Abstract

A method to perform fast 3-D optical reconstruction, based on structured light, in thick samples is demonstrated and experimentally validated. The experimental and reconstruction procedure, based on Finite Elements Method, used to reconstruct absorbing heterogeneities, with arbitrary arrangement in space, is discussed. In particular we demonstrated that a 2D sampling of the source Fourier plane is required to improve the imaging capability.

© 2010 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
A finite-element-based reconstruction method for 3D fluorescence tomography

Alexander X. Cong and Ge Wang
Opt. Express 13(24) 9847-9857 (2005)

Spatial-frequency-compression scheme for diffuse optical tomography with dense sampling dataset

Xiaoqing Zhou, Ying Fan, Qiang Hou, Huijuan Zhao, and Feng Gao
Appl. Opt. 52(9) 1779-1792 (2013)

3D shape based reconstruction of experimental data in Diffuse Optical Tomography

Athanasios Zacharopoulos, Martin Schweiger, Ville Kolehmainen, and Simon Arridge
Opt. Express 17(21) 18940-18956 (2009)

More Recommended Articles

References

  • View by:
  • Article Order
  • |
  • Year
  • |
  • Author
  • |
  • Publication
  1. A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express 15, 6696–6716 (2007).
  2. P. Taroni, D. Comelli, A. Pifferi, A. Torricelli, and R. Cubeddu, “Absorption of collagen: effects on the estimate of breast composition and related diagnostic implications,” J. Biomed. Opt. 12, 014021 (2007).
  3. J. Selb, D. K. Joseph, and D. A. Boas, “Time-gated optical system for depth-resolved functional brain imaging,” J. Biomed. Opt. 11, 044008 (2006).
  4. G. Strangman, D. A. Boas, and J. P. Sutton, “Non-invasive neuroimaging using near-infrared light,” Biol. Psychiatry 52, 679–693 (2002).
  5. R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nature Med. 9, 123–128 (2003).
  6. A. Koenig, L. Herve, V. Josserand, M. Berger, J. Boutet, A. D. Silva, J.-M. Dinten, P. Peltie, J.-L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).
  7. V. A. Markel and J. C. Schotland, “Symmetries, inversion formulas, and image reconstruction for optical tomography,” Phys. Rev. E 70, 056616 (2004).
  8. J. Ripoll, “Hybrid fourier-real space method for diffuse optical tomography,” Opt. Lett. 35, 688–690 (2010).
  9. T. J. Rudge, V. Y. Soloviev, and S. R. Arridge, “Fast image reconstruction in fluoresence optical tomography using data compression,” Opt. Lett. 35, 763–765 (2010).
  10. V. A. Markel, V. Mital, and J. C. Schotland, “Inverse problem in optical diffusion tomography. iii. inversion formulas and singular-value decomposition,” J. Opt. Soc. Am. A 20, 890–902 (2003).
  11. S. D. Konecky, G. Y. Panasyuk, K. Lee, V. Markel, A. G. Yodh, and J. C. Schotland, “Imaging complex structures with diffuse light,” Opt. Express 16, 5048–5060 (2008).
  12. Z.-M. Wang, G. Y. Panasyuk, V. A. Markel, and J. C. Schotland, “Experimental demonstration of an analytic method for image reconstruction in optical diffusion tomography with large data sets,” Opt. Lett. 30, 3338–3340 (2005).
  13. G. Y. Panasyuk, Z.-M. Wang, J. C. Schotland, and V. A. Markel, “Fluorescent optical tomography with large data sets,” Opt. Lett. 33, 1744–1746 (2008).
  14. D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Opt. Lett. 30, 1354–1356 (2005).
  15. N. Dognitz and G. Wagnieres, “Determination of tissue optical properties by steady-state spatial frequency-domain reflectometry,” Lasers Med. Sci. 13, 55–65 (1998).
  16. A. Bassi, C. D’Andrea, G. Valentini, R. Cubeddu, and S. Arridge, “Temporal propagation of spatial information in turbid media,” Opt. Lett. 33, 2836–2838 (2008).
  17. F. L. J. Chen, V. Venugopal, and X. Intes, “Time-resolved diffuse optical tomography with patterned-light illumination and detection,” Opt. Lett. 35, 2121–2123 (2010).
  18. A. Joshi, W. Bangerth, and E. M. Sevick-muraca, “Non-contact fluorescence optical tomography with scanning patterned illumination,” Opt. Express 14, 55–64 (2006).
  19. V. Lukic, V. A. Markel, and J. C. Schotland, “Optical tomography with structured illumination,” Opt. Lett. 34, 983–985 (2009).
  20. A. J. Dutta, S. Ahn, and R. M. Leahy, “Illumination pattern optimization for fluorescence tomography: theory and simulation studies,” Phys. Med. Biol. 10, 2961–2982 (2010).
  21. S. D. Konecky, A. Mazhar, D. Cuccia, A. J. Durkin, J. C. Schotland, and B. J. Tromberg, “Quantitative optical tomography of sub-surface heterogeneities using spatially modulated structured light,” Opt. Express 17, 14780–14790 (2009).
  22. A. Bassi, C. D’Andrea, G. Valentini, R. Cubeddu, and S. Arridge, “Detection of inhomogeneities in diffusive media using spatially modulated light,” Opt. Lett. 34, 2156–2158 (2009).
  23. S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15, 016006 (2010).
  24. A. Bassi, A. Farina, C. D’Andrea, A. Pifferi, G. Valentini, and R. Cubeddu, “Portable, large-bandwidth time-resolved system for diffuse optical spectroscopy,” Opt. Express 15, 14482–14487 (2007).
  25. M. Schweiger and S. R. Arridge, “Toast reconstruction package,” http://web4.cs.ucl.ac.uk/research/vis/toast/.

2010 (5)

J. Ripoll, “Hybrid fourier-real space method for diffuse optical tomography,” Opt. Lett. 35, 688–690 (2010).

T. J. Rudge, V. Y. Soloviev, and S. R. Arridge, “Fast image reconstruction in fluoresence optical tomography using data compression,” Opt. Lett. 35, 763–765 (2010).

F. L. J. Chen, V. Venugopal, and X. Intes, “Time-resolved diffuse optical tomography with patterned-light illumination and detection,” Opt. Lett. 35, 2121–2123 (2010).

A. J. Dutta, S. Ahn, and R. M. Leahy, “Illumination pattern optimization for fluorescence tomography: theory and simulation studies,” Phys. Med. Biol. 10, 2961–2982 (2010).

S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15, 016006 (2010).

2009 (3)

2008 (4)

2007 (3)

2006 (2)

J. Selb, D. K. Joseph, and D. A. Boas, “Time-gated optical system for depth-resolved functional brain imaging,” J. Biomed. Opt. 11, 044008 (2006).

A. Joshi, W. Bangerth, and E. M. Sevick-muraca, “Non-contact fluorescence optical tomography with scanning patterned illumination,” Opt. Express 14, 55–64 (2006).

2005 (2)

2004 (1)

V. A. Markel and J. C. Schotland, “Symmetries, inversion formulas, and image reconstruction for optical tomography,” Phys. Rev. E 70, 056616 (2004).

2003 (2)

2002 (1)

G. Strangman, D. A. Boas, and J. P. Sutton, “Non-invasive neuroimaging using near-infrared light,” Biol. Psychiatry 52, 679–693 (2002).

1998 (1)

N. Dognitz and G. Wagnieres, “Determination of tissue optical properties by steady-state spatial frequency-domain reflectometry,” Lasers Med. Sci. 13, 55–65 (1998).

Abran, M.

S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15, 016006 (2010).

Ahn, S.

A. J. Dutta, S. Ahn, and R. M. Leahy, “Illumination pattern optimization for fluorescence tomography: theory and simulation studies,” Phys. Med. Biol. 10, 2961–2982 (2010).

Arridge, S.

Arridge, S. R.

Bangerth, W.

A. Joshi, W. Bangerth, and E. M. Sevick-muraca, “Non-contact fluorescence optical tomography with scanning patterned illumination,” Opt. Express 14, 55–64 (2006).

Bassi, A.

Bélanger, S.

S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15, 016006 (2010).

Berger, M.

A. Koenig, L. Herve, V. Josserand, M. Berger, J. Boutet, A. D. Silva, J.-M. Dinten, P. Peltie, J.-L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).

Bevilacqua, F.

Boas, D. A.

J. Selb, D. K. Joseph, and D. A. Boas, “Time-gated optical system for depth-resolved functional brain imaging,” J. Biomed. Opt. 11, 044008 (2006).

G. Strangman, D. A. Boas, and J. P. Sutton, “Non-invasive neuroimaging using near-infrared light,” Biol. Psychiatry 52, 679–693 (2002).

Boutet, J.

A. Koenig, L. Herve, V. Josserand, M. Berger, J. Boutet, A. D. Silva, J.-M. Dinten, P. Peltie, J.-L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).

Casanova, C.

S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15, 016006 (2010).

Chen, F. L. J.

Choe, R.

Coll, J.-L.

A. Koenig, L. Herve, V. Josserand, M. Berger, J. Boutet, A. D. Silva, J.-M. Dinten, P. Peltie, J.-L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).

Comelli, D.

P. Taroni, D. Comelli, A. Pifferi, A. Torricelli, and R. Cubeddu, “Absorption of collagen: effects on the estimate of breast composition and related diagnostic implications,” J. Biomed. Opt. 12, 014021 (2007).

Corlu, A.

Cubeddu, R.

Cuccia, D.

Cuccia, D. J.

D’Andrea, C.

Dinten, J.-M.

A. Koenig, L. Herve, V. Josserand, M. Berger, J. Boutet, A. D. Silva, J.-M. Dinten, P. Peltie, J.-L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).

Dognitz, N.

N. Dognitz and G. Wagnieres, “Determination of tissue optical properties by steady-state spatial frequency-domain reflectometry,” Lasers Med. Sci. 13, 55–65 (1998).

Durduran, T.

Durkin, A. J.

Dutta, A. J.

A. J. Dutta, S. Ahn, and R. M. Leahy, “Illumination pattern optimization for fluorescence tomography: theory and simulation studies,” Phys. Med. Biol. 10, 2961–2982 (2010).

Farina, A.

Herve, L.

A. Koenig, L. Herve, V. Josserand, M. Berger, J. Boutet, A. D. Silva, J.-M. Dinten, P. Peltie, J.-L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).

Intes, X.

F. L. J. Chen, V. Venugopal, and X. Intes, “Time-resolved diffuse optical tomography with patterned-light illumination and detection,” Opt. Lett. 35, 2121–2123 (2010).

S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15, 016006 (2010).

Joseph, D. K.

J. Selb, D. K. Joseph, and D. A. Boas, “Time-gated optical system for depth-resolved functional brain imaging,” J. Biomed. Opt. 11, 044008 (2006).

Joshi, A.

A. Joshi, W. Bangerth, and E. M. Sevick-muraca, “Non-contact fluorescence optical tomography with scanning patterned illumination,” Opt. Express 14, 55–64 (2006).

Josserand, V.

A. Koenig, L. Herve, V. Josserand, M. Berger, J. Boutet, A. D. Silva, J.-M. Dinten, P. Peltie, J.-L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).

Koenig, A.

A. Koenig, L. Herve, V. Josserand, M. Berger, J. Boutet, A. D. Silva, J.-M. Dinten, P. Peltie, J.-L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).

Konecky, S. D.

Leahy, R. M.

A. J. Dutta, S. Ahn, and R. M. Leahy, “Illumination pattern optimization for fluorescence tomography: theory and simulation studies,” Phys. Med. Biol. 10, 2961–2982 (2010).

Lee, K.

Lesage, F.

S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15, 016006 (2010).

Lukic, V.

Markel, V.

Markel, V. A.

Mazhar, A.

Mital, V.

Ntziachristos, V.

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

Panasyuk, G. Y.

Peltie, P.

A. Koenig, L. Herve, V. Josserand, M. Berger, J. Boutet, A. D. Silva, J.-M. Dinten, P. Peltie, J.-L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).

Pifferi, A.

P. Taroni, D. Comelli, A. Pifferi, A. Torricelli, and R. Cubeddu, “Absorption of collagen: effects on the estimate of breast composition and related diagnostic implications,” J. Biomed. Opt. 12, 014021 (2007).

A. Bassi, A. Farina, C. D’Andrea, A. Pifferi, G. Valentini, and R. Cubeddu, “Portable, large-bandwidth time-resolved system for diffuse optical spectroscopy,” Opt. Express 15, 14482–14487 (2007).

Ripoll, J.

Rizo, P.

A. Koenig, L. Herve, V. Josserand, M. Berger, J. Boutet, A. D. Silva, J.-M. Dinten, P. Peltie, J.-L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).

Rosen, M. A.

Rudge, T. J.

Schnall, M. D.

Schotland, J. C.

Schweiger, M.

Selb, J.

J. Selb, D. K. Joseph, and D. A. Boas, “Time-gated optical system for depth-resolved functional brain imaging,” J. Biomed. Opt. 11, 044008 (2006).

Sevick-muraca, E. M.

A. Joshi, W. Bangerth, and E. M. Sevick-muraca, “Non-contact fluorescence optical tomography with scanning patterned illumination,” Opt. Express 14, 55–64 (2006).

Silva, A. D.

A. Koenig, L. Herve, V. Josserand, M. Berger, J. Boutet, A. D. Silva, J.-M. Dinten, P. Peltie, J.-L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).

Soloviev, V. Y.

Strangman, G.

G. Strangman, D. A. Boas, and J. P. Sutton, “Non-invasive neuroimaging using near-infrared light,” Biol. Psychiatry 52, 679–693 (2002).

Sutton, J. P.

G. Strangman, D. A. Boas, and J. P. Sutton, “Non-invasive neuroimaging using near-infrared light,” Biol. Psychiatry 52, 679–693 (2002).

Taroni, P.

P. Taroni, D. Comelli, A. Pifferi, A. Torricelli, and R. Cubeddu, “Absorption of collagen: effects on the estimate of breast composition and related diagnostic implications,” J. Biomed. Opt. 12, 014021 (2007).

Torricelli, A.

P. Taroni, D. Comelli, A. Pifferi, A. Torricelli, and R. Cubeddu, “Absorption of collagen: effects on the estimate of breast composition and related diagnostic implications,” J. Biomed. Opt. 12, 014021 (2007).

Tromberg, B. J.

Valentini, G.

Venugopal, V.

Wagnieres, G.

N. Dognitz and G. Wagnieres, “Determination of tissue optical properties by steady-state spatial frequency-domain reflectometry,” Lasers Med. Sci. 13, 55–65 (1998).

Wang, Z.-M.

Weissleder, R.

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

Yodh, A. G.

Biol. Psychiatry (1)

G. Strangman, D. A. Boas, and J. P. Sutton, “Non-invasive neuroimaging using near-infrared light,” Biol. Psychiatry 52, 679–693 (2002).

J. Biomed. Opt. (4)

P. Taroni, D. Comelli, A. Pifferi, A. Torricelli, and R. Cubeddu, “Absorption of collagen: effects on the estimate of breast composition and related diagnostic implications,” J. Biomed. Opt. 12, 014021 (2007).

J. Selb, D. K. Joseph, and D. A. Boas, “Time-gated optical system for depth-resolved functional brain imaging,” J. Biomed. Opt. 11, 044008 (2006).

A. Koenig, L. Herve, V. Josserand, M. Berger, J. Boutet, A. D. Silva, J.-M. Dinten, P. Peltie, J.-L. Coll, and P. Rizo, “In vivo mice lung tumor follow-up with fluorescence diffuse optical tomography,” J. Biomed. Opt. 13, 011008 (2008).

S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15, 016006 (2010).

J. Opt. Soc. Am. A (1)

Lasers Med. Sci. (1)

N. Dognitz and G. Wagnieres, “Determination of tissue optical properties by steady-state spatial frequency-domain reflectometry,” Lasers Med. Sci. 13, 55–65 (1998).

Nature Med. (1)

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

Opt. Express (5)

Opt. Lett. (9)

J. Ripoll, “Hybrid fourier-real space method for diffuse optical tomography,” Opt. Lett. 35, 688–690 (2010).

T. J. Rudge, V. Y. Soloviev, and S. R. Arridge, “Fast image reconstruction in fluoresence optical tomography using data compression,” Opt. Lett. 35, 763–765 (2010).

F. L. J. Chen, V. Venugopal, and X. Intes, “Time-resolved diffuse optical tomography with patterned-light illumination and detection,” Opt. Lett. 35, 2121–2123 (2010).

G. Y. Panasyuk, Z.-M. Wang, J. C. Schotland, and V. A. Markel, “Fluorescent optical tomography with large data sets,” Opt. Lett. 33, 1744–1746 (2008).

A. Bassi, C. D’Andrea, G. Valentini, R. Cubeddu, and S. Arridge, “Temporal propagation of spatial information in turbid media,” Opt. Lett. 33, 2836–2838 (2008).

V. Lukic, V. A. Markel, and J. C. Schotland, “Optical tomography with structured illumination,” Opt. Lett. 34, 983–985 (2009).

A. Bassi, C. D’Andrea, G. Valentini, R. Cubeddu, and S. Arridge, “Detection of inhomogeneities in diffusive media using spatially modulated light,” Opt. Lett. 34, 2156–2158 (2009).

D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Opt. Lett. 30, 1354–1356 (2005).

Z.-M. Wang, G. Y. Panasyuk, V. A. Markel, and J. C. Schotland, “Experimental demonstration of an analytic method for image reconstruction in optical diffusion tomography with large data sets,” Opt. Lett. 30, 3338–3340 (2005).

Phys. Med. Biol. (1)

A. J. Dutta, S. Ahn, and R. M. Leahy, “Illumination pattern optimization for fluorescence tomography: theory and simulation studies,” Phys. Med. Biol. 10, 2961–2982 (2010).

Phys. Rev. E (1)

V. A. Markel and J. C. Schotland, “Symmetries, inversion formulas, and image reconstruction for optical tomography,” Phys. Rev. E 70, 056616 (2004).

Other (1)

M. Schweiger and S. R. Arridge, “Toast reconstruction package,” http://web4.cs.ucl.ac.uk/research/vis/toast/.

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.

(a) Experimental set-up; (b) Phantom; (c) Source patterns along x and y from frequency k x(y) = 0 rad mm -1 to k x(y) = 0.40 rad mm -1.

Download Full Size | PPT Slide | PDF

Fig. 2.
Fig. 2.

Scheme of data analysis and reconstruction procedure.

Download Full Size | PPT Slide | PDF

Fig. 3.
Fig. 3.

Reconstructed absorption at different depths in the case of bi-dimensional sampling of the input sources. Image scale in arbitrary units.

Download Full Size | PPT Slide | PDF

Fig. 4.
Fig. 4.

Reconstructed absorption at different depths in the case of bi-dimensional sampling of the input sources with flipped sample. Image scale in arbitrary units.

Download Full Size | PPT Slide | PDF

Fig. 5.
Fig. 5.

Reconstructed absorption at different depths in the case of horizontal (a) and vertical (b) source spatially modulation, respectively. Image scale in arbitrary units.

Download Full Size | PPT Slide | PDF

Fig. 6.
Fig. 6.

Reconstructed absorption at different depths in the case of bi-dimensional sampling of the input sources and a totally absorbing inclusions. Image scale in arbitrary units.

Download Full Size | PPT Slide | PDF

Tables (1)

Tables Icon

Table 1. Table summarizing the FWHM transversal dimension of the parallel (A) and oblique inclusion (B) by using 1D (horizontal and vertical) and 2D modulation.

View Table

Equations (11)

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

( − D ∇ 2 + μ a ) Φ = 0 in Ω ,
Φ + 2 ζ D ∂ Φ ∂ ν = { J − on Γ 1 0 on ∂ Ω \ Γ 1 ,
J + = − D ∂ Φ ∂ ν on Γ 2 .
J + = Λ J − = 𝓜 𝒢 J −
J + = 𝖬 𝖪 − 1 J −
y = 𝖯 J + .
y ( k m , k s ) = ∫ Γ 2 e i k m · ρ 2 J + ( ρ 2 ) d 2 ρ 2 = 〈 e − i k m · ρ 2 , 𝖯𝖬 𝖪 − 1 e − i k s · ρ 1 〉 Γ 2 .
∂ y ∂ δ μ a = 〈 e − i k m · ρ 2 , 𝖯𝖬𝖪 − 1 ∂ 햪 ∂ δ μ a 햪 − 1 e − i k s · ρ 1 〉 Γ 2 = 〈 𝖪 − 1 𝖬 T 𝖯 T e − i k m · ρ 2 , ∂ 𝖪 ∂ δ μ a 𝖪 − 1 e − i k s · ρ 1 〉 Ω
= 〈 Φ m + , ∂ 𝖪 ∂ δ μ a Φ s 〉 Ω
δ g = g obj − g ref = 𝖩 x
x = 𝖩 T ( 𝖩 𝖩 T + α 𝖨 ) − 1 δ g

Metrics