(110.0113) Imaging systems : Imaging through turbid media

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: January 28, 2008

Revised Manuscript: March 24, 2008

Manuscript Accepted: March 24, 2008

Published: March 28, 2008

**Virtual Issues**

Vol. 3, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Soren D. Konecky, George Y. Panasyuk, Kijoon Lee, Vadim Markel, Arjun G. Yodh, and John C. Schotland, "Imaging complex structures with diffuse light," Opt. Express **16**, 5048-5060 (2008)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-16-7-5048

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### References

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