Abstract

We describe a specific bandpass sampling procedure that provides high efficiency for interferogram sampling. This new approach is able to mitigate the important radiometric and noise disadvantages of Fourier transform spectrometry that recent theoretical investigations have pointed out. Proof of concept is given using simulations and measurements performed with a Sagnac triangular interferometer. Adopting an information-theoretic approach to spectrometry, we demonstrate the existence of important limitations to the radiometric efficiency achieved by any interferential or dispersive multiplex spectrometers. We find an extension to optics of the well-known data processing inequality, confirming that the Fellgett (multiplex) advantage is an inappropriate expectation. We give evidence of radiometric disadvantages implicit in the coded aperture architecture typical of compressive sensing.

© 2011 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  21. M. Harwit, P. G. Phillips, T. Fine, and N. J. A. Sloane, “Doubly multiplexed dispersive spectrometers,” Appl. Opt. 9, 1149–1154(1970).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  25. E. E. Fenimore and T. M. Cannon, “Coded aperture imaging with uniformly redundant arrays,” Appl. Opt. 17, 337–347 (1978).
    [CrossRef] [PubMed]
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    [CrossRef]
  27. M. E. Gehm, R. John, D. J. Brady, R. M. Willet, and T. J. Schulz, “Single-shot compressive spectral imaging with a dual-disperser architecture,” Opt. Express 15, 14013–14027 (2007).
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  28. A. Wagadarikar, R. John, R. Willet, and D. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 47, B44–B51 (2008).
    [CrossRef] [PubMed]
  29. H. Arguello and G. Arce, “Code aperture design for compressive spectral imaging,” Proceedings of the 18th European Signal Processing Conference (EUSIPCO-2010) (EURASIP, 2010), pp. 23–27.

2010 (2)

2008 (2)

2007 (1)

2006 (3)

2003 (1)

P. L. Dragotti and M. Vetterli, “Wavelet footprints: theory, algorithms, and applications,” IEEE Trans. Signal Process. 51, 1306–1323 (2003).
[CrossRef]

2002 (1)

A. Barducci, A. Casini, F. Castagnoli, P. Marcoionni, M. Morandi, and I. Pippi, “Performance assessment of a stationary interferometer for high-resolution remote sensing,” Proc. SPIE 4725, 547–555 (2002).
[CrossRef]

1998 (2)

R. Zamir, “A proof of the Fisher information inequality via a data processing argument,” IEEE Trans. Inf. Theory 44, 1246–1250(1998).
[CrossRef]

F. Cavallini, “The Italian panoramic monochromator,” Astron. Astrophys. 128, 589–598 (1998).
[CrossRef]

1997 (1)

A. Bushboom, H. D. Schotten, and H. Elders-Boll, “Coded aperture imaging with multiple measurements,” J. Opt. Soc. Am. 14, 1058–1065 (1997).
[CrossRef]

1991 (1)

R. J. Vaughan, N. L. Scott, and D. R. White, “The theory of bandpass sampling,” IEEE Trans. Signal Process. 39, 1973–1984(1991).
[CrossRef]

1985 (1)

1983 (1)

S. H. Nawab, T. F. Quatieri, and J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. 31, 986–998 (1983).
[CrossRef]

1978 (1)

1972 (1)

1970 (1)

1969 (1)

1967 (1)

P. B. Fellgett, “Conclusions on multiplex methods,” J. Phys. Colloq. 165–171 (1967).
[CrossRef]

1966 (2)

1958 (1)

P. B. Fellgett, “A propos de la théorie du spectromètre interférentiel multiplex,” J. Phys. Rad. 187–191 (1958).
[CrossRef]

1949 (1)

1948 (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948).

Arce, G.

H. Arguello and G. Arce, “Code aperture design for compressive spectral imaging,” Proceedings of the 18th European Signal Processing Conference (EUSIPCO-2010) (EURASIP, 2010), pp. 23–27.

Arguello, H.

H. Arguello and G. Arce, “Code aperture design for compressive spectral imaging,” Proceedings of the 18th European Signal Processing Conference (EUSIPCO-2010) (EURASIP, 2010), pp. 23–27.

Barducci, A.

Bianco, S. Del

Brady, D.

Brady, D. J.

Bushboom, A.

A. Bushboom, H. D. Schotten, and H. Elders-Boll, “Coded aperture imaging with multiple measurements,” J. Opt. Soc. Am. 14, 1058–1065 (1997).
[CrossRef]

Cannon, T. M.

Casini, A.

A. Barducci, A. Casini, F. Castagnoli, P. Marcoionni, M. Morandi, and I. Pippi, “Performance assessment of a stationary interferometer for high-resolution remote sensing,” Proc. SPIE 4725, 547–555 (2002).
[CrossRef]

Castagnoli, F.

A. Barducci, A. Casini, F. Castagnoli, P. Marcoionni, M. Morandi, and I. Pippi, “Performance assessment of a stationary interferometer for high-resolution remote sensing,” Proc. SPIE 4725, 547–555 (2002).
[CrossRef]

Cavallini, F.

F. Cavallini, “The Italian panoramic monochromator,” Astron. Astrophys. 128, 589–598 (1998).
[CrossRef]

Clark, T. A.

Connes, J.

Connes, P.

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
[CrossRef]

Dragotti, P. L.

P. L. Dragotti and M. Vetterli, “Wavelet footprints: theory, algorithms, and applications,” IEEE Trans. Signal Process. 51, 1306–1323 (2003).
[CrossRef]

Elders-Boll, H.

A. Bushboom, H. D. Schotten, and H. Elders-Boll, “Coded aperture imaging with multiple measurements,” J. Opt. Soc. Am. 14, 1058–1065 (1997).
[CrossRef]

Falorni, P.

Fellgett, P.

P. Fellgett, “The multiplex advantage,” Ph.D. dissertation (University of Cambridge, 1951).

Fellgett, P. B.

P. B. Fellgett, “The nature and origin of multiplex Fourier spectrometry,” Notes Rec. Roy. Soc. 60, 91–93 (2006).
[CrossRef]

P. B. Fellgett, “Conclusions on multiplex methods,” J. Phys. Colloq. 165–171 (1967).
[CrossRef]

P. B. Fellgett, “A propos de la théorie du spectromètre interférentiel multiplex,” J. Phys. Rad. 187–191 (1958).
[CrossRef]

Fenimore, E. E.

Fine, T.

Gehm, M. E.

Golay, M. J. E.

Guzzi, D.

Harwit, M.

Hilliard, R. L.

Jennings, R. E.

John, R.

Kawata, S.

Lastri, C.

Lim, J. S.

S. H. Nawab, T. F. Quatieri, and J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. 31, 986–998 (1983).
[CrossRef]

Marcoionni, P.

Mazzoni, M.

McCain, S. T.

Minami, S.

Morandi, M.

A. Barducci, A. Casini, F. Castagnoli, P. Marcoionni, M. Morandi, and I. Pippi, “Performance assessment of a stationary interferometer for high-resolution remote sensing,” Proc. SPIE 4725, 547–555 (2002).
[CrossRef]

Nardino, V.

Nawab, S. H.

S. H. Nawab, T. F. Quatieri, and J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. 31, 986–998 (1983).
[CrossRef]

Okamoto, T.

Phillips, P. G.

Pippi, I.

Pitsianis, N. P.

Potuluri, P.

Quatieri, T. F.

S. H. Nawab, T. F. Quatieri, and J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. 31, 986–998 (1983).
[CrossRef]

Schotten, H. D.

A. Bushboom, H. D. Schotten, and H. Elders-Boll, “Coded aperture imaging with multiple measurements,” J. Opt. Soc. Am. 14, 1058–1065 (1997).
[CrossRef]

Schulz, T. J.

Scott, N. L.

R. J. Vaughan, N. L. Scott, and D. R. White, “The theory of bandpass sampling,” IEEE Trans. Signal Process. 39, 1973–1984(1991).
[CrossRef]

Shannon, C. E.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948).

Shefferd, G. G.

Sloane, N. J. A.

Sullivan, M. E.

Thomas, J. A.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
[CrossRef]

Vaughan, R. J.

R. J. Vaughan, N. L. Scott, and D. R. White, “The theory of bandpass sampling,” IEEE Trans. Signal Process. 39, 1973–1984(1991).
[CrossRef]

Vetterli, M.

P. L. Dragotti and M. Vetterli, “Wavelet footprints: theory, algorithms, and applications,” IEEE Trans. Signal Process. 51, 1306–1323 (2003).
[CrossRef]

Wagadarikar, A.

Walmsley, D. A.

Wang, Y.

White, D. R.

R. J. Vaughan, N. L. Scott, and D. R. White, “The theory of bandpass sampling,” IEEE Trans. Signal Process. 39, 1973–1984(1991).
[CrossRef]

Willet, R.

Willet, R. M.

Zamir, R.

R. Zamir, “A proof of the Fisher information inequality via a data processing argument,” IEEE Trans. Inf. Theory 44, 1246–1250(1998).
[CrossRef]

Appl. Opt. (8)

N. J. A. Sloane, T. Fine, P. G. Phillips, and M. Harwit, “Codes for multiplex spectrometry,” Appl. Opt. 8, 2103–2106 (1969).
[CrossRef] [PubMed]

M. Harwit, P. G. Phillips, T. Fine, and N. J. A. Sloane, “Doubly multiplexed dispersive spectrometers,” Appl. Opt. 9, 1149–1154(1970).
[CrossRef] [PubMed]

D. A. Walmsley, T. A. Clark, and R. E. Jennings, “Correction of off-center sampled interferograms by a change of origin in the Fourier transform: the important effect of overlapping aliases,” Appl. Opt. 11, 1148–1151 (1972).
[CrossRef] [PubMed]

E. E. Fenimore and T. M. Cannon, “Coded aperture imaging with uniformly redundant arrays,” Appl. Opt. 17, 337–347 (1978).
[CrossRef] [PubMed]

T. Okamoto, S. Kawata, and S. Minami, “Optical method for resolution enhancement in photodiode array Fourier transform spectroscopy,” Appl. Opt. 24, 4221–4225 (1985).
[CrossRef] [PubMed]

M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan, “Static two-dimensional aperture coding for multimodal, multiplex spectroscopy,” Appl. Opt. 45, 2965–2973(2006).
[CrossRef] [PubMed]

A. Wagadarikar, R. John, R. Willet, and D. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 47, B44–B51 (2008).
[CrossRef] [PubMed]

A. Barducci, D. Guzzi, C. Lastri, P. Marcoionni, V. Nardino, and I. Pippi, “Radiometric and SNR properties of multiplex dispersive spectrometry,” Appl. Opt. 49, 5366–5373 (2010).
[CrossRef] [PubMed]

Appl. Spectrosc. (1)

Astron. Astrophys. (1)

F. Cavallini, “The Italian panoramic monochromator,” Astron. Astrophys. 128, 589–598 (1998).
[CrossRef]

Bell Syst. Tech. J. (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948).

IEEE Trans. Acoust. Speech Signal Process. (1)

S. H. Nawab, T. F. Quatieri, and J. S. Lim, “Signal reconstruction from short-time Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process. 31, 986–998 (1983).
[CrossRef]

IEEE Trans. Inf. Theory (1)

R. Zamir, “A proof of the Fisher information inequality via a data processing argument,” IEEE Trans. Inf. Theory 44, 1246–1250(1998).
[CrossRef]

IEEE Trans. Signal Process. (2)

P. L. Dragotti and M. Vetterli, “Wavelet footprints: theory, algorithms, and applications,” IEEE Trans. Signal Process. 51, 1306–1323 (2003).
[CrossRef]

R. J. Vaughan, N. L. Scott, and D. R. White, “The theory of bandpass sampling,” IEEE Trans. Signal Process. 39, 1973–1984(1991).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Phys. Colloq. (1)

P. B. Fellgett, “Conclusions on multiplex methods,” J. Phys. Colloq. 165–171 (1967).
[CrossRef]

J. Phys. Rad. (1)

P. B. Fellgett, “A propos de la théorie du spectromètre interférentiel multiplex,” J. Phys. Rad. 187–191 (1958).
[CrossRef]

Notes Rec. Roy. Soc. (1)

P. B. Fellgett, “The nature and origin of multiplex Fourier spectrometry,” Notes Rec. Roy. Soc. 60, 91–93 (2006).
[CrossRef]

Opt. Express (3)

Proc. SPIE (1)

A. Barducci, A. Casini, F. Castagnoli, P. Marcoionni, M. Morandi, and I. Pippi, “Performance assessment of a stationary interferometer for high-resolution remote sensing,” Proc. SPIE 4725, 547–555 (2002).
[CrossRef]

Other (3)

H. Arguello and G. Arce, “Code aperture design for compressive spectral imaging,” Proceedings of the 18th European Signal Processing Conference (EUSIPCO-2010) (EURASIP, 2010), pp. 23–27.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley, 1991).
[CrossRef]

P. Fellgett, “The multiplex advantage,” Ph.D. dissertation (University of Cambridge, 1951).

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Figures (7)

Fig. 1
Fig. 1

Spectral estimation computed by inverse FT of the sampled interferogram of a source with a broadband spectrum. The orange curves are the replicas (aliases) introduced by sampling, and they can be separated from the baseband as long as the sampling frequency κ S is greater than Nyquist’s limit 2 κ max . The pulse at the zero wavenumber in the baseband spectrum represents the dc contribution FT { i ( κ ) } | x = 0 .

Fig. 2
Fig. 2

Spectral estimation computed from the interferogram of a source with narrowband spectrum after bandpass sampling. Aliases introduced by sampling do not overlap the baseband (blue curve) as long as the sampling frequency κ S is matched to the source spectral characteristics. It is assumed that the constant term FT { i ( κ ) } | x = 0 has been removed before spectral estimation are performed. When this contribution is not completely rejected, possible disturbances can affect spectral estimations utilizing the bandpass sampling scheme.

Fig. 3
Fig. 3

Determination of the optimal sampling frequency κ S . The horizontal line in the lower part of the figure represents the inferior limit for the sampling frequency, which is twice the full bandwidth of the source. The hyperbolic curve represents the first relationship in Eq. (11), which is the basic constraint to avoid aliasing. The sampling frequency in this equation is a function of the sampling gain n. The purpose of bandpass sampling is adopting the n that determines the lowest sampling frequency greater than 2 Δ κ . The vertical axis is in logarithmic scale.

Fig. 4
Fig. 4

Examples of simulation of interferogram bandpass sampling [Eqs. (8, 9, 10)] with variable number of samples. Green (blue) curves and markers indicate bandpass (standard) sampled interferograms and estimated spectra. Aliases generated by bandpass sampling in a wide spectral interval are evident. (a) Raw (blue curve) and bandpass sampling (green markers) interferogram of a Gaussian source centered at 800.0 nm with 15 nm of standard deviation and clipped to the 780 820 nm interval ( OPD max = 200 μm , a = 3 ). (b) Spectral estimations obtained from interferograms in (a). Let us note the evident ringing that limits the spectral resolution of the data and may degrade the spectral estimations obtained with WTS due to long-range aliasing. (c) FTS and WTS spectral estimations for a simulated Gaussian source ( 700 nm central wavelength with 5.0 nm of standard deviation) clipped to the region 680.0 720.0 nm with OPD max = 50 μm . WTS estimation has been obtained with 20 interferogram samples (1024 samples for the FTS simulation) and 50% guard band interval ( a = 1.5 ). (d) FTS and WTS spectral estimations for a simulated Gaussian source ( 1000 nm central wavelength with 5.0 nm of standard deviation) clipped to the region 993.0 1007.0 nm with OPD max = 1500 μm . WTS estimation has been obtained with 93 interferogram samples (15,000 samples for the FTS simulation) and 100% guard band interval ( a = 2 ).

Fig. 5
Fig. 5

Example of interferogram acquired with the ALISEO instrument and its bandpass sampling after removing vignetting effects and the dc offset: solid line, FTS interferogram (1024 samples) of a He–Ne laser source; and filled rhombuses, WTS interferogram of a He–Ne source after bandpass sampling (12 samples).

Fig. 6
Fig. 6

Examples of spectral estimations obtained from interferograms acquired with the ALISEO instrument using FTS and WTS. Blue curves always represent the FTS spectral estimate, which always equals the WTS estimate (green curves). (a) Spectrum of a laser diode ( 780 nm central wavelength) computed from the complete interferogram (1024 samples) and the corresponding bandpass sampling estimate (16 samples). Estimation has been executed taking the modulus of the inverse FT, extended to a broad spectral interval in order to show the aliases originated by controlled interferogram undersampling. The decreasing alias amplitude is a combined effect of the OPD spectral dispersion and the 1 / λ 2 spectral density factor typical of the wavenumber domain. (b) Spectra of three laser sources at 530 (diode), 632 (He–Ne), and 808 nm (diode) of central wavelength, respectively. FTS spectra have been inferred from 1024 samples interferograms, while WTS estimates of the same sources have been obtained considering interferograms made up of 40 (530), 12 (632), and 14 samples ( 808 nm ). All spectra have been calculated as the inverse DCT without any radiometric calibration of the resulting signal; hence, they contain large contributions from phase error. Dark markers (crosses) represent the absolute FTS-WTS difference.

Fig. 7
Fig. 7

Scheme of the instrument and the measurement process using the information theory. Each element necessary for the spectral estimation is considered as a communication channel that obeys the Shannon theory. The figure indicates the signals at the I/O ports of every channel (elements), their capacity, and the information entropy pertaining to the observed source.

Equations (29)

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

I ( x ) = FT { i ( κ ) } + FT { i ( κ ) } | x = 0 2 ,
i ˜ ( κ ) = FT 1 { I S ( x ) } = { m = m = + i ( κ m κ S ) sinc [ ( κ m κ S ) d OPD ] } * sinc ( 2 κ OPD max ) ,
κ S 2 κ max ,
η S = K M .
δ i ( κ ) FTS A FTS 2 FT { i ( κ ) } | OPD = 0 ( δ κ B ) l + 1 = A FTS 2 FT { i ( κ ) } | OPD = 0 1 K l + 1 l 0.
lim OPD + ( B · OPD ) l + 1 I ( B · OPD ) = 0.
lim OPD max + I ( B · OPD max ) A FTS FT { i ( κ ) } | OPD = 0 2 1 ( B · OPD max ) l + 1 = A FTS 2 FT { i ( κ ) } | OPD = 0 ( δ κ B ) l + 1 .
κ s Δ κ = κ max κ min κ min + n κ S κ min κ max + ( n + 1 ) κ s κ max .
κ s Δ κ = κ max κ min κ s 2 κ min n = 2 κ c Δ κ n κ s 2 κ max n + 1 = 2 κ c + Δ κ n + 1 .
κ c ( κ c + n κ s ) = [ κ c + ( n + 1 ) κ s ] κ c .
κ s = 4 κ c 2 n + 1 κ s 2 Δ κ .
2 n + 1 2 κ c Δ κ .
Δ κ = ( 1 + a ) W ,
I ( x ) = FT { H ( κ , κ c , W ) i ( κ ) } = + H ( κ , κ c , W ) i ( κ ) exp ( 2 π j κ x ) d x .
{ M = κ s OPD max = 2 ( 1 + a ) W OPD max K = 2 W δ κ = 2 W OPD max η S = 1 1 + a ,
{ δ i ( κ ) FTS A FTS 2 FT { i ( κ ) } | OPD = 0 ( δ κ B ) l + 1 δ i ( κ ) WTS A FTS 2 FT { i ( κ ) } | OPD = 0 ( δ κ Δ κ ) l + 1 δ i ( κ ) WTS δ i ( κ ) FTS ( B Δ κ ) l + 1 .
h = B log 2 P + N N ,
h s = f s 2 log 2 σ s 2 ( κ ) + { E [ s ( κ ) ] } 2 σ s 2 ( κ ) = f s log 2 σ s 2 ( κ ) + { E [ s ( κ ) ] } 2 σ s ( κ ) f s log 2 SNR E [ s ( κ ) ] = C i ( κ ) C = τ A Ω δ κ ,
γ s h s ,
{ h s = f s 2 log 2 ( 1 + s 2 ( κ ) σ s 2 ( κ ) ) H s = D κ h s ( κ ) d κ γ s = h s Γ s = H s Φ s = Γ s τ { h y = κ s 2 log 2 ( 1 + y 2 ( x ) σ y 2 ( x ) ) H y = D x h y ( x ) d x Γ y = D x γ y ( x ) d x Φ y = Γ y τ γ y h y Γ y H y .
h y h s H y H s + M 2 log 2 M 2 H s .
H y H s = Γ s Γ y H y Γ y Φ s Φ y .
H y nonmult = H s Γ y nonmult = Γ s Φ y nonmult = Φ s .
y ( x ) = y eff ( x ) + y noninf ( x ) y noninf ( x ) y eff ( x ) γ y κ s log 2 ( y eff ( x ) / σ y ( x ) ) .
ξ = H y H s , η = Γ y Γ s .
η ξ .
ξ opt = η .
η ξ = log 2 ( 1 + SNR eff 2 ) log 2 ( 1 + SNR 2 ) , SNR eff 2 = y eff 2 ( x ) σ y 2 ( x ) ,
{ γ s ˜ = f s ˜ 2 log 2 ( 1 + SNR s ˜ 2 ) Γ s ˜ = D κ γ s ˜ ( κ ) d κ = K 2 log 2 ( 1 + SNR s ˜ 2 ) = η s M 2 log 2 ( 1 + SNR s ˜ 2 ) f s ˜ f s , ,

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