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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 21 — Oct. 12, 2009
  • pp: 18433–18448

Application of a wavelet-Galerkin method to the forward problem resolution in fluorescence Diffuse Optical Tomography

Anne Landragin-Frassati, Stéphane Bonnet, Anabela Da Silva, Jean-Marc Dinten, and Didier Georges  »View Author Affiliations


Optics Express, Vol. 17, Issue 21, pp. 18433-18448 (2009)
http://dx.doi.org/10.1364/OE.17.018433


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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

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(100.7410) Image processing : Wavelets
(170.3880) Medical optics and biotechnology : Medical and biological imaging

ToC Category:
Medical Optics and Biotechnology

History
Original Manuscript: July 20, 2009
Revised Manuscript: September 4, 2009
Manuscript Accepted: September 8, 2009
Published: September 28, 2009

Citation
Anne Landragin-Frassati, Stéphane Bonnet, Anabela Da Silva, Jean-Marc Dinten, and Didier Georges, "Application of a wavelet-Galerkin method to the forward problem resolution in fluorescence diffuse optical tomography," Opt. Express 17, 18433-18448 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-21-18433


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References

  1. R. Weissleder, and V. Ntziachristos, "Shedding light onto live molecular targets," Nature Med 9(1), 123-128 (2003).
  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]
  3. D. S. Burnett, Finite Element Concept (Addison-Wesley Publishing Company, 1987).
  4. S. R. Arridge, "Optical tomography in medical imaging," Inverse Problems 15(2), R41-R93 (1999). [CrossRef]
  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).
  6. S. Goedecker, Wavelets and their application for the Partial Differential Equations in Physics, PPUR (1998).
  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.
  10. S. Mallat, A wavelet tour of signal processing (Academic Press, 1999).
  11. S. Mallat, "A Theory of Multiresolution Signal Decomposition: the Wavelet Representation," IEEE Trans. Pattern Anal. Machine Intell. 11(7), 674-693 (1989). [CrossRef]
  12. B. Sahiner, and A. E. Yagle, "Image-Reconstruction From Projections Under Wavelet Constraints," IEEE Trans. Signal Proc. 41(12), 3579-3584 (1993). [CrossRef]
  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). [CrossRef]
  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).
  21. J. Baccou, and J. Liandrat, "On coupling wavelets with fictitious domain approaches," Appl. Math. Lett. 18(12), 1325-1331 (2005). [CrossRef]
  22. 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]
  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]
  25. http://www.comsol.com/
  26. A. C. Kak, and M. Slaney, Principles of computerized tomographic imaging (NY, IEEE Press, 1987).
  27. 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]
  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]

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