Abstract

A new numerical imaging algorithm is presented for reconstruction of optical absorption coefficients from near-infrared light data with a continuous-wave source. As a continuation of our earlier efforts in developing a series of methods called “globally convergent reconstruction methods” [J. Opt. Soc. Am. A 23, 2388 (2006) ], this numerical algorithm solves the inverse problem through solution of a boundary-value problem for a Volterra-type integral partial differential equation. We deal here with the particular issues in solving the inverse problems in an arbitrary convex shape domain. It is demonstrated in numerical studies that this reconstruction technique is highly efficient and stable with respect to the complex distribution of actual unknown absorption coefficients. The method is particularly useful for reconstruction from a large data set obtained from a tissue or organ of particular shape, such as the prostate. Numerical reconstructions of a simulated prostate-shaped phantom with three different settings of absorption-inclusions are presented.

© 2009 Optical Society of America

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  1. 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. Treatment 4, 513-526 (2005).
  2. S. Srinivasan, B. W. Pogue, H. Dehghani, S. Jiang, X. Song, and K. D. Paulsen, “Improved quantification of small objects in near-infrared diffuse optical tomography,” J. Biomed. Opt. 9, 1161-1171 (2004).
    [CrossRef] [PubMed]
  3. C. Schmitz, D. Klemer, R. Hardin, M. Katz, Y. Pei, H. Graber, M. Levin, R. Levina, N. Franco, W. Solomon, and R. Barbour, “Design and implementation of dynamic near-infrared optical tomographic imaging instrumentation for simultaneous dual-breast measurements,” Appl. Opt. 44, 2140-2153 (2005).
    [CrossRef] [PubMed]
  4. Y. Xu, X. Gu, L. Fajardo, and H. Jiang, “In vivo breast imaging with diffuse optical tomography based on higher-order diffusion equations,” Appl. Opt. 42, 3163-3169 (2003).
    [CrossRef] [PubMed]
  5. A. Godavarty, A. B. Thompson, Jr., R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraka, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488-496 (2004).
    [CrossRef] [PubMed]
  6. A. Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev, and A. H. Hielscher, “Three-dimensional optical tomographic brain imaging in small animals, part 1: hypercapnia,” J. Biomed. Opt. 9, 1046-1062 (2004).
    [CrossRef] [PubMed]
  7. S. Gopinath, C. S. Robertson, R. G. Grossman, and B. Chance, “Near-infrared spectroscopic localization of intracranial hematomas,” J. Neurosurg. 79, 43-47 (1993).
    [CrossRef] [PubMed]
  8. G. Zhang, A. Katz, R. R. Alfano, A. D. Kofinas, P. G. Stubblefield, W. Rosenfeld, D. Beyer, D. Maulik, and M. R. Stankovic, “Brain perfusion monitoring with frequency-domain and continuous-wave near-infrared spectroscopy: a cross-correlation study in newborn piglets,” Phys. Med. Biol. 45, 3143-3158 (2000).
    [CrossRef] [PubMed]
  9. A. G. Yodh and D. A. Boas, “Functional imaging with diffusing light,” in Biomedical Photonics Handbook, T.Vo-Dinh, ed. (CRC Press), 21-1-21-45 (2003).
  10. B. Chance, E. Anday, S. Nioka, S. Zhou, L. Hong, K. Worden, C. Li, T. Murray, Y. Ovetsky, D. Pidikiti, and R. Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Express 2, 411-423 (1998).
    [CrossRef] [PubMed]
  11. D. A. Boas, T. Gaudette, G. Strangman, X. Cheng, J. J. A. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13, 76-90 (2001).
    [CrossRef] [PubMed]
  12. T. O. McBride, B. W. Pogue, S. Jiang, U. L. Österberg, and K. D. Paulsen, “Initial studies of in vivo absorbing and scattering heterogeneity in near-infrared tomographic breast imaging,” Opt. Lett. 26, 822-824 (2001).
    [CrossRef]
  13. M. Solonenko, R. Cheung, T. M. Busch, A. Kachur, G. M. Griffin, T. Vulcan, T. C. Zhu, H. W. Wang, S. M. Hahn, and A. G. Yodh, “In vivo reflectance measurement of optical properties, blood oxygenation and motexafin lutetium uptake in canine large bowels, kidneys and prostates,” Phys. Med. Biol. 47(6), 857-73 (2002).
    [PubMed]
  14. K. L. Du, R. Mick, T. M. Busch, T. C. Zhu, J. C. Finlay, G. Yu, A. G. Yodh, S. B. Malkowicz, D. Smith, R. Whittington, D. Stripp, and S. M. Hahn, “Preliminary results of interstitial motexafin lutetium-mediated PDT for prostate cancer,” Lasers Surg. Med. 38, 427-34 (2006).
    [CrossRef] [PubMed]
  15. W. B. Wang, J. H. Ali, J. H. Vitenson, J. M. Lombardo, and R. R. Alfano, “Spectral polarization imaging of human rectum-membrane-prostate tissues,” IEEE J. Sel. Top. Quantum Electron. 9, 288-293 (2003).
    [CrossRef]
  16. W. B. Wang, J. H. Ali, M. Zevallos, and R. R. Alfano, “Near infrared imaging of human prostate cancerous and normal tissues based on water absorption,” in Advances in Optical Imaging and Photon Migration (Optical Society of America, 2004), MF 38; on CD-ROM.
  17. B. Chance, “High sensitivity and specificity in human breast cancer detection with near-infrared imaging,” in Biomedical Topical Meeting, Vol. 71 of OSA Trends in Optics and Photonics (Optical Society of America, 2002), pp. 450-455.
  18. S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modeling and reconstruction,” Phys. Med. Biol. 42, pp. 841-853 (1997).
    [CrossRef] [PubMed]
  19. S. R. Arridge and M. Schweiger, “A general framework for iterative reconstruction algorithms in optical tomography, using a finite-element method,” in Computational Radiology and Imaging: Therapy and Diagnosis, C.Borgers and F.Natterer, eds., Vol. 110 of IMA Volumes in Mathematics and its Applications (Springer-Verlag, 1998), pp. 45-70.
    [CrossRef]
  20. A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imaging 18, 262-271 (1999).
    [CrossRef] [PubMed]
  21. J. C. Schotland, “Continuous-wave diffusion imaging,” J. Opt. Soc. Am. A 14, 275-279 (1997).
    [CrossRef]
  22. M. A. O'Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion-photon tomography,” Opt. Lett. 20, 426-428 (1995).
    [CrossRef] [PubMed]
  23. Y. A. Gryazin, M. V. Klibanov, and T. R. Lucas, “Numerical solution of a subsurface imaging inverse problem,” SIAM J. Appl. Math. 62, 664-683 (2001).
    [CrossRef]
  24. M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications (Brill, 2004).
  25. M. V. Klibanov and A. Timonov, “Numerical studies on the globally convergent convexification algorithm in 2D,” Inverse Probl. 23, 123-138 (2007).
    [CrossRef]
  26. H. B. Keller and D. J. Perozzi, “Fast seismic ray tracing,” SIAM J. Appl. Math. 43, 981-992 (1983).
    [CrossRef]
  27. H. Shan, M. V. Klibanov, H. Liu, N. Pantong, and J. Su, “Numerical implementation of the convexification algorithm for an optical diffusion tomograph,” Inverse Probl. 24, 025006 (2008).
    [CrossRef]
  28. H. Shan, M. V. Klibanov, J. Su, N. Pantong, and H. Liu, “A globally accelerated numerical method for optical tomography with continuous wave source,” J Inv. Ill-Posed Probl. accepted for publication. A preprint can be found online at http://arxiv.org/abs/0809.3910 (date of last access: October 8, 2008).
  29. L. Beilina and M. V. Klibanov, “A globally convergent numerical method for a coefficient inverse problem,” SIAM J. Sci. Comput. (USA) 31, 478-509 (2008).
    [CrossRef]
  30. M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations (Springer-Verlag, 1984).
    [CrossRef]
  31. J. Su, H. Shan, H. Liu, and M. V. Klibanov, “Reconstruction method with data from a multiple-site continuous-wave source for three-dimensional optical tomography,” J. Opt. Soc. Am. A 23, 2388-2395 (2006).
    [CrossRef]
  32. A. N. Tikhonov and V. Ya Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).
  33. O. A. Ladyzhenskaya and N. N. Uralceva, Linear and Quasilinear Elliptic Equations (Academic, 1969).
  34. E. W. Cheney, Introduction to Approximation Theory (Chelsea, 1982).
  35. E. Isaacson and H. B. Keller, Analysis of Numerical Methods (Wiley, 1966).
  36. M. V. Newberry, “Signal-to-noise considerations for sky-subtracted CCD data,” Publ. Astron. Soc. Pac. 163, 122 (1991). Available on website http://adsabs.harvard.edu/abs/1991PASP.103.122N (date of last access: October 8, 2008).
  37. E. E. Lewis and W. F. Miller, Jr., Computational Methods of Neutron Transport (Wiley, 1984).
  38. V. A. Markel and J. C. Schotland, “Inverse problem in optical diffusion tomography. II. Role of boundary conditions,” J. Opt. Soc. Am. A 19, 558-566 (2002).
    [CrossRef]
  39. H. Liu, M. Miwa, B. Beauvoit, N. G. Wang, and B. Chance, “Characterization of absorption and scattering properties of small-volume biological samples using time-resolved spectroscopy,” Anal. Biochem. 213, 378-385 (1993).
    [CrossRef] [PubMed]
  40. V. Ntziachristos, C. Bremer, and R. Weissleder, “Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging,” Eur. Radiol. 13, 195-208 (2003).
    [PubMed]
  41. K. T. Kotz, S. S. Dixit, A. D. Gibbs, J. M. Orduna, Z. Haroon, K. Amin, and G. W. Faris, “Inspiratory contrast for in vivo optical imaging,” Opt. Express 16, 19-31 (2008).
    [CrossRef] [PubMed]
  42. A. L. Bukhgeim and M. V. Klibanov, “Uniqueness in the large class of multidimensional inverse problems,” Sov. Math. Dokl. 17, 244-247 (1981).
  43. M. V. Klibanov, “Inverse problems and Carleman estimates,” Inverse Probl. 8, 575-586 (1992).
    [CrossRef]
  44. L. C. Evans, Partial Differential Equations (American Mathematical Society, 1998).
  45. E. A. Coddington and N. Levinson, Theory of Differential Equations, (Krieger, 1984).

2008 (3)

H. Shan, M. V. Klibanov, H. Liu, N. Pantong, and J. Su, “Numerical implementation of the convexification algorithm for an optical diffusion tomograph,” Inverse Probl. 24, 025006 (2008).
[CrossRef]

L. Beilina and M. V. Klibanov, “A globally convergent numerical method for a coefficient inverse problem,” SIAM J. Sci. Comput. (USA) 31, 478-509 (2008).
[CrossRef]

K. T. Kotz, S. S. Dixit, A. D. Gibbs, J. M. Orduna, Z. Haroon, K. Amin, and G. W. Faris, “Inspiratory contrast for in vivo optical imaging,” Opt. Express 16, 19-31 (2008).
[CrossRef] [PubMed]

2007 (1)

M. V. Klibanov and A. Timonov, “Numerical studies on the globally convergent convexification algorithm in 2D,” Inverse Probl. 23, 123-138 (2007).
[CrossRef]

2006 (2)

J. Su, H. Shan, H. Liu, and M. V. Klibanov, “Reconstruction method with data from a multiple-site continuous-wave source for three-dimensional optical tomography,” J. Opt. Soc. Am. A 23, 2388-2395 (2006).
[CrossRef]

K. L. Du, R. Mick, T. M. Busch, T. C. Zhu, J. C. Finlay, G. Yu, A. G. Yodh, S. B. Malkowicz, D. Smith, R. Whittington, D. Stripp, and S. M. Hahn, “Preliminary results of interstitial motexafin lutetium-mediated PDT for prostate cancer,” Lasers Surg. Med. 38, 427-34 (2006).
[CrossRef] [PubMed]

2005 (2)

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. Treatment 4, 513-526 (2005).

C. Schmitz, D. Klemer, R. Hardin, M. Katz, Y. Pei, H. Graber, M. Levin, R. Levina, N. Franco, W. Solomon, and R. Barbour, “Design and implementation of dynamic near-infrared optical tomographic imaging instrumentation for simultaneous dual-breast measurements,” Appl. Opt. 44, 2140-2153 (2005).
[CrossRef] [PubMed]

2004 (3)

S. Srinivasan, B. W. Pogue, H. Dehghani, S. Jiang, X. Song, and K. D. Paulsen, “Improved quantification of small objects in near-infrared diffuse optical tomography,” J. Biomed. Opt. 9, 1161-1171 (2004).
[CrossRef] [PubMed]

A. Godavarty, A. B. Thompson, Jr., R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraka, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488-496 (2004).
[CrossRef] [PubMed]

A. Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev, and A. H. Hielscher, “Three-dimensional optical tomographic brain imaging in small animals, part 1: hypercapnia,” J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

2003 (3)

Y. Xu, X. Gu, L. Fajardo, and H. Jiang, “In vivo breast imaging with diffuse optical tomography based on higher-order diffusion equations,” Appl. Opt. 42, 3163-3169 (2003).
[CrossRef] [PubMed]

W. B. Wang, J. H. Ali, J. H. Vitenson, J. M. Lombardo, and R. R. Alfano, “Spectral polarization imaging of human rectum-membrane-prostate tissues,” IEEE J. Sel. Top. Quantum Electron. 9, 288-293 (2003).
[CrossRef]

V. Ntziachristos, C. Bremer, and R. Weissleder, “Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging,” Eur. Radiol. 13, 195-208 (2003).
[PubMed]

2002 (2)

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

M. Solonenko, R. Cheung, T. M. Busch, A. Kachur, G. M. Griffin, T. Vulcan, T. C. Zhu, H. W. Wang, S. M. Hahn, and A. G. Yodh, “In vivo reflectance measurement of optical properties, blood oxygenation and motexafin lutetium uptake in canine large bowels, kidneys and prostates,” Phys. Med. Biol. 47(6), 857-73 (2002).
[PubMed]

2001 (3)

Y. A. Gryazin, M. V. Klibanov, and T. R. Lucas, “Numerical solution of a subsurface imaging inverse problem,” SIAM J. Appl. Math. 62, 664-683 (2001).
[CrossRef]

D. A. Boas, T. Gaudette, G. Strangman, X. Cheng, J. J. A. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13, 76-90 (2001).
[CrossRef] [PubMed]

T. O. McBride, B. W. Pogue, S. Jiang, U. L. Österberg, and K. D. Paulsen, “Initial studies of in vivo absorbing and scattering heterogeneity in near-infrared tomographic breast imaging,” Opt. Lett. 26, 822-824 (2001).
[CrossRef]

2000 (1)

G. Zhang, A. Katz, R. R. Alfano, A. D. Kofinas, P. G. Stubblefield, W. Rosenfeld, D. Beyer, D. Maulik, and M. R. Stankovic, “Brain perfusion monitoring with frequency-domain and continuous-wave near-infrared spectroscopy: a cross-correlation study in newborn piglets,” Phys. Med. Biol. 45, 3143-3158 (2000).
[CrossRef] [PubMed]

1999 (1)

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

1998 (1)

1997 (2)

J. C. Schotland, “Continuous-wave diffusion imaging,” J. Opt. Soc. Am. A 14, 275-279 (1997).
[CrossRef]

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modeling and reconstruction,” Phys. Med. Biol. 42, pp. 841-853 (1997).
[CrossRef] [PubMed]

1995 (1)

1993 (2)

S. Gopinath, C. S. Robertson, R. G. Grossman, and B. Chance, “Near-infrared spectroscopic localization of intracranial hematomas,” J. Neurosurg. 79, 43-47 (1993).
[CrossRef] [PubMed]

H. Liu, M. Miwa, B. Beauvoit, N. G. Wang, and B. Chance, “Characterization of absorption and scattering properties of small-volume biological samples using time-resolved spectroscopy,” Anal. Biochem. 213, 378-385 (1993).
[CrossRef] [PubMed]

1992 (1)

M. V. Klibanov, “Inverse problems and Carleman estimates,” Inverse Probl. 8, 575-586 (1992).
[CrossRef]

1983 (1)

H. B. Keller and D. J. Perozzi, “Fast seismic ray tracing,” SIAM J. Appl. Math. 43, 981-992 (1983).
[CrossRef]

1981 (1)

A. L. Bukhgeim and M. V. Klibanov, “Uniqueness in the large class of multidimensional inverse problems,” Sov. Math. Dokl. 17, 244-247 (1981).

Abdoulaev, G. S.

A. Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev, and A. H. Hielscher, “Three-dimensional optical tomographic brain imaging in small animals, part 1: hypercapnia,” J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

Alfano, R. R.

W. B. Wang, J. H. Ali, J. H. Vitenson, J. M. Lombardo, and R. R. Alfano, “Spectral polarization imaging of human rectum-membrane-prostate tissues,” IEEE J. Sel. Top. Quantum Electron. 9, 288-293 (2003).
[CrossRef]

G. Zhang, A. Katz, R. R. Alfano, A. D. Kofinas, P. G. Stubblefield, W. Rosenfeld, D. Beyer, D. Maulik, and M. R. Stankovic, “Brain perfusion monitoring with frequency-domain and continuous-wave near-infrared spectroscopy: a cross-correlation study in newborn piglets,” Phys. Med. Biol. 45, 3143-3158 (2000).
[CrossRef] [PubMed]

W. B. Wang, J. H. Ali, M. Zevallos, and R. R. Alfano, “Near infrared imaging of human prostate cancerous and normal tissues based on water absorption,” in Advances in Optical Imaging and Photon Migration (Optical Society of America, 2004), MF 38; on CD-ROM.

Ali, J. H.

W. B. Wang, J. H. Ali, J. H. Vitenson, J. M. Lombardo, and R. R. Alfano, “Spectral polarization imaging of human rectum-membrane-prostate tissues,” IEEE J. Sel. Top. Quantum Electron. 9, 288-293 (2003).
[CrossRef]

W. B. Wang, J. H. Ali, M. Zevallos, and R. R. Alfano, “Near infrared imaging of human prostate cancerous and normal tissues based on water absorption,” in Advances in Optical Imaging and Photon Migration (Optical Society of America, 2004), MF 38; on CD-ROM.

Amin, K.

Anday, E.

Arridge, S. R.

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modeling and reconstruction,” Phys. Med. Biol. 42, pp. 841-853 (1997).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, “A general framework for iterative reconstruction algorithms in optical tomography, using a finite-element method,” in Computational Radiology and Imaging: Therapy and Diagnosis, C.Borgers and F.Natterer, eds., Vol. 110 of IMA Volumes in Mathematics and its Applications (Springer-Verlag, 1998), pp. 45-70.
[CrossRef]

Barbour, R.

Beauvoit, B.

H. Liu, M. Miwa, B. Beauvoit, N. G. Wang, and B. Chance, “Characterization of absorption and scattering properties of small-volume biological samples using time-resolved spectroscopy,” Anal. Biochem. 213, 378-385 (1993).
[CrossRef] [PubMed]

Beilina, L.

L. Beilina and M. V. Klibanov, “A globally convergent numerical method for a coefficient inverse problem,” SIAM J. Sci. Comput. (USA) 31, 478-509 (2008).
[CrossRef]

Beyer, D.

G. Zhang, A. Katz, R. R. Alfano, A. D. Kofinas, P. G. Stubblefield, W. Rosenfeld, D. Beyer, D. Maulik, and M. R. Stankovic, “Brain perfusion monitoring with frequency-domain and continuous-wave near-infrared spectroscopy: a cross-correlation study in newborn piglets,” Phys. Med. Biol. 45, 3143-3158 (2000).
[CrossRef] [PubMed]

Bluestone, A. Y.

A. Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev, and A. H. Hielscher, “Three-dimensional optical tomographic brain imaging in small animals, part 1: hypercapnia,” J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

Boas, D. A.

D. A. Boas, T. Gaudette, G. Strangman, X. Cheng, J. J. A. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13, 76-90 (2001).
[CrossRef] [PubMed]

M. A. O'Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion-photon tomography,” Opt. Lett. 20, 426-428 (1995).
[CrossRef] [PubMed]

A. G. Yodh and D. A. Boas, “Functional imaging with diffusing light,” in Biomedical Photonics Handbook, T.Vo-Dinh, ed. (CRC Press), 21-1-21-45 (2003).

Bremer, C.

V. Ntziachristos, C. Bremer, and R. Weissleder, “Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging,” Eur. Radiol. 13, 195-208 (2003).
[PubMed]

Brooksby, B.

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. Treatment 4, 513-526 (2005).

Bukhgeim, A. L.

A. L. Bukhgeim and M. V. Klibanov, “Uniqueness in the large class of multidimensional inverse problems,” Sov. Math. Dokl. 17, 244-247 (1981).

Busch, T. M.

K. L. Du, R. Mick, T. M. Busch, T. C. Zhu, J. C. Finlay, G. Yu, A. G. Yodh, S. B. Malkowicz, D. Smith, R. Whittington, D. Stripp, and S. M. Hahn, “Preliminary results of interstitial motexafin lutetium-mediated PDT for prostate cancer,” Lasers Surg. Med. 38, 427-34 (2006).
[CrossRef] [PubMed]

M. Solonenko, R. Cheung, T. M. Busch, A. Kachur, G. M. Griffin, T. Vulcan, T. C. Zhu, H. W. Wang, S. M. Hahn, and A. G. Yodh, “In vivo reflectance measurement of optical properties, blood oxygenation and motexafin lutetium uptake in canine large bowels, kidneys and prostates,” Phys. Med. Biol. 47(6), 857-73 (2002).
[PubMed]

Chance, B.

B. Chance, E. Anday, S. Nioka, S. Zhou, L. Hong, K. Worden, C. Li, T. Murray, Y. Ovetsky, D. Pidikiti, and R. Thomas, “A novel method for fast imaging of brain function, non-invasively, with light,” Opt. Express 2, 411-423 (1998).
[CrossRef] [PubMed]

M. A. O'Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion-photon tomography,” Opt. Lett. 20, 426-428 (1995).
[CrossRef] [PubMed]

S. Gopinath, C. S. Robertson, R. G. Grossman, and B. Chance, “Near-infrared spectroscopic localization of intracranial hematomas,” J. Neurosurg. 79, 43-47 (1993).
[CrossRef] [PubMed]

H. Liu, M. Miwa, B. Beauvoit, N. G. Wang, and B. Chance, “Characterization of absorption and scattering properties of small-volume biological samples using time-resolved spectroscopy,” Anal. Biochem. 213, 378-385 (1993).
[CrossRef] [PubMed]

B. Chance, “High sensitivity and specificity in human breast cancer detection with near-infrared imaging,” in Biomedical Topical Meeting, Vol. 71 of OSA Trends in Optics and Photonics (Optical Society of America, 2002), pp. 450-455.

Cheney, E. W.

E. W. Cheney, Introduction to Approximation Theory (Chelsea, 1982).

Cheng, X.

D. A. Boas, T. Gaudette, G. Strangman, X. Cheng, J. J. A. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13, 76-90 (2001).
[CrossRef] [PubMed]

Cheung, R.

M. Solonenko, R. Cheung, T. M. Busch, A. Kachur, G. M. Griffin, T. Vulcan, T. C. Zhu, H. W. Wang, S. M. Hahn, and A. G. Yodh, “In vivo reflectance measurement of optical properties, blood oxygenation and motexafin lutetium uptake in canine large bowels, kidneys and prostates,” Phys. Med. Biol. 47(6), 857-73 (2002).
[PubMed]

Coddington, E. A.

E. A. Coddington and N. Levinson, Theory of Differential Equations, (Krieger, 1984).

Dehghani, H.

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. Treatment 4, 513-526 (2005).

S. Srinivasan, B. W. Pogue, H. Dehghani, S. Jiang, X. Song, and K. D. Paulsen, “Improved quantification of small objects in near-infrared diffuse optical tomography,” J. Biomed. Opt. 9, 1161-1171 (2004).
[CrossRef] [PubMed]

Dixit, S. S.

Du, K. L.

K. L. Du, R. Mick, T. M. Busch, T. C. Zhu, J. C. Finlay, G. Yu, A. G. Yodh, S. B. Malkowicz, D. Smith, R. Whittington, D. Stripp, and S. M. Hahn, “Preliminary results of interstitial motexafin lutetium-mediated PDT for prostate cancer,” Lasers Surg. Med. 38, 427-34 (2006).
[CrossRef] [PubMed]

Eppstein, M. J.

A. Godavarty, A. B. Thompson, Jr., R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraka, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488-496 (2004).
[CrossRef] [PubMed]

Evans, L. C.

L. C. Evans, Partial Differential Equations (American Mathematical Society, 1998).

Fajardo, L.

Faris, G. W.

Finlay, J. C.

K. L. Du, R. Mick, T. M. Busch, T. C. Zhu, J. C. Finlay, G. Yu, A. G. Yodh, S. B. Malkowicz, D. Smith, R. Whittington, D. Stripp, and S. M. Hahn, “Preliminary results of interstitial motexafin lutetium-mediated PDT for prostate cancer,” Lasers Surg. Med. 38, 427-34 (2006).
[CrossRef] [PubMed]

Franco, N.

Gaudette, T.

D. A. Boas, T. Gaudette, G. Strangman, X. Cheng, J. J. A. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13, 76-90 (2001).
[CrossRef] [PubMed]

Gibbs, A. D.

Godavarty, A.

A. Godavarty, A. B. Thompson, Jr., R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraka, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488-496 (2004).
[CrossRef] [PubMed]

Gopinath, S.

S. Gopinath, C. S. Robertson, R. G. Grossman, and B. Chance, “Near-infrared spectroscopic localization of intracranial hematomas,” J. Neurosurg. 79, 43-47 (1993).
[CrossRef] [PubMed]

Graber, H.

Griffin, G. M.

M. Solonenko, R. Cheung, T. M. Busch, A. Kachur, G. M. Griffin, T. Vulcan, T. C. Zhu, H. W. Wang, S. M. Hahn, and A. G. Yodh, “In vivo reflectance measurement of optical properties, blood oxygenation and motexafin lutetium uptake in canine large bowels, kidneys and prostates,” Phys. Med. Biol. 47(6), 857-73 (2002).
[PubMed]

Grossman, R. G.

S. Gopinath, C. S. Robertson, R. G. Grossman, and B. Chance, “Near-infrared spectroscopic localization of intracranial hematomas,” J. Neurosurg. 79, 43-47 (1993).
[CrossRef] [PubMed]

Gryazin, Y. A.

Y. A. Gryazin, M. V. Klibanov, and T. R. Lucas, “Numerical solution of a subsurface imaging inverse problem,” SIAM J. Appl. Math. 62, 664-683 (2001).
[CrossRef]

Gu, X.

Gurfinkel, M.

A. Godavarty, A. B. Thompson, Jr., R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraka, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488-496 (2004).
[CrossRef] [PubMed]

Hahn, S. M.

K. L. Du, R. Mick, T. M. Busch, T. C. Zhu, J. C. Finlay, G. Yu, A. G. Yodh, S. B. Malkowicz, D. Smith, R. Whittington, D. Stripp, and S. M. Hahn, “Preliminary results of interstitial motexafin lutetium-mediated PDT for prostate cancer,” Lasers Surg. Med. 38, 427-34 (2006).
[CrossRef] [PubMed]

M. Solonenko, R. Cheung, T. M. Busch, A. Kachur, G. M. Griffin, T. Vulcan, T. C. Zhu, H. W. Wang, S. M. Hahn, and A. G. Yodh, “In vivo reflectance measurement of optical properties, blood oxygenation and motexafin lutetium uptake in canine large bowels, kidneys and prostates,” Phys. Med. Biol. 47(6), 857-73 (2002).
[PubMed]

Hanson, K. M.

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

Hardin, R.

Haroon, Z.

Hebden, J. C.

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modeling and reconstruction,” Phys. Med. Biol. 42, pp. 841-853 (1997).
[CrossRef] [PubMed]

Hielscher, A. H.

A. Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev, and A. H. Hielscher, “Three-dimensional optical tomographic brain imaging in small animals, part 1: hypercapnia,” J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

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

Hong, L.

Isaacson, E.

E. Isaacson and H. B. Keller, Analysis of Numerical Methods (Wiley, 1966).

Jiang, H.

Jiang, S.

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. Treatment 4, 513-526 (2005).

S. Srinivasan, B. W. Pogue, H. Dehghani, S. Jiang, X. Song, and K. D. Paulsen, “Improved quantification of small objects in near-infrared diffuse optical tomography,” J. Biomed. Opt. 9, 1161-1171 (2004).
[CrossRef] [PubMed]

T. O. McBride, B. W. Pogue, S. Jiang, U. L. Österberg, and K. D. Paulsen, “Initial studies of in vivo absorbing and scattering heterogeneity in near-infrared tomographic breast imaging,” Opt. Lett. 26, 822-824 (2001).
[CrossRef]

Kachur, A.

M. Solonenko, R. Cheung, T. M. Busch, A. Kachur, G. M. Griffin, T. Vulcan, T. C. Zhu, H. W. Wang, S. M. Hahn, and A. G. Yodh, “In vivo reflectance measurement of optical properties, blood oxygenation and motexafin lutetium uptake in canine large bowels, kidneys and prostates,” Phys. Med. Biol. 47(6), 857-73 (2002).
[PubMed]

Katz, A.

G. Zhang, A. Katz, R. R. Alfano, A. D. Kofinas, P. G. Stubblefield, W. Rosenfeld, D. Beyer, D. Maulik, and M. R. Stankovic, “Brain perfusion monitoring with frequency-domain and continuous-wave near-infrared spectroscopy: a cross-correlation study in newborn piglets,” Phys. Med. Biol. 45, 3143-3158 (2000).
[CrossRef] [PubMed]

Katz, M.

Keller, H. B.

H. B. Keller and D. J. Perozzi, “Fast seismic ray tracing,” SIAM J. Appl. Math. 43, 981-992 (1983).
[CrossRef]

E. Isaacson and H. B. Keller, Analysis of Numerical Methods (Wiley, 1966).

Klemer, D.

Klibanov, M. V.

L. Beilina and M. V. Klibanov, “A globally convergent numerical method for a coefficient inverse problem,” SIAM J. Sci. Comput. (USA) 31, 478-509 (2008).
[CrossRef]

H. Shan, M. V. Klibanov, H. Liu, N. Pantong, and J. Su, “Numerical implementation of the convexification algorithm for an optical diffusion tomograph,” Inverse Probl. 24, 025006 (2008).
[CrossRef]

M. V. Klibanov and A. Timonov, “Numerical studies on the globally convergent convexification algorithm in 2D,” Inverse Probl. 23, 123-138 (2007).
[CrossRef]

J. Su, H. Shan, H. Liu, and M. V. Klibanov, “Reconstruction method with data from a multiple-site continuous-wave source for three-dimensional optical tomography,” J. Opt. Soc. Am. A 23, 2388-2395 (2006).
[CrossRef]

Y. A. Gryazin, M. V. Klibanov, and T. R. Lucas, “Numerical solution of a subsurface imaging inverse problem,” SIAM J. Appl. Math. 62, 664-683 (2001).
[CrossRef]

M. V. Klibanov, “Inverse problems and Carleman estimates,” Inverse Probl. 8, 575-586 (1992).
[CrossRef]

A. L. Bukhgeim and M. V. Klibanov, “Uniqueness in the large class of multidimensional inverse problems,” Sov. Math. Dokl. 17, 244-247 (1981).

M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications (Brill, 2004).

H. Shan, M. V. Klibanov, J. Su, N. Pantong, and H. Liu, “A globally accelerated numerical method for optical tomography with continuous wave source,” J Inv. Ill-Posed Probl. accepted for publication. A preprint can be found online at http://arxiv.org/abs/0809.3910 (date of last access: October 8, 2008).

Klose, A. D.

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

Kofinas, A. D.

G. Zhang, A. Katz, R. R. Alfano, A. D. Kofinas, P. G. Stubblefield, W. Rosenfeld, D. Beyer, D. Maulik, and M. R. Stankovic, “Brain perfusion monitoring with frequency-domain and continuous-wave near-infrared spectroscopy: a cross-correlation study in newborn piglets,” Phys. Med. Biol. 45, 3143-3158 (2000).
[CrossRef] [PubMed]

Kogel, C.

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. Treatment 4, 513-526 (2005).

Kotz, K. T.

Ladyzhenskaya, O. A.

O. A. Ladyzhenskaya and N. N. Uralceva, Linear and Quasilinear Elliptic Equations (Academic, 1969).

Lasker, J.

A. Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev, and A. H. Hielscher, “Three-dimensional optical tomographic brain imaging in small animals, part 1: hypercapnia,” J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

Levin, M.

Levina, R.

Levinson, N.

E. A. Coddington and N. Levinson, Theory of Differential Equations, (Krieger, 1984).

Lewis, E. E.

E. E. Lewis and W. F. Miller, Jr., Computational Methods of Neutron Transport (Wiley, 1984).

Li, C.

Liu, H.

H. Shan, M. V. Klibanov, H. Liu, N. Pantong, and J. Su, “Numerical implementation of the convexification algorithm for an optical diffusion tomograph,” Inverse Probl. 24, 025006 (2008).
[CrossRef]

J. Su, H. Shan, H. Liu, and M. V. Klibanov, “Reconstruction method with data from a multiple-site continuous-wave source for three-dimensional optical tomography,” J. Opt. Soc. Am. A 23, 2388-2395 (2006).
[CrossRef]

H. Liu, M. Miwa, B. Beauvoit, N. G. Wang, and B. Chance, “Characterization of absorption and scattering properties of small-volume biological samples using time-resolved spectroscopy,” Anal. Biochem. 213, 378-385 (1993).
[CrossRef] [PubMed]

H. Shan, M. V. Klibanov, J. Su, N. Pantong, and H. Liu, “A globally accelerated numerical method for optical tomography with continuous wave source,” J Inv. Ill-Posed Probl. accepted for publication. A preprint can be found online at http://arxiv.org/abs/0809.3910 (date of last access: October 8, 2008).

Lombardo, J. M.

W. B. Wang, J. H. Ali, J. H. Vitenson, J. M. Lombardo, and R. R. Alfano, “Spectral polarization imaging of human rectum-membrane-prostate tissues,” IEEE J. Sel. Top. Quantum Electron. 9, 288-293 (2003).
[CrossRef]

Lucas, T. R.

Y. A. Gryazin, M. V. Klibanov, and T. R. Lucas, “Numerical solution of a subsurface imaging inverse problem,” SIAM J. Appl. Math. 62, 664-683 (2001).
[CrossRef]

Malkowicz, S. B.

K. L. Du, R. Mick, T. M. Busch, T. C. Zhu, J. C. Finlay, G. Yu, A. G. Yodh, S. B. Malkowicz, D. Smith, R. Whittington, D. Stripp, and S. M. Hahn, “Preliminary results of interstitial motexafin lutetium-mediated PDT for prostate cancer,” Lasers Surg. Med. 38, 427-34 (2006).
[CrossRef] [PubMed]

Mandeville, J. B.

D. A. Boas, T. Gaudette, G. Strangman, X. Cheng, J. J. A. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13, 76-90 (2001).
[CrossRef] [PubMed]

Markel, V. A.

Marota, J. J. A.

D. A. Boas, T. Gaudette, G. Strangman, X. Cheng, J. J. A. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13, 76-90 (2001).
[CrossRef] [PubMed]

Maulik, D.

G. Zhang, A. Katz, R. R. Alfano, A. D. Kofinas, P. G. Stubblefield, W. Rosenfeld, D. Beyer, D. Maulik, and M. R. Stankovic, “Brain perfusion monitoring with frequency-domain and continuous-wave near-infrared spectroscopy: a cross-correlation study in newborn piglets,” Phys. Med. Biol. 45, 3143-3158 (2000).
[CrossRef] [PubMed]

McBride, T. O.

Mick, R.

K. L. Du, R. Mick, T. M. Busch, T. C. Zhu, J. C. Finlay, G. Yu, A. G. Yodh, S. B. Malkowicz, D. Smith, R. Whittington, D. Stripp, and S. M. Hahn, “Preliminary results of interstitial motexafin lutetium-mediated PDT for prostate cancer,” Lasers Surg. Med. 38, 427-34 (2006).
[CrossRef] [PubMed]

Miller, W. F.

E. E. Lewis and W. F. Miller, Jr., Computational Methods of Neutron Transport (Wiley, 1984).

Miwa, M.

H. Liu, M. Miwa, B. Beauvoit, N. G. Wang, and B. Chance, “Characterization of absorption and scattering properties of small-volume biological samples using time-resolved spectroscopy,” Anal. Biochem. 213, 378-385 (1993).
[CrossRef] [PubMed]

Murray, T.

Newberry, M. V.

M. V. Newberry, “Signal-to-noise considerations for sky-subtracted CCD data,” Publ. Astron. Soc. Pac. 163, 122 (1991). Available on website http://adsabs.harvard.edu/abs/1991PASP.103.122N (date of last access: October 8, 2008).

Nioka, S.

Ntziachristos, V.

V. Ntziachristos, C. Bremer, and R. Weissleder, “Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging,” Eur. Radiol. 13, 195-208 (2003).
[PubMed]

O'Leary, M. A.

Orduna, J. M.

Österberg, U. L.

Ovetsky, Y.

Pantong, N.

H. Shan, M. V. Klibanov, H. Liu, N. Pantong, and J. Su, “Numerical implementation of the convexification algorithm for an optical diffusion tomograph,” Inverse Probl. 24, 025006 (2008).
[CrossRef]

H. Shan, M. V. Klibanov, J. Su, N. Pantong, and H. Liu, “A globally accelerated numerical method for optical tomography with continuous wave source,” J Inv. Ill-Posed Probl. accepted for publication. A preprint can be found online at http://arxiv.org/abs/0809.3910 (date of last access: October 8, 2008).

Paulsen, K. D.

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. Treatment 4, 513-526 (2005).

S. Srinivasan, B. W. Pogue, H. Dehghani, S. Jiang, X. Song, and K. D. Paulsen, “Improved quantification of small objects in near-infrared diffuse optical tomography,” J. Biomed. Opt. 9, 1161-1171 (2004).
[CrossRef] [PubMed]

T. O. McBride, B. W. Pogue, S. Jiang, U. L. Österberg, and K. D. Paulsen, “Initial studies of in vivo absorbing and scattering heterogeneity in near-infrared tomographic breast imaging,” Opt. Lett. 26, 822-824 (2001).
[CrossRef]

Pei, Y.

Perozzi, D. J.

H. B. Keller and D. J. Perozzi, “Fast seismic ray tracing,” SIAM J. Appl. Math. 43, 981-992 (1983).
[CrossRef]

Pidikiti, D.

Pogue, B. W.

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. Treatment 4, 513-526 (2005).

S. Srinivasan, B. W. Pogue, H. Dehghani, S. Jiang, X. Song, and K. D. Paulsen, “Improved quantification of small objects in near-infrared diffuse optical tomography,” J. Biomed. Opt. 9, 1161-1171 (2004).
[CrossRef] [PubMed]

T. O. McBride, B. W. Pogue, S. Jiang, U. L. Österberg, and K. D. Paulsen, “Initial studies of in vivo absorbing and scattering heterogeneity in near-infrared tomographic breast imaging,” Opt. Lett. 26, 822-824 (2001).
[CrossRef]

Poplack, S. P.

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. Treatment 4, 513-526 (2005).

Protter, M. H.

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations (Springer-Verlag, 1984).
[CrossRef]

Robertson, C. S.

S. Gopinath, C. S. Robertson, R. G. Grossman, and B. Chance, “Near-infrared spectroscopic localization of intracranial hematomas,” J. Neurosurg. 79, 43-47 (1993).
[CrossRef] [PubMed]

Rosenfeld, W.

G. Zhang, A. Katz, R. R. Alfano, A. D. Kofinas, P. G. Stubblefield, W. Rosenfeld, D. Beyer, D. Maulik, and M. R. Stankovic, “Brain perfusion monitoring with frequency-domain and continuous-wave near-infrared spectroscopy: a cross-correlation study in newborn piglets,” Phys. Med. Biol. 45, 3143-3158 (2000).
[CrossRef] [PubMed]

Roy, R.

A. Godavarty, A. B. Thompson, Jr., R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraka, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488-496 (2004).
[CrossRef] [PubMed]

Schmitz, C.

Schotland, J. C.

Schweiger, M.

S. R. Arridge and M. Schweiger, “A general framework for iterative reconstruction algorithms in optical tomography, using a finite-element method,” in Computational Radiology and Imaging: Therapy and Diagnosis, C.Borgers and F.Natterer, eds., Vol. 110 of IMA Volumes in Mathematics and its Applications (Springer-Verlag, 1998), pp. 45-70.
[CrossRef]

Sevick-Muraka, E. M.

A. Godavarty, A. B. Thompson, Jr., R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraka, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488-496 (2004).
[CrossRef] [PubMed]

Shan, H.

H. Shan, M. V. Klibanov, H. Liu, N. Pantong, and J. Su, “Numerical implementation of the convexification algorithm for an optical diffusion tomograph,” Inverse Probl. 24, 025006 (2008).
[CrossRef]

J. Su, H. Shan, H. Liu, and M. V. Klibanov, “Reconstruction method with data from a multiple-site continuous-wave source for three-dimensional optical tomography,” J. Opt. Soc. Am. A 23, 2388-2395 (2006).
[CrossRef]

H. Shan, M. V. Klibanov, J. Su, N. Pantong, and H. Liu, “A globally accelerated numerical method for optical tomography with continuous wave source,” J Inv. Ill-Posed Probl. accepted for publication. A preprint can be found online at http://arxiv.org/abs/0809.3910 (date of last access: October 8, 2008).

Smith, D.

K. L. Du, R. Mick, T. M. Busch, T. C. Zhu, J. C. Finlay, G. Yu, A. G. Yodh, S. B. Malkowicz, D. Smith, R. Whittington, D. Stripp, and S. M. Hahn, “Preliminary results of interstitial motexafin lutetium-mediated PDT for prostate cancer,” Lasers Surg. Med. 38, 427-34 (2006).
[CrossRef] [PubMed]

Solomon, W.

Solonenko, M.

M. Solonenko, R. Cheung, T. M. Busch, A. Kachur, G. M. Griffin, T. Vulcan, T. C. Zhu, H. W. Wang, S. M. Hahn, and A. G. Yodh, “In vivo reflectance measurement of optical properties, blood oxygenation and motexafin lutetium uptake in canine large bowels, kidneys and prostates,” Phys. Med. Biol. 47(6), 857-73 (2002).
[PubMed]

Song, X.

S. Srinivasan, B. W. Pogue, H. Dehghani, S. Jiang, X. Song, and K. D. Paulsen, “Improved quantification of small objects in near-infrared diffuse optical tomography,” J. Biomed. Opt. 9, 1161-1171 (2004).
[CrossRef] [PubMed]

Srinivasan, S.

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. Treatment 4, 513-526 (2005).

S. Srinivasan, B. W. Pogue, H. Dehghani, S. Jiang, X. Song, and K. D. Paulsen, “Improved quantification of small objects in near-infrared diffuse optical tomography,” J. Biomed. Opt. 9, 1161-1171 (2004).
[CrossRef] [PubMed]

Stankovic, M. R.

G. Zhang, A. Katz, R. R. Alfano, A. D. Kofinas, P. G. Stubblefield, W. Rosenfeld, D. Beyer, D. Maulik, and M. R. Stankovic, “Brain perfusion monitoring with frequency-domain and continuous-wave near-infrared spectroscopy: a cross-correlation study in newborn piglets,” Phys. Med. Biol. 45, 3143-3158 (2000).
[CrossRef] [PubMed]

Stewart, M.

A. Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev, and A. H. Hielscher, “Three-dimensional optical tomographic brain imaging in small animals, part 1: hypercapnia,” J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

Strangman, G.

D. A. Boas, T. Gaudette, G. Strangman, X. Cheng, J. J. A. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13, 76-90 (2001).
[CrossRef] [PubMed]

Stripp, D.

K. L. Du, R. Mick, T. M. Busch, T. C. Zhu, J. C. Finlay, G. Yu, A. G. Yodh, S. B. Malkowicz, D. Smith, R. Whittington, D. Stripp, and S. M. Hahn, “Preliminary results of interstitial motexafin lutetium-mediated PDT for prostate cancer,” Lasers Surg. Med. 38, 427-34 (2006).
[CrossRef] [PubMed]

Stubblefield, P. G.

G. Zhang, A. Katz, R. R. Alfano, A. D. Kofinas, P. G. Stubblefield, W. Rosenfeld, D. Beyer, D. Maulik, and M. R. Stankovic, “Brain perfusion monitoring with frequency-domain and continuous-wave near-infrared spectroscopy: a cross-correlation study in newborn piglets,” Phys. Med. Biol. 45, 3143-3158 (2000).
[CrossRef] [PubMed]

Su, J.

H. Shan, M. V. Klibanov, H. Liu, N. Pantong, and J. Su, “Numerical implementation of the convexification algorithm for an optical diffusion tomograph,” Inverse Probl. 24, 025006 (2008).
[CrossRef]

J. Su, H. Shan, H. Liu, and M. V. Klibanov, “Reconstruction method with data from a multiple-site continuous-wave source for three-dimensional optical tomography,” J. Opt. Soc. Am. A 23, 2388-2395 (2006).
[CrossRef]

H. Shan, M. V. Klibanov, J. Su, N. Pantong, and H. Liu, “A globally accelerated numerical method for optical tomography with continuous wave source,” J Inv. Ill-Posed Probl. accepted for publication. A preprint can be found online at http://arxiv.org/abs/0809.3910 (date of last access: October 8, 2008).

Thomas, R.

Thompson, A. B.

A. Godavarty, A. B. Thompson, Jr., R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraka, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488-496 (2004).
[CrossRef] [PubMed]

Tikhonov, A. N.

A. N. Tikhonov and V. Ya Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).

Timonov, A.

M. V. Klibanov and A. Timonov, “Numerical studies on the globally convergent convexification algorithm in 2D,” Inverse Probl. 23, 123-138 (2007).
[CrossRef]

M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications (Brill, 2004).

Uralceva, N. N.

O. A. Ladyzhenskaya and N. N. Uralceva, Linear and Quasilinear Elliptic Equations (Academic, 1969).

Vitenson, J. H.

W. B. Wang, J. H. Ali, J. H. Vitenson, J. M. Lombardo, and R. R. Alfano, “Spectral polarization imaging of human rectum-membrane-prostate tissues,” IEEE J. Sel. Top. Quantum Electron. 9, 288-293 (2003).
[CrossRef]

Vulcan, T.

M. Solonenko, R. Cheung, T. M. Busch, A. Kachur, G. M. Griffin, T. Vulcan, T. C. Zhu, H. W. Wang, S. M. Hahn, and A. G. Yodh, “In vivo reflectance measurement of optical properties, blood oxygenation and motexafin lutetium uptake in canine large bowels, kidneys and prostates,” Phys. Med. Biol. 47(6), 857-73 (2002).
[PubMed]

Wang, H. W.

M. Solonenko, R. Cheung, T. M. Busch, A. Kachur, G. M. Griffin, T. Vulcan, T. C. Zhu, H. W. Wang, S. M. Hahn, and A. G. Yodh, “In vivo reflectance measurement of optical properties, blood oxygenation and motexafin lutetium uptake in canine large bowels, kidneys and prostates,” Phys. Med. Biol. 47(6), 857-73 (2002).
[PubMed]

Wang, N. G.

H. Liu, M. Miwa, B. Beauvoit, N. G. Wang, and B. Chance, “Characterization of absorption and scattering properties of small-volume biological samples using time-resolved spectroscopy,” Anal. Biochem. 213, 378-385 (1993).
[CrossRef] [PubMed]

Wang, W. B.

W. B. Wang, J. H. Ali, J. H. Vitenson, J. M. Lombardo, and R. R. Alfano, “Spectral polarization imaging of human rectum-membrane-prostate tissues,” IEEE J. Sel. Top. Quantum Electron. 9, 288-293 (2003).
[CrossRef]

W. B. Wang, J. H. Ali, M. Zevallos, and R. R. Alfano, “Near infrared imaging of human prostate cancerous and normal tissues based on water absorption,” in Advances in Optical Imaging and Photon Migration (Optical Society of America, 2004), MF 38; on CD-ROM.

Weinberger, H. F.

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations (Springer-Verlag, 1984).
[CrossRef]

Weissleder, R.

V. Ntziachristos, C. Bremer, and R. Weissleder, “Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging,” Eur. Radiol. 13, 195-208 (2003).
[PubMed]

Whittington, R.

K. L. Du, R. Mick, T. M. Busch, T. C. Zhu, J. C. Finlay, G. Yu, A. G. Yodh, S. B. Malkowicz, D. Smith, R. Whittington, D. Stripp, and S. M. Hahn, “Preliminary results of interstitial motexafin lutetium-mediated PDT for prostate cancer,” Lasers Surg. Med. 38, 427-34 (2006).
[CrossRef] [PubMed]

Worden, K.

Xu, Y.

Ya Arsenin, V.

A. N. Tikhonov and V. Ya Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).

Yodh, A. G.

K. L. Du, R. Mick, T. M. Busch, T. C. Zhu, J. C. Finlay, G. Yu, A. G. Yodh, S. B. Malkowicz, D. Smith, R. Whittington, D. Stripp, and S. M. Hahn, “Preliminary results of interstitial motexafin lutetium-mediated PDT for prostate cancer,” Lasers Surg. Med. 38, 427-34 (2006).
[CrossRef] [PubMed]

M. Solonenko, R. Cheung, T. M. Busch, A. Kachur, G. M. Griffin, T. Vulcan, T. C. Zhu, H. W. Wang, S. M. Hahn, and A. G. Yodh, “In vivo reflectance measurement of optical properties, blood oxygenation and motexafin lutetium uptake in canine large bowels, kidneys and prostates,” Phys. Med. Biol. 47(6), 857-73 (2002).
[PubMed]

M. A. O'Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion-photon tomography,” Opt. Lett. 20, 426-428 (1995).
[CrossRef] [PubMed]

A. G. Yodh and D. A. Boas, “Functional imaging with diffusing light,” in Biomedical Photonics Handbook, T.Vo-Dinh, ed. (CRC Press), 21-1-21-45 (2003).

Yu, G.

K. L. Du, R. Mick, T. M. Busch, T. C. Zhu, J. C. Finlay, G. Yu, A. G. Yodh, S. B. Malkowicz, D. Smith, R. Whittington, D. Stripp, and S. M. Hahn, “Preliminary results of interstitial motexafin lutetium-mediated PDT for prostate cancer,” Lasers Surg. Med. 38, 427-34 (2006).
[CrossRef] [PubMed]

Zevallos, M.

W. B. Wang, J. H. Ali, M. Zevallos, and R. R. Alfano, “Near infrared imaging of human prostate cancerous and normal tissues based on water absorption,” in Advances in Optical Imaging and Photon Migration (Optical Society of America, 2004), MF 38; on CD-ROM.

Zhang, C.

A. Godavarty, A. B. Thompson, Jr., R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraka, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488-496 (2004).
[CrossRef] [PubMed]

Zhang, G.

G. Zhang, A. Katz, R. R. Alfano, A. D. Kofinas, P. G. Stubblefield, W. Rosenfeld, D. Beyer, D. Maulik, and M. R. Stankovic, “Brain perfusion monitoring with frequency-domain and continuous-wave near-infrared spectroscopy: a cross-correlation study in newborn piglets,” Phys. Med. Biol. 45, 3143-3158 (2000).
[CrossRef] [PubMed]

Zhou, S.

Zhu, T. C.

K. L. Du, R. Mick, T. M. Busch, T. C. Zhu, J. C. Finlay, G. Yu, A. G. Yodh, S. B. Malkowicz, D. Smith, R. Whittington, D. Stripp, and S. M. Hahn, “Preliminary results of interstitial motexafin lutetium-mediated PDT for prostate cancer,” Lasers Surg. Med. 38, 427-34 (2006).
[CrossRef] [PubMed]

M. Solonenko, R. Cheung, T. M. Busch, A. Kachur, G. M. Griffin, T. Vulcan, T. C. Zhu, H. W. Wang, S. M. Hahn, and A. G. Yodh, “In vivo reflectance measurement of optical properties, blood oxygenation and motexafin lutetium uptake in canine large bowels, kidneys and prostates,” Phys. Med. Biol. 47(6), 857-73 (2002).
[PubMed]

Anal. Biochem. (1)

H. Liu, M. Miwa, B. Beauvoit, N. G. Wang, and B. Chance, “Characterization of absorption and scattering properties of small-volume biological samples using time-resolved spectroscopy,” Anal. Biochem. 213, 378-385 (1993).
[CrossRef] [PubMed]

Appl. Opt. (2)

Eur. Radiol. (1)

V. Ntziachristos, C. Bremer, and R. Weissleder, “Fluorescence imaging with near-infrared light: new technological advances that enable in vivo molecular imaging,” Eur. Radiol. 13, 195-208 (2003).
[PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

W. B. Wang, J. H. Ali, J. H. Vitenson, J. M. Lombardo, and R. R. Alfano, “Spectral polarization imaging of human rectum-membrane-prostate tissues,” IEEE J. Sel. Top. Quantum Electron. 9, 288-293 (2003).
[CrossRef]

IEEE Trans. Med. Imaging (1)

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

Inverse Probl. (3)

M. V. Klibanov and A. Timonov, “Numerical studies on the globally convergent convexification algorithm in 2D,” Inverse Probl. 23, 123-138 (2007).
[CrossRef]

H. Shan, M. V. Klibanov, H. Liu, N. Pantong, and J. Su, “Numerical implementation of the convexification algorithm for an optical diffusion tomograph,” Inverse Probl. 24, 025006 (2008).
[CrossRef]

M. V. Klibanov, “Inverse problems and Carleman estimates,” Inverse Probl. 8, 575-586 (1992).
[CrossRef]

J. Biomed. Opt. (3)

A. Godavarty, A. B. Thompson, Jr., R. Roy, M. Gurfinkel, M. J. Eppstein, C. Zhang, and E. M. Sevick-Muraka, “Diagnostic imaging of breast cancer using fluorescence-enhanced optical tomography: phantom studies,” J. Biomed. Opt. 9, 488-496 (2004).
[CrossRef] [PubMed]

A. Y. Bluestone, M. Stewart, J. Lasker, G. S. Abdoulaev, and A. H. Hielscher, “Three-dimensional optical tomographic brain imaging in small animals, part 1: hypercapnia,” J. Biomed. Opt. 9, 1046-1062 (2004).
[CrossRef] [PubMed]

S. Srinivasan, B. W. Pogue, H. Dehghani, S. Jiang, X. Song, and K. D. Paulsen, “Improved quantification of small objects in near-infrared diffuse optical tomography,” J. Biomed. Opt. 9, 1161-1171 (2004).
[CrossRef] [PubMed]

J. Neurosurg. (1)

S. Gopinath, C. S. Robertson, R. G. Grossman, and B. Chance, “Near-infrared spectroscopic localization of intracranial hematomas,” J. Neurosurg. 79, 43-47 (1993).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (3)

Lasers Surg. Med. (1)

K. L. Du, R. Mick, T. M. Busch, T. C. Zhu, J. C. Finlay, G. Yu, A. G. Yodh, S. B. Malkowicz, D. Smith, R. Whittington, D. Stripp, and S. M. Hahn, “Preliminary results of interstitial motexafin lutetium-mediated PDT for prostate cancer,” Lasers Surg. Med. 38, 427-34 (2006).
[CrossRef] [PubMed]

Neuroimage (1)

D. A. Boas, T. Gaudette, G. Strangman, X. Cheng, J. J. A. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13, 76-90 (2001).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (2)

Phys. Med. Biol. (3)

M. Solonenko, R. Cheung, T. M. Busch, A. Kachur, G. M. Griffin, T. Vulcan, T. C. Zhu, H. W. Wang, S. M. Hahn, and A. G. Yodh, “In vivo reflectance measurement of optical properties, blood oxygenation and motexafin lutetium uptake in canine large bowels, kidneys and prostates,” Phys. Med. Biol. 47(6), 857-73 (2002).
[PubMed]

G. Zhang, A. Katz, R. R. Alfano, A. D. Kofinas, P. G. Stubblefield, W. Rosenfeld, D. Beyer, D. Maulik, and M. R. Stankovic, “Brain perfusion monitoring with frequency-domain and continuous-wave near-infrared spectroscopy: a cross-correlation study in newborn piglets,” Phys. Med. Biol. 45, 3143-3158 (2000).
[CrossRef] [PubMed]

S. R. Arridge and J. C. Hebden, “Optical imaging in medicine: II. Modeling and reconstruction,” Phys. Med. Biol. 42, pp. 841-853 (1997).
[CrossRef] [PubMed]

SIAM J. Appl. Math. (2)

H. B. Keller and D. J. Perozzi, “Fast seismic ray tracing,” SIAM J. Appl. Math. 43, 981-992 (1983).
[CrossRef]

Y. A. Gryazin, M. V. Klibanov, and T. R. Lucas, “Numerical solution of a subsurface imaging inverse problem,” SIAM J. Appl. Math. 62, 664-683 (2001).
[CrossRef]

SIAM J. Sci. Comput. (USA) (1)

L. Beilina and M. V. Klibanov, “A globally convergent numerical method for a coefficient inverse problem,” SIAM J. Sci. Comput. (USA) 31, 478-509 (2008).
[CrossRef]

Sov. Math. Dokl. (1)

A. L. Bukhgeim and M. V. Klibanov, “Uniqueness in the large class of multidimensional inverse problems,” Sov. Math. Dokl. 17, 244-247 (1981).

Technol. Cancer Res. Treatment (1)

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. Treatment 4, 513-526 (2005).

Other (15)

A. G. Yodh and D. A. Boas, “Functional imaging with diffusing light,” in Biomedical Photonics Handbook, T.Vo-Dinh, ed. (CRC Press), 21-1-21-45 (2003).

M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications (Brill, 2004).

W. B. Wang, J. H. Ali, M. Zevallos, and R. R. Alfano, “Near infrared imaging of human prostate cancerous and normal tissues based on water absorption,” in Advances in Optical Imaging and Photon Migration (Optical Society of America, 2004), MF 38; on CD-ROM.

B. Chance, “High sensitivity and specificity in human breast cancer detection with near-infrared imaging,” in Biomedical Topical Meeting, Vol. 71 of OSA Trends in Optics and Photonics (Optical Society of America, 2002), pp. 450-455.

A. N. Tikhonov and V. Ya Arsenin, Solutions of Ill-Posed Problems (Winston, 1977).

O. A. Ladyzhenskaya and N. N. Uralceva, Linear and Quasilinear Elliptic Equations (Academic, 1969).

E. W. Cheney, Introduction to Approximation Theory (Chelsea, 1982).

E. Isaacson and H. B. Keller, Analysis of Numerical Methods (Wiley, 1966).

M. V. Newberry, “Signal-to-noise considerations for sky-subtracted CCD data,” Publ. Astron. Soc. Pac. 163, 122 (1991). Available on website http://adsabs.harvard.edu/abs/1991PASP.103.122N (date of last access: October 8, 2008).

E. E. Lewis and W. F. Miller, Jr., Computational Methods of Neutron Transport (Wiley, 1984).

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations (Springer-Verlag, 1984).
[CrossRef]

H. Shan, M. V. Klibanov, J. Su, N. Pantong, and H. Liu, “A globally accelerated numerical method for optical tomography with continuous wave source,” J Inv. Ill-Posed Probl. accepted for publication. A preprint can be found online at http://arxiv.org/abs/0809.3910 (date of last access: October 8, 2008).

S. R. Arridge and M. Schweiger, “A general framework for iterative reconstruction algorithms in optical tomography, using a finite-element method,” in Computational Radiology and Imaging: Therapy and Diagnosis, C.Borgers and F.Natterer, eds., Vol. 110 of IMA Volumes in Mathematics and its Applications (Springer-Verlag, 1998), pp. 45-70.
[CrossRef]

L. C. Evans, Partial Differential Equations (American Mathematical Society, 1998).

E. A. Coddington and N. Levinson, Theory of Differential Equations, (Krieger, 1984).

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Figures (8)

Fig. 1
Fig. 1

Schematic sketch of imaging of prostate cancer. (Drawing is not to scale.)

Fig. 2
Fig. 2

Schematic sketch of the computational domain Ω 0 , which is a 20.1 cm × 20.1 cm box. In the inverse problem, the computational domain Ω is a small rectangular domain of size 3.6 cm × 4.5 cm . The convex-shape domain A represents the phantom under study. The light sources are moving along the right side of Ω. (Drawing is not to scale.)

Fig. 3
Fig. 3

Domain mesh for the forward and inverse problems: (a) domain Ω 0 and its mesh in the full forward problem, (b) domain A and its mesh, (c) domain Ω 0 A and its mesh in the exterior forward problem, (d) domain Ω and its mesh in the inverse problem.

Fig. 4
Fig. 4

The forward problem solution shows no visible difference between the solution of the full forward problem and the solution extrapolated from the boundary-value data at A via an exterior diffusion problem. (a) The diffusion model (full forward problem) is solved for entire domain Ω 0 . (b) The diffusion model is solved in Ω 0 A , the exterior of the convex-shape domain (small gray area left of center).

Fig. 5
Fig. 5

Preprocessing the noise in the measurement data: (a) measurement data + 2 % noise, (b) measurement data after smoothing (superimposed with original and initial smoothing through exterior extrapolation).

Fig. 6
Fig. 6

Numerical results of Example 1. (a) Original distribution of μ a ; its average value of inclusion is 0.3, (b) Reconstruction of μ a from the 2% noisy data; its peak value is 0.2950. (c) Difference between two consecutive reconstructions as a function of m, the number of iterations, (D) Relative error as a function of m, the number of iterations. In (e) Illustration of the reconstructions under different intensities ( 1 8 , 1 4 , 1 2 , 1) of the original light source ( amplitude = 10,000 ) . The peak values are also illustrated. There are no significant changes to the reconstructed inclusions. (f) Noise level is varied from 2% to 4%, 6%, 8%, 10%. The reconstructions have increasingly large errors; however, the locations of the reconstructed inclusions are still correct. (g) We reconstruct by using different mesh sizes, one smaller mesh size (left panel) and one larger mesh size (right panel). Our current choice of mesh size (middle panel) appears to be optimal. Figure continues on next page.

Fig. 7
Fig. 7

Numerical results of Example 2. (a) Original distribution of μ a ; its average value of inclusion is 0.2461, (b) Construction of μ a from the 2% noisy data; its peak value is 0.2087, (c) Difference between two consecutive reconstructions as a function of m, the number of iterations, (d) Relative error as a function of m, the number of iterations.

Fig. 8
Fig. 8

Numerical results of Example 3. (a) Original distribution of μ a ; its average value of inclusion is 0.2519, (b) Construction of μ a from the 2% noisy data; its peak value is 0.2197, (c) Difference between two consecutive reconstruction as a function of m, the number of iterations, (d) Relative error as a function of m, the number of iterations.

Tables (6)

Tables Icon

Table 1 Errors of the Exterior Forward Problem versus the Full Forward Problem

Tables Icon

Table 2 Number of Iterations Required for Convergence

Tables Icon

Table 3 Relative Errors of Reconstructions in Example 1

Tables Icon

Table 4 Detailed Information on Three Different Mesh Sizes Used for Comparison

Tables Icon

Table 5 Relative Errors of Reconstruction in Example 2

Tables Icon

Table 6 Relative Errors of Reconstruction in Example 3

Equations (60)

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[ D ( x , y ) w ( x , y ) ] μ a ( x , y ) w ( x , y ) = δ ( x x m , y r ) ( x , y ) Ω 0 ,
w ( x , y , r ) = φ ( x , y , r ) ( x , y ) A ,
w ( x , y ) n + w ( x , y ) = 0 ( x , y ) Ω 0 ,
Δ w ̃ ( x , y ) a ( x , y ) w ̃ ( x , y ) = 0 .
Δ u ( x , y ) + u ( x , y ) u ( x , y ) a ( x , y ) = 0 .
a ( x , y ) = 1 2 Δ [ ln D ( x , y ) ] + 1 4 [ ln D ( x , y ) ] [ ln D ( x , y ) ] + μ a ( x , y ) D ( x , y ) .
u ( x , y , r ) = r 1 r v ( x , y , τ ) d τ + u ( x , y ) .
Δ v ( x , y , r ) + 2 v ( x , y , r ) [ r 1 r v ( x , y , τ ) d τ + u ( x , y ) ] = 0 .
Δ v ( x , y , r i ) + 2 v ( x , y , r i ) [ u ( x , y ) ] = 0 , i = 1 ,
Δ v ( x , y , r i ) + 2 v ( x , y , r i ) [ 1 2 v ( x , y , r 1 ) Δ r + j = 2 i 1 v ( x , y , r j ) Δ r + 1 2 v ( x , y , r i ) Δ r + u ( x , y ) ] = 0 , i = 2 , 3 , 4 .
[ D ( x , y ) w ( x , y ) ] μ a ( x , y ) w ( x , y ) = δ ( x x m , y r ) , ( x , y ) Ω 0 A ,
w ( x , y , r ) = φ ( x , y , r ) , ( x , y ) A ,
w ( x , y ) n + w ( x , y ) = 0 , ( x , y ) Ω 0 .
RMSE = k = 1 N ( w k w ̂ k ) 2 N max k w k ; MAE = k = 1 N w k w ̂ k N max k w k ;
ME = k = 1 N ( w k w ̂ k ) N max k w k ;
Δ v ( k ) ( x , y , r i ) + 2 v ( k ) ( x , y , r i ) [ u ( x , y ) ] = 0 ,
( x , y ) Ω , i = 1 ,
Δ v ( k ) ( x , y , r i ) + 2 v ( k ) ( x , y , r i ) [ 1 2 v ( x , y , r 1 ) Δ r + j = 2 i 1 v ( x , y , r j ) Δ r + 1 2 v ( k 1 ) ( x , y , r i ) Δ r + u ( x , y ) ] = 0 ,
( x , y ) Ω , i = 2 , 3 , 4 ,
v ( k ) ( x , y , r i ) = ψ i ( x , y ) ( x , y ) Ω ,
ψ i ( x , y ) = 1 Δ r r i r i 1 ψ ( x , y , r ) d r 1 2 [ ψ ( x , y , r i 1 ) + ψ ( x , y , r i ) ] .
v ( k ) ( x , y , r i ) η + 2 v ( k ) ( x , y , r i ) [ u ( x , y , ) ] η = 0 , i = 1 ,
v ( k ) ( x , y , r i ) η + 2 v ( k ) ( x , y , r i ) [ 1 2 v ( x , y , r 1 ) Δ r + j = 2 i 1 v ( x , y , r j ) Δ r + 1 2 v ( k 1 ) ( x , y , r i ) Δ r + u ( x , y ) ] η = 0 , i = 2 , 3 , 4 ,
η w ̃ ( x , y , r i ) a ( x , y ) w ̃ ( x , y , r i ) η = 0 ,
a ( x , y ) = l = 1 N a l η l ( x , y ) ,
u ( x , y , r ¯ ) = k S + 1 2 ln π 2 S + g ( x , y ) + O ( 1 S ) ,
u ( x 0 , y , r j ) = k S ( x , y , r j ) + 1 2 ln π 2 S ( x , y , r j ) + g j ( x 0 , y ) .
g ( x 0 , y ) = 1 3 j = 1 3 g j ( x 0 , y ) .
T 1 ( x , y ) = k S + 1 2 ln π 2 S + 1 2 [ g ( x 0 , y ) + g ( x , y 0 ) ] .
Δ p 1 , m ( x , y ) a 1 , m ( x , y ) p 1 , m ( x , y ) = λ m [ a 1 , m ( x , y ) a 1 , m 1 ( x , y ) ] u 1 , m 1 ( x , y ) ,
( x , y ) Ω , p 1 , m 1 ( x , y ) = 0 , ( x , y ) Ω ,
u 1 , m ( x , y ) = u 1 , m 1 ( x , y ) + p 1 , m ( x , y ) .
η l w ̃ 1 , m ( x , y ) a 1 , m ( x , y ) w ̃ 1 , m ( x , y ) η l = 0
v 0 ( x , y , r i ) = 1 Δ r r i r i 1 V 0 ( x , y , r ) d r ,
ψ i 0 ( x , y ) = 1 Δ r r i r i 1 ψ 0 ( x , y , r ) d r ,
( v 0 V 0 ) ( x , y , r i ) C 2 + α ( D ¯ ) C * Δ r , ψ i 0 ( x , y ) ψ 0 ( x , y , r i ) C 2 + α ( D ) C * Δ r ,
( ψ i ψ 0 i ) C 2 + α ( D ) C * ( Δ r + σ ) .
u appr ( x , y , r ¯ ) u ( x , y ) C 2 + α ( D ¯ ) ξ ,
a rec ( x , y ) a 0 ( x , y ) C α ( A ¯ ) K η .
λ ( x , y , r 1 ) = λ ( x , y , r 2 ) λ ( x , y , r 1 ) r 2 r 1 ,
a 1 , m 1 ( x , y ) a 1 , m 1 1 ( x , y ) = i = 1 , i max , j = 1 , , j max a 1 , m 1 ( x i , y j ) a 1 , m 1 1 ( x i , y j ) 2 N 1 max i , j a 1 , m 1 ( x i , y j ) ε ,
a 1 , m 1 ( x , y ) a 1 , m 1 1 ( x , y ) = i = 1 , i max , j = 1 , , j max a 1 , m 1 ( x i , y j ) a 1 , m 1 1 ( x i , y j ) 2 N 1 max i , j a 1 , m 1 ( x i , y j ) ,
RMSE = i = 1 , i max , j = 1 , , j max a 1 , m 1 ( x i , y j ) a ( x i , y j ) 2 N 1 max i , j a ( x i , y j )
μ a ( x , y ) = { max [ 0.3 cos d ( x , y ) , 0.1 ] inside each circle 0.1 otherwise } ,
μ a ( x , y )
= { max [ 0.3 ( cos d ( x , y ) + 0.1 η ( x , y ) ) , 0.1 ] inside each circle 0.1 otherwise } .
Δ u + j = 1 2 b j ( x , y ) u x j c ( x , y ) u = f ( x , y ) , ( x , y ) D .
u ( x , y ) = g ( x , y ) , ( x , y ) D .
b j , c , f C α ( D ¯ ) , c 0 , g C 2 + α ( D ) .
b j C α ( D ¯ ) M , c C α ( D ¯ ) M .
u C 2 + α ( D ¯ ) P ( f C α ( D ¯ ) + g C 2 + α ( D ) ) .
M * = { [ max 1 j N ( v 0 ( x , y , r j ) C 1 + α ( D ¯ ) ) + 2 u ( x , y , r ¯ ) C 1 + α ( D ¯ ) + v 0 ( x , y , r 1 ) 2 + 1 ] , C * } .
Δ p 1 ( x , y , r 1 ) + 2 p 1 ( x , y , r 1 ) u appr ( x , y , r ¯ ) = 2 v 0 ( x , y , r 1 ) [ ( u appr ( x , y , r ¯ ) u ( x , y ) ) ] ,
p 1 ( x , y , r 1 ) = ( ψ ψ 0 ) ( x , y , r 1 ) , ( x , y ) D .
2 u appr ( x , y , r ¯ ) C α ( D ¯ ) ξ + u ( x , y ) C 1 + α ( D ¯ ) 2 M * .
2 v 0 ( x , y , r 1 ) [ ( u appr ( x , y , r ¯ ) u ( x , y ) ) ] C α ( D ¯ ) 2 M * η ,
p 1 ( x , y , r 1 ) C 2 + α ( D ¯ ) P ( 2 M * η + η ) 3 P M * η .
v 1 ( x , y , r 1 ) C 2 + α ( D ¯ ) = ( p 1 + v 0 ) ( x , y , r 1 ) C 2 + α ( D ¯ ) p 1 ( x , y , r 1 ) C 2 + α ( D ¯ ) + v 0 ( x , y , r 1 ) C 2 + α ( D ¯ ) M * ( 3 P η + 1 ) 2 M *
v k ( x , y , r i ) v 0 ( x , y , r i ) C 2 + α ( D ¯ ) 3 P M * η .
2 v ( x , y , r ) r 1 r v ( x , y , τ ) d τ .

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