## Accelerated image reconstruction in fluorescence molecular tomography using dimension reduction |

Biomedical Optics Express, Vol. 4, Issue 1, pp. 1-14 (2013)

http://dx.doi.org/10.1364/BOE.4.000001

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### Abstract

With the development of charge-coupled device (CCD) camera based non-contact fluorescence molecular tomography (FMT) imaging systems, multi projections and densely sampled fluorescent measurements used in subsequent image reconstruction can be easily obtained. However, challenges still remain in fast image reconstruction because of the large computational burden and memory requirement in the inverse problem. In this work, an accelerated image reconstruction method in FMT using principal components analysis (PCA) is presented to reduce the dimension of the inverse problem. Phantom experiments are performed to verify the feasibility of the proposed method. The results demonstrate that the proposed method can accelerate image reconstruction in FMT almost without quality degradation.

© 2012 OSA

## 1. Introduction

*in vivo*imaging method with the advantage of non invasive and non ionizing imaging [1

1. V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. **23**(3), 313–320 (2005). [CrossRef] [PubMed]

2. E. E. Graves, R. Weissleder, and V. Ntziachristos, “Fluorescence molecular imaging of small animal tumour models,” Curr. Mol. Med. **4**(4), 419–430 (2004). [CrossRef] [PubMed]

3. N. C. Deliolanis, J. Dunham, T. Wurdinger, J. L. Figueiredo, T. Bakhos, and V. Ntziachristos, “In-vivo imaging of murine tumors using complete-angle projection fluorescence molecular tomography,” J. Biomed. Opt. **14**(3), 030509 (2009). [CrossRef] [PubMed]

4. M. Rudin and R. Weissleder, “Molecular imaging in drug discovery and development,” Nat. Rev. Drug Discov. **2**(2), 123–131 (2003). [CrossRef] [PubMed]

5. J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discovery **7**(7), 591–607 (2008). [CrossRef]

6. T. F. Massoud and S. S. Gambhir, “Molecular imaging in living subjects: seeing fundamental biological processes in a new light,” Genes Dev. **17**(5), 545–580 (2003). [CrossRef] [PubMed]

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

8. D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media,” Appl. Opt. **36**(10), 2260–2272 (1997). [CrossRef] [PubMed]

9. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. **20**(2), 299–309 (1993). [CrossRef] [PubMed]

11. J. Ripoll, M. Nieto-Vesperinas, R. Weissleder, and V. Ntziachristos, “Fast analytical approximation for arbitrary geometries in diffuse optical tomography,” Opt. Lett. **27**(7), 527–529 (2002). [CrossRef]

12. N. Deliolanis, T. Lasser, D. Hyde, A. Soubret, J. Ripoll, and V. Ntziachristos, “Free-space fluorescence molecular tomography utilizing 360° geometry projections,” Opt. Lett. **32**(4), 382–384 (2007). [CrossRef] [PubMed]

13. J. Ripoll, “Hybrid Fourier-real space method for diffuse optical tomography,” Opt. Lett. **35**(5), 688–690 (2010). [CrossRef] [PubMed]

14. T. J. Rudge, V. Y. Soloviev, and S. R. Arridge, “Fast image reconstruction in fluoresence optical tomography using data compression,” Opt. Lett. **35**(5), 763–765 (2010). [CrossRef] [PubMed]

15. N. Ducros, C. D. Andrea, G. Valentini, T. Rudge, S. Arridge, and A. Bassi, “Full-wavelet approach for fluorescence diffuse optical tomography with structured illumination,” Opt. Lett. **35**(21), 3676–3678 (2010). [CrossRef] [PubMed]

16. N. Ducros, A. Bassi, G. Valentini, M. Schweiger, S. Arridge, and C. D Andrea, “Multiple-view fluorescence optical tomography reconstruction using compression of experimental data,” Opt. Lett. **36**(8), 1377–1379 (2011). [CrossRef] [PubMed]

17. A. D. Zacharopoulos, P. Svenmarker, J. Axelsson, M. Schweiger, S. R. Arridge, and S. Andersson-Engels, “A matrix-free algorithm for multiple wavelength fluorescence tomography,” Opt. Express **17**(5), 3042–3051 (2009). [CrossRef]

18. A. D. Zacharopoulos, A. Garofalakis, J. Ripoll, S. R. Arridge, and S. Andersson-Engels, “Development of in-vivo fluorescence imaging with the Matrix-Free method,” J. Phys. Conf. Ser. **255**(1), 012006 (2010). [CrossRef]

19. T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal. **11**(4), 389–399 (2007). [CrossRef] [PubMed]

20. D. Wang, X. Liu, and J. Bai, “Analysis of fast full angle fluorescence diffuse optical tomography with beam-forming illumination,” Opt. Exp. **17**(24), 21376–21395 (2009). [CrossRef]

14. T. J. Rudge, V. Y. Soloviev, and S. R. Arridge, “Fast image reconstruction in fluoresence optical tomography using data compression,” Opt. Lett. **35**(5), 763–765 (2010). [CrossRef] [PubMed]

## 2. Method

### 2.1. Forward model

*r*, the diffusion equation with the Robin-type boundary condition [21

_{s}21. M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite elementmethod for the propagation of light in scatteringmedia: Boundary and source conditions,” Med. Phys. **22**(11), 1779–1792 (1995). [CrossRef] [PubMed]

*r*is the position vector in the domain,

*b*is the position vector on the boundary,

*μ*is the absorption coefficient,

_{a}*D*= 1/(3(

*μ*+

_{a}*μ*′

*)) is the diffusion coefficient of the tissue with reduced scattering coefficient*

_{s}*μ*′

*,*

_{s}*n⃗*is the outward normal vector to the surface and

*q*is a constant associated with the ratio of optical reflective index of the inner tissue to that outside the boundary.

*G*(

*r*,

_{s}*r*) is the Green’s function describing the propagation of light from point

*r*to

_{s}*r*, where

*r*is the point located at one transport mean free path

_{s}*ltr*= 1/

*μ*′

*into the medium from the illumination spot.*

_{s}*ϕ*(

*r*,

_{s}*r*) detected at location

_{d}*r*due to an illumination spot located at

_{d}*r*can be formulated using the Born approximation as follows [22

_{s}22. R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imaging **23**(4), 492–500 (2004). [CrossRef] [PubMed]

*G*(

*r*,

_{s}*r*) and

*G*(

*r*,

_{d}*r*) are the Green’s functions of excitation and emission light respectively, Θ is a unit-less constant taking account of the unknown gain and attenuation factors of the system.

*x*(

*r*) is the unknown distribution of fluorescent targets.

*X*is a column vector representing the concentration of fluorescent targets to be reconstructed,

*W*

_{s,d}is a row vector representing the source detector map with the element of where Δ

*v*is the volume of the discrete voxel.

*sth*projection, the sub matrix equation is formed by assembling all the source detector maps and the fluorescent measurements of the

*sth*projection as follows where

*W*= {

_{s}*W*

_{s,d}} is the sub weight matrix constituted by all the source-detector maps of the

*sth*projection and

*ϕ*= {

_{s}*ϕ*(

*r*)} is a column vector constituted by fluorescent measurements acquired from the

_{s}*sth*projection.

*W*= {

*W*} is the weight matrix mapping unknown distributions of the fluorescent markers inside the small animal onto the measured fluorescent data over the surface.

_{s}*ϕ*= {

*ϕ*} is a column vector constituted by fluorescent measurements from all the projections.

_{s}### 2.2. Kirchhoff approximation

11. J. Ripoll, M. Nieto-Vesperinas, R. Weissleder, and V. Ntziachristos, “Fast analytical approximation for arbitrary geometries in diffuse optical tomography,” Opt. Lett. **27**(7), 527–529 (2002). [CrossRef]

*G*represents the total intensity given by the KA,

^{KA}*r*is a point on the the surface,

_{p}*n⃗*is the outward normal vector at point

_{p}*r*, Δ

_{p}*S*(

*r*) is the locally planar discrete area,

_{p}*Cnd*is a constant that takes into account the refractive index mismatch between the diffusive and the non-diffusive media,

*g*is the infinite homogeneous Greens function and the surface values

*∂G*(

^{KA}*r*)/

_{p}*∂n*can be obtained by: where

_{p}*Z*= (

*r⃗*−

_{s}*r⃗*) · (−

_{p}*n⃗*) ·

_{p}*n⃗*. Comparing with many numerical methods for solving the diffusion equation, KA is not only able to handle arbitrary geometries, but also has the advantage of computation efficiency, because it is a linear method that does not involve matrix inversion when solving diffusion equation in arbitrary geometries. Ref. [10] demonstrated that KA performs with an error less than 5% when the average radius of the geometry considered is

_{p}*R*> 3

*κ*and is more than two orders of magnitude faster than accurate numerical methods.

### 2.3. Dimension reduction by PCA

*W*can be treated as a multivariate. The covariance matrix

_{s}*C*of the sub weight matrix

*W*′

*(′ means the transfer operation of a matrix) can be diagonalized as follows where*

_{s}*P*is the matrix of eigenvectors of the covariance matrix

*C*. Λ is a diagonal matrix with the elements of the eigenvalues

*λ*

_{1}≥

*λ*

_{2}≥

*λ*

_{3}⋯ of the covariance matrix

*C*. The principal components Ξ

*of the sub weight matrix*

_{s}*W*can be obtain by Accordingly, the left side of Eq. 5 should be multiplied by

_{s}*P*Then a transformed sub matrix equation for the

*sth*projection is obtained After retaining the first

*t*large principal components and leaving out the rest less significant principal components of Ξ

*and Γ*

_{s}*, a dimension reduced sub matrix equation is given as follows*

_{s}*CPV*) is used to decide the number of retained principal components

_{t}*t*[24

24. D. A. Jackson, “Stopping rules in principal components analysis: a comparison of heuristical and statistical approaches,” Ecology **74**(8), 2204–2214 (1993). [CrossRef]

*t*is determined when the

*CPV*reaches to a preset threshold

_{t}*CPV*.

_{TH}### 2.4. Tikhonov regularization method

25. M. Hanke and C. W. Groetsch, “Nonstationary iterated Tikhonov regularization,” J. Optim. Theor. Appl. **98**(1), 37–53 (1998). [CrossRef]

*α*is the regularization parameter. The initial

*X*

_{0}is set to

*X*

_{0}=

**0**, the number of iterations is set to 30, and the regularization parameter is set to 10

^{−5}

*tr*(Ξ

*Ξ′*

_{t}*) for the following phantom experiments. Note that (Ξ*

_{t}*Ξ′*

_{t}*+*

_{t}*αI*)

^{−1}is the most time-consuming process in Eq.18, which involves matrix multiplication and matrix inversion. The time cost in calculating (Ξ

*Ξ′*

_{t}*+*

_{t}*αI*)

^{−1}is reduced significantly when the row number of original weight matrix

*W*is reduced into Ξ

*. Fortunately, (Ξ*

^{t}*Ξ′*

_{t}*+*

_{t}*αI*)

^{−1}needs to be calculated only once, as it is independent on iteration parameter

*n*. Except for the matrix inversion, the other processes only contain matrix-vector multiplications and cost little time.

## 3. Experimental setup and materials

### 3.1. Experimental setup

28. F. Liu, X. Liu, D. F. Wang, B. Zhang, and J. Bai, “Parallel Excitation Based Fluorescence Molecular Tomography System for Whole-Body Simultaneous Imaging of Small Animals,” Ann. Biomed. Eng. **38**(11), 3440–3448 (2010). [CrossRef] [PubMed]

### 3.2. Phantom studies

*μ*=0.02cm

_{a}^{−1}and

*μ*′

*=10cm*

_{s}^{−1}) [29

29. G. Q. Yu, T. Durduran, C. Zhou, H. W. Wang, M. E. Putt, H. M Saunders, C. M. Sehgal, E. Glatstein, A. G. Yodh, and T. M. Busch, “Noninvasive monitoring of murine tumor blood flow during and after photodynamic therapy provides early assessment of therapeutic efficacy,” Clin Cancer Res. **1**(9), 3543–3552 (2005). [CrossRef]

*μ*L ICG with the concentration of 1.3

*μ*M, was immersed in the phantom at (−0.1cm, 0cm, 1.5cm). For the double fluorescent targets case, two small transparent glass tubes (0.4cm in diameter) filled with 10

*μ*L ICG with the concentration of 6.5

*μ*M were immersed in the phantom with the edge to edge distance of 0.5cm (one was at (−0.1cm, 0.5cm, 1.6cm), the other at (0cm, −0.4cm, 1.6cm)).

## 4. Results and discussion

### 4.1. Phantom results

*CPV*= 90% is used to determine the number of retained large principal components of each sub weight matrix

_{TH}*W*. There are almost no visual differences in both three dimensional (3D) views and sections of the results reconstructed from the reduced weight matrix using PCA and the original weight matrix in the two experiments. The time cost and sizes of dimension reduced and original weight matrices for the two experiments are listed in Table 1. As shown in the table, both the methods need to solve forward problem with the same scale, so the time cost in the calculation of forward problem is the same. In the assembling process, due to the extra compression or dimension reduction operation, assembling by the reduce weight matrix using PCA costs more time than the original weight matrix. In the matrix inversion process, using the reduced weight matrix can save lots of time compared with the original weight matrix. As a result, the total reconstruction time of the proposed method is less than one minute, which is about 1/8 of the time cost by using the original weight matrix. These results demonstrate that the proposed method can effectively accelerate the reconstruction almost without degrading the qualities of the reconstructed images.

_{s}### 4.2. Influence of the number of retained principal components

*W*on the qualities of reconstructed images and computation time, we investigated the reconstructed results using different

_{s}*CPV*. The errors

_{TH}*sqrt*(||

*X*−

_{reduced}*X*||/||

_{original}*X*||) are calculated to quantify the differences between the results reconstructed from the dimension reduced weight matrix and the original weight matrix, where

_{original}*X*and

_{reduced}*X*are the results reconstructed using the dimension reduced weight matrix and the original weight matrix, respectively.

_{original}*CPV*for the single and double fluorescent target cases are shown in Figs. 4 and 5, respectively. Figures 4 (a) and 5 (a) are the sections of reconstructed results using the original weight matrix. Figures 4 (b)–(h) and 5 (b)–(h) are the sections of reconstructed results obtained using the dimension reduction weight matrix when

_{TH}*CPV*are 98%, 95%, 90%, 85%, 80%, 75% and 70%, respectively. In the single fluorescent target case, it is hard to find visual differences in the results reconstructed from the original weight matrix and the dimension reduced weight matrices using different

_{TH}*CPV*. In the double fluorescent targets case, there appears some stains between the targets when

_{TH}*CPV*= 70% but no distinct differences for other larger

_{TH}*CPV*. That is because resolving a complex distribution of multi fluorescent targets usually needs more projected information. To quantify the above results, Figs. 4 (i) and 5 (i) show the errors and computation time, respectively. It can be seen that increasing

_{TH}*CPV*improves the reconstruction image quality and reduces the reconstruction errors at the cost of increasing computation time. A compromise method is to choose a appropriate

_{TH}*CPV*considering both image quality and computation time. The

_{TH}*CPV*of 90% is suggested as a compromise between reconstruction image quality and computation time.

_{TH}### 4.3. Comparison with data compression method

*CPV*of the PCA method and the retained wavelet coefficients of the wavelet transformation method to acquired a same CR of the original weight matrix.

_{TH}## 5. Conclusion

30. V. A. Markel and J. C. Schotland, “Inverse problem in optical diffusion tomography. II. Role of boundary conditions,” J. Opt. Soc. Am. A **19**(3), 558–566 (2002). [CrossRef]

## Acknowledgments

## References and links

1. | V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. |

2. | E. E. Graves, R. Weissleder, and V. Ntziachristos, “Fluorescence molecular imaging of small animal tumour models,” Curr. Mol. Med. |

3. | N. C. Deliolanis, J. Dunham, T. Wurdinger, J. L. Figueiredo, T. Bakhos, and V. Ntziachristos, “In-vivo imaging of murine tumors using complete-angle projection fluorescence molecular tomography,” J. Biomed. Opt. |

4. | M. Rudin and R. Weissleder, “Molecular imaging in drug discovery and development,” Nat. Rev. Drug Discov. |

5. | J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discovery |

6. | T. F. Massoud and S. S. Gambhir, “Molecular imaging in living subjects: seeing fundamental biological processes in a new light,” Genes Dev. |

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

8. | D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media,” Appl. Opt. |

9. | S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. |

10. | J. Ripoll, V. Ntziachristos, R. Carminati, and M. Nieto-Vesperinas, “Kirchhoff approximation for diffusive waves,” Phys. Rev. E |

11. | J. Ripoll, M. Nieto-Vesperinas, R. Weissleder, and V. Ntziachristos, “Fast analytical approximation for arbitrary geometries in diffuse optical tomography,” Opt. Lett. |

12. | N. Deliolanis, T. Lasser, D. Hyde, A. Soubret, J. Ripoll, and V. Ntziachristos, “Free-space fluorescence molecular tomography utilizing 360° geometry projections,” Opt. Lett. |

13. | J. Ripoll, “Hybrid Fourier-real space method for diffuse optical tomography,” Opt. Lett. |

14. | T. J. Rudge, V. Y. Soloviev, and S. R. Arridge, “Fast image reconstruction in fluoresence optical tomography using data compression,” Opt. Lett. |

15. | N. Ducros, C. D. Andrea, G. Valentini, T. Rudge, S. Arridge, and A. Bassi, “Full-wavelet approach for fluorescence diffuse optical tomography with structured illumination,” Opt. Lett. |

16. | N. Ducros, A. Bassi, G. Valentini, M. Schweiger, S. Arridge, and C. D Andrea, “Multiple-view fluorescence optical tomography reconstruction using compression of experimental data,” Opt. Lett. |

17. | A. D. Zacharopoulos, P. Svenmarker, J. Axelsson, M. Schweiger, S. R. Arridge, and S. Andersson-Engels, “A matrix-free algorithm for multiple wavelength fluorescence tomography,” Opt. Express |

18. | A. D. Zacharopoulos, A. Garofalakis, J. Ripoll, S. R. Arridge, and S. Andersson-Engels, “Development of in-vivo fluorescence imaging with the Matrix-Free method,” J. Phys. Conf. Ser. |

19. | T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal. |

20. | D. Wang, X. Liu, and J. Bai, “Analysis of fast full angle fluorescence diffuse optical tomography with beam-forming illumination,” Opt. Exp. |

21. | M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite elementmethod for the propagation of light in scatteringmedia: Boundary and source conditions,” Med. Phys. |

22. | R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imaging |

23. | I. T. Jolliffe, |

24. | D. A. Jackson, “Stopping rules in principal components analysis: a comparison of heuristical and statistical approaches,” Ecology |

25. | M. Hanke and C. W. Groetsch, “Nonstationary iterated Tikhonov regularization,” J. Optim. Theor. Appl. |

26. | V. Faber, A. Manteuffel, A. B. White Jr., and G. M. Wing, “Asymptotic behavior of singular values and functions of certain convolution operators,” Comput. Math. Appl. |

27. | C. R. Rao and S. K. Mitra, |

28. | F. Liu, X. Liu, D. F. Wang, B. Zhang, and J. Bai, “Parallel Excitation Based Fluorescence Molecular Tomography System for Whole-Body Simultaneous Imaging of Small Animals,” Ann. Biomed. Eng. |

29. | G. Q. Yu, T. Durduran, C. Zhou, H. W. Wang, M. E. Putt, H. M Saunders, C. M. Sehgal, E. Glatstein, A. G. Yodh, and T. M. Busch, “Noninvasive monitoring of murine tumor blood flow during and after photodynamic therapy provides early assessment of therapeutic efficacy,” Clin Cancer Res. |

30. | V. A. Markel and J. C. Schotland, “Inverse problem in optical diffusion tomography. II. Role of boundary conditions,” J. Opt. Soc. Am. A |

**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

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

(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: September 12, 2012

Revised Manuscript: November 20, 2012

Manuscript Accepted: November 27, 2012

Published: December 5, 2012

**Citation**

Xu Cao, Xin Wang, Bin Zhang, Fei Liu, Jianwen Luo, and Jing Bai, "Accelerated image reconstruction in fluorescence molecular tomography using dimension reduction," Biomed. Opt. Express **4**, 1-14 (2013)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-4-1-1

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### References

- V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol.23(3), 313–320 (2005). [CrossRef] [PubMed]
- E. E. Graves, R. Weissleder, and V. Ntziachristos, “Fluorescence molecular imaging of small animal tumour models,” Curr. Mol. Med.4(4), 419–430 (2004). [CrossRef] [PubMed]
- N. C. Deliolanis, J. Dunham, T. Wurdinger, J. L. Figueiredo, T. Bakhos, and V. Ntziachristos, “In-vivo imaging of murine tumors using complete-angle projection fluorescence molecular tomography,” J. Biomed. Opt.14(3), 030509 (2009). [CrossRef] [PubMed]
- M. Rudin and R. Weissleder, “Molecular imaging in drug discovery and development,” Nat. Rev. Drug Discov.2(2), 123–131 (2003). [CrossRef] [PubMed]
- J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discovery7(7), 591–607 (2008). [CrossRef]
- T. F. Massoud and S. S. Gambhir, “Molecular imaging in living subjects: seeing fundamental biological processes in a new light,” Genes Dev.17(5), 545–580 (2003). [CrossRef] [PubMed]
- L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Meth. Programs Biomed.47(2), 131–146 (1995). [CrossRef]
- D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light reemitted from random media,” Appl. Opt.36(10), 2260–2272 (1997). [CrossRef] [PubMed]
- S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys.20(2), 299–309 (1993). [CrossRef] [PubMed]
- J. Ripoll, V. Ntziachristos, R. Carminati, and M. Nieto-Vesperinas, “Kirchhoff approximation for diffusive waves,” Phys. Rev. E64(5), 0519172001.
- J. Ripoll, M. Nieto-Vesperinas, R. Weissleder, and V. Ntziachristos, “Fast analytical approximation for arbitrary geometries in diffuse optical tomography,” Opt. Lett.27(7), 527–529 (2002). [CrossRef]
- N. Deliolanis, T. Lasser, D. Hyde, A. Soubret, J. Ripoll, and V. Ntziachristos, “Free-space fluorescence molecular tomography utilizing 360° geometry projections,” Opt. Lett.32(4), 382–384 (2007). [CrossRef] [PubMed]
- J. Ripoll, “Hybrid Fourier-real space method for diffuse optical tomography,” Opt. Lett.35(5), 688–690 (2010). [CrossRef] [PubMed]
- T. J. Rudge, V. Y. Soloviev, and S. R. Arridge, “Fast image reconstruction in fluoresence optical tomography using data compression,” Opt. Lett.35(5), 763–765 (2010). [CrossRef] [PubMed]
- N. Ducros, C. D. Andrea, G. Valentini, T. Rudge, S. Arridge, and A. Bassi, “Full-wavelet approach for fluorescence diffuse optical tomography with structured illumination,” Opt. Lett.35(21), 3676–3678 (2010). [CrossRef] [PubMed]
- N. Ducros, A. Bassi, G. Valentini, M. Schweiger, S. Arridge, and C. D Andrea, “Multiple-view fluorescence optical tomography reconstruction using compression of experimental data,” Opt. Lett.36(8), 1377–1379 (2011). [CrossRef] [PubMed]
- A. D. Zacharopoulos, P. Svenmarker, J. Axelsson, M. Schweiger, S. R. Arridge, and S. Andersson-Engels, “A matrix-free algorithm for multiple wavelength fluorescence tomography,” Opt. Express17(5), 3042–3051 (2009). [CrossRef]
- A. D. Zacharopoulos, A. Garofalakis, J. Ripoll, S. R. Arridge, and S. Andersson-Engels, “Development of in-vivo fluorescence imaging with the Matrix-Free method,” J. Phys. Conf. Ser.255(1), 012006 (2010). [CrossRef]
- T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal.11(4), 389–399 (2007). [CrossRef] [PubMed]
- D. Wang, X. Liu, and J. Bai, “Analysis of fast full angle fluorescence diffuse optical tomography with beam-forming illumination,” Opt. Exp.17(24), 21376–21395 (2009). [CrossRef]
- M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite elementmethod for the propagation of light in scatteringmedia: Boundary and source conditions,” Med. Phys.22(11), 1779–1792 (1995). [CrossRef] [PubMed]
- R. B. Schulz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imaging23(4), 492–500 (2004). [CrossRef] [PubMed]
- I. T. Jolliffe, Principal Component Analysis, 3rd ed. (Springer-Verlag, 2002).
- D. A. Jackson, “Stopping rules in principal components analysis: a comparison of heuristical and statistical approaches,” Ecology74(8), 2204–2214 (1993). [CrossRef]
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