## 3D level set reconstruction of model and experimental data in Diffuse Optical Tomography

Optics Express, Vol. 18, Issue 1, pp. 150-164 (2010)

http://dx.doi.org/10.1364/OE.18.000150

Acrobat PDF (1554 KB)

### Abstract

The level set technique is an implicit shape-based image reconstruction method that allows the recovery of the location, size and shape of objects of distinct contrast with well-defined boundaries embedded in a medium of homogeneous or moderately varying background parameters. In the case of diffuse optical tomography, level sets can be employed to simultaneously recover inclusions that differ in their absorption or scattering parameters from the background medium. This paper applies the level set method to the three-dimensional reconstruction of objects from simulated model data and from experimental frequency-domain data of light transmission obtained from a cylindrical phantom with tissue-like parameters. The shape and contrast of two inclusions, differing in absorption and diffusion parameters from the background, respectively, are reconstructed simultaneously. We compare the performance of level set reconstruction with results from an image-based method using a Gauss-Newton iterative approach, and show that the level set technique can improve the detection and localisation of small, high-contrast targets.

© 2010 Optical Society of America

## 1. Introduction

4. A. Zacharopoulos, M. Schweiger, V. Kolehmainen, and S. R. Arridge, “3D shape based reconstruction of experimental data in diffuse optical tomography,” Opt. Express **17**, 18940–18956 (2009). [CrossRef]

5. M. E. Kilmer, E. L. Miller, A. Barbaro, and D. Boas, “Three-dimensional shape-based imaging of absorption perturbation for diffuse optical tomography,” Appl. Opt. **42**, 3129–3144 (2003). [CrossRef] [PubMed]

*level set*technique, where the zero contours

*ψ*(

**r**) = 0 of a level set function

*ψ*defined over the domain Ω define the boundary between regions with a step-like parameter contrast. Level set methods are a class of implicit shape methods, i.e. the topology of the problem is not predefined and can change during the image reconstruction process. An application for a shape-only recovery in electromagnetic tomography has been previously presented in [6

6. O. Dorn, E. L. Miller, and C. Rappaport, “A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets,” Inverse Probl. **16**, 1119–1156 (2000). [CrossRef]

*et al*[7

7. E. T. Chung, T. F. Chan, and X.-C. Tai, “Electrical impedance tomography using level set representation and total variational regularization,” J. Comp. Phys. **205**, 357–372 (2005). [CrossRef]

8. G. Bal and K. Ren, “Reconstruction of singular surfaces by shape sensitivity analysis and level set method,” Math. Mod. Methods Appl. Sci. **16**, 1347–1373 (2006). [CrossRef]

*et al*[9

9. N. Irishina, M. Moscoso, and O. Dorn, “Microwave imaging for early breast cancer detection using a shape-based strategy,” IEEE Trans. Biomed. Eng. **56**, 1143–1153 (2009). [CrossRef] [PubMed]

*et al*used an iterative approach for recovering brain activation images that alternated between a level set step and an image-based step [10

10. M. Jacob, Y. Bresler, V. Toronov, X. Zhang, and A. Webb, “Level-set algorithm for the reconstruction of functional activation in near-infrared spectroscopic imaging,” J. Biomed. Opt. **11**, 064029-1 – 064029-12 (2006). [CrossRef]

*et al*[11]. A recent review is given in Dorn et al. [12

12. O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Probl. **22**, R67–R131 (2006). [CrossRef]

13. M. Schweiger, S. R. Arridge, O. Dorn, A. Zacharopoulos, and V. Kolehmainen, “Reconstructing absorption and diffusion shape profiles in optical tomography by a level set technique,” Opt. Lett. **31**, 471–473 (2006). [CrossRef] [PubMed]

14. M. Schweiger, O. Dorn, and S. R. Arridge, “3-D shape and contrast reconstruction in optical tomography with level sets,” in “First International Congress of the International Association of Inverse Problems (IPIA),” J. Phys.: Conf. Ser. **124**, 012043. Institute of Physics, 2007. [CrossRef]

## 2. Method

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

16. A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. **26**, 1698–1707 (1999). [CrossRef] [PubMed]

17. K. Ren, G. S. Abdoulaev, G. Bal, and A. Hielscher, “Algorithm for solving the equation of radiative transfer in the frequency domain,” Opt. Lett. **29**, 578–580 (2004). [CrossRef] [PubMed]

18. T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Hybrid radiative-transfer-diffusion model for optical tomography,” Appl. Opt. **44**, 876–886 (2005). [CrossRef] [PubMed]

19. K. D. Paulsen and H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. **22**, 691–701 (1995). [CrossRef] [PubMed]

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

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

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

_{3}, bounded by ∂Ω, the diffusion equation in the frequency domain is given by

*ω*∈ ℝ

^{+}is the modulation frequency, Φ is the photon density field,

*c*=

*c*

_{0}/

*n*is the speed of light in a medium with refractive index

*n*,

*c*

_{0}is the vacuum speed of light,

*q*is a source term formulated as an incident flux,

*ζ*is a boundary term which incorporates the refractive index mismatch at the surface,

*ν*is the outward normal at ∂Ω, and

*κ*and

*μ*are the diffusion and absorption coefficients, respectively. In the following we assume each source

_{a}*q*to be a radio-frequency modulated signal at a frequency

_{i}*ω*

_{0}:

*u*defines a source intensity profile over ∂Ω centered at source position

_{i}**m**

_{i}^{(S)}. The boundary measurements

**M**

*obtained from an experimental data acquisition system are represented in the model by the integral*

_{ij}**y**

*of the outgoing diffuse exitance Γ for source*

_{ij}*i*,

*W*on the boundary, centered at detector position

_{j}**m**

_{j}^{(D)}, where Γ is given by

*x*

_{1}(

**r**),

*x*

_{2}(

**r**)} = {

*κ*(

**r**),

*μ*(

_{a}**r**)} are discretised into a finite-dimensional vector of basis coefficients

**x**= {

**x**

_{1},

**x**

_{2}} ∈ ℝ

^{N}by a suitable basis expansion

*b*, and

_{k}**x**

*= {*

_{l}*x*} is the set of coefficients for the expansion of

_{lk}*x*(

_{l}**r**).

23. M. Schweiger and S. R. Arridge, “Image reconstruction in optical tomography using local basis functions,” J. Electron. Imaging **12**, 583–593 (2003). [CrossRef]

**x**̂, given a set of measurement data

**M**, can then be expressed in terms of the minimisation of an

*objective function*𝓙

**,**

_{M}**that describes the difference between model data**

_{M}**y**and measurement data

**M**

*Q*is a regularisation functional multiplied by the hyperparameter

*τ*.

24. S. R. Arridge and M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express **2**, 213–226 (1998). [CrossRef] [PubMed]

25. R. Roy and E. M. Sevick-Muraca, “Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part I theory and formulation,” Opt. Express **4**, 353–371 (1999). [CrossRef] [PubMed]

26. A. D. Klose and A. H. Hielscher, “Quasi-Newton methods in optical tomographic image reconstruction,” Inverse Probl. **19**, 387–409 (2003). [CrossRef]

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

*Q*is required to impose additional constraints on the solutions. Typically,

*Q*applies local smoothness conditions on the solution. Positivity of the parameter distributions

*x*can be enforced by mapping to logarithmic values.

_{l}*object region S*and a

_{l}*background region*Ω\

*S*for each parameter

_{l}*x*(

_{l}**r**),

*l*∈ {1,2}, where

*S*

_{1}and

*S*

_{2}need not coincide.

*x*is of the form

_{l}*x*

_{l,i}and

*x*

_{l,e}are constants. If their values are known, the task of the reconstruction is to recover the shapes

*S*. If

_{l}*x*

_{l,i,e}are not or only partially known a-priori, they can be added as unknowns to the reconstruction problem. Typically, the background parameters

*x*

_{l,e}may be assumed to be known, while the inclusion contrasts

*x*

_{l,i}are to be reconstructed.

*S*with the level set technique[6

_{l}6. O. Dorn, E. L. Miller, and C. Rappaport, “A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets,” Inverse Probl. **16**, 1119–1156 (2000). [CrossRef]

*ψ*such that

_{l}*D*= ∂

_{l}*S*of object

_{l}*S*is defined by the zero level set of the level set function

_{l}*ψ*. To solve the shape reconstruction problem, we will adopt a time evolution approach [29]. As a consequence,

_{l}*D*and

_{l}*ψ*will be functions of an artificial evolution time

_{l}*t*, i.e.

*D*(

_{l}*t*) = {

**r**:

*ψ*(

_{l}**r**,

*t*) = 0}.

*ψ*that minimise the shape least squares cost functional

_{l}**(**

_{M}*ψ*) = 𝓡

_{l}**(**

_{M}*x*(

_{l}*ψ*)). For a combined shape and object contrast reconstruction, the object contrast also evolves with time,

_{l}*x*

_{l,i}=

*x*

_{l,i}(

*t*), and in addition to level set functions

*ψ*we find interior parameter values

_{l}*x*

_{l,i}that minimise the least squares cost functional

**(**

_{M}*ψ*,

_{l}*x*

_{l,i}) = 𝓡

**(**

_{M}*x*(

_{l}*ψ*,

_{l}*x*

_{l,i})). We want to derive an evolution equation for the unknown parameters

*ψ*

_{1}(

*t*),

*ψ*

_{2}(

*t*), and optionally

*x*

_{1,i}(

*t*) and

*x*

_{2,i}(

*t*) which reduces (and eventually minimizes) the above defined least squares cost functional. We are therefore looking for forcing terms

*f*

_{1}(

**r**,

*t*),

*f*

_{2}(

**r**,

*t*), and optionally

*g*

_{1}(

*t*), and

*g*

_{2}(

*t*) such that the system

13. M. Schweiger, S. R. Arridge, O. Dorn, A. Zacharopoulos, and V. Kolehmainen, “Reconstructing absorption and diffusion shape profiles in optical tomography by a level set technique,” Opt. Lett. **31**, 471–473 (2006). [CrossRef] [PubMed]

_{M}^{(s)}and 𝓙

_{M}^{(s+c)}with respect to the evolution time to be

*(*

_{l}*x*)

_{l}^{*}is the adjoint Fréchet derivative, and

*δ*(

*ψ*) =

_{l}*H*′(

*ψ*) is the one-dimensional Dirac delta distribution. In the following numerical reconstructions the expressions 𝓡

_{l}*′(*

_{l}*x*)

_{l}^{*}𝓡 are determined using an adjoint scheme [31

31. S. R. Arridge, “Photon measurement density functions. Part 1: Analytical forms,” Appl. Opt. **34**, 7395–7409 (1995). [CrossRef] [PubMed]

*δ*(

*ψ*) > 0, we can define descent directions

_{l}*f*

_{l,d}and

*g*

_{l,d}for

*f*and

_{l}*g*as

_{l}*l*= 1, 2 which satisfy the descent flow condition (12) for 𝓙.

*x*

_{l,e}(

**r**) such that

*x*is not in the range of the model ℱ(

_{l}*x*).

_{l}*Q*to the cost term (7), we perform an implicit regularisation by applying a smoothing operator 𝓟 to the forcing terms in Eq. 13 which has the effect of smoothing the boundaries of the shapes during the evolution by projecting the updates towards a smoother subspace[32

32. P. González-Rodriguez, M. Kindelan, M. Moscoso, and O. Dorn, “History matching problem in reservoir engineering using the propagation back-projection method,” Inverse Probl. **21**, 565–590 (2005). [CrossRef]

*α*> 0 and

*β*> 0 can be chosen freely. Discretizing (11), (13) by a straightforward finite difference time-discretization with time-step Δ

*t*> 0 and interpreting

*ψ*

_{l}^{(n)}=

*ψ*(

_{l}*t*),

*ψ*

_{l}^{(n+1)}=

*ψ*(

_{l}*t*+ Δ

*t*),

*f*

_{l,d}

^{(n)}=

*f*

_{l,d}(

*t*) and

*g*

_{l,d}

^{(n)}=

*g*

_{l,d}(

*t*), we arrive at the iteration

## 3. Results

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

*x*,

*y*,

*z*)|(

*x*

^{2}+

*y*

^{2})

^{1/2}≤ 35 mm and |

*z*| ≤ 55 mm}. The background parameters were homogeneous with

*x*

_{1,e}(

**r**) =

*μ*

_{a,e}= 0.0078mm

^{-1}and

*x*

_{2,e}(

**r**) =

*κ*= 0.31 mm. The two inclusions had cylindrical shape with radius 9.5 mm and height 9.5 mm, located in the central plane of the phantom, as shown in Fig. 1. One of the inclusions was centered at position (-17.5,0,0) and had an increased absorption coefficient of

_{e}*x*

_{1,i}= 0.0156mm

^{-1}, the other was centered at position (30.3,-17.5,0) and had a decreased diffusion coefficient of

*x*

_{2,i}= 0.155 mm. The refractive index was homogeneous over the cylinder with

*n*= 1.56.

*z*= ±6 mm. Each ring contained 8 sources and 8 detectors, indicated with ‘

*x*’ and ‘o’ in Fig. 1. For the simulations, both the source profiles

*u*, and detector profiles

_{i}*W*were assumed to be Gaussian with width

_{j}*σ*= 2 mm. The boundary measurements consisted of the logarithmic amplitude and phase shift for a modulation frequency of

*ω*

_{0}= 2

*π*× 10

^{8}Hz.

**M**

^{(sig)}in the presence of the inclusions, reference data

**M**

^{(bkg)}were obtained on a homogeneous object with

*x*=

_{l}*x*

_{l,e}. The reference data were then used to generate synthetic data

**M**̃ with

**M**̃ was used to evaluate the misfit operator (Eq. 8) for the level set reconstructions. Difference reconstructions can eliminate systematic differences between model and experiment. This includes uncertainties in signal amplitude, coupling losses, errors in the outer boundary of the object, or, in general, unknown model errors of any kind. Practical applications often require the reconstruction of parameter differences from measurements taken at different subject states, for example during stimulus and rest, or at different times to follow treatment success, or measurements at different wavelengths to classify the object type.

### 3.1. Reconstructions from simulated data

*σ*= 10%, 20% and 50% of the mean background value. The background distributions were assumed to be identical for signal and reference measurements. Fig. 2 shows cross sections of one sample of absorption and diffusion distribution for each of the noise levels.

_{p}**a) Reconstructions for shape only**. In this section, we present results of level set difference reconstructions from simulated data, where object contrast was assumed known and fixed (shape-only reconstruction). Simulated measurement data were generated for the homogeneous background case, as well as for all 50 samples of each of the three background noise levels indicated in Fig. 2. Level set reconstructions were performed for each data set.

*shape error ε*as the relative volume of mislabeled regions,

**Reconstructions for shape and inclusion contrast**The results of simultaneous reconstructions for shapes and inclusion contrasts

*x*

_{l,i}for level set reconstruction from difference data are shown in Fig. 6. Displayed are the cross sections through the image means and variances over the 50 background noise realisations. Contrast evolutions for one sample of each background noise level are shown in Fig. 7, and cross sections through the objects of those samples are shown in Fig. 8. Again, the inclusion locations and shapes are recovered well for all background noise realisations, with comparable results, although a small artefact can be seen in the diffusion image of the reconstruction from homogeneous background. After 400 level set iterations, the reconstruction from homogeneous background data recovers the absorption contrast very well, but slightly underestimates the diffusion contrast. For noisy backgrounds, the recovered contrast values exhibit more variation, although the shape recovery is little affected.

**c) Comparison with Gauss-Newton-Krylov image-based reconstructions**To estimate the benefits of the level set approach for object identification, we compare the results with reconstructions from an image-based reconstruction of parameter values. The solver in this case employs a damped Gauss-Newton approach using a Krylov linear solver (DGN-K) that supports an implicit definition of the Hessian matrix [22

**50**, 2365–2386 (2005). [CrossRef] [PubMed]

^{2}iterations for the shape-only reconstruction, and 4 × 10

^{2}iterations for the shape and contrast reconstruction.

### 3.2. Reconstructions from experimental phantom data

33. M. Firbank and D. T. Delpy, “A design for a stable and reproducible phantom for use in near infrared imaging and spectroscopy,” Phys. Med. Biol. **38**, 847–853 (1993). [CrossRef]

_{2}particles and an infrared dye to provide scattering and absorption properties. Phantom geometry and parameters, as well as the measurement arrangement, corresponded to the simulated data in Section 3.1. The frequency-domain data acquisition instrument used to collect the measurement data was developed at Helsinki University of Technology [34

34. I. Nissilä, K. Kotilahti, K. Fallström, and T. Katila, “Instrumentation for the accurate measurement of phase and amplitude in optical tomography,” Rev. Sci. Instrum. **73**, 3306–3312 (2002). [CrossRef]

35. I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, “Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography,” Rev. Sci. Instrum. **76** (2005). Art. no. 044302. [CrossRef]

**a) Level set reconstructions.**Signal and reference data were used to reconstruct parameter differences from data differences. Two reconstructions were performed: (i), shape only, assuming a-priori known background parameters and object contrasts, and (ii) shape and inclusion contrast, assuming known background parameters.

**b) Comparison with Gauss-Newton-Krylov image-based reconstructions.**For comparison of the shape-based level set method with a voxel-based method, we performed a reconstruction using the DGN-K solver with the phantom difference data. The right column of images in Figure 11 shows the reconstruction results, displayed with the same gray-scale range as the level set reconstructions. Line profiles through the target objects are shown in Fig. 12. The reconstruction does recover the target locations for the absorption and diffusion inclusions, but underestimates the contrast in particular for absorption. In addition, comparison of the profiles with the level set results shows less well-defined target boundaries and some cross-talk in the diffusion image.

## 4. Conclusions

## Acknowlegments

## References and links

1. | M. Cope and D. T. Delpy, “System for long term measurement of cerebral blood and tissue oxygenation on newborn infants by near infrared transillumination,” Med. Biol. Eng. Comput. |

2. | D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Sig. Proc. Magazine |

3. | B. W. Pogue, K. D. Paulsen, C. Abele, and H. Kaufman, “Calibration of near-infrared frequency-domain tissue spectroscopy for absolute absorption coefficient quantitation in neonatal head-simulating phantoms,” J. Biomed. Opt. |

4. | A. Zacharopoulos, M. Schweiger, V. Kolehmainen, and S. R. Arridge, “3D shape based reconstruction of experimental data in diffuse optical tomography,” Opt. Express |

5. | M. E. Kilmer, E. L. Miller, A. Barbaro, and D. Boas, “Three-dimensional shape-based imaging of absorption perturbation for diffuse optical tomography,” Appl. Opt. |

6. | O. Dorn, E. L. Miller, and C. Rappaport, “A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets,” Inverse Probl. |

7. | E. T. Chung, T. F. Chan, and X.-C. Tai, “Electrical impedance tomography using level set representation and total variational regularization,” J. Comp. Phys. |

8. | G. Bal and K. Ren, “Reconstruction of singular surfaces by shape sensitivity analysis and level set method,” Math. Mod. Methods Appl. Sci. |

9. | N. Irishina, M. Moscoso, and O. Dorn, “Microwave imaging for early breast cancer detection using a shape-based strategy,” IEEE Trans. Biomed. Eng. |

10. | M. Jacob, Y. Bresler, V. Toronov, X. Zhang, and A. Webb, “Level-set algorithm for the reconstruction of functional activation in near-infrared spectroscopic imaging,” J. Biomed. Opt. |

11. | X.-C. Tai and T. F. Chan, “A survey on multiple level set methods with applications for identifying piecewise constant functions,” Int. J. Numer. Anal. Mod. |

12. | O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Probl. |

13. | M. Schweiger, S. R. Arridge, O. Dorn, A. Zacharopoulos, and V. Kolehmainen, “Reconstructing absorption and diffusion shape profiles in optical tomography by a level set technique,” Opt. Lett. |

14. | M. Schweiger, O. Dorn, and S. R. Arridge, “3-D shape and contrast reconstruction in optical tomography with level sets,” in “First International Congress of the International Association of Inverse Problems (IPIA),” J. Phys.: Conf. Ser. |

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

16. | A. D. Klose and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys. |

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

18. | T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Hybrid radiative-transfer-diffusion model for optical tomography,” Appl. Opt. |

19. | K. D. Paulsen and H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. |

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

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

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

23. | M. Schweiger and S. R. Arridge, “Image reconstruction in optical tomography using local basis functions,” J. Electron. Imaging |

24. | S. R. Arridge and M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express |

25. | R. Roy and E. M. Sevick-Muraca, “Truncated Newton’s optimization scheme for absorption and fluorescence optical tomography: Part I theory and formulation,” Opt. Express |

26. | A. D. Klose and A. H. Hielscher, “Quasi-Newton methods in optical tomographic image reconstruction,” Inverse Probl. |

27. | J. Nocedal and S. Wright, |

28. | S. Osher and R. Fedkiw, |

29. | F. Santosa, “A level-set approach for inverse problems involving obstacles,,” in “ESAIM: Control, Optimization and Calculus of Variations,”, vol. 1 (1996), vol. 1, pp. 17–22. |

30. | J. A. Sethian, |

31. | S. R. Arridge, “Photon measurement density functions. Part 1: Analytical forms,” Appl. Opt. |

32. | P. González-Rodriguez, M. Kindelan, M. Moscoso, and O. Dorn, “History matching problem in reservoir engineering using the propagation back-projection method,” Inverse Probl. |

33. | M. Firbank and D. T. Delpy, “A design for a stable and reproducible phantom for use in near infrared imaging and spectroscopy,” Phys. Med. Biol. |

34. | I. Nissilä, K. Kotilahti, K. Fallström, and T. Katila, “Instrumentation for the accurate measurement of phase and amplitude in optical tomography,” Rev. Sci. Instrum. |

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

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

**ToC Category:**

Image Processing

**History**

Original Manuscript: October 20, 2009

Revised Manuscript: December 10, 2009

Manuscript Accepted: December 11, 2009

Published: December 22, 2009

**Virtual Issues**

Vol. 5, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

M. Schweiger, O. Dorn, A. Zacharopoulos, I. Nissila, and S. R. Arridge, "3D level set reconstruction of model and experimental data in Diffuse Optical Tomography," Opt. Express **18**, 150-164 (2010)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-1-150

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### References

- M. Cope and D. T. Delpy, "System for long term measurement of cerebral blood and tissue oxygenation on newborn infants by near infrared transillumination," Med. Biol. Eng. Comput. 26, 289-294 (1988). [CrossRef] [PubMed]
- D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, "Imaging the body with diffuse optical tomography," IEEE Sig. Proc. Magazine 18, 57-75 (2001). [CrossRef]
- B. W. Pogue, K. D. Paulsen, C. Abele, and H. Kaufman, "Calibration of near-infrared frequency-domain tissue spectroscopy for absolute absorption coefficient quantitation in neonatal head-simulating phantoms," J. Biomed. Opt. 5, 185-193 (2000). [CrossRef] [PubMed]
- A. Zacharopoulos, M. Schweiger, V. Kolehmainen, and S. R. Arridge, "3D shape based reconstruction of experimental data in diffuse optical tomography," Opt. Express 17, 18940-18956 (2009). [CrossRef]
- M. E. Kilmer, E. L. Miller, A. Barbaro, and D. Boas, "Three-dimensional shape-based imaging of absorption perturbation for diffuse optical tomography," Appl. Opt. 42, 3129-3144 (2003). [CrossRef] [PubMed]
- O. Dorn, E. L. Miller, and C. Rappaport, "A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets," Inverse Probl. 16, 1119-1156 (2000). [CrossRef]
- E. T. Chung, T. F. Chan, and X.-C. Tai, "Electrical impedance tomography using level set representation and total variational regularization," J. Comp. Phys. 205, 357-372 (2005). [CrossRef]
- G. Bal and K. Ren, "Reconstruction of singular surfaces by shape sensitivity analysis and level set method," Math. Mod. Methods Appl. Sci. 16, 1347-1373 (2006). [CrossRef]
- N. Irishina, M. Moscoso, and O. Dorn, "Microwave imaging for early breast cancer detection using a shape-based strategy," IEEE Trans. Biomed. Eng. 56, 1143-1153 (2009). [CrossRef] [PubMed]
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