Abstract

We investigate the use of the Mellin–Laplace transform for reconstructing optical parameters from time-resolved optical tomography in diffusive media. We present here its definition, its mathematical properties, and its sensitivity to variations of optical properties. The method was validated on two-dimensional reconstructions from simulation in the reflection geometry. We conclude that reconstructions based on the Mellin–Laplace transform are more robust to noise than the methods using first moments.

© 2012 Optical Society of America

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  1. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
    [CrossRef]
  2. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005).
    [CrossRef]
  3. S. R. Arridge and W. R. B. Lionheart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett. 23, 882–884 (1998).
    [CrossRef]
  4. S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azar, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
    [CrossRef]
  5. J. Wang, S. C. Davis, S. Srinivasan, S. Jiang, B. W. Pogue, and K. D. Paulsen, Spectral tomography with diffuse near-infrared light: inclusion of broadband frequency domain spectral data,” J. Biomed. Opt. 13, 041305 (2008).
    [CrossRef]
  6. A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. 44, 2082–2093 (2005).
    [CrossRef]
  7. J. Wang, B. Pogue, S. Jiang, and K. D. Paulsen, “Near-infrared tomography of breast cancer hemoglobin, water, lipid, and scattering using combined frequency domain and CW measurement,” Opt. Lett. 35, 82–84 (2010).
    [CrossRef]
  8. I. Nissilä, J. C. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
    [CrossRef]
  9. M. Schweiger and S. R. Arridge, “Direct calculation with a finite-element method of the Laplace transform of the distribution of photon time-of-flight in tissue,” Appl. Opt. 36, 9042–9049 (1997).
    [CrossRef]
  10. M. Schweiger and S. R. Arridge, “Application of temporal filters to time resolved data in optical tomography,” Phys. Med. Biol. 44, 1699–1717 (1999).
    [CrossRef]
  11. H. Zhao, F. Gao, Y. Tanikawa, K. Homma, and Y. Yamada, “Time-resolved diffuse optical tomographic imaging for the provision of both anatomical and functional information about biological tissue,” Appl. Opt. 44, 1905–1916 (2005).
    [CrossRef]
  12. J. Selb, A. M. Dale, and D. A. Boas, “Linear 3D reconstruction of time-domain diffuse optical imaging differential data: improved depth localization and lateral resolution,” Opt. Express 15, 16400–16412 (2007).
    [CrossRef]
  13. F. Gao, H. Zhao, and Y. Yamada, “Improvement of image quality in diffuse optical tomography by use of full time-resolved data,” Appl. Opt. 41, 778–791 (2002).
    [CrossRef]
  14. F. Gao, H. Zhao, L. Zhang, Y. Tanikawa, A. Marjono, and Y. Yamada, “A self-normalized, full time-resolved method for fluorescence diffuse optical tomography,” Opt. Express 16, 13104–13121 (2008).
    [CrossRef]
  15. Q. Zhao, L. Spinelli, A. Bassi, G. Valentini, D. Contini, A. Torricelli, R. Cubeddu, G. Zaccanti, F. Martelli, and A. Pifferi, “Functional tomography using a time-gated ICCD camera,” Biomed. Opt. Express 2, 705–716 (2011).
    [CrossRef]
  16. F. Nouizi, M. Torregrossa, R. Chabrier, and P. Poulet, “Improvement of absorption and scattering discrimination by selection of sensitive points on temporal profile in diffuse optical tomography,” Opt. Express 19, 12843–12854(2011).
    [CrossRef]
  17. N. Ducros, L. Hervé, A. Da Silva, J. M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part I. Theoretical material,” Phys. Med. Biol. 54, 7089–7105 (2009).
    [CrossRef]
  18. B. Montcel, R. Chabrier, and P. Poulet, “Detection of cortical activation with time-resolved diffuse optical methods,” Appl. Opt. 44, 1942–1947 (2005).
    [CrossRef]
  19. A. Liebert, H. Wabnitz, J. Steinbrink, H. Obrig, M. Möller, R. Macdonald, A. Villringer, and H. Rinneberg, “Time-resolved multidistance near-infrared spectroscopy of the adult head: intracerebral and extracerebral absorption changes from moments of distribution of times of flight of photons,” Appl. Opt. 43, 3037–3047 (2004).
    [CrossRef]
  20. A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Optical tomography of a realistic neonatal head phantom,” Appl. Opt. 42, 3109–3116 (2003).
    [CrossRef]
  21. E. Hillman, J. C. Hebden, M. Schweiger, H. Dehghani, F. E. W. Schmid, D. T. Delpy, and S. R. Arridge, “Time resolved optical tomography of the human forearm,” Phys. Med. Biol. 46, 1117–1130 (2001).
    [CrossRef]
  22. R. Pierrat, L. J. Greffet, and R. Carminati, “Photon diffusion coefficient in scattering and absorbing media,” J. Opt. Soc. Am. A 23, 1106–1110 (2006).
    [CrossRef]
  23. G. B. Arfken, H. J. Weber, and H. J. Weber, Mathematical Methods for Physicists (Academic, 1995).
  24. A. Tosi, A. D. Mora, F. Zappa, A. Gulinatti, D. Contini, A. Pifferi, L. Spinelli, A. Torricelli, and R. Cubeddu, “Fast-gated single-photon counting technique widens dynamic range and speeds up acquisition time in time-resolved measurements,” Opt. Express 19, 10735–10746 (2011).
    [CrossRef]
  25. V. V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis (SPIE, 2007).
  26. R. C. Haskell, L. O. Svaasand, T.-T. Tsay, T.-C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
    [CrossRef]
  27. A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source–detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
    [CrossRef]

2011

2010

2009

N. Ducros, L. Hervé, A. Da Silva, J. M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part I. Theoretical material,” Phys. Med. Biol. 54, 7089–7105 (2009).
[CrossRef]

2008

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source–detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef]

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azar, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef]

J. Wang, S. C. Davis, S. Srinivasan, S. Jiang, B. W. Pogue, and K. D. Paulsen, Spectral tomography with diffuse near-infrared light: inclusion of broadband frequency domain spectral data,” J. Biomed. Opt. 13, 041305 (2008).
[CrossRef]

F. Gao, H. Zhao, L. Zhang, Y. Tanikawa, A. Marjono, and Y. Yamada, “A self-normalized, full time-resolved method for fluorescence diffuse optical tomography,” Opt. Express 16, 13104–13121 (2008).
[CrossRef]

2007

2006

R. Pierrat, L. J. Greffet, and R. Carminati, “Photon diffusion coefficient in scattering and absorbing media,” J. Opt. Soc. Am. A 23, 1106–1110 (2006).
[CrossRef]

I. Nissilä, J. C. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
[CrossRef]

2005

2004

2003

2002

2001

E. Hillman, J. C. Hebden, M. Schweiger, H. Dehghani, F. E. W. Schmid, D. T. Delpy, and S. R. Arridge, “Time resolved optical tomography of the human forearm,” Phys. Med. Biol. 46, 1117–1130 (2001).
[CrossRef]

1999

M. Schweiger and S. R. Arridge, “Application of temporal filters to time resolved data in optical tomography,” Phys. Med. Biol. 44, 1699–1717 (1999).
[CrossRef]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

1998

1997

1994

Arfken, G. B.

G. B. Arfken, H. J. Weber, and H. J. Weber, Mathematical Methods for Physicists (Academic, 1995).

Arridge, S. R.

A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. 44, 2082–2093 (2005).
[CrossRef]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005).
[CrossRef]

A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Optical tomography of a realistic neonatal head phantom,” Appl. Opt. 42, 3109–3116 (2003).
[CrossRef]

E. Hillman, J. C. Hebden, M. Schweiger, H. Dehghani, F. E. W. Schmid, D. T. Delpy, and S. R. Arridge, “Time resolved optical tomography of the human forearm,” Phys. Med. Biol. 46, 1117–1130 (2001).
[CrossRef]

M. Schweiger and S. R. Arridge, “Application of temporal filters to time resolved data in optical tomography,” Phys. Med. Biol. 44, 1699–1717 (1999).
[CrossRef]

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

S. R. Arridge and W. R. B. Lionheart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett. 23, 882–884 (1998).
[CrossRef]

M. Schweiger and S. R. Arridge, “Direct calculation with a finite-element method of the Laplace transform of the distribution of photon time-of-flight in tissue,” Appl. Opt. 36, 9042–9049 (1997).
[CrossRef]

Azar, F.

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azar, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef]

Bassi, A.

Boas, D. A.

Carminati, R.

Chabrier, R.

Choe, R.

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azar, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef]

A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. 44, 2082–2093 (2005).
[CrossRef]

Contini, D.

Corlu, A.

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azar, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef]

A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. 44, 2082–2093 (2005).
[CrossRef]

Cova, S.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source–detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef]

Cubeddu, R.

Da Silva, A.

N. Ducros, L. Hervé, A. Da Silva, J. M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part I. Theoretical material,” Phys. Med. Biol. 54, 7089–7105 (2009).
[CrossRef]

Dale, A. M.

Davis, S. C.

J. Wang, S. C. Davis, S. Srinivasan, S. Jiang, B. W. Pogue, and K. D. Paulsen, Spectral tomography with diffuse near-infrared light: inclusion of broadband frequency domain spectral data,” J. Biomed. Opt. 13, 041305 (2008).
[CrossRef]

Dehghani, H.

A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Optical tomography of a realistic neonatal head phantom,” Appl. Opt. 42, 3109–3116 (2003).
[CrossRef]

E. Hillman, J. C. Hebden, M. Schweiger, H. Dehghani, F. E. W. Schmid, D. T. Delpy, and S. R. Arridge, “Time resolved optical tomography of the human forearm,” Phys. Med. Biol. 46, 1117–1130 (2001).
[CrossRef]

Delpy, D. T.

A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Optical tomography of a realistic neonatal head phantom,” Appl. Opt. 42, 3109–3116 (2003).
[CrossRef]

E. Hillman, J. C. Hebden, M. Schweiger, H. Dehghani, F. E. W. Schmid, D. T. Delpy, and S. R. Arridge, “Time resolved optical tomography of the human forearm,” Phys. Med. Biol. 46, 1117–1130 (2001).
[CrossRef]

Dinten, J. M.

N. Ducros, L. Hervé, A. Da Silva, J. M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part I. Theoretical material,” Phys. Med. Biol. 54, 7089–7105 (2009).
[CrossRef]

Ducros, N.

N. Ducros, L. Hervé, A. Da Silva, J. M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part I. Theoretical material,” Phys. Med. Biol. 54, 7089–7105 (2009).
[CrossRef]

Durduran, T.

Everdell, N.

Feng, T.-C.

Freifelder, R.

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azar, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef]

Gao, F.

Gibson, A.

I. Nissilä, J. C. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
[CrossRef]

A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Optical tomography of a realistic neonatal head phantom,” Appl. Opt. 42, 3109–3116 (2003).
[CrossRef]

Gibson, A. P.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005).
[CrossRef]

Greffet, L. J.

Gulinatti, A.

Hajjioui, N.

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azar, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef]

Haskell, R. C.

Hebden, J. C.

I. Nissilä, J. C. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
[CrossRef]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005).
[CrossRef]

A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Optical tomography of a realistic neonatal head phantom,” Appl. Opt. 42, 3109–3116 (2003).
[CrossRef]

E. Hillman, J. C. Hebden, M. Schweiger, H. Dehghani, F. E. W. Schmid, D. T. Delpy, and S. R. Arridge, “Time resolved optical tomography of the human forearm,” Phys. Med. Biol. 46, 1117–1130 (2001).
[CrossRef]

Heino, J.

I. Nissilä, J. C. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
[CrossRef]

Hervé, L.

N. Ducros, L. Hervé, A. Da Silva, J. M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part I. Theoretical material,” Phys. Med. Biol. 54, 7089–7105 (2009).
[CrossRef]

Hillman, E.

A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. 44, 2082–2093 (2005).
[CrossRef]

E. Hillman, J. C. Hebden, M. Schweiger, H. Dehghani, F. E. W. Schmid, D. T. Delpy, and S. R. Arridge, “Time resolved optical tomography of the human forearm,” Phys. Med. Biol. 46, 1117–1130 (2001).
[CrossRef]

Homma, K.

Järvenpää, S.

I. Nissilä, J. C. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
[CrossRef]

Jennions, D.

I. Nissilä, J. C. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
[CrossRef]

Jiang, S.

J. Wang, B. Pogue, S. Jiang, and K. D. Paulsen, “Near-infrared tomography of breast cancer hemoglobin, water, lipid, and scattering using combined frequency domain and CW measurement,” Opt. Lett. 35, 82–84 (2010).
[CrossRef]

J. Wang, S. C. Davis, S. Srinivasan, S. Jiang, B. W. Pogue, and K. D. Paulsen, Spectral tomography with diffuse near-infrared light: inclusion of broadband frequency domain spectral data,” J. Biomed. Opt. 13, 041305 (2008).
[CrossRef]

Karp, J. S.

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azar, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef]

Katila, T.

I. Nissilä, J. C. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
[CrossRef]

Konecky, S. D.

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azar, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef]

Kotilahti, K.

I. Nissilä, J. C. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
[CrossRef]

Lee, K.

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azar, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef]

A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. 44, 2082–2093 (2005).
[CrossRef]

Liebert, A.

Lionheart, W. R. B.

Lipiäinen, L.

I. Nissilä, J. C. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
[CrossRef]

Macdonald, R.

Marjono, A.

Martelli, F.

Q. Zhao, L. Spinelli, A. Bassi, G. Valentini, D. Contini, A. Torricelli, R. Cubeddu, G. Zaccanti, F. Martelli, and A. Pifferi, “Functional tomography using a time-gated ICCD camera,” Biomed. Opt. Express 2, 705–716 (2011).
[CrossRef]

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source–detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef]

McAdams, M. S.

Möller, M.

Montcel, B.

Mora, A. D.

A. Tosi, A. D. Mora, F. Zappa, A. Gulinatti, D. Contini, A. Pifferi, L. Spinelli, A. Torricelli, and R. Cubeddu, “Fast-gated single-photon counting technique widens dynamic range and speeds up acquisition time in time-resolved measurements,” Opt. Express 19, 10735–10746 (2011).
[CrossRef]

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source–detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef]

Nissilä, I.

I. Nissilä, J. C. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
[CrossRef]

Noponen, T.

I. Nissilä, J. C. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
[CrossRef]

Nouizi, F.

Obrig, H.

Paulsen, K. D.

J. Wang, B. Pogue, S. Jiang, and K. D. Paulsen, “Near-infrared tomography of breast cancer hemoglobin, water, lipid, and scattering using combined frequency domain and CW measurement,” Opt. Lett. 35, 82–84 (2010).
[CrossRef]

J. Wang, S. C. Davis, S. Srinivasan, S. Jiang, B. W. Pogue, and K. D. Paulsen, Spectral tomography with diffuse near-infrared light: inclusion of broadband frequency domain spectral data,” J. Biomed. Opt. 13, 041305 (2008).
[CrossRef]

Peyrin, F.

N. Ducros, L. Hervé, A. Da Silva, J. M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part I. Theoretical material,” Phys. Med. Biol. 54, 7089–7105 (2009).
[CrossRef]

Pierrat, R.

Pifferi, A.

Pogue, B.

Pogue, B. W.

J. Wang, S. C. Davis, S. Srinivasan, S. Jiang, B. W. Pogue, and K. D. Paulsen, Spectral tomography with diffuse near-infrared light: inclusion of broadband frequency domain spectral data,” J. Biomed. Opt. 13, 041305 (2008).
[CrossRef]

Poulet, P.

Richards, R.

Riley, J.

Rinneberg, H.

Saffer, J. R.

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azar, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef]

Schmid, F. E. W.

E. Hillman, J. C. Hebden, M. Schweiger, H. Dehghani, F. E. W. Schmid, D. T. Delpy, and S. R. Arridge, “Time resolved optical tomography of the human forearm,” Phys. Med. Biol. 46, 1117–1130 (2001).
[CrossRef]

Schweiger, M.

I. Nissilä, J. C. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
[CrossRef]

A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. 44, 2082–2093 (2005).
[CrossRef]

A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Optical tomography of a realistic neonatal head phantom,” Appl. Opt. 42, 3109–3116 (2003).
[CrossRef]

E. Hillman, J. C. Hebden, M. Schweiger, H. Dehghani, F. E. W. Schmid, D. T. Delpy, and S. R. Arridge, “Time resolved optical tomography of the human forearm,” Phys. Med. Biol. 46, 1117–1130 (2001).
[CrossRef]

M. Schweiger and S. R. Arridge, “Application of temporal filters to time resolved data in optical tomography,” Phys. Med. Biol. 44, 1699–1717 (1999).
[CrossRef]

M. Schweiger and S. R. Arridge, “Direct calculation with a finite-element method of the Laplace transform of the distribution of photon time-of-flight in tissue,” Appl. Opt. 36, 9042–9049 (1997).
[CrossRef]

Selb, J.

Spinelli, L.

Srinivas, S. M.

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azar, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef]

Srinivasan, S.

J. Wang, S. C. Davis, S. Srinivasan, S. Jiang, B. W. Pogue, and K. D. Paulsen, Spectral tomography with diffuse near-infrared light: inclusion of broadband frequency domain spectral data,” J. Biomed. Opt. 13, 041305 (2008).
[CrossRef]

Steinbrink, J.

Svaasand, L. O.

Tanikawa, Y.

Torregrossa, M.

Torricelli, A.

Tosi, A.

A. Tosi, A. D. Mora, F. Zappa, A. Gulinatti, D. Contini, A. Pifferi, L. Spinelli, A. Torricelli, and R. Cubeddu, “Fast-gated single-photon counting technique widens dynamic range and speeds up acquisition time in time-resolved measurements,” Opt. Express 19, 10735–10746 (2011).
[CrossRef]

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source–detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef]

Tromberg, B. J.

Tsay, T.-T.

Tuchin, V. V.

V. V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis (SPIE, 2007).

Valentini, G.

Villringer, A.

Wabnitz, H.

Wang, J.

J. Wang, B. Pogue, S. Jiang, and K. D. Paulsen, “Near-infrared tomography of breast cancer hemoglobin, water, lipid, and scattering using combined frequency domain and CW measurement,” Opt. Lett. 35, 82–84 (2010).
[CrossRef]

J. Wang, S. C. Davis, S. Srinivasan, S. Jiang, B. W. Pogue, and K. D. Paulsen, Spectral tomography with diffuse near-infrared light: inclusion of broadband frequency domain spectral data,” J. Biomed. Opt. 13, 041305 (2008).
[CrossRef]

Weber, H. J.

G. B. Arfken, H. J. Weber, and H. J. Weber, Mathematical Methods for Physicists (Academic, 1995).

G. B. Arfken, H. J. Weber, and H. J. Weber, Mathematical Methods for Physicists (Academic, 1995).

Wiener, R.

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azar, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef]

Yamada, Y.

Yodh, A. G.

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azar, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef]

A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. 44, 2082–2093 (2005).
[CrossRef]

Yusof, R. M.

Zaccanti, G.

Q. Zhao, L. Spinelli, A. Bassi, G. Valentini, D. Contini, A. Torricelli, R. Cubeddu, G. Zaccanti, F. Martelli, and A. Pifferi, “Functional tomography using a time-gated ICCD camera,” Biomed. Opt. Express 2, 705–716 (2011).
[CrossRef]

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source–detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef]

Zappa, F.

A. Tosi, A. D. Mora, F. Zappa, A. Gulinatti, D. Contini, A. Pifferi, L. Spinelli, A. Torricelli, and R. Cubeddu, “Fast-gated single-photon counting technique widens dynamic range and speeds up acquisition time in time-resolved measurements,” Opt. Express 19, 10735–10746 (2011).
[CrossRef]

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source–detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef]

Zhang, L.

Zhao, H.

Zhao, Q.

Appl. Opt.

M. Schweiger and S. R. Arridge, “Direct calculation with a finite-element method of the Laplace transform of the distribution of photon time-of-flight in tissue,” Appl. Opt. 36, 9042–9049 (1997).
[CrossRef]

F. Gao, H. Zhao, and Y. Yamada, “Improvement of image quality in diffuse optical tomography by use of full time-resolved data,” Appl. Opt. 41, 778–791 (2002).
[CrossRef]

A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, “Optical tomography of a realistic neonatal head phantom,” Appl. Opt. 42, 3109–3116 (2003).
[CrossRef]

A. Liebert, H. Wabnitz, J. Steinbrink, H. Obrig, M. Möller, R. Macdonald, A. Villringer, and H. Rinneberg, “Time-resolved multidistance near-infrared spectroscopy of the adult head: intracerebral and extracerebral absorption changes from moments of distribution of times of flight of photons,” Appl. Opt. 43, 3037–3047 (2004).
[CrossRef]

H. Zhao, F. Gao, Y. Tanikawa, K. Homma, and Y. Yamada, “Time-resolved diffuse optical tomographic imaging for the provision of both anatomical and functional information about biological tissue,” Appl. Opt. 44, 1905–1916 (2005).
[CrossRef]

B. Montcel, R. Chabrier, and P. Poulet, “Detection of cortical activation with time-resolved diffuse optical methods,” Appl. Opt. 44, 1942–1947 (2005).
[CrossRef]

A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. 44, 2082–2093 (2005).
[CrossRef]

Biomed. Opt. Express

Inverse Probl.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

J. Biomed. Opt.

J. Wang, S. C. Davis, S. Srinivasan, S. Jiang, B. W. Pogue, and K. D. Paulsen, Spectral tomography with diffuse near-infrared light: inclusion of broadband frequency domain spectral data,” J. Biomed. Opt. 13, 041305 (2008).
[CrossRef]

I. Nissilä, J. C. Hebden, D. Jennions, J. Heino, M. Schweiger, K. Kotilahti, T. Noponen, A. Gibson, S. Järvenpää, L. Lipiäinen, and T. Katila, “Comparison between a time-domain and a frequency-domain system for optical tomography,” J. Biomed. Opt. 11, 064015 (2006).
[CrossRef]

J. Opt. Soc. Am. A

Med. Phys.

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azar, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Med. Biol.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005).
[CrossRef]

M. Schweiger and S. R. Arridge, “Application of temporal filters to time resolved data in optical tomography,” Phys. Med. Biol. 44, 1699–1717 (1999).
[CrossRef]

N. Ducros, L. Hervé, A. Da Silva, J. M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part I. Theoretical material,” Phys. Med. Biol. 54, 7089–7105 (2009).
[CrossRef]

E. Hillman, J. C. Hebden, M. Schweiger, H. Dehghani, F. E. W. Schmid, D. T. Delpy, and S. R. Arridge, “Time resolved optical tomography of the human forearm,” Phys. Med. Biol. 46, 1117–1130 (2001).
[CrossRef]

Phys. Rev. Lett.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source–detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
[CrossRef]

Other

G. B. Arfken, H. J. Weber, and H. J. Weber, Mathematical Methods for Physicists (Academic, 1995).

V. V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis (SPIE, 2007).

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Figures (11)

Fig. 1.
Fig. 1.

W(p,n) sequence for p=2ns1 and for n=0 to 9 (indicated at the maximum of each curve).

Fig. 2.
Fig. 2.

MLT of Green’s function in the infinite homogeneous 2D case, for μa=0.05cm1, μs=10cm1, c=30/1.4cm/ns, and a distance of 4 cm and for different p values. The larger the value of p, the best it compares with the analytical solution.

Fig. 3.
Fig. 3.

The time-resolved measurement is the convolution product of several physical processes. Here Ss is the source function, Gs is the propagation from the source to the chromophore, Gd is the propagation from the chromophore to the detector, and Dd is the detector function.

Fig. 4.
Fig. 4.

2D diffusive medium (8cm×5cm) used for validation simulations. 21 sources and detectors are used to probe the medium from the top side. For numerical computations, the medium is discretized on a mesh with 1 mm spacing between nodes. μa=0.05cm1 and μs=10cm1 in the background. In the heterogeneous case, two absorption discs (μa=0.1cm1 and μa=0.2cm1) are added to the background 1.0 and 2.0 cm deep. In the homogeneous case, no absorption heterogeneity is added.

Fig. 5.
Fig. 5.

Sensitivity Js,d(p,n)(r⃗)=Ys,d(p,n)δμa(r⃗) of the measurement Ys,d(p,n) to variations of absorption at position r⃗. Each plot is normalized by its maximum. μa=0.05cm1, μs=10cm1, and p=2ns1.

Fig. 6.
Fig. 6.

Ratios of homogeneous case and heterogeneous case measurements [Y1s,d(p,n)/Y0s,d(p,n)] as defined in Subsection 3.B for p=2ns1 and n=0, n=4, and n=8. We see in this case that high orders of the MLT are sensitive to deeply buried absorption inclusions since the figures typically reach 50%.

Fig. 7.
Fig. 7.

Variations of measurements between the homogeneous case and the heterogeneous case for intensity and mean time of flight. (Left) Ratio of intensities. (Right) Ratio of mean time of flight. Same conditions as Fig. 6.

Fig. 8.
Fig. 8.

Ratio of the MLT between the homogeneous and the heterogeneous cases, for s=d. Same conditions as Fig. 6.

Fig. 9.
Fig. 9.

Responses of the medium for extremal source and detector (s=1, d=21) or for source and detector at the same location (s=11, d=11). In the second case, late photon density (at 5 ns) is 7 orders of magnitude weaker than early photons (at 0 ns). Simulations obtained by solving Eq. (5) with Δt=0.1ns.

Fig. 10.
Fig. 10.

Reconstructions of the absorption map by using four data sets and three noise intensities. First row, “M0-M1,” moment-based approach by using the first two moments. Second row, “M0-M1-M2,” moment-based approach by using the first three moments. Third row, “few MLTs case” with 10 orders (p=2ns1 and 0n9). Fourth row, “full time-resolved data case” with 250 orders (p=50ns1 and 0n249). The three columns correspond to levels of noise of 10% or 30% and 80% on simulated measurements. At the bottom right, line used for the profiles of Fig. 11.

Fig. 11.
Fig. 11.

Profiles of the reconstructed μa along the line shown in Fig. 10. M0-M1, M0-M1-M2, few MLTs case, and 10 orders (p=2ns1 and 0n9) are at the top left, at the bottom left, at the top right, and at the bottom right of the figure, respectively.

Equations (25)

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M(p,n)[f(t)]=f(p,n)=+f(t)W(p,n)(t)dt=pnn!0+f(t)tnexp(pt)dt.
t=+tW(p,n)(t)dt/+W(p,n)(t)dt=n+1p.
Δt2(W(p,n))=+(tt)2W(p,n)(t)dt/+W(p,n)(t)dt=+t2W(p,n)(t)dt/+W(p,n)(t)dtt2=(n+2)(n+1)p2(n+1)(n+1)p2=n+1p2.
(a*b*c*d)(p,n)=i+j+k+l=na(p,i)b(p,j)c(p,k)d(p,l).
ϕ(r⃗,t)=4πI(r⃗,Ω⃗,t)dΩ⃗.
1cϕ(r⃗,t)t∇⃗·(D(r⃗)∇⃗ϕ(r⃗,t))+μa(r⃗)ϕ(r⃗,t)=S(r⃗,t),
∇⃗·(D(r⃗)∇⃗ϕ(p,n)(r⃗))+(μa(r⃗)+pc)ϕ(p,n)(r⃗)=S(p,n)(r⃗)+pcϕ(p,n1)(r⃗),
∇⃗·(D(r⃗)∇⃗ϕ(1/Δt,n)(r⃗))+(μa(r⃗)+1cΔt)ϕ(1/Δt,n)(r⃗)=S(1/Δt,n)(r⃗)+ϕ(1/Δt,n1)(r⃗)cΔt,
Gs0(r⃗,t)t∇⃗·(D0(r⃗)∇⃗Gs0(r⃗,t))+μa0(r⃗)Gs0(r⃗,t)=δ(r⃗r⃗s,t),
Gd1(r⃗,t)t∇⃗·(D1(r⃗)∇⃗Gd1(r⃗,t))+μa1(r⃗)Gd1(r⃗,t)=δ(r⃗r⃗d,t).
Gs1(r⃗d,t)Gs0(r⃗d,t)=Ω∇⃗Gs0(r⃗,t)*δD(r⃗,t)*∇⃗Gd1(r⃗,t)dr⃗ΩGs0(r⃗,t)*δμa(r⃗,t)*Gd1(r⃗,t)dr⃗,
δMs,d(t)=Ms,d1(t)Ms,d0(t)=− Ss(t)*[Ω(∇⃗Gs0(r⃗,t)*δD(r⃗,t)*∇⃗Gd1(r⃗,t)+Gs0(r⃗,t)*δμa(r⃗,t)*Gd1(r⃗,t))dr⃗]*Dd(t).
(s,d),δMs,d(p,n)=i+j+k+l=nSs(p,i)[Ω(∇⃗Gs0(p,j)(r⃗)δD(r⃗)∇⃗Gd1(p,k)(r⃗)+Gs0(p,j)(r⃗)δμa(r⃗)Gd1(p,k)(r⃗))dr⃗]Dd(p,l).
cov(M(t),M(t))dtdt=E[M(t)]δ(t,t)dtdt,
i!pij!pjcov(M(p,i),M(p,j))=E[(0M(t)exp(pt)tidt)(0M(t)exp(pt)tjdt)]E[0M(t)exp(pt)tidt]E[0M(t)exp(pt)tjdt]=00(E[M(t)·M(t)]E[M(t)]E[M(t)])titjepteptdtdt=0E[M(t)]ti+je(p+p)tdt=(i+j)!(p+p)i+j·E[M(p+p,i+j)].
cov(M(p,i),M(p,j))=pipj(p+p)i+j(i+ji)E[M(p+p,i+j)].
cov(M(p,i),M(p,j))=12i+j(i+ji)E[M(2p,i+j)].
cov(M(t),M(t))dtdt=α2E[M2(t)]δ(t,t)dtdt,
cov(M(p,i),M(p,j))=α22i+j(i+ji)E[(M2)(2p,i+j)].
Ys,d(p,n)=δMs,d(p,n)i=0n1Is,d(p,ni)Ys,d(p,i)Is,d(p,0).
Ys,d(p,n)Ωj+k=nGs(p,j)(r⃗)δμa(r⃗)Gd(p,k)(r⃗)dr⃗Ωj+k=n∇⃗Gs(p,j)(r⃗)δD(r⃗)∇⃗Gd(p,k)(r⃗)dr⃗.
Js,d(p,n)(r⃗)=Ys,d(p,n)δμa(r⃗)=j+k=nGs(p,j)(r⃗)·Gd(p,k)(r⃗).
χ2(X)=W2XY22,
f(t)=(g*h)(t)=+g(t)·h(tt)dt.
f(p,n)=+(pt)nn!(+g(t)·h(tt)dt)·ept·dt=t=ttpnn!++(t+t)n·g(t)·h(t)·ep(t+t)·dt·dt=pnn!k=0nn!(nk)!k!(+tk·g(t)·ept·dt)·(+tnk·h(t)·ept·dt)=k=0n(pkk!+tk·g(t)·ept·dt)·(pnk(nk)!+tnk·h(t)·ept·dt)=k=0ng(p,k)·h(p,nk).

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