Abstract

Photon-counting integral imaging has been introduced recently, and its applications in three-dimensional (3D) object sensing, visualization, recognition, and classification under photon-starved conditions have been demonstrated. This paper sheds light on the underlying information-theoretic foundation behind the ability of photon-counting integral imaging in performing complex tasks with far fewer photons than conventional imaging systems. A metric for photon-information content is formulated in the context of 3D photon-counting imaging, and its properties are investigated. It is shown that there is an inherent trade-off between imaging fidelity, measured by the entropy-normalized mutual information associated with a given imaging system, and the amount of information in each photon used in the imaging process, as represented by the photon-number–normalized mutual information. The dependence of this trade-off on photon statistics, correlation in the 3D image, and the signal-to-noise ratio of the photon-detection system is also investigated.

© 2012 Optical Society of America

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References

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  2. C. B. Burckhardt, “Optimum parameters and resolution limitation of integral photography,” J. Opt. Soc. Am. A 58, 71–76 (1968).
    [CrossRef]
  3. T. Okoshi, “Three-dimensional displays,” Proc. IEEE 68, 548–564 (1980).
    [CrossRef]
  4. H. Hoshino, F. Okano, H. Isono, and I. Yuyama, “Analysis of resolution limitation of integral photography,” J. Opt. Soc. Am. A 15, 2059–2065 (1998).
    [CrossRef]
  5. A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE 94, 591–607 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. B. Javidi, F. Okano, and J. Y. Son, eds., Three Dimensional Imaging, Visualization, and Display” (Springer, 2008).
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    [CrossRef]
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2011

M. Cho, M. Daneshpanah, I. Moon, and B. Javidi, “Three-dimensional optical sensing and visualization using integral imaging,” Proc. IEEE 99, 556–575 (2011).
[CrossRef]

2010

2009

I. Moon and B. Javidi, “Three-dimensional recognition of photon starved events using computational integral imaging and statistical sampling,” Opt. Lett. 34, 731–733 (2009).
[CrossRef]

M. C. Forman, N. Davies, and M. McCormick, “Continuous parallax in discrete pixelated integral three-dimensional displays,” Proc. IEEE 97, 1067–1077 (2009).
[CrossRef]

A. Mahalanobis and R. Muise, “Object specific image reconstruction using a compressive sensing architecture for application in surveillance,” IEEE Trans. Aerospace Electron. Syst. 45, 1167–1180 (2009).
[CrossRef]

2008

2007

2006

B. Javidi, S.-H. Hong, and O. Matoba, “Multi dimensional optical sensors and imaging systems,” Appl. Opt. 45, 2986–2994 (2006).
[CrossRef]

F. Okano, J. Arai, K. Mitani, and M. Okui, “Real-time integral imaging based on extremely high resolution video system,” Proc. IEEE 94, 490–501 (2006).
[CrossRef]

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE 94, 591–607 (2006).
[CrossRef]

2005

2003

2002

2001

H. Arimoto and B. Javidi, “Integrated three-dimensional imaging with computed reconstruction,” Opt. Lett. 26, 157–159 (2001).
[CrossRef]

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]

1998

1984

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-6, 721–741 (1984).
[CrossRef]

1980

T. Okoshi, “Three-dimensional displays,” Proc. IEEE 68, 548–564 (1980).
[CrossRef]

1979

1968

C. B. Burckhardt, “Optimum parameters and resolution limitation of integral photography,” J. Opt. Soc. Am. A 58, 71–76 (1968).
[CrossRef]

1931

H. E. Ives, “Optical properties of a lippmann lenticulated sheet,” J. Opt. Soc. Am. A 21, 171–176 (1931).
[CrossRef]

1908

G. Lippmann, “Epreuves reversibles donnant la sensation du relief,” J. Phys. (Paris) 7, 821–825 (1908).

Arai, J.

F. Okano, J. Arai, K. Mitani, and M. Okui, “Real-time integral imaging based on extremely high resolution video system,” Proc. IEEE 94, 490–501 (2006).
[CrossRef]

Arimoto, H.

Berthod, M.

E. Volden, G. Giraudon, and M. Berthod, “Information in Markov random fields and image redundancy,” in Selected Papers from the 4th Canadian Workshop on Information Theory and Applications II (Springer-Verlag, 1996), pp. 250–268.

Burckhardt, C. B.

C. B. Burckhardt, “Optimum parameters and resolution limitation of integral photography,” J. Opt. Soc. Am. A 58, 71–76 (1968).
[CrossRef]

Cho, M.

M. Cho, M. Daneshpanah, I. Moon, and B. Javidi, “Three-dimensional optical sensing and visualization using integral imaging,” Proc. IEEE 99, 556–575 (2011).
[CrossRef]

D. Dey, J. Jung, M. Cho, and B. Javidi, “Three-dimensional photon counting integral imaging using Bayesian estimation,” Opt. Lett. 35, 1825–1827 (2010).
[CrossRef]

Cover, T. M.

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

Daneshpanah, M.

M. Cho, M. Daneshpanah, I. Moon, and B. Javidi, “Three-dimensional optical sensing and visualization using integral imaging,” Proc. IEEE 99, 556–575 (2011).
[CrossRef]

Davies, N.

M. C. Forman, N. Davies, and M. McCormick, “Continuous parallax in discrete pixelated integral three-dimensional displays,” Proc. IEEE 97, 1067–1077 (2009).
[CrossRef]

Dey, D.

Forman, M. C.

M. C. Forman, N. Davies, and M. McCormick, “Continuous parallax in discrete pixelated integral three-dimensional displays,” Proc. IEEE 97, 1067–1077 (2009).
[CrossRef]

Geman, D.

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-6, 721–741 (1984).
[CrossRef]

Geman, S.

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-6, 721–741 (1984).
[CrossRef]

Giraudon, G.

E. Volden, G. Giraudon, and M. Berthod, “Information in Markov random fields and image redundancy,” in Selected Papers from the 4th Canadian Workshop on Information Theory and Applications II (Springer-Verlag, 1996), pp. 250–268.

Guillaume, M.

Hayat, M. M.

Hong, S.-H.

Hoshino, H.

Isono, H.

Ives, H. E.

H. E. Ives, “Optical properties of a lippmann lenticulated sheet,” J. Opt. Soc. Am. A 21, 171–176 (1931).
[CrossRef]

Jang, J. S.

Javidi, B.

M. Cho, M. Daneshpanah, I. Moon, and B. Javidi, “Three-dimensional optical sensing and visualization using integral imaging,” Proc. IEEE 99, 556–575 (2011).
[CrossRef]

D. Dey, J. Jung, M. Cho, and B. Javidi, “Three-dimensional photon counting integral imaging using Bayesian estimation,” Opt. Lett. 35, 1825–1827 (2010).
[CrossRef]

S. R. Narravula, M. M. Hayat, and B. Javidi, “Information theoretic approach for assessing image fidelity in photon-counting arrays,” Opt. Express 18, 2449–2466 (2010).
[CrossRef]

I. Moon and B. Javidi, “Three-dimensional recognition of photon starved events using computational integral imaging and statistical sampling,” Opt. Lett. 34, 731–733 (2009).
[CrossRef]

B. Tavakoli, B. Javidi, and E. Watson, “Three-dimensional visualization by photon counting computational integral imaging,” Opt. Express 16, 4426–4436 (2008).
[CrossRef]

S. Yeom, B. Javidi, C. Lee, and E. A. Watson, “Photon-counting passive 3D image sensing for reconstruction and recognition of occluded objects,” Opt. Express 15, 16189–16195(2007).
[CrossRef]

S. Yeom, B. Javidi, and E. Watson, “Three-dimensional distortion-tolerant object recognition using photon-counting integral imaging,” Opt. Express 15, 1513–1533 (2007).
[CrossRef]

B. Javidi, S.-H. Hong, and O. Matoba, “Multi dimensional optical sensors and imaging systems,” Appl. Opt. 45, 2986–2994 (2006).
[CrossRef]

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE 94, 591–607 (2006).
[CrossRef]

S. Yeom, B. Javidi, and E. Watson, “Photon counting passive 3D image sensing for automatic target recognition,” Opt. Express 13, 9310–9330 (2005).
[CrossRef]

R. Martinez-Cuenca, G. Saavedra, M. Martinez-Corral, and B. Javidi, “Progress in 3-d multiperspective display by integral imaging,” J. Opt. Soc. Am. A 20, 411–420 (2003).
[CrossRef]

J. S. Jang and B. Javidi, “3D synthetic aperture integral imaging,” Opt. Lett. 27, 1144–1146 (2002).
[CrossRef]

H. Arimoto and B. Javidi, “Integrated three-dimensional imaging with computed reconstruction,” Opt. Lett. 26, 157–159 (2001).
[CrossRef]

Jung, J.

Lee, C.

Lippmann, G.

G. Lippmann, “Epreuves reversibles donnant la sensation du relief,” J. Phys. (Paris) 7, 821–825 (1908).

Mahalanobis, A.

A. Mahalanobis and R. Muise, “Object specific image reconstruction using a compressive sensing architecture for application in surveillance,” IEEE Trans. Aerospace Electron. Syst. 45, 1167–1180 (2009).
[CrossRef]

Mandel, L.

Martinez-Corral, M.

Martinez-Cuenca, R.

Matoba, O.

McCormick, M.

M. C. Forman, N. Davies, and M. McCormick, “Continuous parallax in discrete pixelated integral three-dimensional displays,” Proc. IEEE 97, 1067–1077 (2009).
[CrossRef]

Melon, P.

Mitani, K.

F. Okano, J. Arai, K. Mitani, and M. Okui, “Real-time integral imaging based on extremely high resolution video system,” Proc. IEEE 94, 490–501 (2006).
[CrossRef]

Moon, I.

M. Cho, M. Daneshpanah, I. Moon, and B. Javidi, “Three-dimensional optical sensing and visualization using integral imaging,” Proc. IEEE 99, 556–575 (2011).
[CrossRef]

I. Moon and B. Javidi, “Three-dimensional recognition of photon starved events using computational integral imaging and statistical sampling,” Opt. Lett. 34, 731–733 (2009).
[CrossRef]

Muise, R.

A. Mahalanobis and R. Muise, “Object specific image reconstruction using a compressive sensing architecture for application in surveillance,” IEEE Trans. Aerospace Electron. Syst. 45, 1167–1180 (2009).
[CrossRef]

Narravula, S. R.

Okano, F.

F. Okano, J. Arai, K. Mitani, and M. Okui, “Real-time integral imaging based on extremely high resolution video system,” Proc. IEEE 94, 490–501 (2006).
[CrossRef]

H. Hoshino, F. Okano, H. Isono, and I. Yuyama, “Analysis of resolution limitation of integral photography,” J. Opt. Soc. Am. A 15, 2059–2065 (1998).
[CrossRef]

Okoshi, T.

T. Okoshi, “Three-dimensional displays,” Proc. IEEE 68, 548–564 (1980).
[CrossRef]

Okui, M.

F. Okano, J. Arai, K. Mitani, and M. Okui, “Real-time integral imaging based on extremely high resolution video system,” Proc. IEEE 94, 490–501 (2006).
[CrossRef]

Rarity, J. G.

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]

Refregier, P.

Ribordy, G.

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]

Saavedra, G.

Sadjadi, F.

F. Sadjadi, Selected papers on automatic target recognition, SPIE-CDROM (1999).

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 2007).

Stefanov, A.

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]

Stern, A.

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE 94, 591–607 (2006).
[CrossRef]

Stucki, D.

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]

Tavakoli, B.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 2007).

Thomas, J. A.

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

Volden, E.

E. Volden, G. Giraudon, and M. Berthod, “Information in Markov random fields and image redundancy,” in Selected Papers from the 4th Canadian Workshop on Information Theory and Applications II (Springer-Verlag, 1996), pp. 250–268.

Wall, T.

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]

Watson, E.

Watson, E. A.

Yeom, S.

Yuyama, I.

Zbinden, H.

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]

Appl. Opt.

IEEE Trans. Aerospace Electron. Syst.

A. Mahalanobis and R. Muise, “Object specific image reconstruction using a compressive sensing architecture for application in surveillance,” IEEE Trans. Aerospace Electron. Syst. 45, 1167–1180 (2009).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell.

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-6, 721–741 (1984).
[CrossRef]

J. Mod. Opt.

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. A

J. Phys. (Paris)

G. Lippmann, “Epreuves reversibles donnant la sensation du relief,” J. Phys. (Paris) 7, 821–825 (1908).

Opt. Express

Opt. Lett.

Proc. IEEE

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE 94, 591–607 (2006).
[CrossRef]

M. Cho, M. Daneshpanah, I. Moon, and B. Javidi, “Three-dimensional optical sensing and visualization using integral imaging,” Proc. IEEE 99, 556–575 (2011).
[CrossRef]

T. Okoshi, “Three-dimensional displays,” Proc. IEEE 68, 548–564 (1980).
[CrossRef]

M. C. Forman, N. Davies, and M. McCormick, “Continuous parallax in discrete pixelated integral three-dimensional displays,” Proc. IEEE 97, 1067–1077 (2009).
[CrossRef]

F. Okano, J. Arai, K. Mitani, and M. Okui, “Real-time integral imaging based on extremely high resolution video system,” Proc. IEEE 94, 490–501 (2006).
[CrossRef]

Other

B. Javidi, F. Okano, and J. Y. Son, eds., Three Dimensional Imaging, Visualization, and Display” (Springer, 2008).

F. Sadjadi, Selected papers on automatic target recognition, SPIE-CDROM (1999).

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

E. Volden, G. Giraudon, and M. Berthod, “Information in Markov random fields and image redundancy,” in Selected Papers from the 4th Canadian Workshop on Information Theory and Applications II (Springer-Verlag, 1996), pp. 250–268.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics(Wiley, 2007).

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Figures (14)

Fig. 1.
Fig. 1.

3D integral imaging. (a) Pickup (image capture) stage (passive sensing), (b) computational image reconstruction.

Fig. 2.
Fig. 2.

Information-theoretic representation of a photon-counting imaging system. Photons from the input scene pass through the medium, resulting in a photon-starved stochastic photon stream in time and space. Photons are detected by a photon-counting array yielding output image with M gray levels. At extremely low photon numbers, as is typically the case with the PCII approach, the output image is a very sparse, binary image.

Fig. 3.
Fig. 3.

Binary asymmetric channel representing the transformation of a gray level to a maximum-likelihood decision based upon photon numbers in quantum and dark/readout noise.

Fig. 4.
Fig. 4.

Photon-information-content metric, Ip (in bpp), versus average number of photons for various levels of quantum efficiency and dark/readout noise.

Fig. 5.
Fig. 5.

Fidelity metric, ρ, versus average number of photons for various levels of quantum efficiency and dark/readout noise.

Fig. 6.
Fig. 6.

2D slice of the a 3D Ising MRF realization exhibiting strong correlation. The correlation structure is symmetric in x, y, and z directions. The 3D MRF described in the text was used to generate the 3D images assuming βc=0.6.

Fig. 7.
Fig. 7.

Fidelity metric as a function of the average number of photons per pixel for three scenarios of correlation: no correlation (case 1), only 2D correlation in the xy plane (case 2), and full 3D correlation (case 3).

Fig. 8.
Fig. 8.

Photon-information-content metric as a function of the average number of photons per pixel for three scenarios of correlation: no correlation (case 1), only 2D correlation in the xy plane (case 2), and full 3D correlation (case 3).

Fig. 9.
Fig. 9.

Fidelity metric as a function of the average number of photons per pixel for three different photon statistics: Poisson, geometric, and binomial.

Fig. 10.
Fig. 10.

Photon-information-content metric as a function of the average number of photons per pixel for three different photon statistics: Poisson, geometric, and binomial.

Fig. 11.
Fig. 11.

Input 500×500 image (a) and one slice [at 370 mm] of the 3D reconstruction (b) using the PCII system. The calculated fidelity metric is ρ=0.2438 and Ip=0.86bpp. The mutual information and entropy are estimated as in [19].

Fig. 12.
Fig. 12.

(a) Input (true) 1664×2496 image used to generate 100 elemental images (with an average of 10 000 photons in each elemental image) and 3D reconstruction based on PCII and (b) one slice at 95 mm.

Fig. 13.
Fig. 13.

Fidelity metric and photon-information content versus number of depth slices used (256 gray levels). The reconstruction was performed using 100 elemental images with each elemental image having, on average, Np×1664×2496 photons.

Fig. 14.
Fig. 14.

Fidelity metric and photon-information content versus number of photons per pixel for different number of depths included in the 3D image (256 gray levels).

Tables (1)

Tables Icon

Table 1. Fidelity Metric (ρ) Versus Photon-Information-Content Metric (Ip) for a Representative Elemental Image (EI) and the Reconstructed Image (RI)

Equations (9)

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

R(x,y,z)=1O(x,y)k=0K1k=0L1Ikl(xkSx,ylSy),
Sx=NxpfcxdandSy=Nypfcyd,
PYi|Xi(yi|xi)=(Npxiϵ)yieNpxiϵyi!,yi=0,1,2,.
H(X(L),X(1))E[log2(PX(L),X(1)(X(L),,X(1)))],
I(Y(L),,Y(1);X(L),,X(1))E[log2(PY(L),,Y(1)|X(L),,X(1)(Y(L),,Y(1)|X(L),,X(1))PY(L),,Y(1)(Y(L),,Y(1)))].
ρI(Y(L),Y(1);X(L),,X(1))H(X(L),,X(1)),
Ip=I(Y(L),,Y(1);X(L),,X(1))/m,
Ip=log2((nk)+1)nklog2((n+1k))nk=log2(n+1)++log2(nk+2)log2(k!)nk(1+(nk)1)log2(n+1),
Vc(X)={βcif the pixel values ofXat the sites incare the sameβcotherwise,

Metrics