## An efficient Jacobian reduction method for diffuse optical image reconstruction

Optics Express, Vol. 15, Issue 24, pp. 15908-15919 (2007)

http://dx.doi.org/10.1364/OE.15.015908

Acrobat PDF (857 KB)

### Abstract

Model based image reconstruction in Diffuse Optical Tomography relies on both the numerical accuracy of the forward model as well as the computational speed and efficiency of the inverse model. Most model based image reconstruction algorithms rely on Newton type inversion methods, whereby the inverse of a large Jacobian is approximated. In this work we present an efficient Jacobian reduction method which takes into account the total sensitivity of the imaging domain to the measured boundary data. It is shown using numerical and phantom data that by removing regions within the inverse model whose contribution to the measured data is less than 1%, it has no significant effect upon the estimated inverse problem, but does provide up to a 14 fold improvement in computational time.

© 2007 Optical Society of America

## 1. Introduction

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

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

4. S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, and K. D. Paulsen, “Spectrally constrained Chromophore and Scattering NIR Tomography provides quantitative and robust reconstruction,” Appl. Opt. **44**, 1858–1869 (2005). [CrossRef] [PubMed]

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

7. B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Natl. Acad. Sci. U. S.A. **104**, 12169–12174 (2007). [CrossRef] [PubMed]

2. H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional nearinfrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt. **42**, 135–145 (2003). [CrossRef] [PubMed]

8. 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]

9. R. Choe, A. Corlu, K. Lee, T. Durduran, S. D. Konecky, M. Grosicka-Koptyra, S. R. Arridge, B. J. Czerniecki, D. L. Fraker, A. DeMichele, B. Chance, M. A. Rosen, and A. G. Yodh, “Diffuse optical tomography of breast cancer during neoadjuvant chemotherapy: A case study with comparison to MRI,” Med. Phys. **32**, 1128–1139 (2005). [CrossRef] [PubMed]

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

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

10. P. K. Yalavarthy, B. W. Pogue, H. Dehghani, C. M. Carpenter, S. Jiang, and K. D. Paulsen, “Structural information within regularization matrices improves near infrared diffuse optical tomography,” Opt. Express , **15**, 8043–8058 (2007). [CrossRef] [PubMed]

12. P. K. Yalavarthy, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography,” Med. Phys. **34**, 2085–2098 (2007). [CrossRef] [PubMed]

13. 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]

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

2. H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional nearinfrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt. **42**, 135–145 (2003). [CrossRef] [PubMed]

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

17. S. R. Arridge and M. Schwieger, “Gradient-based optimisation scheme for optical tomography,” Opt. Express. **2**, 212–226 (1998). [CrossRef]

18. A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time- resolved optical tomography,” IEEE Trans. Med. Imaging **18**, 262–271 (1999). [CrossRef] [PubMed]

12. P. K. Yalavarthy, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography,” Med. Phys. **34**, 2085–2098 (2007). [CrossRef] [PubMed]

19. H. Dehghani, B. W. Pogue, S. Jiang, B. Brooksby, and K. D. Paulsen, “Three dimensional optical tomography: resolution in small object imaging,” Appl. Opt. **42**, 3117–3128 (2003). [CrossRef] [PubMed]

20. 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]

21. A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express **12**, 5402–5417 (2004). [CrossRef] [PubMed]

22. P. K. Yalavarthy, H. Dehghani, B. W. Pogue, and K. D. Paulsen, “Critical computational aspects of near infrared circular tomographic imaging: Analysis of measurement number, mesh resolution and reconstruction basis,” Opt. Express **14**, 6113–6127 (2006). [CrossRef] [PubMed]

23. M. Guven, B. Yazici, K. Kwon, E. Giladi, and X. Intes, “Effect of discretization error and adaptive mesh generation in diffuse optical absorption imaging,” Inverse Probl. **23**, 1135–1160 (2007). [CrossRef]

24. M. Molinari, B. H. Blott, S. J. Cox, and G. J. Daniell, “Optimal imaging with adaptive mesh refinement in electrical impedance tomography,” Physiol. Meas. **23**, 121–128 (2002). [CrossRef] [PubMed]

25. H. Dehghani and M Soleimani, “Numerical modelling errors in electrical impedance tomography,” Phys Meas **28**, S45–S55 (2007). [CrossRef]

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

26. 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]

^{-1}and 1.0 mm

^{-1}for absorption and reduced scatter respectively, Fig. 1, for which the Jacobian for a single source and detector has been calculated. In each instance, the contour line which represents the threshold of the magnitude of the Jacobian at different levels is shown. It is evident that given a source detector combination, the total area and therefore the number of nodes within that contribute to the measured data is highly dependent on the amount of sensitivity from the Jacobian which is required.

27. S. Srinivasan, B. W. Pogue, H. Dehghani, F. Leblond, and X. Intes, “A data subset algorithm for computationally efficient reconstruction of 3-D spectral imaging in diffuse optical tomography,” Opt. Express **14**, 5394–5410 (2006). [CrossRef] [PubMed]

## 2. Theory

*q*

_{0}(

**r**,

*ω*) is an isotropic source, Φ(

**r**,

*ω*) is the photon fluence rate at position

*r*and modulation frequency ω.

*κ*is the diffusion coefficient given by

*µ*and

_{a}*µ*′ are absorption and reduced scattering (or transport scattering) coefficients respectively and

_{s}*c*(

_{m}*r*) is the speed of light in the medium. The air-tissue boundary is derived from a type III condition given by

*A*depends upon the relative refractive index mismatch between the tissue domain and air derived from Fresnel’s law.

*µ*=[

*µ*,

_{a}*µ*′] for each node with the FEM mesh. The inversion is achieved by finding the minimum between the measured data, Φ

_{s}*, and the calculated data, Φ*

^{M}*, using a modified- Levenberg-Marquardt minimization approach given by*

^{C}*NM*is the number of measurements and

*µ*represents the optical properties that need to be reconstructed. For an ill-posed problem like DOT, minimizing Eq. (4) with respect to the optical properties

*µ*, and considering only the linear terms leads to the update equation

*δµ*is now the update vector for the optical properties.

*λ*is implemented similar to Levenberg-Marquardt approach [28

28. D. W. Marquardt, “An algorithm for least squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math **11**, 431–441 (1963). [CrossRef]

*J*and is systematically reduced at each iteration and

^{T}J*J*is the Jacobian matrix.

## 3. Jacobian reduction

*J*is of the size number of measurements,

*NM*, by the number of FEM nodes,

*NN*, and is calculated by using the Adjoint method. For simplicity, limiting the Jacobian to the measured amplitude data and optical absorption, it links a change in log amplitude, at the boundary, with a change in absorption coefficient,

*µ*. The Jacobian thus has the form

_{a}*J*is needed which has an order of

^{T}J*NN*×

*NN*.

^{3}operations and N

^{2}memory [29]. Thus the usage of full Hessian matrix with less sensitive parts being zero does not reduce the computational burden, unless a sparse matrix solver is used. But usage of sparse matrix solvers requires calculation of half band width of the matrix, which is difficult to measure in these cases where the structure of Hessian largely depends on node numbering and the sequence how the measurements are taken. Alternatively we can look at the total sensitivity within the volume and find regions where the value is very small, typically below a certain threshold. These regions can then be removed from the matrix thus reducing the size of the Hessian. In effect this method reduces the volume of interest for the case of the update equation without decreasing the mesh resolution, but the entire volume is used to accurately calculate the forward model.

*J*̃

*, is formed:*

_{ij}*j*corresponds to the node number within the domain. Given a new Jacobian,

*J*̃

*, where the total sensitivity for a given node is equal to zero, the entire column corresponding to that node is removed to produce a much smaller Jacobian matrix drastically reducing the number of operations and memory required for the calculation of the update of optical properties (Eq. 5).*

_{ij}## 4. Method and results

### 4.1 Simulated breast model

### 4.2 Experimental phantom

_{2}for scattering. Layers are constructed by successively hardening gel solutions with different concentrations of ink and TiO

_{2}. The outer layer of thickness 5mm, similar to a typical fatty breast layer, has the optical properties 0.0065 mm

^{-1}and 0.65 mm

^{-1}(at 785 nm) for absorption and reduced scatter respectively. The fibroglandular layer has optical properties of

*µ*=0.01 mm

_{a}^{-1}and

*µ*’=1.0 mm

_{s}^{-1}and has a radius of 38mm. An anomaly, which represents the tumor, of radius 8 mm of cylindrical shape stretching the entire

*z*direction was placed approximately 20 mm from the centre with optical properties

*µ*=0.02 mm

_{a}^{-1}and

*µ*’=1.2 mm

_{s}^{-1}. NIR data was collected using a frequency domain system with a modulation frequency of 100MHz using 16 optical fibers placed circularly along the mid-height of the phantom.

*µ*and

_{a}*κ*. Since both the absorption and scattering parameters determine the sensitivity, each were utilized accordingly using varying levels of threshold of 0%, 1%, 5% and 10% for the reduction scheme. The initial guess for optical absorption and reduced scattering for the iterative image reconstruction was assumed to be that of a homogeneous phantom, together with a uniform pixel basis of 20×20×20 for image reconstruction.

*µ*and

_{a}*µ*’ using varying levels of threshold of 0%, 1%, 5% and 10% for the reduction scheme are shown in Fig. 6. For each method we find a large contrast for the reconstructed images for

_{s}*µ*and as we would expect the contrast for the reduced scattering is small. There is no difference in the quantitative and qualitative accuracy between the reduction method with a tolerance of 1% and the usual reconstruction method (0%). As the tolerance increases there is a large saving in memory usage for the inversion of the Jacobian, table 3. With a tolerance of just 1% the number of nodes within the Jacobian is reduced by 8% saving 0.15Gb of memory for storage alone (18% reduction). At a tolerance 10%, the number of nodes decreases further to just 51% of the total, saving 0.73Gb of memory (74% reduction). The reconstructed images are extremely sensitive to the initial guess of the optical parameters and in this case the method has been unable to reconstruct the true

_{a}*µ*’. However, the difference between the standard method (0%) and a tolerance of either 1% or 5% is negligible and demonstrating that the reduction method gives the same result at improved computational speed.

_{s}## 5. Discussions and conclusions

## Acknowledgments

## References and links

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

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

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

4. | S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, and K. D. Paulsen, “Spectrally constrained Chromophore and Scattering NIR Tomography provides quantitative and robust reconstruction,” Appl. Opt. |

5. | S. Srinivasan, B. W. Pogue, B. Brooksby, S. Jiang, H. Dehghani, C. Kogel, S. P. Poplack, and K. D. Paulsen, “Near-infrared characterization of breast tumors in vivo using spectrally-constrained reconstruction,” Technol. Cancer Res. Treat. |

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

7. | B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Natl. Acad. Sci. U. S.A. |

8. | 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. |

9. | R. Choe, A. Corlu, K. Lee, T. Durduran, S. D. Konecky, M. Grosicka-Koptyra, S. R. Arridge, B. J. Czerniecki, D. L. Fraker, A. DeMichele, B. Chance, M. A. Rosen, and A. G. Yodh, “Diffuse optical tomography of breast cancer during neoadjuvant chemotherapy: A case study with comparison to MRI,” Med. Phys. |

10. | P. K. Yalavarthy, B. W. Pogue, H. Dehghani, C. M. Carpenter, S. Jiang, and K. D. Paulsen, “Structural information within regularization matrices improves near infrared diffuse optical tomography,” Opt. Express , |

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

12. | P. K. Yalavarthy, B. W. Pogue, H. Dehghani, and K. D. Paulsen, “Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography,” Med. Phys. |

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

14. | A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. |

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

16. | S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, “The finite element model of the propagation of light in scattering media: A direct method for domains with nonscattering regions,” Med. Phys. |

17. | S. R. Arridge and M. Schwieger, “Gradient-based optimisation scheme for optical tomography,” Opt. Express. |

18. | A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time- resolved optical tomography,” IEEE Trans. Med. Imaging |

19. | H. Dehghani, B. W. Pogue, S. Jiang, B. Brooksby, and K. D. Paulsen, “Three dimensional optical tomography: resolution in small object imaging,” Appl. Opt. |

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

21. | A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express |

22. | P. K. Yalavarthy, H. Dehghani, B. W. Pogue, and K. D. Paulsen, “Critical computational aspects of near infrared circular tomographic imaging: Analysis of measurement number, mesh resolution and reconstruction basis,” Opt. Express |

23. | M. Guven, B. Yazici, K. Kwon, E. Giladi, and X. Intes, “Effect of discretization error and adaptive mesh generation in diffuse optical absorption imaging,” Inverse Probl. |

24. | M. Molinari, B. H. Blott, S. J. Cox, and G. J. Daniell, “Optimal imaging with adaptive mesh refinement in electrical impedance tomography,” Physiol. Meas. |

25. | H. Dehghani and M Soleimani, “Numerical modelling errors in electrical impedance tomography,” Phys Meas |

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

27. | S. Srinivasan, B. W. Pogue, H. Dehghani, F. Leblond, and X. Intes, “A data subset algorithm for computationally efficient reconstruction of 3-D spectral imaging in diffuse optical tomography,” Opt. Express |

28. | D. W. Marquardt, “An algorithm for least squares estimation of nonlinear parameters,” J. Soc. Ind. Appl. Math |

29. | J. R. Westlake, |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

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

**ToC Category:**

Image Processing

**History**

Original Manuscript: September 10, 2007

Revised Manuscript: November 2, 2007

Manuscript Accepted: November 8, 2007

Published: November 15, 2007

**Virtual Issues**

Vol. 2, Iss. 12 *Virtual Journal for Biomedical Optics*

**Citation**

Matthew E. Eames, Brian W. Pogue, Phaneendra K. Yalavarthy, and Hamid Dehghani, "An efficient Jacobian reduction method for diffuse optical image reconstruction," Opt. Express **15**, 15908-15919 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-24-15908

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### References

- S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999). [CrossRef]
- 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 (2003). [CrossRef] [PubMed]
- A. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005). [CrossRef] [PubMed]
- S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, and K. D. Paulsen, "Spectrally constrained Chromophore and Scattering NIR Tomography provides quantitative and robust reconstruction," Appl. Opt. 44, 1858-1869 (2005). [CrossRef] [PubMed]
- S. Srinivasan, B. W. Pogue, B. Brooksby, S. Jiang, H. Dehghani, C. Kogel, S. P. Poplack, and K. D. Paulsen, "Near-infrared characterization of breast tumors in vivo using spectrally-constrained reconstruction," Technol. Cancer Res. Treat. 5, 513-526 (2005).
- A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. M. C. Hillman, A. G. Yodh, "Diffuse optical tomography with spectral constraints and wavelength optimization," Appl. Opt. 44, 2082-2093 (2005). [CrossRef] [PubMed]
- B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, "Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography," Proc. Natl. Acad. Sci. U. S.A. 104, 12169-12174 (2007). [CrossRef] [PubMed]
- S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, 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]
- R. Choe, A. Corlu, K. Lee, T. Durduran, S. D. Konecky, M. Grosicka-Koptyra, S. R. Arridge, B. J. Czerniecki, D. L. Fraker, A. DeMichele, B. Chance, M. A. Rosen, and A. G. Yodh, "Diffuse optical tomography of breast cancer during neoadjuvant chemotherapy: A case study with comparison to MRI," Med. Phys. 32, 1128-1139 (2005). [CrossRef] [PubMed]
- P. K. Yalavarthy, B. W. Pogue, H. Dehghani, C. M. Carpenter, S. Jiang, and K. D. Paulsen, "Structural information within regularization matrices improves near infrared diffuse optical tomography," Opt. Express 15, 8043-8058 (2007). [CrossRef] [PubMed]
- A. D. Klose and A. H. Hielscher, "Quasi Newton methods in optical tomographic image reconstruction," Inverse Probl. 19, 387-409 (2003). [CrossRef]
- P. K. Yalavarthy, B. W. Pogue, H. Dehghani, and K. D. Paulsen, "Weight-matrix structured regularization provides optimal generalized least-squares estimate in diffuse optical tomography," Med. Phys. 34, 2085-2098 (2007). [CrossRef] [PubMed]
- 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]
- A. D. Klose and E. W. Larsen, "Light transport in biological tissue based on the simplified spherical harmonics equations," J. Comput. Phys. 220, 441-470 (2006). [CrossRef]
- O. Dorn, "A transport-backtransport method for optical tomography," Inverse Probl. 14, 1107-1130 (1998). [CrossRef]
- S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, "The finite element model of the propagation of light in scattering media: A direct method for domains with nonscattering regions," Med. Phys. 27, 252-264 (2000).
- S. R. Arridge and M. Schwieger, "Gradient-based optimisation scheme for optical tomography," Opt. Express. 2, 212-226 (1998). [CrossRef]
- A. H. Hielscher, A. D. Klose, and K. M. Hanson, "Gradient-based iterative image reconstruction scheme for time- resolved optical tomography," IEEE Trans. Med. Imaging 18, 262-271 (1999). [CrossRef] [PubMed]
- H. Dehghani, B. W. Pogue, S. Jiang, B. Brooksby, and K. D. Paulsen, "Three dimensional optical tomography: resolution in small object imaging," Appl. Opt. 42, 3117-3128 (2003). [CrossRef] [PubMed]
- 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]
- A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, "Adaptive finite element based tomography for fluorescence optical imaging in tissue," Opt. Express 12, 5402-5417 (2004). [CrossRef] [PubMed]
- P. K. Yalavarthy, H. Dehghani, B. W. Pogue, and K. D. Paulsen, "Critical computational aspects of near infrared circular tomographic imaging: Analysis of measurement number, mesh resolution and reconstruction basis," Opt. Express 14, 6113-6127 (2006). [CrossRef] [PubMed]
- M. Guven, B. Yazici, K. Kwon, E. Giladi, and X. Intes, "Effect of discretization error and adaptive mesh generation in diffuse optical absorption imaging," Inverse Probl. 23, 1135-1160 (2007). [CrossRef]
- M. Molinari, B. H. Blott, S. J. Cox, G. J. Daniell, "Optimal imaging with adaptive mesh refinement in electrical impedance tomography," Physiol. Meas. 23, 121-128 (2002). [CrossRef] [PubMed]
- H. Dehghani and M , Soleimani, "Numerical modelling errors in electrical impedance tomography," Physiol Meas 28, S45-S55 (2007). [CrossRef]
- 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]
- S. Srinivasan, B. W. Pogue, H. Dehghani, F. Leblond, and X. Intes, "A data subset algorithm for computationally efficient reconstruction of 3-D spectral imaging in diffuse optical tomography," Opt. Express 14, 5394-5410 (2006). [CrossRef] [PubMed]
- D. W. Marquardt, "An algorithm for least squares estimation of nonlinear parameters," Appl. Math 11, 431-441 (1963). [CrossRef]
- J. R. Westlake, A handbook of numerical matrix inversion and solution of linear equations (John Wiley & Sons Inc, New York, 1968).

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