Abstract

In this paper, we propose photon counting three-dimensional (3D) passive sensing and object recognition using integral imaging. The application of this approach to 3D automatic target recognition (ATR) is investigated using both linear and nonlinear matched filters. We find there is significant potential of the proposed system for 3D sensing and recognition with a low number of photons. The discrimination capability of the proposed system is quantified in terms of discrimination ratio, Fisher ratio, and receiver operating characteristic (ROC) curves. To the best of our knowledge, this is the first report on photon counting 3D passive sensing and ATR with integral imaging.

© 2005 Optical Society of America

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  1. F. Sadjadi, �??Improved target classification using optimum polarimetric SAR signatures,�?? IEEE Trans. On Aerosp. Electron. Syst. 38, 38-49 (2002).
    [CrossRef]
  2. A. Mahalanobis, R. R. Muise, S. R. Stanfill, and A. V. Nevel, �??Design and application of quadratic correlation filters for target detection,�?? IEEE Trans. on Aerosp. Electron. Syst. 40, 837-850 (2004).
    [CrossRef]
  3. H. Sjoberg, F. Goudail, and P. Refregier, �??Optimal algorithms for target location in nonhomogeneous binary images,�?? J. Opt. Soc. Am. A. 15, 2976-2985 (1998).
    [CrossRef]
  4. H. Kwon and N. M. Nasrabadi, �??Kernel RX-algorithm: a nonlinear anomaly detector for hyperspectral imagery,�?? IEEE Trans. on Geosci. Remote Sens. 43, 388-397 (2005).
    [CrossRef]
  5. F. Sadjadi, ed., Selected Papers on Automatic Target Recognition, (SPIE-CDROM, 1999).
  6. B. Javidi, ed., Image Recognition and Classification: Algorithms, Systems, and Applications, (Marcel Dekker, New York, 2002).
    [CrossRef]
  7. B. Javidi and F. Okano, eds., Three-dimensional television, video, and display technologies, (Springer, New York, 2002).
  8. R. O. Duda, P. E. Hart, and D. G. Stork, Pattern classification 2nd, (Wiley Interscience, New York, 2001).
  9. C. M. Bishop, Neural networks for pattern recognition, (Oxford University Press, New York, 1995).
  10. E. Hecht, Optics 4th Edition, (Addison Wesley, 2001).
  11. G. M. Morris, �??Scene matching using photon-limited images,�?? J. Opt. Soc. Am. A. 1, 482-488 (1984).
    [CrossRef]
  12. G. M. Morris, �??Image correlation at low light levels: a computer simulation,�?? Appl. Opt. 23, 3152-3159 (1984).
    [CrossRef] [PubMed]
  13. . E. A. Watson and G. M. Morris, �??Comparison of infrared upconversion methods for photon-limited imaging,�?? J. Appl. Phys. 67, 6075-6084 (1990).
    [CrossRef]
  14. E. A. Watson and G. M. Morris, �??Imaging thermal objects with photon-counting detector,�?? Appl. Opt. 31, 4751-4757 (1992).
    [CrossRef] [PubMed]
  15. D. Stucki, G. Ribordy, A. Stefanov, H. Zbinden, J. G. Rarity, and T. Wall, �??Photon counting for quantum key distribution with Peltier cooled InGaAs/InP APDs,�?? J. Mod. Opt. 48, 1967-1981 (2001).
    [CrossRef]
  16. P. A. Hiskett, G. S. Buller, A. Y. Loudon, J. M. Smith, I Gontijo, A. C. Walker, P. D. Townsend, and M. J. Robertson, �??Performance and design of InGaAs/InP photodiodes for single-photon counting at 1.55 um,�?? Appl. Opt. 39, 6818-6829 (2000).
    [CrossRef]
  17. L. Duraffourg, J.-M. Merolla, J.-P. Goedgebuer, N. Butterlin, and W. Rhods, �??Photon Counting in the 1540-nm Wavelength Region with a Germanium Avalanche photodiode,�?? IEEE J. Quantum Electron. 37, 75-79 (2001).
    [CrossRef]
  18. J. G. Rarity, T. E. Wall, K. D. Ridley, P. C. M. Owens, and P. R. Tapster, �??Single-photon counting for the 1300-1600-nm range by use of Peltier-cooled and passively quenched InGaAs avalanche photodiodes,�?? Appl. Opt. 39, 6746-6753 (2000).
    [CrossRef]
  19. M. G. Lippmann, �??Epreuves reversibles donnant la sensation du relief,�?? J. Phys. (Paris) 7, 821-825 (1908).
  20. H. E. Ives, �??Optical properties of a Lippmann lenticulated sheet,�?? J. Opt. Soc. Am. 21, 171-176 (1931).
    [CrossRef]
  21. T. Okoshi, �??Three-dimensional displays,�?? Proceedings of the IEEE 68, 548-564 (1980).
    [CrossRef]
  22. J.-S. Jang and B. Javidi, �??Time-multiplexed integral imaging for 3D sensing and display,�?? Optics and Photonics News 15, 36-43 (2004), <a href="http://www.osa-opn.org/abstract.cfm?URI=OPN-15-4-36">http://www.osa-opn.org/abstract.cfm?URI=OPN-15-4-36</a>
  23. O. Matoba, E. Tajahuerce, and B. Javidi, �??Real-time three-dimensional object recognition with multiple perspectives imaging,�?? Appl. Opt. 40, 3318-3325 (2001).
    [CrossRef]
  24. Y. Frauel and B. Javidi, �??Digital three-dimensional image correlation by use of computer-reconstructed integral imaging,�?? Appl. Opt. 41, 5488-5496 (2002).
    [CrossRef] [PubMed]
  25. S. Kishk and B. Javidi, �??Improved resolution 3D object sensing and recognition using time multiplexed computational integral imaging,�?? Opt. Express 11, 3528-3541 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3528">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3528</a>.
    [CrossRef] [PubMed]
  26. S. Yeom and B. Javidi, �??Three-dimensional distortion tolerant object recognition using integral imaging,�?? Opt. Express 12, 5795-5809 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-23-5795">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-23-5795</a>
    [CrossRef] [PubMed]
  27. M. Martínez-Corral, B. Javidi, R. Martínez-Cuenca, and G. Saavedra, �??Multifacet structure of observed reconstructed integral images,�?? J. Opt. Soc. Am. A. 22, 597-603 (2005).
    [CrossRef]
  28. H. Arimoto and B. Javidi, �??Integrate three-dimensional imaging with computed reconstruction,�?? Opt. Lett. 26, 157-159 (2001).
    [CrossRef]
  29. A. Stern and B. Javidi, �??Shannon number and information capacity of integral imaging,�?? J. Opt. Soc. Am. A. 21, 1602-1612 (2004).
    [CrossRef]
  30. J. W. Goodman, Statistical optics, (John Wiley & Sons, inc., 1985), Chap 9.
  31. B. Javidi, �??Nonlinear joint power spectrum based optical correlation,�?? Appl. Opt. 28, 2358-2367 (1989).
    [CrossRef] [PubMed]
  32. F. Sadjadi, ed., Milestones in performance evaluations of signal and image processing systems, (SPIE Press, 1993).
  33. A. Papoulis, Probability, random variables, and stochastic processes 3rd, (McGraw-Hill, Inc. 1991).

Appl. Opt. (7)

IEEE J. Quantum Electron. (1)

L. Duraffourg, J.-M. Merolla, J.-P. Goedgebuer, N. Butterlin, and W. Rhods, �??Photon Counting in the 1540-nm Wavelength Region with a Germanium Avalanche photodiode,�?? IEEE J. Quantum Electron. 37, 75-79 (2001).
[CrossRef]

IEEE Trans. On Aerosp. Electron. Syst. (1)

F. Sadjadi, �??Improved target classification using optimum polarimetric SAR signatures,�?? IEEE Trans. On Aerosp. Electron. Syst. 38, 38-49 (2002).
[CrossRef]

A. Mahalanobis, R. R. Muise, S. R. Stanfill, and A. V. Nevel, �??Design and application of quadratic correlation filters for target detection,�?? IEEE Trans. on Aerosp. Electron. Syst. 40, 837-850 (2004).
[CrossRef]

IEEE Trans. on Geosci. Remote Sens. (1)

H. Kwon and N. M. Nasrabadi, �??Kernel RX-algorithm: a nonlinear anomaly detector for hyperspectral imagery,�?? IEEE Trans. on Geosci. Remote Sens. 43, 388-397 (2005).
[CrossRef]

J. Appl. Phys. (1)

. E. A. Watson and G. M. Morris, �??Comparison of infrared upconversion methods for photon-limited imaging,�?? J. Appl. Phys. 67, 6075-6084 (1990).
[CrossRef]

J. Mod. Opt. (1)

D. Stucki, G. Ribordy, A. Stefanov, H. Zbinden, J. G. Rarity, and T. Wall, �??Photon counting for quantum key distribution with Peltier cooled InGaAs/InP APDs,�?? J. Mod. Opt. 48, 1967-1981 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A. (4)

G. M. Morris, �??Scene matching using photon-limited images,�?? J. Opt. Soc. Am. A. 1, 482-488 (1984).
[CrossRef]

H. Sjoberg, F. Goudail, and P. Refregier, �??Optimal algorithms for target location in nonhomogeneous binary images,�?? J. Opt. Soc. Am. A. 15, 2976-2985 (1998).
[CrossRef]

M. Martínez-Corral, B. Javidi, R. Martínez-Cuenca, and G. Saavedra, �??Multifacet structure of observed reconstructed integral images,�?? J. Opt. Soc. Am. A. 22, 597-603 (2005).
[CrossRef]

A. Stern and B. Javidi, �??Shannon number and information capacity of integral imaging,�?? J. Opt. Soc. Am. A. 21, 1602-1612 (2004).
[CrossRef]

J. Phys. (Paris) (1)

M. G. Lippmann, �??Epreuves reversibles donnant la sensation du relief,�?? J. Phys. (Paris) 7, 821-825 (1908).

Opt. Express (1)

Opt. Lett. (1)

Optics and Photonics News (1)

J.-S. Jang and B. Javidi, �??Time-multiplexed integral imaging for 3D sensing and display,�?? Optics and Photonics News 15, 36-43 (2004), <a href="http://www.osa-opn.org/abstract.cfm?URI=OPN-15-4-36">http://www.osa-opn.org/abstract.cfm?URI=OPN-15-4-36</a>

Proceedings of the IEEE, 1980 (1)

T. Okoshi, �??Three-dimensional displays,�?? Proceedings of the IEEE 68, 548-564 (1980).
[CrossRef]

Other (10)

S. Kishk and B. Javidi, �??Improved resolution 3D object sensing and recognition using time multiplexed computational integral imaging,�?? Opt. Express 11, 3528-3541 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3528">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3528</a>.
[CrossRef] [PubMed]

J. W. Goodman, Statistical optics, (John Wiley & Sons, inc., 1985), Chap 9.

F. Sadjadi, ed., Milestones in performance evaluations of signal and image processing systems, (SPIE Press, 1993).

A. Papoulis, Probability, random variables, and stochastic processes 3rd, (McGraw-Hill, Inc. 1991).

F. Sadjadi, ed., Selected Papers on Automatic Target Recognition, (SPIE-CDROM, 1999).

B. Javidi, ed., Image Recognition and Classification: Algorithms, Systems, and Applications, (Marcel Dekker, New York, 2002).
[CrossRef]

B. Javidi and F. Okano, eds., Three-dimensional television, video, and display technologies, (Springer, New York, 2002).

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern classification 2nd, (Wiley Interscience, New York, 2001).

C. M. Bishop, Neural networks for pattern recognition, (Oxford University Press, New York, 1995).

E. Hecht, Optics 4th Edition, (Addison Wesley, 2001).

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Figures (12)

Fig. 1.
Fig. 1.

Schematic diagram of photon counting integral imaging system.

Fig. 2.
Fig. 2.

(a) Experimental set-up for the integral imaging, (b) three toy cars used in the experiments; car 1, 2 and 3 are shown from right to left.

Fig. 3.
Fig. 3.

Three sets of elemental images for intensity information, (a) car 1, (b) car 2, (c) car 3.

Fig. 4.
Fig. 4.

Probability map that no photon occurs for the car 1 [Fig. 3(a)] when NP = 1,000.

Fig. 5.
Fig. 5.

Photon-limited image when NP =1,000 of (a) car 1 corresponding to Fig. 3(a), (b) car 2 corresponding to Fig. 3(b), (c) car 3 corresponding to Fig. 3(c).

Fig. 6.
Fig. 6.

(a) Magnified elemental images of car 1, magnified photon-limited elemental images corresponding to Fig. 6(a) when (b) Np = 10,000, (c) Np = 1,000.

Fig. 7.
Fig. 7.

Mean and variance of Crs (0;0); (a) sample mean and theoretical prediction, (b) sample variance and theoretical prediction.

Fig. 8.
Fig. 8.

Mean and variance of Crs (0;0.5); (a) sample mean, (b) sample variance.

Fig. 9.
Fig. 9.

Mean and variance of Crs (0;1); (a) sample mean and theoretical prediction, (b) sample variance and theoretical prediction.

Fig. 10.
Fig. 10.

ROC curve of Crs (0;0); (a) reference is car 1 and false object is car 2, (b) reference is car 1 and false object is car 3.

Fig. 11.
Fig. 11.

ROC curve of Crs (0;0.5); (a) reference is car 1 and false object is car 2, (b) reference is car 1 and false object is car 3.

Fig. 12.
Fig. 12.

ROC curve of Crs (0;1); (a) reference is car 1 and false object is car 2, (b) reference is car 1 and false object is car 3.

Tables (2)

Tables Icon

Table 1. Discimination ratios.

Tables Icon

Table 2. Fisher ratios.

Equations (59)

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P d ( k ; x , τ ) = [ a ( x ) τ ] k e a ( x ) τ k ! , k = 0,1,2 , ,
a = ηE h ν ¯ ,
n p ( x ) k ( x ) = a ( x ) τ = ηE ( x ) h ν ¯ τ ,
n p ( x i ) = N P S ( x i ) j = 1 N r S ( x j ) ,
P ( 0 ; x i ) = e n p ( x i ) .
P ( 1 ; x i ) = 1 P ( 0 ; x i ) = 1 e n p ( x i ) .
S ̂ ( x i ) = { 0 , If rand ( x i ) P ( 0 ; x i ) 1 , otherwise ,
S ̂ ( x i ) = b i , i = 1 , , N T ,
P ( b i = 1 ) = 1 e n p ( x i ) n p ( x i ) ,
P ( b i = 1 ) = e n p ( x i ) 1 n p ( x i ) ,
S ̂ ( x i ) n p ( x i ) ,
i = 1 N T S ̂ ( x i ) i = 1 N T n p ( x i ) = N p .
S ̂ ( x i ) = k = 1 N δ ( x i x k ) , i = 1 , , N T ,
C rs ( x j ; v ) = i = 1 N T R ( x i + x j ) S ̂ ( x i ) ( i = 1 N T R 2 ( x i ) ) 1 2 ( i = 1 N T S ̂ ( x i ) ) v = k = 1 N R ( x k + x j ) A ( i = 1 N T S ̂ ( x i ) ) v ,
A = ( i = 1 N T R 2 ( x i ) ) 1 2 ,
i = 1 N T R ( x i ) = i = 1 N T S ( x i ) = 1 .
C rs ( 0 ; v ) = max x j C rs ( x j , v ) .
C rs ( 0 ; 0 ) N p A i = 1 N T R ( x i ) S ( x i ) ,
var ( C rs ( 0 ; 0 ) ) N P A 2 i = 1 N T { R 2 ( x i ) S ( x i ) [ 1 N P S ( x i ) ] } ,
C rs ( 0 ; 0 ) 1 A i = 1 N T R ( x i ) S ( x i ) ,
var ( C rs ( 0 ; 1 ) ) 1 A 2 ( 1 N p i = 1 N T R 2 ( x i ) S ( x i ) i = 1 N T R 2 ( x i ) S 2 ( x i ) ) .
DR ( r , s ) m rr m rs ,
FR ( r , s ) [ m rr m rs ] 2 σ rr 2 + σ rs 2 ,
C rs ( 0 ; 0 ) = 1 A i = 1 N T R ( x i ) S ̂ ( x i ) 1 A i = 1 N T R ( x i ) n p ( x i )
= N P A i = 1 N T R ( x i ) S ( x i ) j = 1 N T S ( x j ) = N P A i = 1 N T R ( x i ) S ( x i ) ,
var ( C rs ( 0 ; 0 ) ) = 1 A 2 i = 1 N T R 2 ( x i ) var ( S ̂ ( x i ) ) 1 A 2 i = 1 N T { R 2 ( x i ) [ n p ( x i ) n p 2 ( x i ) ] }
= N P A 2 i = 1 N T { R 2 ( x i ) S ( x i ) j = 1 N T S ( x j ) [ 1 N P S ( x i ) j = 1 N T S ( x i ) ] }
= N P A 2 i = 1 N T { R 2 ( x i ) S ( x i ) [ 1 N P S ( x i ) ] } ,
C rs ( 0 ; 1 ) = i = 1 N T R ( x i ) S ̂ ( x i ) A i = 1 N T S ̂ ( x i ) = 1 A i = 1 N T R ( x i ) b i N = 1 A i = 1 N T R ( x i ) b i b i + N i ,
N i = N b i .
Φ N i ( s ) = e s N i = j = 1 , j i N T ( p j e s + q j ) j = 1 , j i N T e p j ( e s 1 ) = e α i ( e s 1 ) ,
P j = P ( b j = 1 ) = 1 e n p ( x j ) n p ( x j ) = N p S ( x j ) ,
q j = P ( b j = 0 ) = e n p ( x j ) 1 n p ( x j ) = 1 N p S ( x j ) ,
α i = j = 1 , j i N T p j N p ( 1 S ( x i ) ) ,
P ( N i ) α i N i e α i N i ! , N i = 0,1,2 ,
y i = b i b i + N i .
Φ y i ( s ) = e s y i = N i = 0 ( p i e s 1 + N i + q i ) P ( N i ) .
y i = Φ y i ( 0 ) = N i = 0 p i e s 1 + N i P ( N i ) 1 + N i s = 0 = N i = 0 p i P ( N i ) 1 + N i
N i = 0 p i e α i ( α i ) N i ( 1 + N i ) N i ! = p i α i ( 1 e α i ) N p S ( x i ) N p ( 1 S ( x i ) ) ( 1 e α i ) S ( x i ) .
S ( x i ) 1 S ( x i ) S ( x i ) ,
1 e α i 1 ,
C rs ( 0 ; 1 ) = 1 A i = 1 N T R ( x i ) y i 1 A i = 1 N T R ( x i ) S ( x i ) .
var ( y i ) = y i 2 y i 2 ,
cov ( y i , y j ) = y i y j y i y i .
y i 2 = Φ y i ( 0 ) = N i = 0 p i e s 1 + N i P ( N i ) ( 1 + N i ) 2 s = 0 = N i = 0 p i P ( N i ) ( 1 + N i )
N i = 0 p i e α i ( α i ) N i ( 1 + N i ) 2 = N i = 0 p i e α i ( α i ) N i ( 2 + N i ) ( 2 + N i ) ! ( 1 + N i ) p i α i 2 ( 1 e α i α i e α i )
N p S ( x i ) N P 2 ( 1 S ( x i ) ) 2 ( 1 e α i α i e α i ) S ( x i ) N p ,
y i = b i b i + b j + N ij ,
N ij = N b i b j .
P ( N ij ) α ij N ji e α ji N ij ! , N ij = 0,1,2 , ,
α ij = k = 1 , k i , k j N T p k = N p ( 1 S ( x i ) S ( x j ) ) .
Φ y i y j ( s i , s 2 ) = e s 1 y i + s 2 y j = N ij = 0 ( p i p j e s 1 2 + N ji + s 2 2 + N ij + p i q j e s 1 1 + N ij + q i p j e s 2 1 + N ij + q i q j ) P ( N ij ) .
y i y j = 2 Φ y i y j ( s 1 , s 2 ) s 1 s 2 s 1 = s 2 = 0 = N ij = 0 p i p j ( 2 + N ij ) 2 P ( N ij )
N ij = 0 p i p j e α ij ( α ij ) N ij ( 2 + N ij ) 2 N ij ! = N ij = 0 p i p j e α ij ( α ij ) N ij ( 1 + N ij ) ( 2 + N ij ) ! ( 2 + N ij )
p i p j ( α ij ) 2 ( 1 e α ij α ij e α ij ) N P 2 S ( x i ) S ( x j ) N P 2 ( 1 S ( x i ) S ( x j ) ) 2 S ( x i ) S ( x j ) .
cov ( y i , y j ) = y i y j y i y i 0
var ( C rs ( 0 ; 1 ) ) = 1 A 2 [ i = 1 N T R 2 ( x i ) var ( y i ) + i = 1 N T j = 1 , i j N T R ( x i ) R ( x j ) cov ( y i , y j ) ]
1 A 2 i = 1 N T R 2 ( x i ) [ y i 2 y i 2 ]
1 A 2 ( 1 N p i = 1 N T R 2 ( x i ) S ( x i ) i = 1 N T R 2 ( x i ) S 2 ( x i ) ) .

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