Abstract

Optical systems based on rotating reticles were invented to determine the polar coordinates of a primarily IR optical source. Such systems fail when several optical sources are present in their field of view simultaneously. It is demonstrated experimentally that this drawback can be overcome by the application of a blind-signal-separation algorithm on the output signals of a modified optical system. The separation of the modified optical system responses into independent components yields modulating functions that carry information concerning the polar coordinates of the corresponding single optical sources.

© 1999 Optical Society of America

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  1. R. D. Hudson, “Optical modulation,” in Infrared System Engineering (Wiley, New York, 1969), Chap. 6, pp. 235–263.
  2. G. F. Aroyan, “The technique of spatial filtering,” Proc. Inst. Radio Eng. 47, 1561–1568 (1959).
  3. T. B. Buttweiler, “Optimum modulation characteristics for amplitude-modulated and frequency-modulated infrared systems,” J. Opt. Soc. Am. 51, 1011–1015 (1961).
    [CrossRef]
  4. A. F. Nicholson, “Error signals and discrimination in optical trackers that see several sources,” Proc. IEEE 53, 56–71 (1965).
    [CrossRef]
  5. H. Taub, D. L. Schilling, “Frequency-modulation systems,” in Principles of Communication Systems (McGraw-Hill, New York, 1987), Chap. 4, pp. 142–182.
  6. J. Singh, “Optoelectronic detectors,” in Semiconductor Optoelectronics—Physics and Technology (McGraw-Hill, New York, 1995), Chap. 7, pp. 336–398.
  7. X. R. Cao, R. W. Liu, “General approach to blind source separation,” IEEE Trans. Signal Process. 44, 562–571 (1996).
    [CrossRef]
  8. K. Torkkola, “Blind separation of convolved sources based on information maximization,” in IEEE Workshop on Neural Networks for Signal Processing, Kyoto, Japan, 4–6 September, 1996 (Institute of Electrical and Electronics Engineers, New York, 1996).
  9. D. Yellin, E. Weinstein, “Criteria for multichannel signal separation,” IEEE Trans. Signal Process. 42, 2158–2168 (1994).
    [CrossRef]
  10. D. Yellin, E. Weinstein, “Multichannel signal separation: methods and analysis,” IEEE Trans. Signal Process. 44, 106–118 (1996).
    [CrossRef]
  11. J. Wang, H. Zhenya, “Blind identification and separation of convolutively mixed independent sources,” IEEE Trans. Aerospace Electron. Syst. 33, 997–1002 (1997).
    [CrossRef]
  12. D. R. Brillinger, “Foundations,” in Time Series Data Analysis and Theory (McGraw-Hill, New York, 1981), Chap. 2, pp. 16–44.
  13. J. M. Mendel, “Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications,” Proc. IEEE 79, 278–305 (1991).
    [CrossRef]
  14. P. McCullagh, “Elementary theory of cumulants,” in Tensor Methods in Statistics (Chapman & Hall, London, 1987, 1995), Chap. 2, pp. 24–46.
  15. E. Weinstein, A. V. Oppenheim, M. Feder, J. R. Buck, “Iterative and sequential algorithms for multisensor signal enhancement,” IEEE Trans. Signal Process. 42, 846–859 (1994).
    [CrossRef]
  16. S. Van Gerven, D. Van Compernolle, “Signal separation by symmetric adaptive decorrelation: stability, convergence, and uniqueness,” IEEE Trans. Signal Process. 43, 1602–1612 (1995).
    [CrossRef]
  17. H. L. Nguyen Thi, C. Jutten, J. Caelen, “Speech enhancement: analysis and comparison of methods on various real situations,” in Signal Processing VI: Theories and Applications, J. Vandewalle, R. Boite, A. Oosterlick, eds. (Elsevier, New York, 1992), pp. 303–306.
  18. A. J. Bell, T. J. Sejnowski, “An information-maximization approach to blind separation and blind deconvolution,” Neural Comput. 7, 1129–1159 (1995).
    [CrossRef] [PubMed]
  19. P. Common, “Independent component analysis, a new concept?” Signal Process. 36, 287–314 (1994).
    [CrossRef]
  20. E. Sorouchyari, “Blind separation of sources, Part III: Stability analysis,” Signal Process. 24, 21–29 (1991).
    [CrossRef]
  21. P. J. Smaragdis, “Blind separation of convolved mixtures in the frequency domain,” in International Workshop on Independency and Artificial Neural Networks, Tenerife, Spain, 9–10 February, 1998 (University of Laguna, Tenerife, Spain, 1998).

1997

J. Wang, H. Zhenya, “Blind identification and separation of convolutively mixed independent sources,” IEEE Trans. Aerospace Electron. Syst. 33, 997–1002 (1997).
[CrossRef]

1996

D. Yellin, E. Weinstein, “Multichannel signal separation: methods and analysis,” IEEE Trans. Signal Process. 44, 106–118 (1996).
[CrossRef]

X. R. Cao, R. W. Liu, “General approach to blind source separation,” IEEE Trans. Signal Process. 44, 562–571 (1996).
[CrossRef]

1995

S. Van Gerven, D. Van Compernolle, “Signal separation by symmetric adaptive decorrelation: stability, convergence, and uniqueness,” IEEE Trans. Signal Process. 43, 1602–1612 (1995).
[CrossRef]

A. J. Bell, T. J. Sejnowski, “An information-maximization approach to blind separation and blind deconvolution,” Neural Comput. 7, 1129–1159 (1995).
[CrossRef] [PubMed]

1994

P. Common, “Independent component analysis, a new concept?” Signal Process. 36, 287–314 (1994).
[CrossRef]

E. Weinstein, A. V. Oppenheim, M. Feder, J. R. Buck, “Iterative and sequential algorithms for multisensor signal enhancement,” IEEE Trans. Signal Process. 42, 846–859 (1994).
[CrossRef]

D. Yellin, E. Weinstein, “Criteria for multichannel signal separation,” IEEE Trans. Signal Process. 42, 2158–2168 (1994).
[CrossRef]

1991

J. M. Mendel, “Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications,” Proc. IEEE 79, 278–305 (1991).
[CrossRef]

E. Sorouchyari, “Blind separation of sources, Part III: Stability analysis,” Signal Process. 24, 21–29 (1991).
[CrossRef]

1965

A. F. Nicholson, “Error signals and discrimination in optical trackers that see several sources,” Proc. IEEE 53, 56–71 (1965).
[CrossRef]

1961

1959

G. F. Aroyan, “The technique of spatial filtering,” Proc. Inst. Radio Eng. 47, 1561–1568 (1959).

Aroyan, G. F.

G. F. Aroyan, “The technique of spatial filtering,” Proc. Inst. Radio Eng. 47, 1561–1568 (1959).

Bell, A. J.

A. J. Bell, T. J. Sejnowski, “An information-maximization approach to blind separation and blind deconvolution,” Neural Comput. 7, 1129–1159 (1995).
[CrossRef] [PubMed]

Brillinger, D. R.

D. R. Brillinger, “Foundations,” in Time Series Data Analysis and Theory (McGraw-Hill, New York, 1981), Chap. 2, pp. 16–44.

Buck, J. R.

E. Weinstein, A. V. Oppenheim, M. Feder, J. R. Buck, “Iterative and sequential algorithms for multisensor signal enhancement,” IEEE Trans. Signal Process. 42, 846–859 (1994).
[CrossRef]

Buttweiler, T. B.

Caelen, J.

H. L. Nguyen Thi, C. Jutten, J. Caelen, “Speech enhancement: analysis and comparison of methods on various real situations,” in Signal Processing VI: Theories and Applications, J. Vandewalle, R. Boite, A. Oosterlick, eds. (Elsevier, New York, 1992), pp. 303–306.

Cao, X. R.

X. R. Cao, R. W. Liu, “General approach to blind source separation,” IEEE Trans. Signal Process. 44, 562–571 (1996).
[CrossRef]

Common, P.

P. Common, “Independent component analysis, a new concept?” Signal Process. 36, 287–314 (1994).
[CrossRef]

Feder, M.

E. Weinstein, A. V. Oppenheim, M. Feder, J. R. Buck, “Iterative and sequential algorithms for multisensor signal enhancement,” IEEE Trans. Signal Process. 42, 846–859 (1994).
[CrossRef]

Hudson, R. D.

R. D. Hudson, “Optical modulation,” in Infrared System Engineering (Wiley, New York, 1969), Chap. 6, pp. 235–263.

Jutten, C.

H. L. Nguyen Thi, C. Jutten, J. Caelen, “Speech enhancement: analysis and comparison of methods on various real situations,” in Signal Processing VI: Theories and Applications, J. Vandewalle, R. Boite, A. Oosterlick, eds. (Elsevier, New York, 1992), pp. 303–306.

Liu, R. W.

X. R. Cao, R. W. Liu, “General approach to blind source separation,” IEEE Trans. Signal Process. 44, 562–571 (1996).
[CrossRef]

McCullagh, P.

P. McCullagh, “Elementary theory of cumulants,” in Tensor Methods in Statistics (Chapman & Hall, London, 1987, 1995), Chap. 2, pp. 24–46.

Mendel, J. M.

J. M. Mendel, “Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications,” Proc. IEEE 79, 278–305 (1991).
[CrossRef]

Nguyen Thi, H. L.

H. L. Nguyen Thi, C. Jutten, J. Caelen, “Speech enhancement: analysis and comparison of methods on various real situations,” in Signal Processing VI: Theories and Applications, J. Vandewalle, R. Boite, A. Oosterlick, eds. (Elsevier, New York, 1992), pp. 303–306.

Nicholson, A. F.

A. F. Nicholson, “Error signals and discrimination in optical trackers that see several sources,” Proc. IEEE 53, 56–71 (1965).
[CrossRef]

Oppenheim, A. V.

E. Weinstein, A. V. Oppenheim, M. Feder, J. R. Buck, “Iterative and sequential algorithms for multisensor signal enhancement,” IEEE Trans. Signal Process. 42, 846–859 (1994).
[CrossRef]

Schilling, D. L.

H. Taub, D. L. Schilling, “Frequency-modulation systems,” in Principles of Communication Systems (McGraw-Hill, New York, 1987), Chap. 4, pp. 142–182.

Sejnowski, T. J.

A. J. Bell, T. J. Sejnowski, “An information-maximization approach to blind separation and blind deconvolution,” Neural Comput. 7, 1129–1159 (1995).
[CrossRef] [PubMed]

Singh, J.

J. Singh, “Optoelectronic detectors,” in Semiconductor Optoelectronics—Physics and Technology (McGraw-Hill, New York, 1995), Chap. 7, pp. 336–398.

Smaragdis, P. J.

P. J. Smaragdis, “Blind separation of convolved mixtures in the frequency domain,” in International Workshop on Independency and Artificial Neural Networks, Tenerife, Spain, 9–10 February, 1998 (University of Laguna, Tenerife, Spain, 1998).

Sorouchyari, E.

E. Sorouchyari, “Blind separation of sources, Part III: Stability analysis,” Signal Process. 24, 21–29 (1991).
[CrossRef]

Taub, H.

H. Taub, D. L. Schilling, “Frequency-modulation systems,” in Principles of Communication Systems (McGraw-Hill, New York, 1987), Chap. 4, pp. 142–182.

Torkkola, K.

K. Torkkola, “Blind separation of convolved sources based on information maximization,” in IEEE Workshop on Neural Networks for Signal Processing, Kyoto, Japan, 4–6 September, 1996 (Institute of Electrical and Electronics Engineers, New York, 1996).

Van Compernolle, D.

S. Van Gerven, D. Van Compernolle, “Signal separation by symmetric adaptive decorrelation: stability, convergence, and uniqueness,” IEEE Trans. Signal Process. 43, 1602–1612 (1995).
[CrossRef]

Van Gerven, S.

S. Van Gerven, D. Van Compernolle, “Signal separation by symmetric adaptive decorrelation: stability, convergence, and uniqueness,” IEEE Trans. Signal Process. 43, 1602–1612 (1995).
[CrossRef]

Wang, J.

J. Wang, H. Zhenya, “Blind identification and separation of convolutively mixed independent sources,” IEEE Trans. Aerospace Electron. Syst. 33, 997–1002 (1997).
[CrossRef]

Weinstein, E.

D. Yellin, E. Weinstein, “Multichannel signal separation: methods and analysis,” IEEE Trans. Signal Process. 44, 106–118 (1996).
[CrossRef]

D. Yellin, E. Weinstein, “Criteria for multichannel signal separation,” IEEE Trans. Signal Process. 42, 2158–2168 (1994).
[CrossRef]

E. Weinstein, A. V. Oppenheim, M. Feder, J. R. Buck, “Iterative and sequential algorithms for multisensor signal enhancement,” IEEE Trans. Signal Process. 42, 846–859 (1994).
[CrossRef]

Yellin, D.

D. Yellin, E. Weinstein, “Multichannel signal separation: methods and analysis,” IEEE Trans. Signal Process. 44, 106–118 (1996).
[CrossRef]

D. Yellin, E. Weinstein, “Criteria for multichannel signal separation,” IEEE Trans. Signal Process. 42, 2158–2168 (1994).
[CrossRef]

Zhenya, H.

J. Wang, H. Zhenya, “Blind identification and separation of convolutively mixed independent sources,” IEEE Trans. Aerospace Electron. Syst. 33, 997–1002 (1997).
[CrossRef]

IEEE Trans. Aerospace Electron. Syst.

J. Wang, H. Zhenya, “Blind identification and separation of convolutively mixed independent sources,” IEEE Trans. Aerospace Electron. Syst. 33, 997–1002 (1997).
[CrossRef]

IEEE Trans. Signal Process.

D. Yellin, E. Weinstein, “Criteria for multichannel signal separation,” IEEE Trans. Signal Process. 42, 2158–2168 (1994).
[CrossRef]

D. Yellin, E. Weinstein, “Multichannel signal separation: methods and analysis,” IEEE Trans. Signal Process. 44, 106–118 (1996).
[CrossRef]

E. Weinstein, A. V. Oppenheim, M. Feder, J. R. Buck, “Iterative and sequential algorithms for multisensor signal enhancement,” IEEE Trans. Signal Process. 42, 846–859 (1994).
[CrossRef]

S. Van Gerven, D. Van Compernolle, “Signal separation by symmetric adaptive decorrelation: stability, convergence, and uniqueness,” IEEE Trans. Signal Process. 43, 1602–1612 (1995).
[CrossRef]

X. R. Cao, R. W. Liu, “General approach to blind source separation,” IEEE Trans. Signal Process. 44, 562–571 (1996).
[CrossRef]

J. Opt. Soc. Am.

Neural Comput.

A. J. Bell, T. J. Sejnowski, “An information-maximization approach to blind separation and blind deconvolution,” Neural Comput. 7, 1129–1159 (1995).
[CrossRef] [PubMed]

Proc. IEEE

A. F. Nicholson, “Error signals and discrimination in optical trackers that see several sources,” Proc. IEEE 53, 56–71 (1965).
[CrossRef]

J. M. Mendel, “Tutorial on higher-order statistics (spectra) in signal processing and system theory: theoretical results and some applications,” Proc. IEEE 79, 278–305 (1991).
[CrossRef]

Proc. Inst. Radio Eng.

G. F. Aroyan, “The technique of spatial filtering,” Proc. Inst. Radio Eng. 47, 1561–1568 (1959).

Signal Process.

P. Common, “Independent component analysis, a new concept?” Signal Process. 36, 287–314 (1994).
[CrossRef]

E. Sorouchyari, “Blind separation of sources, Part III: Stability analysis,” Signal Process. 24, 21–29 (1991).
[CrossRef]

Other

P. J. Smaragdis, “Blind separation of convolved mixtures in the frequency domain,” in International Workshop on Independency and Artificial Neural Networks, Tenerife, Spain, 9–10 February, 1998 (University of Laguna, Tenerife, Spain, 1998).

R. D. Hudson, “Optical modulation,” in Infrared System Engineering (Wiley, New York, 1969), Chap. 6, pp. 235–263.

K. Torkkola, “Blind separation of convolved sources based on information maximization,” in IEEE Workshop on Neural Networks for Signal Processing, Kyoto, Japan, 4–6 September, 1996 (Institute of Electrical and Electronics Engineers, New York, 1996).

H. Taub, D. L. Schilling, “Frequency-modulation systems,” in Principles of Communication Systems (McGraw-Hill, New York, 1987), Chap. 4, pp. 142–182.

J. Singh, “Optoelectronic detectors,” in Semiconductor Optoelectronics—Physics and Technology (McGraw-Hill, New York, 1995), Chap. 7, pp. 336–398.

P. McCullagh, “Elementary theory of cumulants,” in Tensor Methods in Statistics (Chapman & Hall, London, 1987, 1995), Chap. 2, pp. 24–46.

H. L. Nguyen Thi, C. Jutten, J. Caelen, “Speech enhancement: analysis and comparison of methods on various real situations,” in Signal Processing VI: Theories and Applications, J. Vandewalle, R. Boite, A. Oosterlick, eds. (Elsevier, New York, 1992), pp. 303–306.

D. R. Brillinger, “Foundations,” in Time Series Data Analysis and Theory (McGraw-Hill, New York, 1981), Chap. 2, pp. 16–44.

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Figures (21)

Fig. 1
Fig. 1

Optical tracker. BP, bandpass.

Fig. 2
Fig. 2

Modulating disk with fan-bladed pattern.

Fig. 3
Fig. 3

Reticle coordinate system.

Fig. 4
Fig. 4

Frequency-modulated signal generation principle (taken from Ref. 1).

Fig. 5
Fig. 5

Modified optical tracker.

Fig. 6
Fig. 6

Modified optical tracker signal model.

Fig. 7
Fig. 7

Feedback separation network.

Fig. 8
Fig. 8

Relative spectral responsivity of the first optical source.

Fig. 9
Fig. 9

Relative spectral responsivity of the second optical source.

Fig. 10
Fig. 10

Ge photodetector spectral responsivity in relative units.

Fig. 11
Fig. 11

Beam-splitter transmission coefficient.

Fig. 12
Fig. 12

Power spectrum of the first observed signal.

Fig. 13
Fig. 13

Power spectrum of the second observed signal.

Fig. 14
Fig. 14

Power spectrum of the first recovered signal.

Fig. 15
Fig. 15

Power spectrum of the second recovered signal.

Fig. 16
Fig. 16

FM demodulation of the first source signal.

Fig. 17
Fig. 17

FM demodulation of the first recovered signal.

Fig. 18
Fig. 18

Cross-correlation between observed signals.

Fig. 19
Fig. 19

Cross-correlation between recovered signal.

Fig. 20
Fig. 20

Fourth-order cross-cumulant C 22 between observed signals.

Fig. 21
Fig. 21

Fourth-order cross-cumulant C 22 between recovered signals.

Equations (64)

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

Ψ=arctanr sin φ+r0 sin ΩMtr cos φ+r0 cos ΩMt=arctan Δ sin φ+sin ΩMtΔ cos φ+cos ΩMt,
ωt=dΨdt=ΩM1+Δ cos ΩMt cos φ+Δ sin ΩMt sin φ1+Δ2+2Δcos ΩMt cos φ+sin ΩMt sin φ.
ωt=ΩM1-Δ2cos ΩMt cos φ+sin ΩMt sin φ21+Δcos ΩMt cos φ+sin ΩMt sin φ.
ωt=ΩM1-Δ cosΩMt-φ.
ωˆt=ω0-Δωm cosΩMt-φ,
ω0=nΩM,  Δωm=ΔnΩM.
sr, φ, t=cosω0t-β sinΩMt-φ,
Φˆλ, t=Φλ, t×sr, φ, t.
β=Δf0fM=ΔnfMfM=Δn=n rr0=k×r,
iλ=A chλ Φˆλ, tRλ,
it=Ahc λRλΦλ, tλdλsr, φ, t,
xt=gt * it=0t gτit-τdτ.
xt=Ahc 0t gτλRλΦλ, t-τλdλsr, φ, t-τdτ.
xt=Ahc 0tgτλRλΦλ, τλdλsr, φ, t-τdτ,
xt=gˆt * sr, φ, t,
gˆt=gtAhc λRλΦλ, tλdλ.
Φˆλ, t=Φ1λ, ts1r, φ, t+Φ2λ, ts2r, φ, t,
s1r, φ, t=sr1, φ1, t,  s2r, φ, t=sr2, φ2, t,
xt=g1t * s1r, φ, t+g2t * s2r, φ, t,
g1t=AhcgtλRλΦ1λ, tλdλ,  g2t=AhcgtλRλΦ2λ, tλdλ.
e=12β1+β2,
i1t=i11ts1r, φ, t+i12ts2r, φ, t,  i2t=i21ts1r, φ, t+i22ts2r, φ, t,
i11t=A1hc λτλR1λΦ1λ, tλdλ,  i12t=A1hc λτλR1λΦ2λ, tλdλ,  i21t=A2hc λρλR2λΦ1λ, tλdλ,  i22t=A2hc λρλR2λΦ2λ, tλdλ.
x1t=g1t * i1t,  x2t=g2t * i2t.
x1t=g11t * s1r, φ, t+g12t * s2r, φ, t,  x2t=g21t * s1r, φ, t+g22t * s2r, φ, t,
g11t=A1hcg1tλτλR1λΦ1λ, tλdλ,  g12t=A1hcg1tλτλR1λΦ2λ, tλdλ,  g21t=A2hcg2tλρλR2λΦ1λ, tλdλ,  g22t=A2hcg2tλρλR2λΦ2λ, tλdλ.
x=G * s,
Qz=Q11z00Q22z
Qz=0Q12zQ21z0.
fs=i=1n fisi,
κx=C4xC22x,
Xω=Gω×Sω,
det Gω0,  ωω1, ω2,
Gijω=Giω×kij,  i, j1, 2,
G1ωG2ωk11k22-k21k120, ωω1, ω2
k11k22-k21k120,
λτλRλΦ1λλdλ λRλΦ2λλdλλτλRλΦ2λλdλ λRλΦ1λλdλ.
τλconst
SSiSiSiSiω1, ω2, ω30,  ω1, ω2, ω3i=1, 2
SSiSjSkSlω1, ω2, ω3=0,  ω1, ω2, ω3i, j, k, l1, 2 except i=j=k=l,
Sy1y1y1y2ω1, ω2, ω3=0,  ω1, ω2, ω3,  Sy2y2y2y1ω1, ω2, ω3=0,  ω1, ω2, ω3.
cumy1k, y1k+τ1, y1k+τ2, y2k+τ3=0,  τ1, τ2, τ3,cumy2k, y2k+τ1, y2k+τ2, y1k+τ3=0,  τ1, τ2, τ3.
y1k=x1k-i=1M12 w12iy2k-i,  y2k=x2k-i=1M21 w21iy1k-i.
Iz, y=Hz-Hz/y,
maxwijIz, y =maxwijHz.
MIz=δfz, i=1n fizi=- fzlogfzi=1n fizidz,
MIz=-Hz+i=1n Hzi,
Hz=-Elog fz=-- fzlog fzdz,  Hzi=-Elog fzi=-- fzlog fizidz.
Hz=i=1n Hzi-MIz.
fz=fx|J|,
J=detz1x1z1xn··znx1znxn.
Hz=-Eln fz=Eln|J|-Eln fx.
maxwij Hz =maxwijln|J|.
|J|=detzixiij=z1y1z2y2.
Δwijk, m=Hzwijk, m= ln|J|wijk, m=wijk, m lnziyi,  Δwijk, m=ziyi-1wijk, mziyi,
Δwijk, m=2zikyjk-m=2 tanhyikyjk-m.
wijk+1, m=wijk, m+μΔwijk, m=wijk, m+2μ tanhyiyjk-m,
EΔwijk, m=0.
EΔwijk, m=2Etanhyikyjk-m=0,
EΔwijk, m=El=0 c2l+1yi2l+1kyjk-m=l=0 c2l+1Eyi2l+1kyjk-m.
Eyi2l+1kyjk-m=Eyi2l+1kEyjk-m,
Eyi2l+1kyjk-m=0,
cx1x2τ=Ex1tx2t+τ,
cxixjxkxlτ1, τ2, τ3=Exitxjt+τ1xkt+τ2xlt+τ3-Exitxjt+τ1Exkt+τ3×xlt+τ2-Exitxkt+τ2×Exlt+τ1xjt+τ3-Exitxlt+τ3Exjt+τ2×xkt+τ1,

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