Abstract

The method of photon-counting integral imaging has been introduced recently for three-dimensional object sensing, visualization, recognition and classification of scenes under photon-starved conditions. This paper presents an information-theoretic model for the photon-counting imaging (PCI) method, thereby providing a rigorous foundation for the merits of PCI in terms of image fidelity. This, in turn, can facilitate our understanding of the demonstrated success of photon-counting integral imaging in compressive imaging and classification. The mutual information between the source and photon-counted images is derived in a Markov random field setting and normalized by the source-image’s entropy, yielding a fidelity metric that is between zero and unity, which respectively corresponds to complete loss of information and full preservation of information. Calculations suggest that the PCI fidelity metric increases with spatial correlation in source image, from which we infer that the PCI method is particularly effective for source images with high spatial correlation; the metric also increases with the reduction in photon-number uncertainty. As an application to the theory, an image-classification problem is considered showing a congruous relationship between the fidelity metric and classifier’s performance.

© 2010 Optical Society of America

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References

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2009 (2)

R. Martinez-Cuenca, G. Saavedra, M. Martinez-Corral, and B. Javidi, “Progress in 3-d multiperspective display by integral imaging,” Proc. IEEE J. 97, 1067–1077 (June 2009).
[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] [PubMed]

2008 (1)

2007 (2)

2006 (3)

A. Stern and B. Javidi, “3d image sensing, visualization, and processing using integral imaging,” Proc. IEEE J. 94, 591–608 (2006).
[CrossRef]

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

Y. Frauel, T. Naughton, O. Matoba, E. Tahajuerce, and B. Javidi, “Three dimensional imaging and display using computational holographic imaging,” Proc. IEEE J. 94, 636–654 (2006).
[CrossRef]

2004 (1)

2002 (1)

2001 (2)

1999 (1)

F. Okano, J. Arai, H. Hoshino, and I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

1998 (1)

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]

1997 (1)

A. R. L. Travis, “The display of three dimensional video images,” Proc. IEEE J. 85, 1817–1832 (1997).
[CrossRef]

1996 (1)

M. Levoy and P. Hanrahan, “Light field rendering,” Proc. ACM Siggarph, ACM Press NEEDS TO BE CHANGED 15, 31–42 (1996).

1990 (1)

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

1988 (1)

1984 (3)

K. Lange and R. Carson, “Em reconstruction algorithms for emission and transmission tomography,” J. Comput. Assist. Tomogr. 8, 306–316 (1984).
[PubMed]

S. Geman and D. Geman, “Stochastic relaxation, gibbs distributions, and the bayesian restoration of images,” IEEE Trans. Pattern Analysis and Machine Intelligence 6, 721–741 (1984).
[CrossRef]

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

1980 (1)

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

1979 (1)

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]

1968 (1)

1931 (1)

1908 (1)

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

Arai, J.

F. Okano, J. Arai, H. Hoshino, and I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

Arimoto, H.

Benton, S.

S. Benton, “Selected papers on 3d displays,” SPIE Press Book (2001).

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, London, UK, 1996), pp. 250–268.

Burckhardt, C. B.

Carson, R.

K. Lange and R. Carson, “Em reconstruction algorithms for emission and transmission tomography,” J. Comput. Assist. Tomogr. 8, 306–316 (1984).
[PubMed]

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory (John Wiley & sons, 1991).
[CrossRef]

Davies, N.

Erdmann, L.

Frauel, Y.

Y. Frauel, T. Naughton, O. Matoba, E. Tahajuerce, and B. Javidi, “Three dimensional imaging and display using computational holographic imaging,” Proc. IEEE J. 94, 636–654 (2006).
[CrossRef]

Gabriel, K. J.

Geman, D.

S. Geman and D. Geman, “Stochastic relaxation, gibbs distributions, and the bayesian restoration of images,” IEEE Trans. Pattern Analysis and Machine Intelligence 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 Analysis and Machine Intelligence 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, London, UK, 1996), pp. 250–268.

Goodman, J. W.

J. W. Goodman, Statistical optics (John Wiley & sons, 1985).

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]

Hanrahan, P.

M. Levoy and P. Hanrahan, “Light field rendering,” Proc. ACM Siggarph, ACM Press NEEDS TO BE CHANGED 15, 31–42 (1996).

Hong, S.-H.

Hoshino, H.

F. Okano, J. Arai, H. Hoshino, and I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

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, J. S.

Javidi, B.

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] [PubMed]

R. Martinez-Cuenca, G. Saavedra, M. Martinez-Corral, and B. Javidi, “Progress in 3-d multiperspective display by integral imaging,” Proc. IEEE J. 97, 1067–1077 (June 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] [PubMed]

R. Martnez-Cuenca, H. Navarro, G. Saavedra, B. Javidi, and M. Martnez-Corral, “Enhanced viewing-angle integral imaging by multiple-axis telecentric relay system,” Opt. Express 15, 16255–16260 (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] [PubMed]

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

A. Stern and B. Javidi, “3d image sensing, visualization, and processing using integral imaging,” Proc. IEEE J. 94, 591–608 (2006).
[CrossRef]

Y. Frauel, T. Naughton, O. Matoba, E. Tahajuerce, and B. Javidi, “Three dimensional imaging and display using computational holographic imaging,” Proc. IEEE J. 94, 636–654 (2006).
[CrossRef]

J. S. Jang and B. Javidi, “Three-dimensional integral imaging of micro-objects,” Opt. Lett. 29, 1230–1232 (2004).
[CrossRef] [PubMed]

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

Jung, S.

Lange, K.

K. Lange and R. Carson, “Em reconstruction algorithms for emission and transmission tomography,” J. Comput. Assist. Tomogr. 8, 306–316 (1984).
[PubMed]

Lee, B.

Levoy, M.

M. Levoy and P. Hanrahan, “Light field rendering,” Proc. ACM Siggarph, ACM Press NEEDS TO BE CHANGED 15, 31–42 (1996).

Lippmann, G.

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

Mandel, L.

Martinez-Corral, M.

R. Martinez-Cuenca, G. Saavedra, M. Martinez-Corral, and B. Javidi, “Progress in 3-d multiperspective display by integral imaging,” Proc. IEEE J. 97, 1067–1077 (June 2009).
[CrossRef]

Martinez-Cuenca, R.

R. Martinez-Cuenca, G. Saavedra, M. Martinez-Corral, and B. Javidi, “Progress in 3-d multiperspective display by integral imaging,” Proc. IEEE J. 97, 1067–1077 (June 2009).
[CrossRef]

Martnez-Corral, M.

Martnez-Cuenca, R.

Matoba, O.

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

Y. Frauel, T. Naughton, O. Matoba, E. Tahajuerce, and B. Javidi, “Three dimensional imaging and display using computational holographic imaging,” Proc. IEEE J. 94, 636–654 (2006).
[CrossRef]

McCornick, M.

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]

Moon, I.

Morris, G. M.

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

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

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]

Naughton, T.

Y. Frauel, T. Naughton, O. Matoba, E. Tahajuerce, and B. Javidi, “Three dimensional imaging and display using computational holographic imaging,” Proc. IEEE J. 94, 636–654 (2006).
[CrossRef]

Navarro, H.

Okano, F.

F. Okano, J. Arai, H. Hoshino, and I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

Okoshi, T.

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

Park, J.-H.

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]

Saavedra, G.

R. Martinez-Cuenca, G. Saavedra, M. Martinez-Corral, and B. Javidi, “Progress in 3-d multiperspective display by integral imaging,” Proc. IEEE J. 97, 1067–1077 (June 2009).
[CrossRef]

R. Martnez-Cuenca, H. Navarro, G. Saavedra, B. Javidi, and M. Martnez-Corral, “Enhanced viewing-angle integral imaging by multiple-axis telecentric relay system,” Opt. Express 15, 16255–16260 (2007).
[CrossRef]

Saleh, B. E. A.

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

Stern, A.

A. Stern and B. Javidi, “3d image sensing, visualization, and processing using integral imaging,” Proc. IEEE J. 94, 591–608 (2006).
[CrossRef]

Tahajuerce, E.

Y. Frauel, T. Naughton, O. Matoba, E. Tahajuerce, and B. Javidi, “Three dimensional imaging and display using computational holographic imaging,” Proc. IEEE J. 94, 636–654 (2006).
[CrossRef]

Tavakoli, B.

Teich, M. C.

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

Thomas, J. A.

T. M. Cover and J. A. Thomas, Elements of Information Theory (John Wiley & sons, 1991).
[CrossRef]

Travis, A. R. L.

A. R. L. Travis, “The display of three dimensional video images,” Proc. IEEE J. 85, 1817–1832 (1997).
[CrossRef]

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]

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, London, UK, 1996), pp. 250–268.

Watson, E.

Watson, E. A.

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

Yang, L.

Yeom, S.

Yuyama, I.

F. Okano, J. Arai, H. Hoshino, and I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

Appl. Opt. (3)

IEEE Trans. Pattern Analysis and Machine Intelligence (1)

S. Geman and D. Geman, “Stochastic relaxation, gibbs distributions, and the bayesian restoration of images,” IEEE Trans. Pattern Analysis and Machine Intelligence 6, 721–741 (1984).
[CrossRef]

J. Appl. Phys. (1)

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

J. Comput. Assist. Tomogr. (1)

K. Lange and R. Carson, “Em reconstruction algorithms for emission and transmission tomography,” J. Comput. Assist. Tomogr. 8, 306–316 (1984).
[PubMed]

J. Opt. Soc. Am. (2)

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 (Paris) (1)

G. Lippmann, “Epreuves reversibles donnant la sensation du relief,” J. Phys (Paris) 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. Eng. (1)

F. Okano, J. Arai, H. Hoshino, and I. Yuyama, “Three-dimensional video system based on integral photography,” Opt. Eng. 38, 1072–1077 (1999).
[CrossRef]

Opt. Express (3)

Opt. Lett. (5)

Proc. ACM Siggarph, ACM Press NEEDS TO BE CHANGED (1)

M. Levoy and P. Hanrahan, “Light field rendering,” Proc. ACM Siggarph, ACM Press NEEDS TO BE CHANGED 15, 31–42 (1996).

Proc. IEEE J. (5)

A. Stern and B. Javidi, “3d image sensing, visualization, and processing using integral imaging,” Proc. IEEE J. 94, 591–608 (2006).
[CrossRef]

R. Martinez-Cuenca, G. Saavedra, M. Martinez-Corral, and B. Javidi, “Progress in 3-d multiperspective display by integral imaging,” Proc. IEEE J. 97, 1067–1077 (June 2009).
[CrossRef]

A. R. L. Travis, “The display of three dimensional video images,” Proc. IEEE J. 85, 1817–1832 (1997).
[CrossRef]

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

Y. Frauel, T. Naughton, O. Matoba, E. Tahajuerce, and B. Javidi, “Three dimensional imaging and display using computational holographic imaging,” Proc. IEEE J. 94, 636–654 (2006).
[CrossRef]

Other (6)

S. Benton, “Selected papers on 3d displays,” SPIE Press Book (2001).

B. Javidi, F. Okano, and J. Y. S. eds, “Three dimensional imaging, visualization, and display,” Springer (2009).

J. W. Goodman, Statistical optics (John Wiley & sons, 1985).

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

T. M. Cover and J. A. Thomas, Elements of Information Theory (John Wiley & sons, 1991).
[CrossRef]

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, London, UK, 1996), pp. 250–268.

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Figures (11)

Fig. 1.
Fig. 1.

Schematic of the PCI system considered in this paper. The figure shows a source image, the transformation rule and the output image. These components of the PCI system can be identified as source, channel and output of a communication system. In most scenarios of interest, the output image is a sparse, binary array since the photon stream is very weak.

Fig. 2.
Fig. 2.

This figure shows Poisson, geometric and binomial probability mass functions. The various parameters used are λ = 4 for Poisson, p = 0.2 for geometric, and n = 20, p = 0.2 for binomial. In all three cases the mean value is 4.

Fig. 3.
Fig. 3.

Figure showing the output 128×128 photon-counted array, where the photon counts are represented by the symbols ‘*,’ and the best-fit estimate of the line from the photon counts. The value of k is 5 and the corresponding noise variance is 2.08. Here Npε = 0.3.

Fig. 4.
Fig. 4.

Figure showing the fidelity metric, ρ, and the classification error versus the noise variance in the input image. The classifier is run for 10,000 times for each value of noise variance to average out the classification error. Here Npε = 0.3.

Fig. 5.
Fig. 5.

Representative spatial conditional probability mass function, P{X i+1 = x i+1|Xi = 4}, for different values of the correlation index m, which determines the amount of correlation present among source-image pixels.

Fig. 6.
Fig. 6.

Normalized mutual information for different possible channel distributions while using a Markov-chain model for P X (x), with a specific correlation index, m = 0.01. Here we used Np = 3.

Fig. 7.
Fig. 7.

Family of curves showing that presence of spatial correlation in the source image results in an increase in the normalized mutual information for (a) the Poisson photon-counting channel and (b) the geometric photon-counting channel. Here we used Np = 3.

Fig. 8.
Fig. 8.

8(a) First-order and second-order neighborhoods (of the center pixel) considered in this paper. 8(b) Cliques of various sizes up to four-point sites. The first three cliques shown (from top-left to top-right) correspond to a first-order neighborhood; all ten cliques correspond to the second-order neighborhood system.

Fig. 9.
Fig. 9.

Sample MRF images generated according to a generalized Ising model with potential energies given by (18) using various values for the parameter β. We employed a Metropolis sampler using 1000 iterations. The values used for the parameter β are (a)β = -2, (b)β = -1, (c)β = -0.975, (d)β = -0.95, and (e)β = -0.9, (f)β = -0.4.

Fig. 10.
Fig. 10.

Figure showing the relation between correlation metric, and the spatial correlation parameter β.

Fig. 11.
Fig. 11.

Figure showing the relation between correlation metric, ρ and the mean number of photon counts at the out put per pixel per unit integration time.

Tables (2)

Tables Icon

Table 1. Table showing the relationship between the variance of the uniform noise in the input image, the normalized mutual information, ρ, and the average error of classification.

Tables Icon

Table 2. Dependence of the spatial-correlation and fidelity metrics on the parameter β for the case N p ε = 3.

Equations (36)

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P Y i X i ( y i x i ) = ( N p x i ε ) y i e N p x i ε y i ! , y i = 0,1,2 , .
P Y X ( y x ) = i = 1 n P Y i X i ( y i x i ) ,
P Y ( y ) = x P X ( X ) P Y X ( y x ) ,
P Y i X i ( 0 x i ) = e N p x i ε ,
P Y i X i ( 1 x i ) = 1 e N p x i ε .
P Y i X i ( y i x i ) = n y i p y i ( 1 p ) n y i , y i = 0,1,2 , . , n ,
P Y i X i ( 0 x i ) = ( 1 p ) N p x i ε p 1
P Y I X i ( 1 x i ) = 1 ( 1 p ) N p x i ε p 1 .
P Y i X i ( y i x i ) = 1 N p x i ε + 1 ( N p x i ε N p x i ε + 1 ) y i , y i = 0,1,2 , .
P Y i X i ( 0 x i ) = 1 1 + N p x i ε ,
P Y i X i ( 1 x i ) = N p x i ε 1 + N p x i ε .
ρ = I ( Y ; X ) H ( X ) .
P X | s ( X | s ) = i = 1 128 j = 1 128 P X i , j | s ( x i , j | s ) .
P Y | s ( Y | s ) = i = 1 128 j = 1 128 P Y i , j | s ( y i , j | s ) .
P { X i + 1 = x i + 1 X i = x i , X 1 = x 1 } = P { X i + 1 = x i + 1 X i = x i } , i = 1 , , n 1 .
P { X i + 1 = x i + 1 X i = x i } = { max ( m x i + 1 m x i + c x i , 0 ) , for x i + 1 = 0,1 , x i max ( m x i + 1 m x i + c x i , 0 ) , for x i + 1 = x i + 1 , , I ,
c x i = 1 I + 1 ( m x i 2 m I x i + m 2 I ( I + 1 ) + 1 ) .
P x ( x ) = P { X n = x n X n 1 = x n 1 } P { X n 1 = x n 1 X n 2 = x n 2 } P { X 2 = x 2 X 1 = x 1 } P X 1 ( x 1 ) .
P { X = x } = 1 Z exp ( U ( x ) T ) ,
U ( x ) = c C V c ( x ) ,
V c ( X ) = { β c if the pixel values of X at the site s in c are same β c otherwise .
X = { X i , j , ( i , j ) S } ,
S = { ( i , j ) | 1 i N , 1 j N } .
N i , j = { ( k , l ) S : D [ ( i , j ) , ( k , l ) ] K , ( k , l ) ( i , j ) } ,
N = { N i , j , ( i , j ) S } .
X i , j c { X k , l , ( k , l ) S \ ( i , j ) } ,
P { X i , j = x i , j X i , j c } = P { X i , j = x i , j X k , l = x k , l , ( k , l ) N i , j } .
P x ( x ) = 1 Z exp ( U ( x ) T ) ,
Z = x Ω exp ( U ( x ) T ) ,
U ( x ) = c C V c ( x )
P { X i , j = x i , j X k , l = x k , l , ( k , l ) N i , j } = 1 Z i , j exp ( U ( x i , j ) T ) ,
U ( x i , j ) = c : ( i , j ) C V c ( X c ) ,
H ( X i , j X k , l , ( k , l ) N i , j ) = x i , j α x P { A X = α x , X i , j = x i , j } log 2 ( P { X i , j = x i , j A X = α x } ) ,
H ( X i , j X k , l , ( k , l ) N i , j ) = x i , j α x f A X X i , j ( α x , x i , j ) log 2 ( f A X X i , j ( α x , x i , j ) f A X ( α x ) ) .
I ( X i , j ; Y i , j X k , l , Y k , l , ( k , l ) N i , j ) =
x i , j y i , j α x α y f A X X i , j A Y Y i , j ( α x , x i , j , α y , y i , j ) log 2 ( f A X X i , j A Y Y i , j ( α x , x i , j , α y , y i , j ) f A X ( α x ) f A Y ( α y ) f A X A Y ( α x , α y ) f A X X i , j ( α x , x i , j ) f A Y Y i , j ( α y , y i , j ) )

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