Abstract

We suggest adopting an efficient digital signal processing algorithm, which includes several fast Fourier transforms, to efficiently reconstruct the impulse response of a diffusive medium from its amplitude spectrum. It is also demonstrated that the singularities, which appear in the phase spectrum reconstruction, can be easily eliminated through the implementation of at least two types of data padding.

© 2010 Optical Society of America

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References

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  1. R. Kronig, “On the theory of dispersion of X-rays',” J. Opt. Soc. Am. 12, 547 (1926).
    [CrossRef]
  2. H. A. Kramers, Estratto dagli Atti del Congresso Internazionale di Fisici Como (Nicolo Zonichello, 1927).
  3. V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).
  4. E. Granot and S. Sternklar, “Reconstructing the impulse response of a diffusive medium with the Kramers-Kronig relations,” J. Opt. Soc. Am. B 24, 1620-1626 (2007).
    [CrossRef]
  5. Y. Ben-Aderet, E. Granot, S. Sternklar, and N. S. Kopeika, “Spectral analysis of a one-dimensional scattering medium with the differential multiply subtractive Kramers-Kronig method,” J. Opt. Soc. Am. B 26, 125-128 (2009).
    [CrossRef]
  6. E. Granot and S. Sternklar, “Spectral ballistic imaging: a novel technique for viewing through turbid or obstructing media,” J. Opt. Soc. Am. A 20, 1595-1599 (2003).
    [CrossRef]
  7. E. Granot, S. Sternklar, D. Schermann, Y. Ben-Aderet, and M. H. Itzhaq, “200 femtosecond impulse response of a Fabry-Perot etalon with the spectral ballistic imaging technique,” Appl. Phys. B: Photophys. Laser Chem. 82, 359-362 (2006).
    [CrossRef]
  8. E. Granot, S. Sternklar, Y. Ben-Aderet, and D. Schermann, “Quasi-ballistic imaging through a dynamic scattering medium with optical-field averaging using spectral-ballistic-imaging,” Opt. Express 14, 8598-8603 (2006).
    [CrossRef] [PubMed]
  9. A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography--principles and applications,” Rep. Prog. Phys. 66, 239-303 (2003).
    [CrossRef]
  10. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice-Hall, 1999).
  11. C. W. Peterson and B. W. Knight, “Causality calculations in the time domain: An efficient alternative to the Kramers-Kronig method,” J. Opt. Soc. Am. 63, 1238 (1973).
    [CrossRef]
  12. L. Wang, P. P. Ho, F. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769-771 (1991).
    [CrossRef] [PubMed]
  13. L. Wang, X. Liang, P. Galland, P. P. Ho, and R. R. Alfano, “True scattering coefficients of turbid matter measured by early-time gating,” Opt. Lett. 20, 913-915 (1995).
    [CrossRef] [PubMed]

2009 (1)

2007 (1)

2006 (2)

E. Granot, S. Sternklar, D. Schermann, Y. Ben-Aderet, and M. H. Itzhaq, “200 femtosecond impulse response of a Fabry-Perot etalon with the spectral ballistic imaging technique,” Appl. Phys. B: Photophys. Laser Chem. 82, 359-362 (2006).
[CrossRef]

E. Granot, S. Sternklar, Y. Ben-Aderet, and D. Schermann, “Quasi-ballistic imaging through a dynamic scattering medium with optical-field averaging using spectral-ballistic-imaging,” Opt. Express 14, 8598-8603 (2006).
[CrossRef] [PubMed]

2003 (2)

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography--principles and applications,” Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

E. Granot and S. Sternklar, “Spectral ballistic imaging: a novel technique for viewing through turbid or obstructing media,” J. Opt. Soc. Am. A 20, 1595-1599 (2003).
[CrossRef]

1995 (1)

1991 (1)

L. Wang, P. P. Ho, F. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769-771 (1991).
[CrossRef] [PubMed]

1973 (1)

1926 (1)

Alfano, R. R.

L. Wang, X. Liang, P. Galland, P. P. Ho, and R. R. Alfano, “True scattering coefficients of turbid matter measured by early-time gating,” Opt. Lett. 20, 913-915 (1995).
[CrossRef] [PubMed]

L. Wang, P. P. Ho, F. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769-771 (1991).
[CrossRef] [PubMed]

Ben-Aderet, Y.

Buck, J. R.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice-Hall, 1999).

Drexler, W.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography--principles and applications,” Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Fercher, A. F.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography--principles and applications,” Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Galland, P.

Granot, E.

Hitzenberger, C. K.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography--principles and applications,” Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Ho, P. P.

L. Wang, X. Liang, P. Galland, P. P. Ho, and R. R. Alfano, “True scattering coefficients of turbid matter measured by early-time gating,” Opt. Lett. 20, 913-915 (1995).
[CrossRef] [PubMed]

L. Wang, P. P. Ho, F. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769-771 (1991).
[CrossRef] [PubMed]

Itzhaq, M. H.

E. Granot, S. Sternklar, D. Schermann, Y. Ben-Aderet, and M. H. Itzhaq, “200 femtosecond impulse response of a Fabry-Perot etalon with the spectral ballistic imaging technique,” Appl. Phys. B: Photophys. Laser Chem. 82, 359-362 (2006).
[CrossRef]

Knight, B. W.

Kopeika, N. S.

Kramers, H. A.

H. A. Kramers, Estratto dagli Atti del Congresso Internazionale di Fisici Como (Nicolo Zonichello, 1927).

Kronig, R.

Lasser, T.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography--principles and applications,” Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Liang, X.

Liu, F.

L. Wang, P. P. Ho, F. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769-771 (1991).
[CrossRef] [PubMed]

Lucarini, V.

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice-Hall, 1999).

Peiponen, K.-E.

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).

Peterson, C. W.

Saarinen, J. J.

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice-Hall, 1999).

Schermann, D.

E. Granot, S. Sternklar, D. Schermann, Y. Ben-Aderet, and M. H. Itzhaq, “200 femtosecond impulse response of a Fabry-Perot etalon with the spectral ballistic imaging technique,” Appl. Phys. B: Photophys. Laser Chem. 82, 359-362 (2006).
[CrossRef]

E. Granot, S. Sternklar, Y. Ben-Aderet, and D. Schermann, “Quasi-ballistic imaging through a dynamic scattering medium with optical-field averaging using spectral-ballistic-imaging,” Opt. Express 14, 8598-8603 (2006).
[CrossRef] [PubMed]

Sternklar, S.

Vartiainen, E. M.

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).

Wang, L.

L. Wang, X. Liang, P. Galland, P. P. Ho, and R. R. Alfano, “True scattering coefficients of turbid matter measured by early-time gating,” Opt. Lett. 20, 913-915 (1995).
[CrossRef] [PubMed]

L. Wang, P. P. Ho, F. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769-771 (1991).
[CrossRef] [PubMed]

Zhang, G.

L. Wang, P. P. Ho, F. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769-771 (1991).
[CrossRef] [PubMed]

Appl. Phys. B: Photophys. Laser Chem. (1)

E. Granot, S. Sternklar, D. Schermann, Y. Ben-Aderet, and M. H. Itzhaq, “200 femtosecond impulse response of a Fabry-Perot etalon with the spectral ballistic imaging technique,” Appl. Phys. B: Photophys. Laser Chem. 82, 359-362 (2006).
[CrossRef]

J. Opt. Soc. Am. (2)

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

J. Opt. Soc. Am. B (2)

Opt. Express (1)

Opt. Lett. (1)

Rep. Prog. Phys. (1)

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography--principles and applications,” Rep. Prog. Phys. 66, 239-303 (2003).
[CrossRef]

Science (1)

L. Wang, P. P. Ho, F. Liu, G. Zhang, and R. R. Alfano, “Ballistic 2-D imaging through scattering walls using an ultrafast optical Kerr gate,” Science 253, 769-771 (1991).
[CrossRef] [PubMed]

Other (3)

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing (Prentice-Hall, 1999).

H. A. Kramers, Estratto dagli Atti del Congresso Internazionale di Fisici Como (Nicolo Zonichello, 1927).

V. Lucarini, J. J. Saarinen, K.-E. Peiponen, and E. M. Vartiainen, Kramers-Kronig Relations in Optical Materials Research (Springer-Verlag, 2005).

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Figures (2)

Fig. 1
Fig. 1

A comparison between the reconstructed impulse response (solid curve) and the exact impulse response (dashed curve) of a simulation of a diffusive medium. The lower two figures are a zoom-in of the upper one.

Fig. 2
Fig. 2

A comparison between the time delay reconstruction of the mirror padding (dash-dotted curve), the linear padding (solid curve), the exact (dashed), and the no-padding techniques (dotted). There is a small time lag between the exact and the reconstructed results, which is absent in this figure to aid in the comparison.

Equations (22)

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h ( t ) = h ( t ) u ( t ) ,
u ( t ) = { 1 t 0 0 t < 0 }
H ( ω ) = H ( ω ) U ( ω ) ,
R H ( ω ) = 1 π P d Ω I H ( Ω ) ω Ω ,
I H ( ω ) = 1 π P d Ω R H ( Ω ) ω Ω .
ln H ( ω ) = ln | H ( ω ) | + i θ ( ω ) ,
θ ( ω ) = 1 π P d Ω ln | H ( Ω ) | ω Ω .
h ( t ) = F 1 { | H ( ω ) | exp [ i θ ( ω ) ] } .
h ( t ) = h e ( t ) + h o ( t ) ,
h ( t ) = 2 h e ( t ) u ( t ) ,
h ( t ) = 2 F 1 { R H ( ω ) } u ( t ) .
F 1 { ln H ( ω ) } = 2 F 1 { ln | H ( ω ) | } u ( t ) .
H ( ω ) = exp 2 F { F 1 { ln | H ( ω ) | } u ( t ) } .
h ( t ) = F 1 { exp 2 F { F 1 { ln | H ( ω ) | } u ( t ) } } .
h ( n ) = h e ( n ) u ( n ) ,
u ( n ) = { 1 n = 0 , N 2 2 n = 1 , 2 , ( N 2 ) 1 0 ( N 2 ) + 1 , N 1 } .
h e ( n ) = 1 N k = 0 N 1 R H ( k ) exp ( 2 π i k n N ) = IFFT { R H ( k ) } .
[ IFFT { ln | H ( ω k ) | } u ( n ) ] = IFFT { ln H ( ω k ) } ,
H ( ω k ) = exp ( FFT [ IFFT { ln | H ( ω k ) | } u ( n ) ] )
h ( t n ) = IFFT { H ( ω k ) } = IFFT { exp ( FFT [ IFFT { ln | H ( ω k ) | } u ( n ) ] ) } .
| H ̃ ( ω k ) | = { | H ( ω k ) | 0 k N 1 ( k N ) M 1 | H ( ω 0 ) | ( k N M + 1 ) M 1 | H ( ω N 1 ) | N k N + M 1 } .
| H ̃ ( ω k ) | = { | H ( ω k ) | 0 k N 1 | H ( ω 2 N 1 k ) | N k 2 N 1 } .

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