Abstract

In this paper, we present three dimensional (3D) object reconstruction using photon-counted elemental images acquired by a passive 3D Integral Imaging (II) system. The maximum likelihood (ML) estimator is derived to reconstruct the irradiance of the 3D scene pixels and the reliability of the estimator is described by confidence intervals. For applications in photon scarce environments, our proposed technique provides 3D reconstruction for better visualization as well as significant reduction in the computational burden and required bandwidth for transmission of integral images. The performance of the reconstruction is illustrated qualitatively and compared quantitatively with Peak to Signal to Noise Ratio (PSNR) criterion.

© 2008 Optical Society of America

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  1. Y. Frauel, T. Naughton, O. Matoba, E. Tahajuerce, and B. Javidi, "Three Dimensional Imaging and Display Using Computational Holographic Imaging," Proc. IEEE Journal 94 636-654, (2006).
    [CrossRef]
  2. Javidi and F. Okano, eds., Three Dimensional Television, Video, and Display Technologies (Springer, Berlin, 2002).
  3. M. Levoy and P. Hanrahan. "Light field rendering" Proc. ACM Siggarph, ACM Press, 31-42 (1996).
  4. B.  Javidi, S.-H.  Hong, and O.  Matoba, "Multi dimensional optical sensors and imaging systems," Appl. Opt.  45, 2986-2994 (2006).
    [CrossRef] [PubMed]
  5. Stern and B. Javidi, "Three-dimensional image sensing, visualization, and processing using integral imaging," Proc. IEEE  94, 591-607 (2006).
    [CrossRef]
  6. H. Arimoto and B. Javidi, "Integrate three-dimensional imaging with computed reconstruction," Opt. Lett. 26, 157-159 (2001).
    [CrossRef]
  7. Stern and B. Javidi, "3-D computational synthetic aperture integral imaging (COMPSAII)," Opt. Express  11, 2446-2451 (2003).
    [CrossRef] [PubMed]
  8. 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]
  9. M. G. Lippmann, "Epreuves reversibles donnant la sensation du relief," J. Phys. 7, 821-825 (1908).
  10. H. E. Ives, "Optical properties of a Lippmann lenticuled sheet," J. Opt. Soc. Am.  21, 171-176 (1931).
    [CrossRef]
  11. S. Jang and B. Javidi, "Three-dimensional synthetic aperture integral imaging," Opt. Lett. 27, 1144-1146 (2002).
    [CrossRef]
  12. T. Okoshi, "Three-dimensional displays," Proc. IEEE 68, 548-564 (1980).
    [CrossRef]
  13. Y. Igarishi, H. Murata, and M. Ueda, "3D display system using a computer-generated integral photograph," Jpn. J. Appl. Phys. 17, 1683-1684 (1978).
    [CrossRef]
  14. Tavakoli, M. Danesh Panah, B. Javidi, and E. Watson, "Performance of 3D integral imaging with position uncertainty," Opt. Express 15, 11889-11902 (2007).
    [CrossRef] [PubMed]
  15. Wilburn, N.  Joshi, V. Vaish, A. Barth, A. Adams, M. Horowitz, and M. Levoy, "High performance imaging using large camera arrays," Proc. of the ACM 24, 765-776 (2005).
  16. M.  Martínez-Corral, B.  Javidi, R.  Martínez-Cuenca, and G.  Saavedra, "Integral imaging with improved depth of field by use of amplitude modulated microlens array," Appl. Opt. 43, 5806-5813 (2004).
    [CrossRef] [PubMed]
  17. O. Matoba, E. Tajahuerce, and B. Javidi, "Real-time three-dimensional object recognition with multiple perspectives imaging," Appl. Opt. 40, 3318-3325 (2001).
    [CrossRef]
  18. 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]
  19. F. A. Sadjadi and A. Mahalanobis, "Target-adaptive polarimetric synthetic aperture radar target discrimination using maximum average correlation height filters," Appl. Opt. 45, 3063-3070 (2006).
    [CrossRef] [PubMed]
  20. Erdmann and K. J. Gabriel, "High resolution digital photography by use of a scanning microlens array," Appl. Opt. 40, 5592-5599 (2001).
    [CrossRef]
  21. K. Nitta, R. Shogenji, S. Miyatake, and J. Tanida, "Image reconstruction for thin observation module by bound optics by using the iterative backprojection method," Appl. Opt. 45, 2893-2900 (2006).
    [CrossRef] [PubMed]
  22. G. M. Morris, "Scene matching using photon-limited images," J. Opt. Soc. Am. A. 1, 482-488 (1984).
    [CrossRef]
  23. G. M. Morris, "Image correlation at low light levels: a computer simulation," Appl. Opt. 23, 3152-3159 (1984).
    [CrossRef] [PubMed]
  24. Watson and G. M. Morris, "Comparison of infrared up conversion methods for photon-limited imaging," J. Appl. Phys. 67, 6075-6084 (1990).
    [CrossRef]
  25. Watson and G. M. Morris, "Imaging thermal objects with photon-counting detector," Appl. Opt. 31, 4751-4757 (1992).
    [CrossRef] [PubMed]
  26. 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]
  27. 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]
  28. 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]
  29. K. Lange and R. Carson, "EM reconstruction algorithms for emission and transmission tomography," Proc. IEEE J. Comput. Assist. Tomogr. 8, 306-316 (1984).
  30. M.  Guillaume, P.  Melon, and P.  Refregier, "Maximum-likelihood estimation of an astronomical image from a sequence at low photon levels," J. Opt. Soc. Am. A. 15, 2841-2848 (1998).
    [CrossRef]
  31. J. W. Goodman, Statistical optics (John Wiley & Sons, inc., 1985), Chap 9.
  32. E. Kolaczyk, "Bayesian multi-scale models for Poisson processes," J. Amer. Stat. Assoc. 94, 920-933 (1999).
    [CrossRef]
  33. 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] [PubMed]
  34. N.  Mukhopadhyay, Probability and Statistical Inference (Marcel Dekker, Inc. New York, 2000).
  35. N. Otsu, "A Threshold Selection Method from Gray-Level Histograms," IEEE Trans. Syst. Man. Cybern 9, 62-66 (1979).
    [CrossRef]

2007 (2)

2006 (4)

2004 (1)

2003 (1)

2002 (2)

2001 (5)

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

O. Matoba, E. Tajahuerce, and B. Javidi, "Real-time three-dimensional object recognition with multiple perspectives imaging," Appl. Opt. 40, 3318-3325 (2001).
[CrossRef]

Erdmann and K. J. Gabriel, "High resolution digital photography by use of a scanning microlens array," Appl. Opt. 40, 5592-5599 (2001).
[CrossRef]

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]

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]

2000 (1)

1999 (1)

E. Kolaczyk, "Bayesian multi-scale models for Poisson processes," J. Amer. Stat. Assoc. 94, 920-933 (1999).
[CrossRef]

1998 (2)

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]

M.  Guillaume, P.  Melon, and P.  Refregier, "Maximum-likelihood estimation of an astronomical image from a sequence at low photon levels," J. Opt. Soc. Am. A. 15, 2841-2848 (1998).
[CrossRef]

1992 (1)

1990 (1)

Watson and G. M. Morris, "Comparison of infrared up conversion methods for photon-limited imaging," J. Appl. Phys. 67, 6075-6084 (1990).
[CrossRef]

1984 (2)

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

G. M. Morris, "Image correlation at low light levels: a computer simulation," Appl. Opt. 23, 3152-3159 (1984).
[CrossRef] [PubMed]

1980 (1)

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

1979 (1)

N. Otsu, "A Threshold Selection Method from Gray-Level Histograms," IEEE Trans. Syst. Man. Cybern 9, 62-66 (1979).
[CrossRef]

1978 (1)

Y. Igarishi, H. Murata, and M. Ueda, "3D display system using a computer-generated integral photograph," Jpn. J. Appl. Phys. 17, 1683-1684 (1978).
[CrossRef]

1931 (1)

1908 (1)

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

Arimoto, H.

Buller, G. S.

Butterlin, N.

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]

Duraffourg, L.

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]

Frauel, Y.

Goedgebuer, J.-P.

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]

Gontijo, I

Guillaume, M.

M.  Guillaume, P.  Melon, and P.  Refregier, "Maximum-likelihood estimation of an astronomical image from a sequence at low photon levels," J. Opt. Soc. Am. A. 15, 2841-2848 (1998).
[CrossRef]

Hiskett, P. A.

Hong, S.-H.

Hoshino,

Igarishi, Y.

Y. Igarishi, H. Murata, and M. Ueda, "3D display system using a computer-generated integral photograph," Jpn. J. Appl. Phys. 17, 1683-1684 (1978).
[CrossRef]

Ives, H. E.

Jang, S.

Javidi, B.

Kolaczyk, E.

E. Kolaczyk, "Bayesian multi-scale models for Poisson processes," J. Amer. Stat. Assoc. 94, 920-933 (1999).
[CrossRef]

Lippmann, M. G.

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

Loudon, A. Y.

Mahalanobis, A.

Martínez-Corral, M.

Martínez-Cuenca, R.

Matoba, O.

Melon, P.

M.  Guillaume, P.  Melon, and P.  Refregier, "Maximum-likelihood estimation of an astronomical image from a sequence at low photon levels," J. Opt. Soc. Am. A. 15, 2841-2848 (1998).
[CrossRef]

Merolla, J.-M.

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]

Miyatake, S.

Morris, G. M.

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

G. M. Morris, "Image correlation at low light levels: a computer simulation," Appl. Opt. 23, 3152-3159 (1984).
[CrossRef] [PubMed]

Murata, H.

Y. Igarishi, H. Murata, and M. Ueda, "3D display system using a computer-generated integral photograph," Jpn. J. Appl. Phys. 17, 1683-1684 (1978).
[CrossRef]

Nitta, K.

Okoshi, T.

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

Otsu, N.

N. Otsu, "A Threshold Selection Method from Gray-Level Histograms," IEEE Trans. Syst. Man. Cybern 9, 62-66 (1979).
[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.

M.  Guillaume, P.  Melon, and P.  Refregier, "Maximum-likelihood estimation of an astronomical image from a sequence at low photon levels," J. Opt. Soc. Am. A. 15, 2841-2848 (1998).
[CrossRef]

Rhods, W.

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]

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]

Robertson, M. J.

Saavedra, G.

Sadjadi, F. A.

Shogenji, R.

Smith, J. M.

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]

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]

Tajahuerce, E.

Tanida, J.

Tavakoli,

Townsend, P. D.

Ueda, M.

Y. Igarishi, H. Murata, and M. Ueda, "3D display system using a computer-generated integral photograph," Jpn. J. Appl. Phys. 17, 1683-1684 (1978).
[CrossRef]

Walker, A. C.

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.

Yeom, S.

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. (10)

G. M. Morris, "Image correlation at low light levels: a computer simulation," Appl. Opt. 23, 3152-3159 (1984).
[CrossRef] [PubMed]

Watson and G. M. Morris, "Imaging thermal objects with photon-counting detector," Appl. Opt. 31, 4751-4757 (1992).
[CrossRef] [PubMed]

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]

O. Matoba, E. Tajahuerce, and B. Javidi, "Real-time three-dimensional object recognition with multiple perspectives imaging," Appl. Opt. 40, 3318-3325 (2001).
[CrossRef]

Erdmann and K. J. Gabriel, "High resolution digital photography by use of a scanning microlens array," Appl. Opt. 40, 5592-5599 (2001).
[CrossRef]

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]

M.  Martínez-Corral, B.  Javidi, R.  Martínez-Cuenca, and G.  Saavedra, "Integral imaging with improved depth of field by use of amplitude modulated microlens array," Appl. Opt. 43, 5806-5813 (2004).
[CrossRef] [PubMed]

K. Nitta, R. Shogenji, S. Miyatake, and J. Tanida, "Image reconstruction for thin observation module by bound optics by using the iterative backprojection method," Appl. Opt. 45, 2893-2900 (2006).
[CrossRef] [PubMed]

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

F. A. Sadjadi and A. Mahalanobis, "Target-adaptive polarimetric synthetic aperture radar target discrimination using maximum average correlation height filters," Appl. Opt. 45, 3063-3070 (2006).
[CrossRef] [PubMed]

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. Syst. Man. Cybern (1)

N. Otsu, "A Threshold Selection Method from Gray-Level Histograms," IEEE Trans. Syst. Man. Cybern 9, 62-66 (1979).
[CrossRef]

J. Amer. Stat. Assoc. (1)

E. Kolaczyk, "Bayesian multi-scale models for Poisson processes," J. Amer. Stat. Assoc. 94, 920-933 (1999).
[CrossRef]

J. Appl. Phys. (1)

Watson and G. M. Morris, "Comparison of infrared up conversion 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 (1)

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

M.  Guillaume, P.  Melon, and P.  Refregier, "Maximum-likelihood estimation of an astronomical image from a sequence at low photon levels," J. Opt. Soc. Am. A. 15, 2841-2848 (1998).
[CrossRef]

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

J. Phys. (1)

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

Jpn. J. Appl. Phys. (1)

Y. Igarishi, H. Murata, and M. Ueda, "3D display system using a computer-generated integral photograph," Jpn. J. Appl. Phys. 17, 1683-1684 (1978).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Proc. IEEE (2)

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

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

Other (7)

Y. Frauel, T. Naughton, O. Matoba, E. Tahajuerce, and B. Javidi, "Three Dimensional Imaging and Display Using Computational Holographic Imaging," Proc. IEEE Journal 94 636-654, (2006).
[CrossRef]

Javidi and F. Okano, eds., Three Dimensional Television, Video, and Display Technologies (Springer, Berlin, 2002).

M. Levoy and P. Hanrahan. "Light field rendering" Proc. ACM Siggarph, ACM Press, 31-42 (1996).

Wilburn, N.  Joshi, V. Vaish, A. Barth, A. Adams, M. Horowitz, and M. Levoy, "High performance imaging using large camera arrays," Proc. of the ACM 24, 765-776 (2005).

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

K. Lange and R. Carson, "EM reconstruction algorithms for emission and transmission tomography," Proc. IEEE J. Comput. Assist. Tomogr. 8, 306-316 (1984).

N.  Mukhopadhyay, Probability and Statistical Inference (Marcel Dekker, Inc. New York, 2000).

Supplementary Material (2)

» Media 1: AVI (442 KB)     
» Media 2: AVI (613 KB)     

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Figures (8)

Fig. 1.
Fig. 1.

Pickup process of three dimensional synthetic aperture integral imaging. The imaging device (Lens and sensor) translates on the pick up plane to capture elemental images.

Fig. 2.
Fig. 2.

Arrangement of elemental images at pickup plane (left), Schematic of II reconstruction process at two different distances (right).

Fig. 3.
Fig. 3.

Reconstruction using irradiance elemental images, (a) reconstruction at =240mm, (b) reconstruction at z=360mm.

Fig. 4.
Fig. 4.

Reconstruction using photon-counted elemental images, (a) central elemental image with Np=103, (b) corresponding reconstruction at =240mm (c), reconstruction at z=360mm, (d) central elemental image with Np=105 (e), corresponding reconstruction at z=240 mm (f), reconstruction at z=360mm.

Fig. 5.
Fig. 5.

Movie of Reconstruction at z=240 mm to z=360mm using photon-counted elemental images with (a) Np=103 (The movie file size: 442 KB) and (b) with Np=105 (The movie file size: 613 KB). [Media 1][Media 2]

Fig. 6.
Fig. 6.

PSNR of the reconstruction with photon-counted elemental images versus the total number of photons per elemental image

Fig. 7.
Fig. 7.

(a) Binerized central irradiance elemental image by thresholding method, (b) corresponding reconstruction at z=240 mm.

Fig. 8.
Fig. 8.

Cropped image of reconstruction at z=240mm, (a) using binary elemental images, (b) photon-counted elemental images with Np=103 (c) grayscale irradiance elemental images.

Equations (17)

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

I ( x , y , z 0 ) = 1 KL k = 0 K 1 l = 0 L 1 I kl ( x + ( 1 M 0 ) S x k , y + ( 1 M 0 ) S y l ) ,
Pr ( C x | I x ) = [ I x ] C x e I x C x ! , C x = 0,1,2 , . . . .
N p = E [ x = 1 N T C x ] = x = 1 N T N p I x = N p x = 1 N T I x
{ p + Δ p kl } { ( x + S x g k z 0 , y + S y g l z 0 ) } for k = 0 , . . . , K 1 , l = 0 , . . . , L 1
L ( I p z 0 { C kl ( p + Δ p kl ) } ) = Π k = 0 K 1 Π l = 0 L 1 Pr ( C kl ( p + Δ p ) I p z 0 )
l ( I p z 0 ) = k = 0 K 1 l = 0 L 1 log [ Pr ( C kl ( p + Δ p kl ) I p z 0 ) ]
C kl I p z 0 ~ Poisson ( N p I p z 0 )
l ( I p z 0 ) = k K l L ( N p I p z 0 + C kl ( p + Δ p kl ) log ( N p I p z 0 ) log ( C kl ( p + Δ p kl ) ! ) )
MLE { I p z 0 } = I ~ p z 0 = 1 N p K L k = 0 K 1 l = 0 L 1 C kl ( p + Δ p kl )
I ~ p z 0 ± z α 2 ( KLN p ) 1 2 I ~ p z 0 ,
I ~ p z 0 ± z α 2 ( KLN p ) 1 2 γ I ~ p z 0
PSNR = 10 log 10 ( I max MSE ( I , I ~ ) ) ,
C kl | I p z 0 ~ Poisson ( N p I p z 0 ) for k = 0 , . . . , K 1 ; l = 0 , . . . , L 1
K L ( N p I ~ p z 0 N p I p z 0 ) N p I ~ p z 0 N ( 0 , 1 ) a s K L
Pr { KLN p ( I ~ p z 0 I p z 0 ) I ~ p z 0 < z α 2 } 1 α
I ~ p z 0 ± z α 2 ( K L N p ) 1 2 I ~ p z 0 ,
z α 2 1 2 π exp ( x 2 2 ) d x = α

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