Abstract

We solve direct and inverse obstacle-scattering problems in a half-space composed of a uniform absorbing and scattering medium. Scattering is sharply forward-peaked, so we use the modified Fokker–Planck approximation to the radiative transport equation. The obstacle is an absorbing inhomogeneity that is thin with respect to depth. Using the first Born approximation, we derive a method to recover the depth and shape of the absorbing obstacle. This method requires only plane-wave illumination at two incidence angles and a detector with a fixed numerical aperture. First we recover the depth of the obstacle through solution of a simple nonlinear least-squares problem. Using that depth, we compute a point-spread function explicitly. We use that point-spread function in a standard deconvolution algorithm to reconstruct the shape of the obstacle. Numerical results show the utility of this method even in the presence of measurement noise.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. J. P. Van Hounten, D. A. Benaron, S. Spilman, and D. K. Stevenson, "Imaging brain injury using time-resolved near infrared light scanning," Pediatr. Res. 39, 470-476 (1996).
    [CrossRef]
  2. S. R. Hintz, W. F. Cheong, J. P. Van Hounten, D. K. Stevenson, and D. A. Benaron, "Bedside imaging of intracranial hemorrhage in the neonate using light: comparison with ultrasound, computed tomography, and magnetic resonance imaging," Pediatr. Res. 45, 54-59 (1999).
    [CrossRef] [PubMed]
  3. S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-93 (1999).
    [CrossRef]
  4. X. F. Cheng and D. A. Boas, "Systematic diffuse optical image errors resulting from uncertainty in the background optical properties," Opt. Express 4, 299-307 (1999).
    [CrossRef] [PubMed]
  5. D. A. Benaron, S. R. Hintz, A. Villringer, D. A. Boas, A. Klein-Schmidt, J. Frahm, C. Hirth, H. Obrig, J. P. Van Hounten, E. L. Kermit, W. Cheong, and D. K. Stevenson, "Noninvasive functional imaging of human brain using light," J. Cereb. Blood Flow Metab. 20, 469-477 (2000).
    [CrossRef] [PubMed]
  6. A. Y. Bluestone, G. Abdoulaev, C. H. Schmitz, R. L. Barbour, and A. H. Hielscher, "Three-dimensional optical tomography of hemodynamics in the human head," Opt. Express 9, 272-286 (2001).
    [CrossRef] [PubMed]
  7. J. C. Hebden, A. Gibson, T. Austin, R. M. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek, and J. S. Wyatt, "Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography," Phys. Med. Biol. 49, 1117-1130 (2004).
    [CrossRef] [PubMed]
  8. B. Monsees, J. M. Destouet, and W. G. Totty, "Light scanning versus mammography in breast-cancer detection," Radiology 163, 463-465 (1987).
    [PubMed]
  9. D. Grosenick, H. Wabnitz, H. H. Rinneberg, K. T. Moesta, and P. M. Schlaq, "Development of a time-domain optical mammograph and first in vivo applications," Appl. Opt. 38, 2927-2943 (1999).
    [CrossRef]
  10. J. C. Hebden, H. Veenstra, H. Dehghani, E. M. C. Hillman, M. Scweinger, S. R. Arridge and D. T. Delpy, "Three-dimensional time-resolved optical tomography of a conical breast phantom," Appl. Opt. 40, 3278-3287 (2001).
    [CrossRef]
  11. J. E. Bugaj, S. Achilefu, R. B. Dorshow, and R. Rajagopalan, "Novel fluorescent contrast agents for optical imaging of in vivo tumors based on a receptor-targeted dye-peptide conjugate platform," J. Biomed. Opt. 6, 122-133 (2001).
    [CrossRef] [PubMed]
  12. V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, "In-vivo tomographic imaging of near infrared fluorescent probes," Mol. Imaging 1, 82-88 (2002).
    [CrossRef]
  13. E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, "A submillimeiter resolution fluorescence molecular imaging system for small animal imaging," Med. Phys. 30, 901-911 (2003).
    [CrossRef] [PubMed]
  14. R. Weissleder and U. Mahmood, "Molecular Imaging," Radiology 219, 316-333 (2001).
    [PubMed]
  15. A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997).
  16. A. D. Kim and J. B. Keller, "Light propagation in biological tissue," J. Opt. Soc. Am. A 20, 92-98 (2003).
    [CrossRef]
  17. Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, "Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer," IEEE J. Sel. Top. Quantum Electron. 9, 243-257 (2003).
    [CrossRef]
  18. O. Dorn, "A transport-backtransport method for optical tomography," Inverse Probl. 14, 1107-1130 (1998).
    [CrossRef]
  19. O. Dorn, "Shape reconstruction in scattering media with voids using a transport model and level sets," Can. Appl. Math. Quart. 10, 239-275 (2002).
  20. A. D. Klose, V. Ntziachristos, and A. H. Hielscher, "The inverse source problem based on the radiative transfer equation in optical molecular imaging," J. Comput. Phys. 202, 323-345 (2005).
    [CrossRef]
  21. A. D. Kim and M. Moscoso, "Radiative transport theory for optical molecular imaging," Inverse Probl. 22, 23-42 (2006).
    [CrossRef]
  22. A. D. Kim, C. Hayakawa, and V. Venugopalan, "Estimating tissue optical properties using the Born approximation of the transport equation," Opt. Lett. 31, 1088-1090 (2006).
    [CrossRef] [PubMed]
  23. A. D. Kim and M. Moscoso, "Beam propagation in sharply peaked forward scattering media," J. Opt. Soc. Am. A 21, 797-803 (2004).
    [CrossRef]
  24. L.-H. Wang and S. L. Jacques, "Use of a laser beam with an oblique angle of incidence to measure the reduced scattering coefficient of a turbid medium," Appl. Opt. 34, 2362-2366 (1995).
    [CrossRef] [PubMed]
  25. S.-P. Lin, L.-H. Wang, S. L. Jacques, and F. K. Tittel, "Measurement of tissue optical properties using oblique incidence optical fiber reflectometry," Appl. Opt. 36, 136-143 (1997).
    [CrossRef] [PubMed]
  26. G. Marquez and L.-H. Wang, "White light oblique incidence reflectometer for measuring absorption and reduced scattering spectra of tissue-like turbid media," Opt. Express 1, 454-460 (1997).
    [CrossRef] [PubMed]
  27. M. Mehrubeoglu, N. Kehtarnavaz, G. Marquez, M. Duvic, and L.-H. Wang, "Skin lesion classification using diffuse reflectance spectroscopic imaging with oblique incidence," Appl. Opt. 41, 182-192 (2002).
    [CrossRef] [PubMed]
  28. A. Garcia-Uribe, N. Kehtarnavaz, G. Marquez, V. Prieto, M. Duvic, and L.-H. Wang, "Skin cancer detection using spectroscopic oblique-incidence reflectometry: classification and physiological origins," Appl. Opt. 43, 2643-2650 (2004).
    [CrossRef] [PubMed]
  29. A. D. Kim, "Transport theory for light propagation in biological tissue," J. Opt. Soc. Am. A 21, 820-827 (2004).
    [CrossRef]
  30. A. K. Dunn and D. A. Boas, "Transport-based image reconstruction in turbid media with small source-detector separations," Opt. Lett. 25, 1777-1779 (2000).
    [CrossRef]
  31. E. M. C. Hillman, D. A. Boas, A. M. Dale and A. K. Dunn, "Laminar optical tomography: demonstration of millimeter-scale depth-resolved imaging in turbid media," Opt. Lett. 29, 1650-1652 (2004).
    [CrossRef] [PubMed]
  32. M. Schweiger, S. R. Arridge, O. Dorn, A. Zacharopoulos, and V. Kolehmainen, "Reconstructing absorption and diffusion shape profiles in optical tomography using a level set technique," Opt. Lett. 31, 471-473 (2006).
    [CrossRef] [PubMed]
  33. S. R. Arridge, O. Dorn, J. P. Kaipio, V. Kolehmainen, M. Schweiger, T. Tarvainen, M. Vauhkonen, and A. Zacharopoulos, "Reconstruction of subdomain boundaries of piecewise constant coefficients of the radiative transfer equation from optical tomography data," Inverse Probl. 22, 2175-2196 (2006).
    [CrossRef]

2006

A. D. Kim and M. Moscoso, "Radiative transport theory for optical molecular imaging," Inverse Probl. 22, 23-42 (2006).
[CrossRef]

S. R. Arridge, O. Dorn, J. P. Kaipio, V. Kolehmainen, M. Schweiger, T. Tarvainen, M. Vauhkonen, and A. Zacharopoulos, "Reconstruction of subdomain boundaries of piecewise constant coefficients of the radiative transfer equation from optical tomography data," Inverse Probl. 22, 2175-2196 (2006).
[CrossRef]

M. Schweiger, S. R. Arridge, O. Dorn, A. Zacharopoulos, and V. Kolehmainen, "Reconstructing absorption and diffusion shape profiles in optical tomography using a level set technique," Opt. Lett. 31, 471-473 (2006).
[CrossRef] [PubMed]

A. D. Kim, C. Hayakawa, and V. Venugopalan, "Estimating tissue optical properties using the Born approximation of the transport equation," Opt. Lett. 31, 1088-1090 (2006).
[CrossRef] [PubMed]

2005

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, "The inverse source problem based on the radiative transfer equation in optical molecular imaging," J. Comput. Phys. 202, 323-345 (2005).
[CrossRef]

2004

2003

A. D. Kim and J. B. Keller, "Light propagation in biological tissue," J. Opt. Soc. Am. A 20, 92-98 (2003).
[CrossRef]

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, "A submillimeiter resolution fluorescence molecular imaging system for small animal imaging," Med. Phys. 30, 901-911 (2003).
[CrossRef] [PubMed]

Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, "Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer," IEEE J. Sel. Top. Quantum Electron. 9, 243-257 (2003).
[CrossRef]

2002

V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, "In-vivo tomographic imaging of near infrared fluorescent probes," Mol. Imaging 1, 82-88 (2002).
[CrossRef]

M. Mehrubeoglu, N. Kehtarnavaz, G. Marquez, M. Duvic, and L.-H. Wang, "Skin lesion classification using diffuse reflectance spectroscopic imaging with oblique incidence," Appl. Opt. 41, 182-192 (2002).
[CrossRef] [PubMed]

2001

J. C. Hebden, H. Veenstra, H. Dehghani, E. M. C. Hillman, M. Scweinger, S. R. Arridge and D. T. Delpy, "Three-dimensional time-resolved optical tomography of a conical breast phantom," Appl. Opt. 40, 3278-3287 (2001).
[CrossRef]

A. Y. Bluestone, G. Abdoulaev, C. H. Schmitz, R. L. Barbour, and A. H. Hielscher, "Three-dimensional optical tomography of hemodynamics in the human head," Opt. Express 9, 272-286 (2001).
[CrossRef] [PubMed]

R. Weissleder and U. Mahmood, "Molecular Imaging," Radiology 219, 316-333 (2001).
[PubMed]

J. E. Bugaj, S. Achilefu, R. B. Dorshow, and R. Rajagopalan, "Novel fluorescent contrast agents for optical imaging of in vivo tumors based on a receptor-targeted dye-peptide conjugate platform," J. Biomed. Opt. 6, 122-133 (2001).
[CrossRef] [PubMed]

2000

D. A. Benaron, S. R. Hintz, A. Villringer, D. A. Boas, A. Klein-Schmidt, J. Frahm, C. Hirth, H. Obrig, J. P. Van Hounten, E. L. Kermit, W. Cheong, and D. K. Stevenson, "Noninvasive functional imaging of human brain using light," J. Cereb. Blood Flow Metab. 20, 469-477 (2000).
[CrossRef] [PubMed]

A. K. Dunn and D. A. Boas, "Transport-based image reconstruction in turbid media with small source-detector separations," Opt. Lett. 25, 1777-1779 (2000).
[CrossRef]

1999

D. Grosenick, H. Wabnitz, H. H. Rinneberg, K. T. Moesta, and P. M. Schlaq, "Development of a time-domain optical mammograph and first in vivo applications," Appl. Opt. 38, 2927-2943 (1999).
[CrossRef]

X. F. Cheng and D. A. Boas, "Systematic diffuse optical image errors resulting from uncertainty in the background optical properties," Opt. Express 4, 299-307 (1999).
[CrossRef] [PubMed]

S. R. Hintz, W. F. Cheong, J. P. Van Hounten, D. K. Stevenson, and D. A. Benaron, "Bedside imaging of intracranial hemorrhage in the neonate using light: comparison with ultrasound, computed tomography, and magnetic resonance imaging," Pediatr. Res. 45, 54-59 (1999).
[CrossRef] [PubMed]

S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-93 (1999).
[CrossRef]

1998

O. Dorn, "A transport-backtransport method for optical tomography," Inverse Probl. 14, 1107-1130 (1998).
[CrossRef]

1997

1996

J. P. Van Hounten, D. A. Benaron, S. Spilman, and D. K. Stevenson, "Imaging brain injury using time-resolved near infrared light scanning," Pediatr. Res. 39, 470-476 (1996).
[CrossRef]

1995

1987

B. Monsees, J. M. Destouet, and W. G. Totty, "Light scanning versus mammography in breast-cancer detection," Radiology 163, 463-465 (1987).
[PubMed]

Appl. Opt.

IEEE J. Sel. Top. Quantum Electron.

Y. L. Kim, Y. Liu, R. K. Wali, H. K. Roy, M. J. Goldberg, A. K. Kromin, K. Chen, and V. Backman, "Simultaneous measurement of angular and spectral properties of light scattering for characterization of tissue microarchitecture and its alteration in early precancer," IEEE J. Sel. Top. Quantum Electron. 9, 243-257 (2003).
[CrossRef]

Inverse Probl.

O. Dorn, "A transport-backtransport method for optical tomography," Inverse Probl. 14, 1107-1130 (1998).
[CrossRef]

A. D. Kim and M. Moscoso, "Radiative transport theory for optical molecular imaging," Inverse Probl. 22, 23-42 (2006).
[CrossRef]

S. R. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-93 (1999).
[CrossRef]

S. R. Arridge, O. Dorn, J. P. Kaipio, V. Kolehmainen, M. Schweiger, T. Tarvainen, M. Vauhkonen, and A. Zacharopoulos, "Reconstruction of subdomain boundaries of piecewise constant coefficients of the radiative transfer equation from optical tomography data," Inverse Probl. 22, 2175-2196 (2006).
[CrossRef]

J. Biomed. Opt.

J. E. Bugaj, S. Achilefu, R. B. Dorshow, and R. Rajagopalan, "Novel fluorescent contrast agents for optical imaging of in vivo tumors based on a receptor-targeted dye-peptide conjugate platform," J. Biomed. Opt. 6, 122-133 (2001).
[CrossRef] [PubMed]

J. Cereb. Blood Flow Metab.

D. A. Benaron, S. R. Hintz, A. Villringer, D. A. Boas, A. Klein-Schmidt, J. Frahm, C. Hirth, H. Obrig, J. P. Van Hounten, E. L. Kermit, W. Cheong, and D. K. Stevenson, "Noninvasive functional imaging of human brain using light," J. Cereb. Blood Flow Metab. 20, 469-477 (2000).
[CrossRef] [PubMed]

J. Comput. Phys.

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, "The inverse source problem based on the radiative transfer equation in optical molecular imaging," J. Comput. Phys. 202, 323-345 (2005).
[CrossRef]

J. Opt. Soc. Am. A

Med. Phys.

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, "A submillimeiter resolution fluorescence molecular imaging system for small animal imaging," Med. Phys. 30, 901-911 (2003).
[CrossRef] [PubMed]

Mol. Imaging

V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, "In-vivo tomographic imaging of near infrared fluorescent probes," Mol. Imaging 1, 82-88 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Pediatr. Res.

J. P. Van Hounten, D. A. Benaron, S. Spilman, and D. K. Stevenson, "Imaging brain injury using time-resolved near infrared light scanning," Pediatr. Res. 39, 470-476 (1996).
[CrossRef]

S. R. Hintz, W. F. Cheong, J. P. Van Hounten, D. K. Stevenson, and D. A. Benaron, "Bedside imaging of intracranial hemorrhage in the neonate using light: comparison with ultrasound, computed tomography, and magnetic resonance imaging," Pediatr. Res. 45, 54-59 (1999).
[CrossRef] [PubMed]

Phys. Med. Biol.

J. C. Hebden, A. Gibson, T. Austin, R. M. Yusof, N. Everdell, D. T. Delpy, S. R. Arridge, J. H. Meek, and J. S. Wyatt, "Imaging changes in blood volume and oxygenation in the newborn infant brain using three-dimensional optical tomography," Phys. Med. Biol. 49, 1117-1130 (2004).
[CrossRef] [PubMed]

Radiology

B. Monsees, J. M. Destouet, and W. G. Totty, "Light scanning versus mammography in breast-cancer detection," Radiology 163, 463-465 (1987).
[PubMed]

R. Weissleder and U. Mahmood, "Molecular Imaging," Radiology 219, 316-333 (2001).
[PubMed]

Other

A. Ishimaru, Wave Propagation and Scattering in Random Media (IEEE Press, 1997).

O. Dorn, "Shape reconstruction in scattering media with voids using a transport model and level sets," Can. Appl. Math. Quart. 10, 239-275 (2002).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Procedure to solve the inverse obstacle-scattering problem: (a) Two separate plane waves in directions Ω 1 and Ω 2 illuminate a half-space composed of a uniformly absorbing and scattering medium except for an absorbing obstacle. (b) Detectors with a fixed numerical aperture (NA) collect the reflectances R ( ρ ; Ω 1 , 2 ) corresponding to each plane wave from which we determine the depth z 0 assuming that the obstacle is negligibly thin. (c) Using the recovered value for z 0 , we reconstruct the cross-sectional shape of the absorbing obstacle.

Fig. 2
Fig. 2

Contour plot of the function A ( ρ ) given in Eq. (5.1). In this plot, the black regions correspond to A = 0 and the white regions correspond to A = 1 .

Fig. 3
Fig. 3

Direct (a)–(c) and reconstructed (d)–(f) images of the shape of the absorbing obstacle at different depths: (a) and (d) z 0 = 0.5 l * ; (b) and (e) z 0 = 1.0 l * ; and (c) and (f) z 0 = 2.0 l * .

Fig. 4
Fig. 4

Reconstructed images with varying levels of measurement noise: (a) 0.50% noise, (b) 1.00% noise, (c) 1.25% noise.

Fig. 5
Fig. 5

Reconstructed images with 1% noise using (a) no denoising with a regularized filter, (b) denoising with a regularized filter, and (c) denoising with a Wiener filter.

Fig. 6
Fig. 6

Reconstructed images with 2% noise using (a) no denoising with a regularized filter, (b) denoising with a regularized filter, and (c) denoising with a Wiener filter.

Tables (2)

Tables Icon

Table 1 Results of Solving the Nonlinear Least-Squares Problem Given by Eq. (4.4) for z 0 and a ̂ ( 0 )

Tables Icon

Table 2 Recovered Values for the Strength and Depth of the Absorber with Measurement Noise Present a

Equations (52)

Equations on this page are rendered with MathJax. Learn more.

Ω I + μ a I μ s L I = 0 .
L I = I + S 2 f ( Ω Ω ) I ( Ω , r ) d Ω .
L I α Δ Ω ( I β Δ Ω ) 1 I ,
f ( Ω Ω ) = 1 4 π 1 g 2 ( 1 + g 2 2 g Ω Ω ) 3 2 ,
β = 2 3 g + g 2 6 g ( 1 g ) ,
α = 1 2 ( 1 g ) ( 1 + 2 β ) .
I z = 0 = F δ ( Ω Ω 0 ) , Ω z ̂ > 0 .
μ a ( ρ , z ) = μ ¯ a + δ μ a ( ρ , z ) .
Ω I + μ ¯ a I μ ¯ s L I = δ μ a I , z > 0 .
R ( ρ ; Ω 0 ) = NA I ( Ω , ρ , 0 ) Ω z d Ω ,
Ω I 0 + μ ¯ a I 0 μ ¯ s L I 0 = 0 , z > 0 ,
I 0 ( Ω , z ; Ω 0 ) = Ω 0 z G ̂ ( Ω , z ; Ω 0 , 0 , q = 0 ) .
I ( Ω , ρ , z ; Ω 0 ) = I 0 ( Ω , z ; Ω 0 ) 0 R 2 S 2 G ( Ω , ρ , z ; Ω , ρ , z ) I ( Ω , ρ , z ; Ω 0 ) δ μ a ( ρ , z ) d Ω d ρ d z .
I = I 0 + n = 1 I n ,
Ω I n + μ ¯ a I n μ ¯ s L I n = δ μ a I n 1 , z > 0 , n > 0 ,
I n z = 0 = 0 , Ω z ̂ > 0 , n > 0 .
I ( Ω , ρ , z ; Ω 0 ) I 0 ( Ω , z ; Ω 0 ) 0 R 2 S 2 G ( Ω , ρ , z ; Ω , ρ , z ) I 0 ( Ω , z ; Ω 0 ) δ μ a ( ρ , z ) d Ω d ρ d z .
I ( Ω , ρ , 0 ; Ω 0 ) I 0 ( Ω , 0 ; Ω 0 ) 0 R 2 S 2 G ( Ω , ρ , 0 ; Ω , ρ , z ) I 0 ( Ω , z ; Ω 0 ) δ μ a ( ρ , z ) d Ω d ρ d z .
δ R ( ρ ; Ω 0 ) 0 R 2 K ( ρ ρ , z ; Ω 0 ) δ μ a ( ρ , z ) d ρ d z ,
δ R ( ρ ; Ω 0 ) = NA [ I ( Ω , ρ , 0 ; Ω 0 ) I 0 ( Ω , 0 ; Ω 0 ) ] Ω z d Ω ,
K ( ρ ρ , z ; Ω 0 ) = NA S 2 G ( Ω , ρ , 0 ; Ω , ρ , z ) I 0 ( Ω , z ; Ω 0 ) d Ω Ω z d Ω .
δ R ̂ ( q ; Ω 0 ) 0 K ̂ ( z ; q , Ω 0 ) δ μ ̂ a ( z ; q ) d z ,
K ̂ ( z ; q , Ω 0 ) = j > 0 v j ( q ) exp [ λ j ( q ) z ] k < 0 A j k ( q ) exp [ λ k ( 0 ) z ] c k ( Ω 0 ) j < 0 v j ( q ) k > 0 Y j k ( q ) exp [ λ k ( q ) z ] l < 0 A k l ( q ) exp [ λ l ( 0 ) z ] c l ( Ω 0 ) ,
v j ( q ) = NA V j ( Ω ; q ) Ω z d Ω ,
A j k ( q ) = S 2 V j ( Ω ; q ) V k ( Ω ; 0 ) d Ω .
δ μ a ( ρ , z ) = a ( ρ ) δ ( z z 0 ) ,
δ R ( ρ ; Ω 1 , 2 ) R 2 K ( ρ ρ , z 0 ; Ω 1 , 2 ) a ( ρ ) d ρ .
δ R ̂ ( q ; Ω 1 , 2 ) K ̂ ( z 0 ; q , Ω 1 , 2 ) a ̂ ( q ) .
δ R ̂ ( 0 ; Ω 1 , 2 ) a ̂ ( 0 ) j > 0 v j ( 0 ) exp [ λ j ( 0 ) z 0 ] k < 0 A j k ( 0 ) exp [ λ k ( 0 ) z 0 ] c k ( Ω 1 , 2 ) j < 0 v j ( 0 ) k > 0 Y j k ( 0 ) exp [ λ k ( 0 ) z 0 ] l < 0 A k l ( 0 ) exp [ λ l ( 0 ) z 0 ] c l ( Ω 1 , 2 ) .
OTF ( q ) = K ̂ ( z 0 , q ; Ω 1 ) .
δ R ( ρ ; Ω 1 ) = PSF a ( ρ ) ,
δ μ a ( ρ , z ) = { ϵ A ( ρ ) , z 0 Δ z 2 < z < z 0 + Δ z 2 , 0 , otherwise } .
δ R ̂ ( 0 ; Ω 1 , 2 ) a ̂ ( 0 ) [ v 1 ( 0 ) v 1 ( 0 ) Y 1 , 1 ( 0 ) ] A 1 , 1 ( 0 ) × exp [ 2 λ 1 ( 0 ) z 0 ] c 1 ( Ω 1 , 2 ) , z .
Ω I + μ ¯ a I μ ¯ s L I = 0
I ( Ω , ρ , z ) = V ( Ω ) exp [ i q ρ + λ z ] .
λ Ω z V + i q Ω V + μ ¯ a V μ ¯ s L V = 0 ,
( λ λ ) S 2 V ( Ω ) V ( Ω ) Ω z d Ω = 0 .
λ ν V + i 1 ν 2 ( q x cos φ + q y sin φ ) V + μ ¯ a V μ ¯ s L V = 0 ,
Re [ λ M 2 ( q ) ] < < Re [ λ 1 ( q ) ] < Re [ λ 1 ( q ) ] < < Re [ λ M 2 ( q ) ] .
S 2 V j ( Ω ; q ) V j ( Ω ; q ) Ω z d Ω = sgn ( j ) , j = ± 1 , , ± N 2 .
Ω G + μ ¯ a G μ ¯ s L G = δ ( Ω Ω ) δ ( r r ) , z , z > 0 ,
G z = 0 = 0 , Ω z ̂ > 0 .
G ( Ω , ρ , z ; Ω , ρ , z ) = ( 2 π ) 2 R 2 G ̂ ( Ω , z ; Ω , z , q ) exp [ i q ( ρ ρ ) ] d q ,
G ̂ ( Ω , z ; Ω , z , q ) = j 0 { V j ( Ω ; q ) exp [ λ j ( q ) ( z z ) ] V j ( Ω ; q ) } j < 0 { V j ( Ω ; q ) exp [ λ j ( q ) z ] k > 0 { Y j k ( q ) exp [ λ k ( q ) z ] V k ( Ω ; q ) } } , z z .
j < 0 V j ( Ω ; q ) Y j k ( q ) = V k ( Ω ; q ) , Ω z ̂ > 0 , k > 0 .
Ω I + μ ¯ a I μ ¯ s L I = Q , z > 0 ,
I ( Ω , ρ , 0 ) = b ( Ω , ρ ) , Ω z ̂ > 0 .
I ( Ω , ρ , z ) = 0 R 2 S 2 G ( Ω , ρ , z ; Ω , ρ , z ) Q ( Ω , ρ , z ) d Ω d ρ d z + R 2 Ω z ̂ > 0 Ω z ̂ G ( Ω , ρ , z ; Ω , ρ , 0 ) b ( Ω , ρ ) d Ω d ρ .
I ( Ω , ρ , z ) = R 2 Ω 0 z ̂ G ( Ω , ρ , z ; Ω 0 , ρ , 0 ) d ρ ,
= Ω 0 z G ̂ ( Ω , z ; Ω 0 , 0 , q = 0 ) ,
= j < 0 V j ( Ω ; 0 ) exp [ λ j ( 0 ) z ] c j ( Ω 0 ) ,
c j ( Ω 0 ) = Ω 0 z [ V j ( Ω 0 ; 0 ) k > 0 Y j k ( 0 ) V k ( Ω 0 ; 0 ) ] .

Metrics