Abstract

We propose a new estimation method for 3D object reconstruction using photon-counting integral imaging. Earlier studies used maximum likelihood estimation (MLE) as a classical statistical method to reconstruct 3D images from photon-counting elemental images. We use an alternative statistical method known as the Bayesian method, which is more flexible and may perform better than MLE in terms of the mean square error (MSE) metric. The performance of the new reconstruction method is illustrated and compared with MLE by using the MSE. To the best of our knowledge, this is the first report to use the Bayesian method for 3D reconstruction of photon-counting integral imaging.

© 2010 Optical Society of America

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2010 (1)

2009 (2)

J. Kohoutek, I. Y. L. Wan, O. G. Memis, and H. Mohseni, Opt. Express 17, 14458 (2009).
[CrossRef] [PubMed]

R. Martinez-Cuenca, G. Saavedra, M. Martinez-Corral, and B. Javidi, Proc. IEEE 97, 1067 (2009).
[CrossRef]

2008 (1)

2006 (2)

F. Okano, J. Arai, K. Mitani, and M. Okui, Proc. IEEE 94, 490 (2006).
[CrossRef]

A. Stern and B. Javidi, Proc. IEEE 94, 591 (2006).
[CrossRef]

2005 (1)

B. Wilburn, N. Joshi, V. Vaish, E.-V. Talvala, E. Antunez, A. Barth, A. Adams, M. Horowitz, and M. Levoy, ACM Trans. Graphics 24, 765 (2005).
[CrossRef]

2004 (1)

2002 (1)

1998 (1)

1980 (1)

T. Okoshi, Proc. IEEE 68, 548 (1980).
[CrossRef]

1968 (1)

1908 (1)

G. Lippmann, C. R. Acad. Sci. Paris Ser. IV 146, 446 (1908).

Adams, A.

B. Wilburn, N. Joshi, V. Vaish, E.-V. Talvala, E. Antunez, A. Barth, A. Adams, M. Horowitz, and M. Levoy, ACM Trans. Graphics 24, 765 (2005).
[CrossRef]

Antunez, E.

B. Wilburn, N. Joshi, V. Vaish, E.-V. Talvala, E. Antunez, A. Barth, A. Adams, M. Horowitz, and M. Levoy, ACM Trans. Graphics 24, 765 (2005).
[CrossRef]

Arai, J.

F. Okano, J. Arai, K. Mitani, and M. Okui, Proc. IEEE 94, 490 (2006).
[CrossRef]

Barth, A.

B. Wilburn, N. Joshi, V. Vaish, E.-V. Talvala, E. Antunez, A. Barth, A. Adams, M. Horowitz, and M. Levoy, ACM Trans. Graphics 24, 765 (2005).
[CrossRef]

Berger, J. O.

J. O. Berger, Statistical Decision Theory and Bayesian Analysis (Springer, 1985).

Burckhardt, C. B.

Chen, M.-H.

M.-H. Chen, Q.-M. Shao, and J. G. Ibrahim, Monte Carlo Methods in Bayesian Computation (Springer-Verlag, 2000).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, 1985).

Guillanume, M.

Hayat, M. M.

Hong, S.-H.

Horowitz, M.

B. Wilburn, N. Joshi, V. Vaish, E.-V. Talvala, E. Antunez, A. Barth, A. Adams, M. Horowitz, and M. Levoy, ACM Trans. Graphics 24, 765 (2005).
[CrossRef]

Ibrahim, J. G.

M.-H. Chen, Q.-M. Shao, and J. G. Ibrahim, Monte Carlo Methods in Bayesian Computation (Springer-Verlag, 2000).
[CrossRef]

Jang, J.-S

Jang, J.-S.

Javidi, B.

Joshi, N.

B. Wilburn, N. Joshi, V. Vaish, E.-V. Talvala, E. Antunez, A. Barth, A. Adams, M. Horowitz, and M. Levoy, ACM Trans. Graphics 24, 765 (2005).
[CrossRef]

Kohoutek, J.

Levoy, M.

B. Wilburn, N. Joshi, V. Vaish, E.-V. Talvala, E. Antunez, A. Barth, A. Adams, M. Horowitz, and M. Levoy, ACM Trans. Graphics 24, 765 (2005).
[CrossRef]

Lippmann, G.

G. Lippmann, C. R. Acad. Sci. Paris Ser. IV 146, 446 (1908).

Martinez-Corral, M.

R. Martinez-Cuenca, G. Saavedra, M. Martinez-Corral, and B. Javidi, Proc. IEEE 97, 1067 (2009).
[CrossRef]

Martinez-Cuenca, R.

R. Martinez-Cuenca, G. Saavedra, M. Martinez-Corral, and B. Javidi, Proc. IEEE 97, 1067 (2009).
[CrossRef]

Melon, P.

Memis, O. G.

Mitani, K.

F. Okano, J. Arai, K. Mitani, and M. Okui, Proc. IEEE 94, 490 (2006).
[CrossRef]

Mohseni, H.

Narravula, S. R.

Okano, F.

F. Okano, J. Arai, K. Mitani, and M. Okui, Proc. IEEE 94, 490 (2006).
[CrossRef]

Okoshi, T.

T. Okoshi, Proc. IEEE 68, 548 (1980).
[CrossRef]

Okui, M.

F. Okano, J. Arai, K. Mitani, and M. Okui, Proc. IEEE 94, 490 (2006).
[CrossRef]

Réfrégier, P.

Saavedra, G.

R. Martinez-Cuenca, G. Saavedra, M. Martinez-Corral, and B. Javidi, Proc. IEEE 97, 1067 (2009).
[CrossRef]

Shao, Q.-M.

M.-H. Chen, Q.-M. Shao, and J. G. Ibrahim, Monte Carlo Methods in Bayesian Computation (Springer-Verlag, 2000).
[CrossRef]

Stern, A.

A. Stern and B. Javidi, Proc. IEEE 94, 591 (2006).
[CrossRef]

Talvala, E.-V.

B. Wilburn, N. Joshi, V. Vaish, E.-V. Talvala, E. Antunez, A. Barth, A. Adams, M. Horowitz, and M. Levoy, ACM Trans. Graphics 24, 765 (2005).
[CrossRef]

Tavakoli, B.

Vaish, V.

B. Wilburn, N. Joshi, V. Vaish, E.-V. Talvala, E. Antunez, A. Barth, A. Adams, M. Horowitz, and M. Levoy, ACM Trans. Graphics 24, 765 (2005).
[CrossRef]

Wan, I. Y. L.

Watson, E.

Wilburn, B.

B. Wilburn, N. Joshi, V. Vaish, E.-V. Talvala, E. Antunez, A. Barth, A. Adams, M. Horowitz, and M. Levoy, ACM Trans. Graphics 24, 765 (2005).
[CrossRef]

ACM Trans. Graphics (1)

B. Wilburn, N. Joshi, V. Vaish, E.-V. Talvala, E. Antunez, A. Barth, A. Adams, M. Horowitz, and M. Levoy, ACM Trans. Graphics 24, 765 (2005).
[CrossRef]

C. R. Acad. Sci. Paris Ser. IV (1)

G. Lippmann, C. R. Acad. Sci. Paris Ser. IV 146, 446 (1908).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Express (4)

Opt. Lett. (1)

Proc. IEEE (4)

A. Stern and B. Javidi, Proc. IEEE 94, 591 (2006).
[CrossRef]

T. Okoshi, Proc. IEEE 68, 548 (1980).
[CrossRef]

R. Martinez-Cuenca, G. Saavedra, M. Martinez-Corral, and B. Javidi, Proc. IEEE 97, 1067 (2009).
[CrossRef]

F. Okano, J. Arai, K. Mitani, and M. Okui, Proc. IEEE 94, 490 (2006).
[CrossRef]

Other (4)

S.A.Benton, ed. Selected Papers on Three-Dimensional Displays (SPIE Optical Engineering Press, 2001).

J. W. Goodman, Statistical Optics (Wiley, 1985).

M.-H. Chen, Q.-M. Shao, and J. G. Ibrahim, Monte Carlo Methods in Bayesian Computation (Springer-Verlag, 2000).
[CrossRef]

J. O. Berger, Statistical Decision Theory and Bayesian Analysis (Springer, 1985).

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Figures (3)

Fig. 1
Fig. 1

3D Integral imaging. (a) pickup stage (passive sensing); (b) computational image reconstruction.

Fig. 2
Fig. 2

3D image reconstruction process using the Bayesian method.

Fig. 3
Fig. 3

Test results. (a) Original 3D scene. (b)–(d) Reconstructed images at a distance of z 0 = 74 mm for the original scene using various sensing and reconstruction algorithms: (b) conventional sensing with reconstruction algorithm in Eqs. (1, 2); (c) photon-counting sensing with MLE reconstruction algorithm; (d) photon-counting sensing with Bayesian reconstruction algorithm. (e) Sample of photon-counting elemental image with 10,000 photons. (f)–(h) Histograms of reconstructed images in (b)–(d): (f) histogram of original scene reconstruction; (g) histogram of photon-counting scene reconstruction with MLE algorithm; (h) histogram of photon- counting scene reconstruction with Bayesian algorithm.

Equations (8)

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S x = N x × p × f c x × d , S y = N y × p × f c y × d ,
I ( x , y , d ) = 1 O ( x , y ) k = 0 K 1 l = 0 L 1 I k l ( x k S x , y l S y ) ,
C k l ( x , y ) Poisson ( λ k l ( x , y ) ) ,
λ k l ( x , y ) = N p I k l ( x , y ) x = 1 N x y = 1 N y I k l ( x , y ) .
L ( λ ) = Π k = 1 K Π l = 1 L λ k l C k l e λ k l C k l ! .
π ( λ k l ) = β α Γ ( α ) λ k l α 1 e β λ k l , λ k l > 0.
λ ^ k l Bayes ( x , y ) = C k l ( x , y ) + α 1 + β .
I ( x , y , z 0 ) = R ( λ ^ k l Bayes ( x , y ) ) min [ R ( λ ^ k l Bayes ( x , y ) ) ] .

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