Abstract

Fluorescence diffuse optical tomography is a powerful tool for the investigation of molecular events in studies for new therapeutic developments. Here, the emphasis is put on the mathematical problem of tomography, which can be formulated in terms of an estimation of physical parameters appearing as a set of Partial Differential Equations (PDEs). The standard polynomial Finite Element Method (FEM) is a method of choice to solve the diffusion equation because it has no restriction in terms of neither the geometry nor the homogeneity of the system, but it is time consuming. In order to speed up computation time, this paper proposes an alternative numerical model, describing the diffusion operator in orthonormal basis of compactly supported wavelets. The discretization of the PDEs yields to matrices which are easily computed from derivative wavelet product integrals. Due to the shape of the wavelet basis, the studied domain is included in a regular fictitious domain. A validation study and a comparison with the standard FEM are conducted on synthetic data.

© 2009 Optical Society of America

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  2. S. R. Arridge, M. Cope, and D. T. Delpy, “The Theoretical Basis For The Determination Of Optical Pathlengths In Tissue - Temporal And Frequency-Analysis,” Phys. Med. Bio. 37(7), 1531–1560 (1992).
    [Crossref]
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  5. G. I. Marchuk, V. I. Agoshkov, and V. P. Shutyaev, “Adjoint equations and perturbation algorithms in nonlinear problems,” CRC Press, Boca Raton, FL (1996).
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  7. O. V. Vasilyev, S. Paolucci, and M. Sen, “A Multilevel Wavelet Collocation Method For Solving Partial-Differential Equations In A Finite Domain,” J.Comp. Phys. 120(1), 33–47 (1995).
    [Crossref]
  8. S. Jaffard, “Wavelet Methods For Fast Resolution Of Elliptic Problems,” SIAM J. Num. Anal. 29(4), 965–986 (1992).
    [Crossref]
  9. G. Oberschmidt, G. Schneider, and A. F. Jacob, “A priori size estimation of wavelet-based Galerkin matrices,” Proceeding of 25th International Conference of Infrared and Millimeter Waves, 241–242, 2000.
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  13. S. Bonnet, F. Peyrin, F. Turjman, and R. Prost, “Multiresolution Reconstruction in Fan-Beam Tomography,” IEEE Trans. Image Proc. 11(3), 169–176 (2002).
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  14. S. K. Nath, R. M. Vasu, and M. Pandit, “Wavelet based compression and denoising of optical tomography data,” Opt. Commun. 167(1–6), 37–46 (1999).
    [Crossref]
  15. W. Zhu, Y. Wang, Y. N. Deng, Y. Q. Yao, and R. L. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imaging 16(2), 210–217 (1997).
    [Crossref]
  16. B. Kanmani, P. Bansal, and R. M. Vasu, “Computationally efficient optical tomographic reconstruction through waveletizing the normalized quadratic perturbation equation,” Proc. SPIE 6164, R1–R10 (2006).
  17. B. Kanmani, “Wavelet-Galerkin solution to the diffusion equation of diffuse optical tomography,” Internet Report, Saratov Fall Meeting: Optical Technologies in Biophysics ans Medicine IX, Russia (2007).
  18. J. C. Baritaux, S. C. Sekhar, and M. Unser, “A Spline-based forward model for Optical Diffuse Tomography,” Biomedical Imaging, ISBI, 384–387 (2008).
  19. D. Georges, “A fast Method for the solution of some Tomography Problems,” Decision and Control, CDC IEEE Conf. (2008).
  20. I. Daubechies, “Ten lectures on Wavelets,” SIAM, 167–213 (1992).
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    [Crossref]
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  23. R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, and M. S. McAdams, “Boundary-Conditions For The Diffusion Equation In Radiative-Transfer,” J. Opt. Soc. Am. A 11(10), 2727–2741 (1994).
    [Crossref]
  24. A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express 12(22), 5402–5417 (2004).
    [Crossref]
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    [Crossref]
  28. S. Dumont and F. Lebon, “Representation of plane elastostatics operators in Daubechies wavelets,” Comp. Structures 60(4), 561–569 (1996).
    [Crossref]
  29. A. Ern and J. Guermond, Eléments Finis: théorie, applications, mise en oeuvre, Chap. 4 (Springer, 2002).
  30. G. Beylkin, “On The Representation Of Operators In Bases Of Compactly Supported Wavelets,” SIAM J. Numerical Anal. 29(6), 1716–1740 (1992).
    [Crossref]
  31. M. Q. Chen, C. I. Hwang, and Y. P. Shih, “The computation of wavelet-Galerkin approximation on a bounded interval,” Int. J. Num. Meth. Engin. 39(17), 2921–2944 (1996).
    [Crossref]
  32. P. Charton and V. Perrier, “Produits rapides matrice-vecteur en bases d’ondelettes: application à la résolution numérique d’équations aux dérivées partielles,” Math. Modelling Numerical Anal. 29(6), 701–747 (1995).
  33. R. Glowinski, T. W. Pan, R. O. Wells, and X. D. Zhou, “Wavelet and finite element solutions for the Neumann problem using fictitious domains,” J. Comput. Phys. 126(1), 40–51 (1996).
    [Crossref]
  34. T. Vo-Dinh, Biomedical Photonics Handbook (CRC Press, 2003).
    [Crossref]
  35. R. Elaloufi, R. Carminati, and J. J. Greffet, “Definition of the diffusion coefficient in scattering and absorbing media,” J. Opt. Soc. Am. A 20(4), 678–685, (2003).
    [Crossref]

2006 (1)

B. Kanmani, P. Bansal, and R. M. Vasu, “Computationally efficient optical tomographic reconstruction through waveletizing the normalized quadratic perturbation equation,” Proc. SPIE 6164, R1–R10 (2006).

2005 (1)

J. Baccou and J. Liandrat, “On coupling wavelets with fictitious domain approaches,” Appl. Math. Lett. 18(12), 1325–1331 (2005).
[Crossref]

2004 (1)

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express 12(22), 5402–5417 (2004).
[Crossref]

2003 (3)

F. Fedele, J. P. Laible, and M. J. Eppstein, “Coupled complex adjoint sensitivities for frequency-domain fluorescence tomography: theory and vectorized implementation,” J. Comp. Phys. 187(2), 597–619 (2003).
[Crossref]

R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nature Med 9(1), 123–128 (2003).

R. Elaloufi, R. Carminati, and J. J. Greffet, “Definition of the diffusion coefficient in scattering and absorbing media,” J. Opt. Soc. Am. A 20(4), 678–685, (2003).
[Crossref]

2002 (1)

S. Bonnet, F. Peyrin, F. Turjman, and R. Prost, “Multiresolution Reconstruction in Fan-Beam Tomography,” IEEE Trans. Image Proc. 11(3), 169–176 (2002).
[Crossref]

1999 (2)

S. K. Nath, R. M. Vasu, and M. Pandit, “Wavelet based compression and denoising of optical tomography data,” Opt. Commun. 167(1–6), 37–46 (1999).
[Crossref]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15(2), R41–R93 (1999).
[Crossref]

1997 (1)

W. Zhu, Y. Wang, Y. N. Deng, Y. Q. Yao, and R. L. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imaging 16(2), 210–217 (1997).
[Crossref]

1996 (3)

S. Dumont and F. Lebon, “Representation of plane elastostatics operators in Daubechies wavelets,” Comp. Structures 60(4), 561–569 (1996).
[Crossref]

M. Q. Chen, C. I. Hwang, and Y. P. Shih, “The computation of wavelet-Galerkin approximation on a bounded interval,” Int. J. Num. Meth. Engin. 39(17), 2921–2944 (1996).
[Crossref]

R. Glowinski, T. W. Pan, R. O. Wells, and X. D. Zhou, “Wavelet and finite element solutions for the Neumann problem using fictitious domains,” J. Comput. Phys. 126(1), 40–51 (1996).
[Crossref]

1995 (2)

P. Charton and V. Perrier, “Produits rapides matrice-vecteur en bases d’ondelettes: application à la résolution numérique d’équations aux dérivées partielles,” Math. Modelling Numerical Anal. 29(6), 701–747 (1995).

O. V. Vasilyev, S. Paolucci, and M. Sen, “A Multilevel Wavelet Collocation Method For Solving Partial-Differential Equations In A Finite Domain,” J.Comp. Phys. 120(1), 33–47 (1995).
[Crossref]

1994 (1)

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, and M. S. McAdams, “Boundary-Conditions For The Diffusion Equation In Radiative-Transfer,” J. Opt. Soc. Am. A 11(10), 2727–2741 (1994).
[Crossref]

1993 (1)

B. Sahiner and A. E. Yagle, “Image-Reconstruction From Projections Under Wavelet Constraints,” IEEE Trans. SignalProc. 41(12), 3579–3584 (1993).
[Crossref]

1992 (3)

G. Beylkin, “On The Representation Of Operators In Bases Of Compactly Supported Wavelets,” SIAM J. Numerical Anal. 29(6), 1716–1740 (1992).
[Crossref]

S. Jaffard, “Wavelet Methods For Fast Resolution Of Elliptic Problems,” SIAM J. Num. Anal. 29(4), 965–986 (1992).
[Crossref]

S. R. Arridge, M. Cope, and D. T. Delpy, “The Theoretical Basis For The Determination Of Optical Pathlengths In Tissue - Temporal And Frequency-Analysis,” Phys. Med. Bio. 37(7), 1531–1560 (1992).
[Crossref]

1989 (1)

S. Mallat, “A Theory of Multiresolution Signal Decomposition: the Wavelet Representation,” IEEE Trans. Pattern Anal. Machine Intell. 11(7), 674–693 (1989).
[Crossref]

1984 (1)

A. Grossmann and J. Morlet, “Decomposition Of Hardy Functions Into Square Integrable Wavelets Of Constant Shape,” SIAM J. Math. Anal. 15(4), 723–736 (1984).
[Crossref]

Agoshkov, V. I.

G. I. Marchuk, V. I. Agoshkov, and V. P. Shutyaev, “Adjoint equations and perturbation algorithms in nonlinear problems,” CRC Press, Boca Raton, FL (1996).

Arridge, S. R.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Problems 15(2), R41–R93 (1999).
[Crossref]

S. R. Arridge, M. Cope, and D. T. Delpy, “The Theoretical Basis For The Determination Of Optical Pathlengths In Tissue - Temporal And Frequency-Analysis,” Phys. Med. Bio. 37(7), 1531–1560 (1992).
[Crossref]

Baccou, J.

J. Baccou and J. Liandrat, “On coupling wavelets with fictitious domain approaches,” Appl. Math. Lett. 18(12), 1325–1331 (2005).
[Crossref]

Bangerth, W.

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express 12(22), 5402–5417 (2004).
[Crossref]

Bansal, P.

B. Kanmani, P. Bansal, and R. M. Vasu, “Computationally efficient optical tomographic reconstruction through waveletizing the normalized quadratic perturbation equation,” Proc. SPIE 6164, R1–R10 (2006).

Barbour, R. L.

W. Zhu, Y. Wang, Y. N. Deng, Y. Q. Yao, and R. L. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imaging 16(2), 210–217 (1997).
[Crossref]

Baritaux, J. C.

J. C. Baritaux, S. C. Sekhar, and M. Unser, “A Spline-based forward model for Optical Diffuse Tomography,” Biomedical Imaging, ISBI, 384–387 (2008).

Beylkin, G.

G. Beylkin, “On The Representation Of Operators In Bases Of Compactly Supported Wavelets,” SIAM J. Numerical Anal. 29(6), 1716–1740 (1992).
[Crossref]

Bonnet, S.

S. Bonnet, F. Peyrin, F. Turjman, and R. Prost, “Multiresolution Reconstruction in Fan-Beam Tomography,” IEEE Trans. Image Proc. 11(3), 169–176 (2002).
[Crossref]

Burnett, D. S.

D. S. Burnett, Finite Element Concept (Addison-Wesley Publishing Company, 1987).

Carminati, R.

R. Elaloufi, R. Carminati, and J. J. Greffet, “Definition of the diffusion coefficient in scattering and absorbing media,” J. Opt. Soc. Am. A 20(4), 678–685, (2003).
[Crossref]

Charton, P.

P. Charton and V. Perrier, “Produits rapides matrice-vecteur en bases d’ondelettes: application à la résolution numérique d’équations aux dérivées partielles,” Math. Modelling Numerical Anal. 29(6), 701–747 (1995).

Chen, M. Q.

M. Q. Chen, C. I. Hwang, and Y. P. Shih, “The computation of wavelet-Galerkin approximation on a bounded interval,” Int. J. Num. Meth. Engin. 39(17), 2921–2944 (1996).
[Crossref]

Cope, M.

S. R. Arridge, M. Cope, and D. T. Delpy, “The Theoretical Basis For The Determination Of Optical Pathlengths In Tissue - Temporal And Frequency-Analysis,” Phys. Med. Bio. 37(7), 1531–1560 (1992).
[Crossref]

Daubechies, I.

I. Daubechies, “Ten lectures on Wavelets,” SIAM, 167–213 (1992).

Delpy, D. T.

S. R. Arridge, M. Cope, and D. T. Delpy, “The Theoretical Basis For The Determination Of Optical Pathlengths In Tissue - Temporal And Frequency-Analysis,” Phys. Med. Bio. 37(7), 1531–1560 (1992).
[Crossref]

Deng, Y. N.

W. Zhu, Y. Wang, Y. N. Deng, Y. Q. Yao, and R. L. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imaging 16(2), 210–217 (1997).
[Crossref]

Dumont, S.

S. Dumont and F. Lebon, “Representation of plane elastostatics operators in Daubechies wavelets,” Comp. Structures 60(4), 561–569 (1996).
[Crossref]

Elaloufi, R.

R. Elaloufi, R. Carminati, and J. J. Greffet, “Definition of the diffusion coefficient in scattering and absorbing media,” J. Opt. Soc. Am. A 20(4), 678–685, (2003).
[Crossref]

Eppstein, M. J.

F. Fedele, J. P. Laible, and M. J. Eppstein, “Coupled complex adjoint sensitivities for frequency-domain fluorescence tomography: theory and vectorized implementation,” J. Comp. Phys. 187(2), 597–619 (2003).
[Crossref]

Ern, A.

A. Ern and J. Guermond, Eléments Finis: théorie, applications, mise en oeuvre, Chap. 4 (Springer, 2002).

Fedele, F.

F. Fedele, J. P. Laible, and M. J. Eppstein, “Coupled complex adjoint sensitivities for frequency-domain fluorescence tomography: theory and vectorized implementation,” J. Comp. Phys. 187(2), 597–619 (2003).
[Crossref]

Feng, T. C.

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, and M. S. McAdams, “Boundary-Conditions For The Diffusion Equation In Radiative-Transfer,” J. Opt. Soc. Am. A 11(10), 2727–2741 (1994).
[Crossref]

Georges, D.

D. Georges, “A fast Method for the solution of some Tomography Problems,” Decision and Control, CDC IEEE Conf. (2008).

Glowinski, R.

R. Glowinski, T. W. Pan, R. O. Wells, and X. D. Zhou, “Wavelet and finite element solutions for the Neumann problem using fictitious domains,” J. Comput. Phys. 126(1), 40–51 (1996).
[Crossref]

Goedecker, S.

S. Goedecker, Wavelets and their application for the Partial Differential Equations in Physics, PPUR (1998).

Greffet, J. J.

R. Elaloufi, R. Carminati, and J. J. Greffet, “Definition of the diffusion coefficient in scattering and absorbing media,” J. Opt. Soc. Am. A 20(4), 678–685, (2003).
[Crossref]

Grossmann, A.

A. Grossmann and J. Morlet, “Decomposition Of Hardy Functions Into Square Integrable Wavelets Of Constant Shape,” SIAM J. Math. Anal. 15(4), 723–736 (1984).
[Crossref]

Guermond, J.

A. Ern and J. Guermond, Eléments Finis: théorie, applications, mise en oeuvre, Chap. 4 (Springer, 2002).

Haskell, R. C.

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, and M. S. McAdams, “Boundary-Conditions For The Diffusion Equation In Radiative-Transfer,” J. Opt. Soc. Am. A 11(10), 2727–2741 (1994).
[Crossref]

Hwang, C. I.

M. Q. Chen, C. I. Hwang, and Y. P. Shih, “The computation of wavelet-Galerkin approximation on a bounded interval,” Int. J. Num. Meth. Engin. 39(17), 2921–2944 (1996).
[Crossref]

Jacob, A. F.

G. Oberschmidt, G. Schneider, and A. F. Jacob, “A priori size estimation of wavelet-based Galerkin matrices,” Proceeding of 25th International Conference of Infrared and Millimeter Waves, 241–242, 2000.

Jaffard, S.

S. Jaffard, “Wavelet Methods For Fast Resolution Of Elliptic Problems,” SIAM J. Num. Anal. 29(4), 965–986 (1992).
[Crossref]

Joshi, A.

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express 12(22), 5402–5417 (2004).
[Crossref]

Kak, A. C.

A. C. Kak and M. Slaney, Principles of computerized tomographic imaging (NY, IEEE Press, 1987).

Kanmani, B.

B. Kanmani, P. Bansal, and R. M. Vasu, “Computationally efficient optical tomographic reconstruction through waveletizing the normalized quadratic perturbation equation,” Proc. SPIE 6164, R1–R10 (2006).

B. Kanmani, “Wavelet-Galerkin solution to the diffusion equation of diffuse optical tomography,” Internet Report, Saratov Fall Meeting: Optical Technologies in Biophysics ans Medicine IX, Russia (2007).

Laible, J. P.

F. Fedele, J. P. Laible, and M. J. Eppstein, “Coupled complex adjoint sensitivities for frequency-domain fluorescence tomography: theory and vectorized implementation,” J. Comp. Phys. 187(2), 597–619 (2003).
[Crossref]

Lebon, F.

S. Dumont and F. Lebon, “Representation of plane elastostatics operators in Daubechies wavelets,” Comp. Structures 60(4), 561–569 (1996).
[Crossref]

Liandrat, J.

J. Baccou and J. Liandrat, “On coupling wavelets with fictitious domain approaches,” Appl. Math. Lett. 18(12), 1325–1331 (2005).
[Crossref]

Mallat, S.

S. Mallat, “A Theory of Multiresolution Signal Decomposition: the Wavelet Representation,” IEEE Trans. Pattern Anal. Machine Intell. 11(7), 674–693 (1989).
[Crossref]

S. Mallat, A wavelet tour of signal processing (Academic Press, 1999).

Marchuk, G. I.

G. I. Marchuk, V. I. Agoshkov, and V. P. Shutyaev, “Adjoint equations and perturbation algorithms in nonlinear problems,” CRC Press, Boca Raton, FL (1996).

McAdams, M. S.

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, and M. S. McAdams, “Boundary-Conditions For The Diffusion Equation In Radiative-Transfer,” J. Opt. Soc. Am. A 11(10), 2727–2741 (1994).
[Crossref]

Morlet, J.

A. Grossmann and J. Morlet, “Decomposition Of Hardy Functions Into Square Integrable Wavelets Of Constant Shape,” SIAM J. Math. Anal. 15(4), 723–736 (1984).
[Crossref]

Nath, S. K.

S. K. Nath, R. M. Vasu, and M. Pandit, “Wavelet based compression and denoising of optical tomography data,” Opt. Commun. 167(1–6), 37–46 (1999).
[Crossref]

Ntziachristos, V.

R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nature Med 9(1), 123–128 (2003).

Oberschmidt, G.

G. Oberschmidt, G. Schneider, and A. F. Jacob, “A priori size estimation of wavelet-based Galerkin matrices,” Proceeding of 25th International Conference of Infrared and Millimeter Waves, 241–242, 2000.

Pan, T. W.

R. Glowinski, T. W. Pan, R. O. Wells, and X. D. Zhou, “Wavelet and finite element solutions for the Neumann problem using fictitious domains,” J. Comput. Phys. 126(1), 40–51 (1996).
[Crossref]

Pandit, M.

S. K. Nath, R. M. Vasu, and M. Pandit, “Wavelet based compression and denoising of optical tomography data,” Opt. Commun. 167(1–6), 37–46 (1999).
[Crossref]

Paolucci, S.

O. V. Vasilyev, S. Paolucci, and M. Sen, “A Multilevel Wavelet Collocation Method For Solving Partial-Differential Equations In A Finite Domain,” J.Comp. Phys. 120(1), 33–47 (1995).
[Crossref]

Perrier, V.

P. Charton and V. Perrier, “Produits rapides matrice-vecteur en bases d’ondelettes: application à la résolution numérique d’équations aux dérivées partielles,” Math. Modelling Numerical Anal. 29(6), 701–747 (1995).

Peyrin, F.

S. Bonnet, F. Peyrin, F. Turjman, and R. Prost, “Multiresolution Reconstruction in Fan-Beam Tomography,” IEEE Trans. Image Proc. 11(3), 169–176 (2002).
[Crossref]

Prost, R.

S. Bonnet, F. Peyrin, F. Turjman, and R. Prost, “Multiresolution Reconstruction in Fan-Beam Tomography,” IEEE Trans. Image Proc. 11(3), 169–176 (2002).
[Crossref]

Sahiner, B.

B. Sahiner and A. E. Yagle, “Image-Reconstruction From Projections Under Wavelet Constraints,” IEEE Trans. SignalProc. 41(12), 3579–3584 (1993).
[Crossref]

Schneider, G.

G. Oberschmidt, G. Schneider, and A. F. Jacob, “A priori size estimation of wavelet-based Galerkin matrices,” Proceeding of 25th International Conference of Infrared and Millimeter Waves, 241–242, 2000.

Sekhar, S. C.

J. C. Baritaux, S. C. Sekhar, and M. Unser, “A Spline-based forward model for Optical Diffuse Tomography,” Biomedical Imaging, ISBI, 384–387 (2008).

Sen, M.

O. V. Vasilyev, S. Paolucci, and M. Sen, “A Multilevel Wavelet Collocation Method For Solving Partial-Differential Equations In A Finite Domain,” J.Comp. Phys. 120(1), 33–47 (1995).
[Crossref]

Sevick-Muraca, E. M.

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, “Adaptive finite element based tomography for fluorescence optical imaging in tissue,” Opt. Express 12(22), 5402–5417 (2004).
[Crossref]

Shih, Y. P.

M. Q. Chen, C. I. Hwang, and Y. P. Shih, “The computation of wavelet-Galerkin approximation on a bounded interval,” Int. J. Num. Meth. Engin. 39(17), 2921–2944 (1996).
[Crossref]

Shutyaev, V. P.

G. I. Marchuk, V. I. Agoshkov, and V. P. Shutyaev, “Adjoint equations and perturbation algorithms in nonlinear problems,” CRC Press, Boca Raton, FL (1996).

Slaney, M.

A. C. Kak and M. Slaney, Principles of computerized tomographic imaging (NY, IEEE Press, 1987).

Svaasand, L. O.

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, and M. S. McAdams, “Boundary-Conditions For The Diffusion Equation In Radiative-Transfer,” J. Opt. Soc. Am. A 11(10), 2727–2741 (1994).
[Crossref]

Tsay, T. T.

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, and M. S. McAdams, “Boundary-Conditions For The Diffusion Equation In Radiative-Transfer,” J. Opt. Soc. Am. A 11(10), 2727–2741 (1994).
[Crossref]

Turjman, F.

S. Bonnet, F. Peyrin, F. Turjman, and R. Prost, “Multiresolution Reconstruction in Fan-Beam Tomography,” IEEE Trans. Image Proc. 11(3), 169–176 (2002).
[Crossref]

Unser, M.

J. C. Baritaux, S. C. Sekhar, and M. Unser, “A Spline-based forward model for Optical Diffuse Tomography,” Biomedical Imaging, ISBI, 384–387 (2008).

Vasilyev, O. V.

O. V. Vasilyev, S. Paolucci, and M. Sen, “A Multilevel Wavelet Collocation Method For Solving Partial-Differential Equations In A Finite Domain,” J.Comp. Phys. 120(1), 33–47 (1995).
[Crossref]

Vasu, R. M.

B. Kanmani, P. Bansal, and R. M. Vasu, “Computationally efficient optical tomographic reconstruction through waveletizing the normalized quadratic perturbation equation,” Proc. SPIE 6164, R1–R10 (2006).

S. K. Nath, R. M. Vasu, and M. Pandit, “Wavelet based compression and denoising of optical tomography data,” Opt. Commun. 167(1–6), 37–46 (1999).
[Crossref]

Vo-Dinh, T.

T. Vo-Dinh, Biomedical Photonics Handbook (CRC Press, 2003).
[Crossref]

Wang, Y.

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Figures (4)

Fig. 1.
Fig. 1.

Daubechies scaling function of order M=3.

Fig. 2.
Fig. 2.

(a) Schema of a fictitious domain (rectangle Ω’) larger than the studied domain Ω in 2D. (b) Schema of the discretized approximation of edges ∂Ω. (c) Schema of a fictitious domain in 1D and corresponding scaling functions Φ j,k =2-j/2Φ)./2 j -k) with j=0 and k=-4 to 7.

Fig. 3.
Fig. 3.

(a)(b) Schemas of the 2D geometries: black points: positions of the source; gray points: detectors; ellipses: 2 simulated fluorophores; gray grid: regular grid for a scale 2–3; large rectangle: fictitious larger domain; (c) schemas of the 3D geometry: ellipsoid: simulated fluorophore. (d) Example of result for the forward problem (ux for the source #8 (surrounded in (a)), in the first simulation), by wavelet-Galerkin method, analytical modelling and FEM. (e) (h) Results of the rectangular simulation. (f)(i) Results of the irregular geometry simulation. g) (j) Results of the cube simulation. (e) (f) (g) Results after reconstruction, with a forward modelling by wavelet-Galerkin method and a reconstruction by ART; (h) (i) (j) Results with a forward modelling by standard FEM method and a reconstruction by ART.

Fig. 4.
Fig. 4.

Reconstruction of the second simulation (a) with connection coefficients accurately calculated on Ω; (b) without accurate connection coefficients on Ω.

Tables (5)

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Table 1. Numerical methods by the Finite Elements Methods.

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Table 2. Description of the 3 simulations

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Table A-1. Daubechies scaling filter coefficients, order M=3.

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Table A-2. Values of Γ1,1 k,l [-∞;+∞]=∫+∞-∞Φ(1)(x-k) Φ(1)(x-l)dx, with Φ the Daubechies scaling function of order M=3. This filter is symmetrical, only the value |k-l| is considered.

Tables Icon

Table A-3. Values of Γ0,0 k,l [0;1]=∫1 0Φ(x-k)Φ(x-l)dx, with Φ the Daubechies scaling function of order M=3.

Equations (31)

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{.(D(r)ux(r,rs))+μa(r)ux(r,rs)=q(rs).(D(r)um(r,r,rs))+μa(r)um(r,r,rs)=β(r)ux(r,rs) .
{n.(Dux(r,rs))+ξxux(r,rs)rΩn.(Dum(r,r,rs))+ξmum(r,r,rs)=0rΩ
umobs (rd,rs)=Ωum(rd,r,rs)dr=Ωgm(r,rd)β(r)ux(r,rs)dr
. (Dgm(r,rd))+μagm(r,rd)=q(rd)
. (Du)+μau=q
u˜ = k=1Nukfkinterp.
q˜ = k=1Nqkfkinterp.
Ω . (Du˜)+μau˜q˜).flproj=0l=1,,N.
A u = Pq q
umobs=W β ,
Φj,k = 2j2 Φ (.2jk)
Φ = 2 nZh(n)Φ(2.n).
D k=1NukΩ(2fkinterp)flproj+μak=1NukΩfkinterpflproj=k=1NqkΩfkinterpflprojl=1,,N
ζ k=1NukΩfkfl+Dk=1NukΩfkfl+μak=1NukΩfkfl=k=1NqkΩfkfll=1,,N
u˜j = k=0Nxpuj,kΦj,k;q˜=k=0Nxpqj,kΦj,k
qj,k = Ωq Φj,k .
ζ k=0Nxpuj,kΩΦj,kΦj,l+Dk=0Nxpuj,kΩΦj,kΦj,l+μak=0Nxpuj,kΩΦj,kΦj,l
= k=pNxpqj,kΩΦj,kΦj,l,l=1,,N.
Γj,k,lm1,m2 [;+]=2j(1+m1+m2)+Φj,k(m1)(x)Φj,l(m2)(x)dx
Γk,l0,0 [;+]=+Φ(xk)Φ(xl)dx=δk,l.
Γk,lm1,m2 [0;n]=0nΦ(m1)(xk)Φ(m2)(xl)dx.
Γk,l0,0 [0;n]=p=0n1Γkp,lp0,0[0;1].
Φj,kx,ky (x,y)=Φj,kx(x)Φj,ky(y)
= 2j Φ (2jxkx)Φ(2jyky),
kx = 0 , , Nx p ,
ky = 0 , , Ny p .
ΩΦkx,ky(x,y)Φlx,ly(x,y)dxdy=Φ(kx)Φ(lx)y=0LyΦky(y)Φly(y)dy
+ Φ (Lxkx)Φ(Lxlx)y=0LyΦky(y)Φly(y)dy
+ Φ (ky)Φ(ly)x=0LxΦkx(x)Φlx(x)dx
+ Φ (Lyky)Φ(Lyly)x=0LxΦkx(x)Φlx(x)dx.
ε = umobsumrec2umobs2

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