## Parametric level set reconstruction methods for hyperspectral diffuse optical tomography |

Biomedical Optics Express, Vol. 3, Issue 5, pp. 1006-1024 (2012)

http://dx.doi.org/10.1364/BOE.3.001006

Acrobat PDF (1117 KB)

### Abstract

A parametric level set method (PaLS) is implemented for image reconstruction for hyperspectral diffuse optical tomography (DOT). Chromophore concentrations and diffusion amplitude are recovered using a linearized Born approximation model and employing data from over 100 wavelengths. The images to be recovered are taken to be piecewise constant and a newly introduced, shape-based model is used as the foundation for reconstruction. The PaLS method significantly reduces the number of unknowns relative to more traditional level-set reconstruction methods and has been show to be particularly well suited for ill-posed inverse problems such as the one of interest here. We report on reconstructions for multiple chromophores from simulated and experimental data where the PaLS method provides a more accurate estimation of chromophore concentrations compared to a pixel-based method.

© 2012 OSA

## 1. Introduction

1. S. Fantini, E. L. Heffer, V. E. Pera, A. Sassaroli, and N. Liu, “Spatial and spectral information in optical mammography,” Technol. Cancer Res. Treat. **4**, 471–482 (2005). [PubMed]

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6. S. Kukreti, A. E. Cerussi, W. Tanamai, D. Hsiang, B. J. Tromberg, and E. Gratton, “Characterization of metabolic differences between benign and malignant tumors: high-spectral-resolution diffuse optical spectroscopy,” Radiology **254**, 277–284 (2010). [CrossRef]

8. H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt. **42**, 135–145 (2004). [CrossRef]

3. 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 Signal Process. Mag. **18**(6), 57–75 (2001). [CrossRef]

7. A. Li, G. Boverman, Y. Zhang, D. Brooks, E. L. Miller, M. E. Kilmer, Q. Zhang, E. M. C. Hillman, and D. A. Boas, “Optimal linear inverse solution with multiple priors in diffuse optical tomography,” Appl. Opt. **44**, 1948–1956 (2005). [CrossRef] [PubMed]

9. J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: a singular-value analysis,” Opt. Lett. **26**, 701–703 (2001). [CrossRef]

10. F. Larusson, S. Fantini, and E. L. Miller, “Hyperspectral image reconstruction for diffuse optical tomography,” Biomed. Opt. Express **2**, 947–965 (2011). [CrossRef]

12. K. D. Paulsen and H. Jiang, “Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization,” Appl. Opt. **35**, 3447–3458, (1996). [CrossRef] [PubMed]

8. H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt. **42**, 135–145 (2004). [CrossRef]

13. G. Boverman, Q. Fang, S. A. Carp, E. L. Miller, D. H. Brooks, J. Selb, R. H. Moore, D. B. Kopans, and D. A. Boas, “Spatio-temporal imaging of the hemoglobin in the compressed breast with diffuse optical tomography,” Phys. Med. Biol. **52**, 3619–3641 (2007). [CrossRef] [PubMed]

8. H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt. **42**, 135–145 (2004). [CrossRef]

10. F. Larusson, S. Fantini, and E. L. Miller, “Hyperspectral image reconstruction for diffuse optical tomography,” Biomed. Opt. Express **2**, 947–965 (2011). [CrossRef]

11. B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and K. D. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. **38**, 2950–2961 (1999). [CrossRef]

10. F. Larusson, S. Fantini, and E. L. Miller, “Hyperspectral image reconstruction for diffuse optical tomography,” Biomed. Opt. Express **2**, 947–965 (2011). [CrossRef]

**2**, 947–965 (2011). [CrossRef]

14. D. A. Boas, “A fundamental limitation of linearized algorithms for diffuse optical tomography,” Opt. Express **1**, 404–413 (1997). [CrossRef] [PubMed]

15. A. Aghasi, M. Kilmer, and E. L. Miller “Parametric level set methods for inverse problems,” SIAM J. Imaging Sci. **4**, 618–650 (2011). [CrossRef]

19. M. Schweiger, O. Dorn, and S. R. Arridge, “3-D shape and contrast reconstruction in optical tomography with level sets,” J. Phys.: Conf. Ser. **124**, 012043 (2008). [CrossRef]

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

21. S. R. Arridge, O. Dorn, V. Kolehmainen, M. Schweiger, and A. Zacharopoulos, “Parameter and structure reconstruction in optical tomography,” J. Phys.: Conf. Ser. **135**, 012001 (2008). [CrossRef]

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

19. M. Schweiger, O. Dorn, and S. R. Arridge, “3-D shape and contrast reconstruction in optical tomography with level sets,” J. Phys.: Conf. Ser. **124**, 012043 (2008). [CrossRef]

21. S. R. Arridge, O. Dorn, V. Kolehmainen, M. Schweiger, and A. Zacharopoulos, “Parameter and structure reconstruction in optical tomography,” J. Phys.: Conf. Ser. **135**, 012001 (2008). [CrossRef]

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

21. S. R. Arridge, O. Dorn, V. Kolehmainen, M. Schweiger, and A. Zacharopoulos, “Parameter and structure reconstruction in optical tomography,” J. Phys.: Conf. Ser. **135**, 012001 (2008). [CrossRef]

11. B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and K. D. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. **38**, 2950–2961 (1999). [CrossRef]

19. M. Schweiger, O. Dorn, and S. R. Arridge, “3-D shape and contrast reconstruction in optical tomography with level sets,” J. Phys.: Conf. Ser. **124**, 012043 (2008). [CrossRef]

15. A. Aghasi, M. Kilmer, and E. L. Miller “Parametric level set methods for inverse problems,” SIAM J. Imaging Sci. **4**, 618–650 (2011). [CrossRef]

15. A. Aghasi, M. Kilmer, and E. L. Miller “Parametric level set methods for inverse problems,” SIAM J. Imaging Sci. **4**, 618–650 (2011). [CrossRef]

**4**, 618–650 (2011). [CrossRef]

4. M. Schweiger and S. R. Arridge, “Optical tomographic reconstruction in a complex head model using a priori region boundary information,” Phys. Med. Biol. **44**, 2703–2721 (1999). [CrossRef] [PubMed]

**2**, 947–965 (2011). [CrossRef]

## 2. Forward problem

2. R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol **45**, 1051–1069 (2000). [CrossRef] [PubMed]

*λ*through tissue can be adequately approximated using a diffusion model [23

23. M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. **20**, 426–428 (1995). [CrossRef]

24. B. Brooksby, B. W. Pogue, S. Jiang, H. Dehghani, S. Srinivasan, C. Kogel, T. D. Tosteson, J. Weaver, S. P. Poplack, and K. D. Paulsen, “Imaging breast adipose and fibroglandular tissue molecular signatures by using hybrid mri-guided near-infrared spectral tomography,” Proc. Natl. Acad. Sci. U.S.A. **103**, 8828–8833 (2006). [CrossRef] [PubMed]

**r**,

*λ*) is the photon fluence rate at position

**r**due to light of wavelength

*λ*injected into the medium,

*v*is the electromagnetic propagation velocity in the medium,

*S*(

**r**,

*λ*) is the photon source with units of optical energy per unit time per unit volume. For the work in this paper the sources are considered to be delta sources in space and can be written as

*S*(

**r**,

*λ*) =

*S*

_{0}(

*λ*)

*δ*(

**r**−

**r**

*) with*

_{s}*S*

_{0}(

*λ*) the source power at wavelength

*λ*. For spatially varying scattering we assume that the diffusion coefficient

*D*

^{0}(

*r*,

*λ*) follows Mie scattering theory where a scattering prefactor Ψ depends on the size and density of scatterers while a scattering exponent

*b*depends on the size of the scatterers. Using this, we write the perturbation in the diffusion coefficient as The arbitrarily chosen reference wavelength

*λ*

_{0}is introduced to achieve a form of the Mie model where Ψ has the units of mm

^{−1}and Ψ′ has units of mm. In this paper, for simplicity we consider the case where the background medium is infinite and homogeneous. Generalization to the more practical case where there are boundaries is straightforward in theory though somewhat more complex in terms of implementation [25

25. D. Boas, M. A. O’Leary, B. Chance, and A. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A. **91**, 4887–4891 (1994). [CrossRef] [PubMed]

*jω*/

*D*(

*λ*) term on the right hand side, in our case we consider

*ω*= 0 giving us the form shown in Eq. (1).

*D*

^{0}(

**r**,

*λ*), as the sum of a constant background absorption,

*μ*(

_{a}*λ*), and a spatially varying perturbation Δ

*μ*(

_{a}**r**,

*λ*) as well as constant background diffusion

*D*(

*λ*) and a diffusion perturbation Δ

*D*(

**r**,

*λ*). To obtain a linear relationship between the measurements and the chromophore concentrations, we subtract Eq. (1) from the perturbed version of the diffusion equation which gives where

*k*

^{2}(

**r**,

*λ*) = (

*v*/

*D*(

*λ*))Δ

*μ*(

_{a}**r**,

*λ*). Assuming the availability of a Green’s function,

*G*(

**r**,

**r**′,

*λ*) for the solution of Eq. (3) as is the case for an unbounded medium as well as range of bounded problems [26], we can change Eq. (3) into an integral Eq. [27

27. B. Brendel, R. Ziegler, and T. Nielsen “Algebraic reconstruction techniques for spectral reconstruction in diffuse optical tomography,” Appl. Opt. **47**, 6392–6403 (2008). [CrossRef] [PubMed]

**r**

*is the location of the detector and (with a small abuse of notation) Φ*

_{d}*(*

_{i}**r**,

**r**

*,*

_{s}*λ*) is used here to denote the incident field at position

**r**and wavelength

*λ*due to a delta-source located at

**r**

*. It should also be noted that we obtain this equation under the assumption that the total fluence rate, Φ, can be approximated as the incident field, Φ*

_{s}*, since Φ*

_{i}*≫ Φ*

_{i}*[2*

_{s}2. R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol **45**, 1051–1069 (2000). [CrossRef] [PubMed]

*μ*is decomposed as follows [28

_{a}28. A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. M. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. **44**, 2082–2093 (2005). [CrossRef] [PubMed]

*N*is the number of absorbing species for the problem under investigation,

_{s}*ε*(

_{k}*λ*) is the extinction coefficient for the

*k*species at wavelength

^{th}*λ*, and

*c*(

_{k}**r**) is the concentration of species

*k*at location

**r**. To obtain the fully discrete form of the Born model used in Section 4, we expand each

*c*(

_{k}**r**) where

*c*

_{k,j}is the value of the concentration for species

*k*in

*V*, the

_{j}*j*“voxel”. The

^{th}*φ*(

**r**) function is an indicator function where After using Eqs. (5) and (6) in Eq. (4) and performing some algebra we obtain We approximate Eq. (4) as the value at the center of each pixel multiplied by the area or volume of each pixel or voxel, so in Eq. (8) we use

*a*as the area of a pixel. This setup is illustrated in Fig. 1(a).

**c**

*∈ *

_{k}^{Nv}as the vector obtained by lexicographically ordering the unknown concentrations associated with the

*k*chromophore and

^{th}**c**

_{k+1}= ΔΨ′ and Φ

*(*

_{s}*λ*) is the vector of observed scattered fluence rate associated with all source-detector pairs collecting data at wavelength

*λ*. Now, with

*N*the number of wavelengths used in a given experiment, Eq. (8) is written in matrix-vector notation as

_{λ}*m*,

*j*)

*element of the*

^{th}*v*/

*D*(

*λ*))

_{l}*G*(

**r**

*,*

_{m}**r**

*,*

_{j}*λ*)Φ

_{l}*(*

_{i}**r**

*,*

_{j}**r**

*,*

_{m}*λ*), where

_{l}*m*represents the

*m*source-detector pair and as before

^{th}*j*represents the

*j*voxel. For

^{th}*v*/

*D*(

*λ*))∇

_{l}*G*(

**r**

*,*

_{m}**r**

*,*

_{j}*λ*) ·∇Φ

*(*

_{i}**r**

*,*

_{j}**r**

*,*

_{m}*λ*)Δ

*D*(

_{j}*λ*).

*N*matrices

_{λ}**K**

*and*

^{a}**K**

*and store it along with the*

^{d}*N*×

_{λ}*N*extinction coefficients. This reduces the amount of memory required for the reconstruction.

_{c}## 3. Parametric level-set method

29. A. Cerussi, D. Hsiang, N. Shah, R. Mehta, A. Durkin, J. Butler, and B. J. Tromberg “Predicting response to breast cancer neoadjuvant chemotherapy using diffuse optical spectroscopy,” Proc. Nat. Acad. Sci. U.S.A. **104**, 4014–4019 (2007). [CrossRef]

31. Q. Zhu, P. U. Hegde, A. Ricci, M. Kane, E. B. Cronin, Y. Ardeshirpour, C. Xu, A. Aguirre, S. H. Kurtzman, P. J. Deckers, and S. H. Tannenbaum, “Early-stage invasive breast cancers: potential role of optical tomography with US localization in assisting diagnosis,” Radiology **256**, 367–378 (2010). [CrossRef] [PubMed]

*k*= 1, 2,...,

*N*+ 1. In this formulation the unknown values are the constant concentration values of the anomaly and background,

_{c}*χ*(

*x*,

*y*) is defined to be the zero level set of a Lipschitz continuous object function 𝒪 : → such that 𝒪 > 0 in Ω(

*x*,

*y*), 𝒪 < 0 in Ω\ and 𝒪(

*x*,

*y*) = 0 in

*∂*Ω. Using 𝒪(

*x*,

*y*),

*χ*(

*x*,

*y*) is written as where

*H*is the step function. In practice we use smooth approximations of the step function and the Dirac delta function denoted as

*H*and

_{ε}*δ*respectively where

_{ε}*H*is computed as and

_{ε}*δ*is computed as the derivative of

_{ε}*H*[32

_{ε}32. T. Chan and L. Vese “Active contours without edges,” IEEE Trans. Image Process. **10**, 266–277 (2001). [CrossRef]

33. H. K. Zhao, S. Osher, B. Merriman, and M. Kang, “Implicit, nonparametric shape reconstruction from unorganized points using a variational level set method,” Comput. Vision Image Understanding **80**, 295–314 (2000). [CrossRef]

*x*,

*y*) parametrically, so instead of using a dense collection of pixel or voxel values [34], we represent it by using basis functions where

*a*’s are the weight coefficients whereas

_{i}*p*(

_{i}*x*,

*y*) are the functions which belong to the basis set of 𝒫 = {

*p*

_{1},

*p*

_{2},...,

*p*}. Possible choices for the 𝒫 basis set include polynomial or radial basis functions. For the purpose of this paper we use Gaussian basis function. The width and number of the Gaussians determines how coarse or fine the reconstruction will be, where a choice of few basis functions will, on the one hand, result in a reduced number of unknowns, it will on the other hand, give a coarser estimation of the shape, which can be a problem for imaging finer more complex structures. In the DOT case, where the physics in the forward model will only allow for a coarse reconstruction of the underlying structure, the Gaussian approach is sufficient, especially for the relatively simple geometries and concentrations presented in this paper. When we will move on to a more complicated non-linear model, the choice of the number and type of basis functions will be more important and should be based on a rigid mathematical framework. This is discussed further in Section 8.

_{l}**a**= [

*a*

_{1},...,

*a*]

_{L}*. Now our forward model in Eq. (9) can be expressed as The forward model has now been parametrized with a vector containing all of the unknowns, which are far fewer then what a pixel based method would attempt to estimate.*

^{T}## 4. Inverse problem

*to recover the value of*

_{s}**c**, is solved as a Levenberg-Marquardt optimization problem of the form The

**W**matrix reflects the structure of the noise corrupting the data [2

2. R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol **45**, 1051–1069 (2000). [CrossRef] [PubMed]

35. M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. **50**, 2837–2858 (2005). [CrossRef] [PubMed]

11. B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and K. D. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. **38**, 2950–2961 (1999). [CrossRef]

*m*elements of Φ,

^{th}**W**is constructed as a diagonal matrix with 1/

*σ*the

_{m}*m*element along the diagonal. For the experimental and simulated data the variance is calculated from where Ω(

^{th}*m*) corresponds to the photon count for each source-detector pair. The SNR for each element of Φ is then calculated from In experimental data

*ε*as

**J**is required. The Jacobian contains derivatives of

*ε*with respect to each element in the parameter vector

*θ*The solution is then obtained by updating

*θ*at each iteration as

*θ*

^{n+1}=

*θ*+

^{n}**h**where

**h**is the solution to the following linear system where

**I**is the identity matrix,

*ρ*is the damping parameter affecting the size and direction of

**h**and found via and appropriate line search algorithm [36]. The stopping criteria used when iterating Eq. (21) is the discrepancy principle [37

37. C. R. Vogel, *Computational Methods for Inverse Problems* (SIAM, 2002) [CrossRef]

38. T. T. Wu and K. Lange, “Coordinate descent algorithms for lasso penalized regression,” Ann. Appl. Stat. **2**, 224–244 (2008). [CrossRef]

**J**

*and*

_{v}**J**

*denote the Jacobian strictly for the concentration value and shape, respectively, and*

_{s}*τ*represents a tolerance for the stopping critera.

_{i}## 5. Simulation analysis

28. A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. M. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. **44**, 2082–2093 (2005). [CrossRef] [PubMed]

_{2}and HbR respectively, along with lipid and water concentration and diffusion amplitude. These chromophores are chosen since they mainly cause near-infrared absorption in breast tissue [39

39. B. Brendel and T. Nielsen, “Selection of optimal wavelengths for spectral reconstruction in diffuse optical tomography,” J. Biomed. Opt. **14**, 034041 (2009). [CrossRef] [PubMed]

_{2}and HbR concentrations than normal tissue [40

40. S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. **100**, 12349–12354 (2003). [CrossRef] [PubMed]

*b*are obtained from [41

41. P. Taroni, A. Pifferi, A. Torricelli, D. Comelli, and R. Cubeddu, “In vivo absorption and scattering spectroscopy of biological tissues,” Photochem. Photobiol. Sci. **2**, 124–129 (2003). [CrossRef] [PubMed]

*μ*are calculated from extinction coefficients, which are in the units cm

_{a}^{−1}/mM and are obtained from data tabulated by Scott Prahl [42

42. S. Prahl, “Tabulated molar extinction coefficient for hemoglobin in water” (Oregon Medical Laser Center, 2007), http://omlc.ogi.edu/spectra/hemoglobin/summary.html.

_{2}and HbR. For water and lipid the concentrations are in percent by volume and the diffusion amplitude is measured in units of millimeter. The background has HbR concentration of 0.01 mM, HbO

_{2}concentration of 0.01 mM, lipid concentration of 32%, water concentration of 13% and Ψ′ is set to 1.6 mm. The target concentration of the object of interest is set to 0.015 mM, 0.012 mM, 50 %, 20 % and 0.25 mm for Hb0

_{2}, HbR, lipid, water and ΔΨ′, respectively.

29. A. Cerussi, D. Hsiang, N. Shah, R. Mehta, A. Durkin, J. Butler, and B. J. Tromberg “Predicting response to breast cancer neoadjuvant chemotherapy using diffuse optical spectroscopy,” Proc. Nat. Acad. Sci. U.S.A. **104**, 4014–4019 (2007). [CrossRef]

30. H. Soliman, A. Gunasekara, M. Rycroft, J. Zubovits, R. Dent, J. Spayne, M. J. Yaffe, and G. Czarnota, “Functional imaging using diffuse optical spectroscopy of neoadjuvant chemotherapy response in women with locally advanced breast cancer,” Clin. Cancer Res. **16**, 2605–2614 (2010). [CrossRef] [PubMed]

*a*’s weight coefficients are initialized to 1.

_{i}**c**are the simulated concentration images for all chromophores and diffusion amplitude, whereas

**n**represents additive noise. Specifically, as in [2

**45**, 1051–1069 (2000). [CrossRef] [PubMed]

**n**is a vector of zero mean, independent Gaussian random variables with variances

*k*chromophore, the mean square error is computed by using the following Eq.

^{th}43. A. A. Joshi, A. J. Chaudhari, D. W. Shattuck, J. Dutta, R. M. Leahy, and A. W. Toga, “Posture matching and elastic registration of a mouse atlas to surface topography range data,” in IEEE International Symposium on Biomedical Imaging: from Nano to Macro, 2009. ISBI ’09 (IEEE, 2009), pp. 366–369 (2009).

*S*is the reconstructed image and

*G*is the ground truth created for each set, the Dice coefficient between

*S*and

*G*can be defined as |

*S*∩

*G|*contains all pixels that belong to the detected segment as well as the ground truth segment, so that if

*S*and

*G*are equal the Dice coefficient is equal to one, indicating an accurate reconstruction. To compute the

*D*(

*S*,

*G*) we use the characteristic function,

*χ*, which essentially works as a binary map of the reconstructed anomaly where the object of interest is represented by 1’s.

## 6. Experimental analysis

**2**, 947–965 (2011). [CrossRef]

*μ*= 0.029 cm

_{a}^{−1}, at 600 nm, which is in the range of optical absorption of the female breast [45

45. T. Durduran, R. Choe, J. P. culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol. **47**, 2847–2861 (2002). [CrossRef] [PubMed]

46. L. Spinelli, A. Torricelli, A. Pifferi, P. Taroni, G. M. Danesini, and R. Cubeddu, “Bulk optical properties and tissue components in the female breast from multiwavelength time-resolved optical mammography,” J. Biomed. Opt. **9**, 1137–1142 (2004). [CrossRef] [PubMed]

47. 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. **30**, 235–247 (2003). [CrossRef] [PubMed]

48. N. Liu, A. Sassaroli, and S. Fantini, “Paired-wavelength spectral approach to measuring the relative concentrations of two localized chromophores in turbid media: an experimental study,” J. Biomed. Opt. **12**, 051602 (2007). [CrossRef] [PubMed]

*μ*and

_{a}*D*(

*λ*) the background has to be known. In the experimental measurements we assume constant scattering, therefore we are not trying to estimate the perturbation, Δ

*D*(

**r**,

*λ*), as was done in simulations. Therefore we have the unperturbed representaion of the reduced scattering coefficient,

*μ*′

*, is given by The diffusion coefficient relates to the reduced scattering coefficient through,*

_{s}*D*(

*λ*) =

*v*/3

*μ*′

*. Phase, amplitude and average intensity data are obtained at two wavelengths using a frequency-domain tissue spectrometer to estimate the Ψ and*

_{s}*b*parameters in Eq. (25) as Ψ = 6.5 cm

^{−1}and

*b*= 0.4. This allows us to extrapolate values for

*μ*′

*at any wavelength [49*

_{s}49. E. Gratton, S. Fantini, M. A. Franceschini, C. Gratton, and M. Fabiani, “Measurements of scattering and absorption changes in muscle and brain,” Philos. Trans. R. Soc. Lond. B. Biol. Sci. **352**, 727–735 (1997). [CrossRef] [PubMed]

*μ*does not follow a law like

_{a}*μ*′

*, values are estimated using extinction coefficient data for ink, dye, milk and water. These extinction coefficient are measured in a standard spectrophotometer.*

_{s}50. B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, S. Srinivasan, X. Song, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Characterization of hemoglobin, water, and nir scattering in breast tissue: analysis of intersubject variability and menstrual cycles changes,” J. Biomed. Opt. **9**, 541–552 (2004). [CrossRef] [PubMed]

*λ*= [480, 550, 610, 630, 650, 690] nm. The wavelengths are chosen around the isosbestic point, where the contrast between the chromophores is the highest and where each chromophore has highest absorption. The absorption spectra and the contrast over the spectrum for set 1 and set 2 are shown in Fig. 3 and Fig. 4, respectively.

^{2}. For three different locations, at x ± 1 cm, a 5 mm diameter collection optical glass fiber bundle is placed on the opposite side of the inclusions (at a y-axis separation of 5 cm). Experiments are made with the light source placed in succession at 8 positions with 1 cm increments for total of 24 source-detector pairs. To ensure that approximately the same amount of photons are collected for both hyperspectral and multispectral reconstructions, two exposure times are used for the CCD camera, a longer one of 10 s for the 6 wavelength case and 500 ms for the 126 wavelength case. Since the goal of this paper is to demonstrate the improvement of including hyperspectral information, we present an ideal case where the signal to noise ratio is large, thereby providing a best-case scenario for the few wavelength reconstruction against which we compare our approach as well as using realistic absorption contrasts for the inclusions. Further details on the experimental setup can be found at [10

**2**, 947–965 (2011). [CrossRef]

*in vivo*measurements it is possible to use

*a priori*structural information from other modalities, e.g. MRI, to estimate the incident field by determining the optical properties of the assumed piecewise constant chromophore distribution over these segments [51

51. G. Boverman, E. L. Miller, A. Li, Q. Zhang, T. Chaves, D. H. Brooks, and D. A. Boas, “Quantitative spectroscopic diffuse optical tomography of the breast guided by imperfect a priori structural information,” Phys. Med. Biol. **50**, 3941–3956 (2005). [CrossRef] [PubMed]

52. S. Fantini, M. A. Franceschini, J. S. Maier, S. A. Walker, B. Barbieri, and E. Gratton, “Frequency-domain multi-channel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng. **34**, 32–42 (1995). [CrossRef]

53. M. A. Franceschini, V. Toronov, M. E. Filiaci, E. Gratton, and S. Fantini, “On-line optical imaging of the human brain with 160-ms temporal resolution,” Opt. Express **6**, 49–57 (2000). [CrossRef] [PubMed]

## 7. Results

### 7.1. Simulations

*λ*= [660, 734, 760, 808, 826, 850, 930, 980] nm and hyperspectral reconstruction using 176 wavelengths, which are equally spaced over the 650–1000 nm range. In the 8 wavelength case the 6 first wavelengths are optimally chosen according to [54

54. M. E. Eames, B. W. Pogue, and H. Dehghani, “Wavelength band optimization in spectral near-infrared optical tomography improves accuracy while reducing data acquisition and computational burden,” J. Biomed. Opt. **13**, 054037 (2008). [CrossRef] [PubMed]

*x*axis but rather diffuse results in

*y*. We also see noticeable artifacts in the reconstructions. Considering the concentration values, the values for HbO

_{2}, HbR and water concentration come close to the actual value. Moving to hyperspectral information, the reconstruction becomes more accurate, estimating the shape close to the ground truth. It should also be noted that the runtime for each reconstruction for the PaLS method is significantly shorter compared to the pixel-based method. A PaLS reconstruction takes around 30 seconds, which is 3–4 times faster than a pixel-based method. Additionally, we do not employ any regularization parameters, freeing us from finding the optimal reconstruction using regularization. This is a major improvement in moving from a pixel-based approach to the PaLS method.

*D*(

*S*,

*G*) for the pixel based reconstructions using a threshold of 80% to

*D*(

*S*,

*G*) of the PaLS reconstructions. The improvement of the PaLS method is confirmed quantitatively through

*D*(

*S*,

*G*) and MSE displayed in Table 2 and Table 1, respectively. The Dice coefficient, shown in Table 2, gives a clear view of how the shape estimation improves by added wavelengths, where

*D*(

*S*,

*G*) approaches 1 for the hyperspectral case and the PaLS method shows superior performance in the MSE values.

### 7.2. Experimental validation

**2**, 947–965 (2011). [CrossRef]

*x*axis. This is somewhat unexpected since in DOT resolving depth information, on the

*y*axis, is usually the more difficult problem. This is noticeable for both the pixel based and PaLS methods, although the PaLS method outperforms the pixel based method, especially in removing edge artifacts. This streaking in the

*x*direction is most likely a combination of how the Gaussian basis are placed within the imaging medium, and measurement error in placing the source and detectors when taking the reference measurement.

*D*(

*S*,

*G*) between pixel based reconstructions and the PaLS method. For the experimental reconstructions we use a threshold of 50% to compare

*D*(

*S*,

*G*) of the PaLS reconstructions. This demonstrates the usefulness of the PaLS method correctly and accurately localizing the anomaly. The PaLS method does very well with eliminating edge artifacts that were severe when doing pixel-based reconstructions for the same data set. These effects are very noticeable in Fig. 7(b) and (d), where, especially in the multispectral case, the edge artifacts were significant. Comparing that to the same data in Fig. 8(b) and (d) it is obvious that the improvement is significant.

## 8. Conclusion

## Acknowledgments

## References and links

1. | S. Fantini, E. L. Heffer, V. E. Pera, A. Sassaroli, and N. Liu, “Spatial and spectral information in optical mammography,” Technol. Cancer Res. Treat. |

2. | R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol |

3. | 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 Signal Process. Mag. |

4. | M. Schweiger and S. R. Arridge, “Optical tomographic reconstruction in a complex head model using a priori region boundary information,” Phys. Med. Biol. |

5. | S. Fantini, D. Hueber, M. A. Franceschini, E. Gratton, W. Rosenfeld, P. G. Stubblefield, D. Maulik, and M. R. Stankovic, “Non-invasive optical monitoring of the newborn piglet brain using continuous-wave and frequency-domain spectroscopy,” Phys. Med. Biol. |

6. | S. Kukreti, A. E. Cerussi, W. Tanamai, D. Hsiang, B. J. Tromberg, and E. Gratton, “Characterization of metabolic differences between benign and malignant tumors: high-spectral-resolution diffuse optical spectroscopy,” Radiology |

7. | A. Li, G. Boverman, Y. Zhang, D. Brooks, E. L. Miller, M. E. Kilmer, Q. Zhang, E. M. C. Hillman, and D. A. Boas, “Optimal linear inverse solution with multiple priors in diffuse optical tomography,” Appl. Opt. |

8. | H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt. |

9. | J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: a singular-value analysis,” Opt. Lett. |

10. | F. Larusson, S. Fantini, and E. L. Miller, “Hyperspectral image reconstruction for diffuse optical tomography,” Biomed. Opt. Express |

11. | B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and K. D. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. |

12. | K. D. Paulsen and H. Jiang, “Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization,” Appl. Opt. |

13. | G. Boverman, Q. Fang, S. A. Carp, E. L. Miller, D. H. Brooks, J. Selb, R. H. Moore, D. B. Kopans, and D. A. Boas, “Spatio-temporal imaging of the hemoglobin in the compressed breast with diffuse optical tomography,” Phys. Med. Biol. |

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

15. | A. Aghasi, M. Kilmer, and E. L. Miller “Parametric level set methods for inverse problems,” SIAM J. Imaging Sci. |

16. | A. Aghasi, E. L. Miller, and L. M. Abriola “Characterization of source zone architecture: a joint electrical and hydrological inversion approach,” presented at 2011 Fall Meeting, AGU, San Francisco, Calif., 5–9 Dec. 2011. |

17. | F. Larusson, S. Fantini, and E. L. Miller, “Parametric level-set approach for hyperspectral diffuse optical tomography,” in 2011 IEEE International Symposium on Biomedical Imaging: from Nano to Macro (IEEE, 2011), pp. 949–955. |

18. | O. Dorn, “A shape reconstruction method for diffuse optical tomography using a transport model and level sets,” in 2002 IEEE International Symposium on Biomedical Imaging, 2002. Proceedings (IEEE, 2002), pp. 1015–1018. |

19. | M. Schweiger, O. Dorn, and S. R. Arridge, “3-D shape and contrast reconstruction in optical tomography with level sets,” J. Phys.: Conf. Ser. |

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

21. | S. R. Arridge, O. Dorn, V. Kolehmainen, M. Schweiger, and A. Zacharopoulos, “Parameter and structure reconstruction in optical tomography,” J. Phys.: Conf. Ser. |

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

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

24. | B. Brooksby, B. W. Pogue, S. Jiang, H. Dehghani, S. Srinivasan, C. Kogel, T. D. Tosteson, J. Weaver, S. P. Poplack, and K. D. Paulsen, “Imaging breast adipose and fibroglandular tissue molecular signatures by using hybrid mri-guided near-infrared spectral tomography,” Proc. Natl. Acad. Sci. U.S.A. |

25. | D. Boas, M. A. O’Leary, B. Chance, and A. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A. |

26. | A. Mandelis, |

27. | B. Brendel, R. Ziegler, and T. Nielsen “Algebraic reconstruction techniques for spectral reconstruction in diffuse optical tomography,” Appl. Opt. |

28. | A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. M. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. |

29. | A. Cerussi, D. Hsiang, N. Shah, R. Mehta, A. Durkin, J. Butler, and B. J. Tromberg “Predicting response to breast cancer neoadjuvant chemotherapy using diffuse optical spectroscopy,” Proc. Nat. Acad. Sci. U.S.A. |

30. | H. Soliman, A. Gunasekara, M. Rycroft, J. Zubovits, R. Dent, J. Spayne, M. J. Yaffe, and G. Czarnota, “Functional imaging using diffuse optical spectroscopy of neoadjuvant chemotherapy response in women with locally advanced breast cancer,” Clin. Cancer Res. |

31. | Q. Zhu, P. U. Hegde, A. Ricci, M. Kane, E. B. Cronin, Y. Ardeshirpour, C. Xu, A. Aguirre, S. H. Kurtzman, P. J. Deckers, and S. H. Tannenbaum, “Early-stage invasive breast cancers: potential role of optical tomography with US localization in assisting diagnosis,” Radiology |

32. | T. Chan and L. Vese “Active contours without edges,” IEEE Trans. Image Process. |

33. | H. K. Zhao, S. Osher, B. Merriman, and M. Kang, “Implicit, nonparametric shape reconstruction from unorganized points using a variational level set method,” Comput. Vision Image Understanding |

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

35. | M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. |

36. | K. Madsen, H. Bruun, and O. Tingleff “Methods for non-linear least squares problems,” lecture notes (2004). |

37. | C. R. Vogel, |

38. | T. T. Wu and K. Lange, “Coordinate descent algorithms for lasso penalized regression,” Ann. Appl. Stat. |

39. | B. Brendel and T. Nielsen, “Selection of optimal wavelengths for spectral reconstruction in diffuse optical tomography,” J. Biomed. Opt. |

40. | S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. |

41. | P. Taroni, A. Pifferi, A. Torricelli, D. Comelli, and R. Cubeddu, “In vivo absorption and scattering spectroscopy of biological tissues,” Photochem. Photobiol. Sci. |

42. | S. Prahl, “Tabulated molar extinction coefficient for hemoglobin in water” (Oregon Medical Laser Center, 2007), http://omlc.ogi.edu/spectra/hemoglobin/summary.html. |

43. | A. A. Joshi, A. J. Chaudhari, D. W. Shattuck, J. Dutta, R. M. Leahy, and A. W. Toga, “Posture matching and elastic registration of a mouse atlas to surface topography range data,” in IEEE International Symposium on Biomedical Imaging: from Nano to Macro, 2009. ISBI ’09 (IEEE, 2009), pp. 366–369 (2009). |

44. | J. D. Vylder and W. Philips, “A computational efficient external energy for active contour segmentation using edge propagation,” in IEEE 2100 International Conference on Image Processing (ICIP 2010) (IEEE, 2010), pp. 661–664. |

45. | T. Durduran, R. Choe, J. P. culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol. |

46. | L. Spinelli, A. Torricelli, A. Pifferi, P. Taroni, G. M. Danesini, and R. Cubeddu, “Bulk optical properties and tissue components in the female breast from multiwavelength time-resolved optical mammography,” J. Biomed. Opt. |

47. | 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. |

48. | N. Liu, A. Sassaroli, and S. Fantini, “Paired-wavelength spectral approach to measuring the relative concentrations of two localized chromophores in turbid media: an experimental study,” J. Biomed. Opt. |

49. | E. Gratton, S. Fantini, M. A. Franceschini, C. Gratton, and M. Fabiani, “Measurements of scattering and absorption changes in muscle and brain,” Philos. Trans. R. Soc. Lond. B. Biol. Sci. |

50. | B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, S. Srinivasan, X. Song, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Characterization of hemoglobin, water, and nir scattering in breast tissue: analysis of intersubject variability and menstrual cycles changes,” J. Biomed. Opt. |

51. | G. Boverman, E. L. Miller, A. Li, Q. Zhang, T. Chaves, D. H. Brooks, and D. A. Boas, “Quantitative spectroscopic diffuse optical tomography of the breast guided by imperfect a priori structural information,” Phys. Med. Biol. |

52. | S. Fantini, M. A. Franceschini, J. S. Maier, S. A. Walker, B. Barbieri, and E. Gratton, “Frequency-domain multi-channel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng. |

53. | M. A. Franceschini, V. Toronov, M. E. Filiaci, E. Gratton, and S. Fantini, “On-line optical imaging of the human brain with 160-ms temporal resolution,” Opt. Express |

54. | M. E. Eames, B. W. Pogue, and H. Dehghani, “Wavelength band optimization in spectral near-infrared optical tomography improves accuracy while reducing data acquisition and computational burden,” J. Biomed. Opt. |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

(170.3660) Medical optics and biotechnology : Light propagation in tissues

(170.3830) Medical optics and biotechnology : Mammography

(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

(290.1990) Scattering : Diffusion

(290.7050) Scattering : Turbid media

**ToC Category:**

Image Reconstruction and Inverse Problems

**History**

Original Manuscript: January 27, 2012

Manuscript Accepted: March 15, 2012

Published: April 18, 2012

**Citation**

Fridrik Larusson, Sergio Fantini, and Eric L. Miller, "Parametric level set reconstruction methods for hyperspectral diffuse optical tomography," Biomed. Opt. Express **3**, 1006-1024 (2012)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-3-5-1006

Sort: Year | Journal | Reset

### References

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- R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol45, 1051–1069 (2000). [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 Signal Process. Mag.18(6), 57–75 (2001). [CrossRef]
- M. Schweiger and S. R. Arridge, “Optical tomographic reconstruction in a complex head model using a priori region boundary information,” Phys. Med. Biol.44, 2703–2721 (1999). [CrossRef] [PubMed]
- S. Fantini, D. Hueber, M. A. Franceschini, E. Gratton, W. Rosenfeld, P. G. Stubblefield, D. Maulik, and M. R. Stankovic, “Non-invasive optical monitoring of the newborn piglet brain using continuous-wave and frequency-domain spectroscopy,” Phys. Med. Biol.44, 1543–1563 (1999). [CrossRef] [PubMed]
- S. Kukreti, A. E. Cerussi, W. Tanamai, D. Hsiang, B. J. Tromberg, and E. Gratton, “Characterization of metabolic differences between benign and malignant tumors: high-spectral-resolution diffuse optical spectroscopy,” Radiology254, 277–284 (2010). [CrossRef]
- A. Li, G. Boverman, Y. Zhang, D. Brooks, E. L. Miller, M. E. Kilmer, Q. Zhang, E. M. C. Hillman, and D. A. Boas, “Optimal linear inverse solution with multiple priors in diffuse optical tomography,” Appl. Opt.44, 1948–1956 (2005). [CrossRef] [PubMed]
- H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt.42, 135–145 (2004). [CrossRef]
- J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: a singular-value analysis,” Opt. Lett.26, 701–703 (2001). [CrossRef]
- F. Larusson, S. Fantini, and E. L. Miller, “Hyperspectral image reconstruction for diffuse optical tomography,” Biomed. Opt. Express2, 947–965 (2011). [CrossRef]
- B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and K. D. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt.38, 2950–2961 (1999). [CrossRef]
- K. D. Paulsen and H. Jiang, “Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization,” Appl. Opt.35, 3447–3458, (1996). [CrossRef] [PubMed]
- G. Boverman, Q. Fang, S. A. Carp, E. L. Miller, D. H. Brooks, J. Selb, R. H. Moore, D. B. Kopans, and D. A. Boas, “Spatio-temporal imaging of the hemoglobin in the compressed breast with diffuse optical tomography,” Phys. Med. Biol.52, 3619–3641 (2007). [CrossRef] [PubMed]
- D. A. Boas, “A fundamental limitation of linearized algorithms for diffuse optical tomography,” Opt. Express1, 404–413 (1997). [CrossRef] [PubMed]
- A. Aghasi, M. Kilmer, and E. L. Miller “Parametric level set methods for inverse problems,” SIAM J. Imaging Sci.4, 618–650 (2011). [CrossRef]
- A. Aghasi, E. L. Miller, and L. M. Abriola “Characterization of source zone architecture: a joint electrical and hydrological inversion approach,” presented at 2011 Fall Meeting, AGU, San Francisco, Calif., 5–9 Dec. 2011.
- F. Larusson, S. Fantini, and E. L. Miller, “Parametric level-set approach for hyperspectral diffuse optical tomography,” in 2011 IEEE International Symposium on Biomedical Imaging: from Nano to Macro (IEEE, 2011), pp. 949–955.
- O. Dorn, “A shape reconstruction method for diffuse optical tomography using a transport model and level sets,” in 2002 IEEE International Symposium on Biomedical Imaging, 2002. Proceedings (IEEE, 2002), pp. 1015–1018.
- M. Schweiger, O. Dorn, and S. R. Arridge, “3-D shape and contrast reconstruction in optical tomography with level sets,” J. Phys.: Conf. Ser.124, 012043 (2008). [CrossRef]
- M. E. Kilmer, E. L. Miller, A. Barbaro, and David Boas, “Three-dimensional shape-based imaging of absorption perturbation for diffuse optical tomography,” Appl. Opt.42, 3129–3144 (2003). [CrossRef] [PubMed]
- S. R. Arridge, O. Dorn, V. Kolehmainen, M. Schweiger, and A. Zacharopoulos, “Parameter and structure reconstruction in optical tomography,” J. Phys.: Conf. Ser.135, 012001 (2008). [CrossRef]
- O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Probl.22, R67–R131 (2006). [CrossRef]
- M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett.20, 426–428 (1995). [CrossRef]
- B. Brooksby, B. W. Pogue, S. Jiang, H. Dehghani, S. Srinivasan, C. Kogel, T. D. Tosteson, J. Weaver, S. P. Poplack, and K. D. Paulsen, “Imaging breast adipose and fibroglandular tissue molecular signatures by using hybrid mri-guided near-infrared spectral tomography,” Proc. Natl. Acad. Sci. U.S.A.103, 8828–8833 (2006). [CrossRef] [PubMed]
- D. Boas, M. A. O’Leary, B. Chance, and A. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A.91, 4887–4891 (1994). [CrossRef] [PubMed]
- A. Mandelis, Diffusion-Wave Fields: Mathematical Methods and Green Functions (Springer, 2001).
- B. Brendel, R. Ziegler, and T. Nielsen “Algebraic reconstruction techniques for spectral reconstruction in diffuse optical tomography,” Appl. Opt.47, 6392–6403 (2008). [CrossRef] [PubMed]
- A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. M. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt.44, 2082–2093 (2005). [CrossRef] [PubMed]
- A. Cerussi, D. Hsiang, N. Shah, R. Mehta, A. Durkin, J. Butler, and B. J. Tromberg “Predicting response to breast cancer neoadjuvant chemotherapy using diffuse optical spectroscopy,” Proc. Nat. Acad. Sci. U.S.A.104, 4014–4019 (2007). [CrossRef]
- H. Soliman, A. Gunasekara, M. Rycroft, J. Zubovits, R. Dent, J. Spayne, M. J. Yaffe, and G. Czarnota, “Functional imaging using diffuse optical spectroscopy of neoadjuvant chemotherapy response in women with locally advanced breast cancer,” Clin. Cancer Res.16, 2605–2614 (2010). [CrossRef] [PubMed]
- Q. Zhu, P. U. Hegde, A. Ricci, M. Kane, E. B. Cronin, Y. Ardeshirpour, C. Xu, A. Aguirre, S. H. Kurtzman, P. J. Deckers, and S. H. Tannenbaum, “Early-stage invasive breast cancers: potential role of optical tomography with US localization in assisting diagnosis,” Radiology256, 367–378 (2010). [CrossRef] [PubMed]
- T. Chan and L. Vese “Active contours without edges,” IEEE Trans. Image Process.10, 266–277 (2001). [CrossRef]
- H. K. Zhao, S. Osher, B. Merriman, and M. Kang, “Implicit, nonparametric shape reconstruction from unorganized points using a variational level set method,” Comput. Vision Image Understanding80, 295–314 (2000). [CrossRef]
- S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, 2002)
- M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol.50, 2837–2858 (2005). [CrossRef] [PubMed]
- K. Madsen, H. Bruun, and O. Tingleff “Methods for non-linear least squares problems,” lecture notes (2004).
- C. R. Vogel, Computational Methods for Inverse Problems (SIAM, 2002) [CrossRef]
- T. T. Wu and K. Lange, “Coordinate descent algorithms for lasso penalized regression,” Ann. Appl. Stat.2, 224–244 (2008). [CrossRef]
- B. Brendel and T. Nielsen, “Selection of optimal wavelengths for spectral reconstruction in diffuse optical tomography,” J. Biomed. Opt.14, 034041 (2009). [CrossRef] [PubMed]
- S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A.100, 12349–12354 (2003). [CrossRef] [PubMed]
- P. Taroni, A. Pifferi, A. Torricelli, D. Comelli, and R. Cubeddu, “In vivo absorption and scattering spectroscopy of biological tissues,” Photochem. Photobiol. Sci.2, 124–129 (2003). [CrossRef] [PubMed]
- S. Prahl, “Tabulated molar extinction coefficient for hemoglobin in water” (Oregon Medical Laser Center, 2007), http://omlc.ogi.edu/spectra/hemoglobin/summary.html .
- A. A. Joshi, A. J. Chaudhari, D. W. Shattuck, J. Dutta, R. M. Leahy, and A. W. Toga, “Posture matching and elastic registration of a mouse atlas to surface topography range data,” in IEEE International Symposium on Biomedical Imaging: from Nano to Macro, 2009. ISBI ’09 (IEEE, 2009), pp. 366–369 (2009).
- J. D. Vylder and W. Philips, “A computational efficient external energy for active contour segmentation using edge propagation,” in IEEE 2100 International Conference on Image Processing (ICIP 2010) (IEEE, 2010), pp. 661–664.
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