OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 16, Iss. 7 — Jul. 1, 1999
  • pp: 1814–1826

Modified distorted Born iterative method with an approximate Fréchet derivative for optical diffusion tomography

J. C. Ye, K. J. Webb, R. P. Millane, and T. J. Downar  »View Author Affiliations


JOSA A, Vol. 16, Issue 7, pp. 1814-1826 (1999)
http://dx.doi.org/10.1364/JOSAA.16.001814


View Full Text Article

Enhanced HTML    Acrobat PDF (640 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

In frequency-domain optical diffusion imaging, the magnitude and the phase of modulated light propagated through a highly scattering medium are used to reconstruct an image of the scattering and absorption coefficients in the medium. Although current reconstruction algorithms have been applied with some success, there are opportunities for improving both the accuracy of the reconstructions and the speed of convergence. In particular, conventional integral equation approaches such as the Born iterative method and the distorted Born iterative method can suffer from slow convergence, especially for large spatial variations in the constitutive parameters. We show that slow convergence of conventional algorithms is due to the linearized integral equations’ not being the correct Fréchet derivative with respect to the absorption and scattering coefficients. The correct Fréchet derivative operator is derived here. However, the Fréchet derivative suffers from numerical instability because it involves gradients of both the Green’s function and the optical flux near singularities, a result of the use of near-field imaging data. To ameliorate these effects we derive an approximation to the Fréchet derivative and implement it in an inversion algorithm. Simulation results show that this inversion algorithm outperforms conventional iterative methods.

© 1999 Optical Society of America

OCIS Codes
(100.3010) Image processing : Image reconstruction techniques
(100.3190) Image processing : Inverse problems
(100.6950) Image processing : Tomographic image processing
(170.3010) Medical optics and biotechnology : Image reconstruction techniques
(290.3200) Scattering : Inverse scattering

History
Original Manuscript: August 18, 1998
Revised Manuscript: February 8, 1999
Manuscript Accepted: February 8, 1999
Published: July 1, 1999

Citation
J. C. Ye, K. J. Webb, R. P. Millane, and T. J. Downar, "Modified distorted Born iterative method with an approximate Fréchet derivative for optical diffusion tomography," J. Opt. Soc. Am. A 16, 1814-1826 (1999)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-16-7-1814


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. M. S. Patterson, B. Chance, B. Wilson, “Time-resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989). [CrossRef] [PubMed]
  2. T. J. Farrell, M. S. Patterson, B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992). [CrossRef] [PubMed]
  3. H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson, “Optical image reconstruction using frequency-domain data: simulation and experiment,” J. Opt. Soc. Am. A 13, 253–266 (1996). [CrossRef]
  4. J. S. Reynolds, S. Yeung, A. Przadka, K. J. Webb, “Optical diffusion imaging: a comparative numerical and experimental study,” Appl. Opt. 35, 3671–3679 (1996). [CrossRef] [PubMed]
  5. U. Tautenhahn, “On a general regularization scheme for nonlinear ill-posed problems,” Inverse Probl. 13, 1427–1437 (1997). [CrossRef]
  6. S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near infrared absorption and scattering imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. Alfano, B. Chance, eds., Proc. SPIE1888, 360–371 (1993). [CrossRef]
  7. Y. Yao, Y. Wang, Y. Pei, W. Zhu, R. L. Barbour, “Frequency domain optical imaging of absorption and scattering distributions by a Born iterative method,” J. Opt. Soc. Am. A 14, 325–342 (1997). [CrossRef]
  8. A. Tikhonov, V. Arsenin, Solutions of Ill-Posed Problems (Winston, New York, 1977).
  9. J. E. Dennis, Numerical Methods for Unconstrained Optimization and Nonlinear Equation (Prentice-Hall, Englewood Cliffs, N.J., 1983).
  10. R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, New York, 1987).
  11. W. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).
  12. N. Joachimowicz, C. Pichot, J. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991). [CrossRef]
  13. J. C. Ye, R. P. Millane, K. J. Webb, T. J. Downar, “Importance of the ∇D term in frequency-resolved optical diffusion imaging,” Opt. Lett. 23, 1423–1425 (1998). [CrossRef]
  14. S. S. Saquib, K. M. Hanson, G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging 1997: Image Processing, K. M. Hanson, ed., Proc. SPIE3034, 369–380 (1997). [CrossRef]
  15. S. R. Arridge, M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express 2, 213–226 (1998). [CrossRef] [PubMed]
  16. M. O’Leary, D. Boas, B. Chance, A. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion-photon tomography,” Opt. Lett. 20, 426–428 (1995). [CrossRef]
  17. A. W. Naylor, G. R. Sell, Linear Operator Theory in Engineering and Science, 2nd ed., Vol. 40 of Applied Mathematical Science (Springer-Verlag, New York, 1982). [CrossRef]
  18. T. J. Connolly, D. J. Wall, “On Fréchet differentiability of some nonlinear operators occurring in inverse problems: an implicit function theorem approach,” Inverse Probl. 6, 949–966 (1990). [CrossRef]
  19. S. R. Arridge, M. Schweiger, “Photon-measurement density functions. 2. Finite-element calculations,” Appl. Opt. 34, 8026–8037 (1995). [CrossRef] [PubMed]
  20. Q. H. Liu, “Nonlinear inversion of electrode-type resistivity measurements,” IEEE Trans. Geosci. Remote Sens. 32, 499–507 (1994). [CrossRef]
  21. O. Arikan, “Regularized inversion of a two-dimensional integral equation with applications in borehole induction measurement,” Radio Sci. 29, 519–538 (1994). [CrossRef]
  22. J. C. Ye, K. J. Webb, T. J. Downar, R. P. Millane, “Weighted cost function reconstruction in optical diffusion imaging,” in Computational, Experimental and Numerical Methods for Solving Ill-Posed Inverse Imaging Problems: Medical and Nonmedical Application, R. L. Barber, M. J. Carvin, M. A. Fiddy, eds., Proc. SPIE3171, 118–127 (1997). [CrossRef]
  23. E. Zeidler, Applied Functional Analysis, Vol. 108 of Applied Mathematical Sciences (Springer-Verlag, New York, 1995).
  24. H. V. Poor, An Introduction of Signal Detection and Estimation, 2nd ed. (Springer-Verlag, New York, 1994).
  25. C. A. Bouman, K. Sauer, “A unified approach to statistical tomography using coordinate descent optimization,” IEEE Trans. Image Process. 5, 480–492 (1996). [CrossRef] [PubMed]
  26. V. A. Morozov, Methods for Solving Incorrectly Posed Problems (Springer-Verlag, New York, 1984).
  27. J. C. Adams, “mudpack: multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations,” Appl. Math. Comput. 34, 113–146 (1989). [CrossRef]
  28. J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).
  29. C. A. Thompson, K. J. Webb, A. M. Weiner, “Diffusive media characterization using laser speckle,” Appl. Opt. 36, 3726–3734 (1997). [CrossRef] [PubMed]
  30. T. J. Connolly, D. J. Wall, “On an inverse problem, with boundary measurements, for the steady state diffusion equation,” Inverse Probl. 4, 995–1012 (1988). [CrossRef]
  31. R. A. Adams, Sobolev Spaces, Vol. 65 of Pure and Applied Mathematics (Academic, New York, 1975).
  32. A. Friedman, Partial Differential Equations (Rinehart and Winston, New York, 1969).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited