Abstract

In this paper, a statistical approach is presented for three-dimensional (3D) visualization and recognition of objects having very small number of photons based on a parametric estimator. A truncated Poisson probability density function is assumed for modeling the distribution of small number of photons count observation. For 3D visualization and recognition of photon-limited objects, an integral imaging system is employed. We utilize virtual geometrical ray propagation for 3D reconstruction of objects. A maximum likelihood estimator (MLE) and statistical inference algorithms are applied to small number of photons counted elemental images captured with integral imaging. We have demonstrated that the MLE using a truncated Poisson model for estimating the average number of photon for each voxel of a photon starved 3D object has a small estimation error compared with the MLE using a Poisson model. Also, we present experiments to investigate the effect of 3D sensing parallax on the recognition performance under a fixed mean number of photons.

© 2009 OSA

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References

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  1. B. Javidi, F. Okano, and J. Sun, 3D Imaging, Visualization, and Display Technologies (Springer, 2008).
  2. G. Lippmann, “La photographie intégrale,” C. R. Acad. Sci. 146, 446–451 (1908).
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    [CrossRef]
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  12. J. Goodman, ed., Statistical Optics, (John Wiley & Sons, Inc. 1985).
  13. M. Hollander, and D. Wolfe, eds., Nonparametric Statistical Methods (Wiley, 1999).

2009

2007

2006

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE 94(3), 591–607 (2006).
[CrossRef]

2001

1998

1908

G. Lippmann, “La photographie intégrale,” C. R. Acad. Sci. 146, 446–451 (1908).

Guillaume, M.

Hoshino, H.

Isono, H.

Javidi, B.

Lippmann, G.

G. Lippmann, “La photographie intégrale,” C. R. Acad. Sci. 146, 446–451 (1908).

Llebaria, A.

Matoba, O.

Melon, P.

Moon, I.

Okano, F.

Refregier, P.

Stern, A.

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE 94(3), 591–607 (2006).
[CrossRef]

Tajahuerce, E.

Watson, E.

Yeom, S.

Yuyama, I.

Appl. Opt.

C. R. Acad. Sci.

G. Lippmann, “La photographie intégrale,” C. R. Acad. Sci. 146, 446–451 (1908).

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Proc. IEEE

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE 94(3), 591–607 (2006).
[CrossRef]

Other

B. Javidi, F. Okano, and J. Sun, 3D Imaging, Visualization, and Display Technologies (Springer, 2008).

Proceedings of IEEE Journal, special issue on 3D Technologies for Imaging and Displays 94, (2006).

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

V. Rohatgi, and A. Ehsanes Saleh, eds., An Introduction to Probability and Statistics (Wiley, 2000).

J. Goodman, ed., Statistical Optics, (John Wiley & Sons, Inc. 1985).

M. Hollander, and D. Wolfe, eds., Nonparametric Statistical Methods (Wiley, 1999).

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Figures (4)

Fig. 1
Fig. 1

Toy car used in the experiments. (a) Front view. (b) Rear view.

Fig. 2
Fig. 2

Sectional images of the 3D scenes reconstructed from very small number of photons-counted elemental images set by using the proposed truncated photon counting model. The total number of the elemental images (parallax) Ne was either 4 or 25. (a) Reconstructed front view car with Ne =4, (b) reconstructed rear view car with Ne =4, (c) reconstructed front view car with Ne =25, (d) reconstructed rear view car with Ne =25.

Fig. 3
Fig. 3

Experimental results of K-S test for the equality of two histograms of the front view and rear view cars for random variable σ versus the sample size n. “PC” and “TPC” refer to photon counting and truncated photon counting, respectively. (a) Ne =4 and (b) Ne =25.

Fig. 4
Fig. 4

Cumulative density function plots of the statistical sampling distributions of the toy car. The front view and rear view of toy car are used as reference and unknown input data for the random variable σ. (a) Poisson distribution method and (b) truncated photon counting method.

Equations (7)

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Pr(C|ρ,T)=[ρT]CeρTC!,
Pr(Cpt=cp)=Pr(Cp=cp)Pr(Cp<k)=exp(N˜Ip)(N˜Ip)cpcp!i=0k1exp(N˜Ip)(N˜Ip)ii!,
E[Cpt]=aE[Cp],V[Cpt]=aE[Cp]+(aa2){E[Cp]}2,
L(N˜Ip|cp(1),...,cp(NxNy))=logn=1NxNyexp(N˜Ip)(N˜Ip)cp(n)cp(n)!i=0k1exp(N˜Ip)(N˜Ip)ii!,
MLE(N˜Ip)=argmaxN˜IpL(N˜Ip|cp(1),...,cp(NxNy)).
MLE(N˜Ip)=IIp(x,y,z0)=1NxNyn=1NxNycp(n)/(11NxNyn=1NxNycp(n))=MLE¯(N˜Ip)1MLE¯(N˜Ip),
[I^p±zα/2(1NxNyN˜)2I^pγ],,

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