Abstract

A sampling-based framework for finding the optimal representation of a finite energy optical field using a finite number of bits is presented. For a given bit budget, we determine the optimum number and spacing of the samples in order to represent the field with as low error as possible. We present the associated performance bounds as trade-off curves between the error and the cost budget. In contrast to common practice, which often treats sampling and quantization separately, we explicitly focus on the interplay between limited spatial resolution and limited amplitude accuracy, such as whether it is better to take more samples with lower amplitude accuracy or fewer samples with higher accuracy. We illustrate that in certain cases sampling at rates different from the Nyquist rate is more efficient.

© 2013 Optical Society of America

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2012 (3)

R. Konsbruck, E. Telatar, and M. Vetterli, “On sampling and coding for distributed acoustic sensing,” IEEE Trans. Inf. Theory 58, 3198–3214 (2012).
[CrossRef]

B. Dulek and S. Gezici, “Cost minimization of measurement devices under estimation accuracy constraints in the presence of Gaussian noise,” Digit. Signal Process. 22, 828–840 (2012).
[CrossRef]

A. Özçelikkale and H. M. Ozaktas, “Representation of optical fields using finite numbers of bits,” Opt. Lett. 37, 2193–2195 (2012).
[CrossRef]

2011 (2)

J. J. Healy and J. T. Sheridan, “Space–bandwidth ratio as a means of choosing between Fresnel and other linear canonical transform algorithms,” J. Opt. Soc. Am. A 28, 786–790 (2011).
[CrossRef]

B. Dulek and S. Gezici, “Average Fisher information maximisation in presence of cost-constrained measurements,” Electron. Lett. 47, 654–656 (2011).
[CrossRef]

2010 (3)

2009 (2)

E. D. Micheli and G. A. Viano, “Inverse optical imaging viewed as a backward channel communication problem,” J. Opt. Soc. Am. A 26, 1393–1402 (2009).
[CrossRef]

F. Oktem and H. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727–730 (2009).
[CrossRef]

2008 (3)

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

M. Migliore, “On electromagnetics and information theory,” IEEE Trans. Antennas Propag. 56, 3188–3200 (2008).
[CrossRef]

J. J. Healy, B. M. Hennelly, and J. T. Sheridan, “Additional sampling criterion for the linear canonical transform,” Opt. Lett. 33, 2599–2601 (2008).
[CrossRef]

2007 (1)

2006 (3)

2004 (3)

A. Stern, and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
[CrossRef]

J. Buescu, “Positive integral operators in unbounded domains,” J. Math. Anal. Appl. 296, 244–255 (2004).
[CrossRef]

A. Stern and B. Javidi, “Shannon number and information capacity of three-dimensional integral imaging,” J. Opt. Soc. Am. A 21, 1602–1612 (2004).
[CrossRef]

2003 (1)

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003).
[CrossRef]

2002 (2)

2001 (2)

G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. 48, 1735–1741 (2001).
[CrossRef]

F. Garcia, I. Lourtie, and J. Buescu, “L2(R) nonstationary processes and the sampling theorem,” IEEE Signal Process. Lett. 8, 117–119 (2001).
[CrossRef]

2000 (2)

1998 (3)

M. A. Neifeld, “Information, resolution, and space-bandwidth product,” Opt. Lett. 23, 1477–1479 (1998).
[CrossRef]

R. Barakat, “Some entropic aspects of optical diffraction imagery,” Opt. Commun. 156, 235–239 (1998).
[CrossRef]

R. Pierri and F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (2)

1993 (2)

M. S. Hughes, “Analysis of digitized wave-forms using Shannon entropy,” J. Acoust. Soc. Am. 93, 892–906 (1993).
[CrossRef]

D. Blacknell and C. J. Oliver, “Information-content of coherent images,” J. Phys. D 26, 1364–1370 (1993).
[CrossRef]

1991 (1)

P. Jixiong, “Waist location and Rayleigh range for Gaussian Schell-model beams,” J. Opt. 22, 157–159 (1991).
[CrossRef]

1989 (1)

O. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

1988 (1)

1986 (1)

1985 (1)

1982 (3)

A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian–Schell model sources and of their radiation fields,” J. Opt. Soc. Am. A 72, 923–928 (1982).
[CrossRef]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

A. Starikov, “Effective number of degrees of freedom of partially coherent sources,” J. Opt. Soc. Am. 72, 1538–1544 (1982).
[CrossRef]

1981 (1)

W. H. Carter and E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, scalar sources,” Opt. Acta 28, 245–259 (1981).
[CrossRef]

1980 (2)

E. Collett and E. Wolf, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Commun. 32, 27–31 (1980).
[CrossRef]

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

1979 (1)

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. A 69, 1710–1716 (1979).
[CrossRef]

1973 (2)

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

T. W. Barret, “Structural information theory,” J. Acoust. Soc. Am. 54, 1092–1098 (1973).
[CrossRef]

1972 (1)

W. A. Gardner, “A sampling theorem for nonstationary random processes,” IEEE Trans. Inf. Theory 18, 808–809 (1972).
[CrossRef]

1971 (2)

1969 (1)

1967 (1)

1966 (1)

1957 (1)

A. Balakrishnan, “A note on the sampling principle for continuous signals,” IEEE Trans. Inf. Theory 3, 143–146 (1957).
[CrossRef]

1955 (1)

1953 (1)

D. MacKay, “Quantal aspects of scientific information,” Trans. IRE Prof. Group Inf. Theory 1, 60–80 (1953).
[CrossRef]

Arikan, E.

A. Özçelikkale, H. M. Ozaktas, and E. Arıkan, “Signal recovery with cost constrained measurements,” IEEE Trans. Signal Process. 58, 3607–3617 (2010).
[CrossRef]

Balakrishnan, A.

A. Balakrishnan, “A note on the sampling principle for continuous signals,” IEEE Trans. Inf. Theory 3, 143–146 (1957).
[CrossRef]

Barakat, R.

Barret, T. W.

T. W. Barret, “Structural information theory,” J. Acoust. Soc. Am. 54, 1092–1098 (1973).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “Uncertainty principle and informational entropy for partially coherent light,” J. Opt. Soc. Am. A 3, 1243–1246 (1986).
[CrossRef]

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. A 69, 1710–1716 (1979).
[CrossRef]

Blacknell, D.

D. Blacknell and C. J. Oliver, “Information-content of coherent images,” J. Phys. D 26, 1364–1370 (1993).
[CrossRef]

Bucci, O.

O. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

Buescu, J.

J. Buescu, “Positive integral operators in unbounded domains,” J. Math. Anal. Appl. 296, 244–255 (2004).
[CrossRef]

F. Garcia, I. Lourtie, and J. Buescu, “L2(R) nonstationary processes and the sampling theorem,” IEEE Signal Process. Lett. 8, 117–119 (2001).
[CrossRef]

Cai, Y.

Candan, C.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Carter, W. H.

W. H. Carter and E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, scalar sources,” Opt. Acta 28, 245–259 (1981).
[CrossRef]

Collett, E.

E. Collett and E. Wolf, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Commun. 32, 27–31 (1980).
[CrossRef]

Cover, T. M.

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

Di Francia, G. T.

Dorsch, R.

Dulek, B.

B. Dulek and S. Gezici, “Cost minimization of measurement devices under estimation accuracy constraints in the presence of Gaussian noise,” Digit. Signal Process. 22, 828–840 (2012).
[CrossRef]

B. Dulek and S. Gezici, “Average Fisher information maximisation in presence of cost-constrained measurements,” Electron. Lett. 47, 654–656 (2011).
[CrossRef]

Ferreira, C.

Franceschetti, G.

O. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

Francia, G. T. D.

Friberg, A. T.

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003).
[CrossRef]

A. T. Friberg and J. Turunen, “Imaging of Gaussian–Schell model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
[CrossRef]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

Gabor, D.

D. Gabor, “Light and information,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1961), Vol. I, pp. 109–153.

Garcia, F.

F. Garcia, I. Lourtie, and J. Buescu, “L2(R) nonstationary processes and the sampling theorem,” IEEE Signal Process. Lett. 8, 117–119 (2001).
[CrossRef]

Gardner, W. A.

W. A. Gardner, “A sampling theorem for nonstationary random processes,” IEEE Trans. Inf. Theory 18, 808–809 (1972).
[CrossRef]

Gbur, G.

G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. 48, 1735–1741 (2001).
[CrossRef]

Gezici, S.

B. Dulek and S. Gezici, “Cost minimization of measurement devices under estimation accuracy constraints in the presence of Gaussian noise,” Digit. Signal Process. 22, 828–840 (2012).
[CrossRef]

B. Dulek and S. Gezici, “Average Fisher information maximisation in presence of cost-constrained measurements,” Electron. Lett. 47, 654–656 (2011).
[CrossRef]

Gori, F.

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

F. Gori and G. Guattari, “Effects of coherence on the degrees of freedom of an image,” J. Opt. Soc. Am. 61, 36–39 (1971).
[CrossRef]

F. Gori, Advanced Topics in Shannon Sampling and Interpolation Theory (Springer-Verlag, 1993), pp. 37–83.

Guattari, G.

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

F. Gori and G. Guattari, “Effects of coherence on the degrees of freedom of an image,” J. Opt. Soc. Am. 61, 36–39 (1971).
[CrossRef]

Healy, J. J.

Hennelly, B. M.

Horn, R. A.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 1990).

Hughes, M. S.

M. S. Hughes, “Analysis of digitized wave-forms using Shannon entropy,” J. Acoust. Soc. Am. 93, 892–906 (1993).
[CrossRef]

Janaswamy, R.

J. Xu and R. Janaswamy, “Electromagnetic degrees of freedom in 2-d scattering environments,” IEEE Trans. Antennas Propag. 54, 3882–3894 (2006).
[CrossRef]

Javidi, B.

A. Stern and B. Javidi, “Shannon number and information capacity of three-dimensional integral imaging,” J. Opt. Soc. Am. A 21, 1602–1612 (2004).
[CrossRef]

A. Stern, and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
[CrossRef]

Jixiong, P.

P. Jixiong, “Waist location and Rayleigh range for Gaussian Schell-model beams,” J. Opt. 22, 157–159 (1991).
[CrossRef]

Johnson, C. R.

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 1990).

Karelin, M.

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003).
[CrossRef]

Koç, A.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35–37 (2006).
[CrossRef]

Konsbruck, R.

R. Konsbruck, E. Telatar, and M. Vetterli, “On sampling and coding for distributed acoustic sensing,” IEEE Trans. Inf. Theory 58, 3198–3214 (2012).
[CrossRef]

Korotkova, O.

Kutay, M. A.

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

H. M. Ozaktas, A. Koç, I. Sari, and M. A. Kutay, “Efficient computation of quadratic-phase integrals in optics,” Opt. Lett. 31, 35–37 (2006).
[CrossRef]

H. M. Ozaktas, S. Yüksel, and M. A. Kutay, “Linear algebraic theory of partial coherence: discrete fields and measures of partial coherence,” J. Opt. Soc. Am. A 19, 1563–1571 (2002).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Lin, Q.

Lohmann, A.

Lohmann, A. W.

Lourtie, I.

F. Garcia, I. Lourtie, and J. Buescu, “L2(R) nonstationary processes and the sampling theorem,” IEEE Signal Process. Lett. 8, 117–119 (2001).
[CrossRef]

Lukozs, W.

MacKay, D.

D. MacKay, “Quantal aspects of scientific information,” Trans. IRE Prof. Group Inf. Theory 1, 60–80 (1953).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Martinsson, P.

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003).
[CrossRef]

Medina, R.

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Stern, A.

A. Stern, and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
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A. Stern and B. Javidi, “Shannon number and information capacity of three-dimensional integral imaging,” J. Opt. Soc. Am. A 21, 1602–1612 (2004).
[CrossRef]

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A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
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R. Konsbruck, E. Telatar, and M. Vetterli, “On sampling and coding for distributed acoustic sensing,” IEEE Trans. Inf. Theory 58, 3198–3214 (2012).
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A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003).
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G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. 48, 1735–1741 (2001).
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A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian–Schell model sources and of their radiation fields,” J. Opt. Soc. Am. A 72, 923–928 (1982).
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W. H. Carter and E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, scalar sources,” Opt. Acta 28, 245–259 (1981).
[CrossRef]

E. Collett and E. Wolf, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Commun. 32, 27–31 (1980).
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L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

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J. Xu and R. Janaswamy, “Electromagnetic degrees of freedom in 2-d scattering environments,” IEEE Trans. Antennas Propag. 54, 3882–3894 (2006).
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Yu, F. T.

F. T. Yu, Entropy and Information Optics (Marcel Dekker, 2000).

F. T. Yu, Optics and Information Theory (Wiley, 1976).

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Zalevsky, Z.

A. Lohmann, R. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space-bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
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H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

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Appl. Opt. (3)

Digit. Signal Process. (1)

B. Dulek and S. Gezici, “Cost minimization of measurement devices under estimation accuracy constraints in the presence of Gaussian noise,” Digit. Signal Process. 22, 828–840 (2012).
[CrossRef]

Electron. Lett. (1)

B. Dulek and S. Gezici, “Average Fisher information maximisation in presence of cost-constrained measurements,” Electron. Lett. 47, 654–656 (2011).
[CrossRef]

IEEE Signal Process. Lett. (2)

F. Garcia, I. Lourtie, and J. Buescu, “L2(R) nonstationary processes and the sampling theorem,” IEEE Signal Process. Lett. 8, 117–119 (2001).
[CrossRef]

F. Oktem and H. Ozaktas, “Exact relation between continuous and discrete linear canonical transforms,” IEEE Signal Process. Lett. 16, 727–730 (2009).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

J. Xu and R. Janaswamy, “Electromagnetic degrees of freedom in 2-d scattering environments,” IEEE Trans. Antennas Propag. 54, 3882–3894 (2006).
[CrossRef]

O. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989).
[CrossRef]

M. Migliore, “On electromagnetics and information theory,” IEEE Trans. Antennas Propag. 56, 3188–3200 (2008).
[CrossRef]

IEEE Trans. Inf. Theory (3)

R. Konsbruck, E. Telatar, and M. Vetterli, “On sampling and coding for distributed acoustic sensing,” IEEE Trans. Inf. Theory 58, 3198–3214 (2012).
[CrossRef]

A. Balakrishnan, “A note on the sampling principle for continuous signals,” IEEE Trans. Inf. Theory 3, 143–146 (1957).
[CrossRef]

W. A. Gardner, “A sampling theorem for nonstationary random processes,” IEEE Trans. Inf. Theory 18, 808–809 (1972).
[CrossRef]

IEEE Trans. Signal Process. (2)

A. Özçelikkale, H. M. Ozaktas, and E. Arıkan, “Signal recovery with cost constrained measurements,” IEEE Trans. Signal Process. 58, 3607–3617 (2010).
[CrossRef]

A. Koç, H. M. Ozaktas, C. Candan, and M. A. Kutay, “Digital computation of linear canonical transforms,” IEEE Trans. Signal Process. 56, 2383–2394 (2008).
[CrossRef]

Inverse Probl. (1)

R. Pierri and F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998).
[CrossRef]

J. Acoust. Soc. Am. (2)

T. W. Barret, “Structural information theory,” J. Acoust. Soc. Am. 54, 1092–1098 (1973).
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M. S. Hughes, “Analysis of digitized wave-forms using Shannon entropy,” J. Acoust. Soc. Am. 93, 892–906 (1993).
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J. Buescu, “Positive integral operators in unbounded domains,” J. Math. Anal. Appl. 296, 244–255 (2004).
[CrossRef]

J. Mod. Opt. (1)

G. Gbur and E. Wolf, “The Rayleigh range of Gaussian Schell-model beams,” J. Mod. Opt. 48, 1735–1741 (2001).
[CrossRef]

J. Opt. (1)

P. Jixiong, “Waist location and Rayleigh range for Gaussian Schell-model beams,” J. Opt. 22, 157–159 (1991).
[CrossRef]

J. Opt. A (1)

A. Thaning, P. Martinsson, M. Karelin, and A. T. Friberg, “Limits of diffractive optics by communication modes,” J. Opt. A 5, 153–158 (2003).
[CrossRef]

J. Opt. Soc. Am. (7)

J. Opt. Soc. Am. A (15)

G. Newsam and R. Barakat, “Essential dimension as a well-defined number of degrees of freedom of finite-convolution operators appearing in optics,” J. Opt. Soc. Am. A 2, 2040–2045 (1985).
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R. Piestun and D. A. B. Miller, “Electromagnetic degrees of freedom of an optical system,” J. Opt. Soc. Am. A 17, 892–902 (2000).
[CrossRef]

F. S. Oktem and H. M. Ozaktas, “Equivalence of linear canonical transform domains to fractional Fourier domains and the bicanonical width product: a generalization of the space–bandwidth product,” J. Opt. Soc. Am. A 27, 1885–1895 (2010).
[CrossRef]

A. Lohmann, R. Dorsch, D. Mendlovic, Z. Zalevsky, and C. Ferreira, “Space-bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996).
[CrossRef]

D. Mendlovic and A. W. Lohmann, “Space-bandwidth product adaptation and its application to superresolution: fundamentals,” J. Opt. Soc. Am. A 14, 558–562 (1997).
[CrossRef]

L. Onural, “Exact analysis of the effects of sampling of the scalar diffraction field,” J. Opt. Soc. Am. A 24, 359–367 (2007).
[CrossRef]

J. J. Healy and J. T. Sheridan, “Space–bandwidth ratio as a means of choosing between Fresnel and other linear canonical transform algorithms,” J. Opt. Soc. Am. A 28, 786–790 (2011).
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M. J. Bastiaans, “Uncertainty principle and informational entropy for partially coherent light,” J. Opt. Soc. Am. A 3, 1243–1246 (1986).
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H. M. Ozaktas, S. Yüksel, and M. A. Kutay, “Linear algebraic theory of partial coherence: discrete fields and measures of partial coherence,” J. Opt. Soc. Am. A 19, 1563–1571 (2002).
[CrossRef]

A. Stern and B. Javidi, “Shannon number and information capacity of three-dimensional integral imaging,” J. Opt. Soc. Am. A 21, 1602–1612 (2004).
[CrossRef]

P. Réfrégier and J. Morio, “Shannon entropy of partially polarized and partially coherent light with Gaussian fluctuations,” J. Opt. Soc. Am. A 23, 3036–3044 (2006).
[CrossRef]

E. D. Micheli and G. A. Viano, “Inverse optical imaging viewed as a backward channel communication problem,” J. Opt. Soc. Am. A 26, 1393–1402 (2009).
[CrossRef]

A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian–Schell model sources and of their radiation fields,” J. Opt. Soc. Am. A 72, 923–928 (1982).
[CrossRef]

A. T. Friberg and J. Turunen, “Imaging of Gaussian–Schell model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988).
[CrossRef]

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. A 69, 1710–1716 (1979).
[CrossRef]

J. Phys. D (1)

D. Blacknell and C. J. Oliver, “Information-content of coherent images,” J. Phys. D 26, 1364–1370 (1993).
[CrossRef]

Opt. Acta (1)

W. H. Carter and E. Wolf, “Correlation theory of wavefields generated by fluctuating, three-dimensional, scalar sources,” Opt. Acta 28, 245–259 (1981).
[CrossRef]

Opt. Commun. (5)

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[CrossRef]

E. Collett and E. Wolf, “Beams generated by Gaussian quasi-homogeneous sources,” Opt. Commun. 32, 27–31 (1980).
[CrossRef]

F. Gori, “Collett-Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

R. Barakat, “Some entropic aspects of optical diffraction imagery,” Opt. Commun. 156, 235–239 (1998).
[CrossRef]

F. Gori and G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973).
[CrossRef]

Opt. Eng. (1)

A. Stern, and B. Javidi, “Analysis of practical sampling and reconstruction from Fresnel fields,” Opt. Eng. 43, 239–250 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Trans. IRE Prof. Group Inf. Theory (1)

D. MacKay, “Quantal aspects of scientific information,” Trans. IRE Prof. Group Inf. Theory 1, 60–80 (1953).
[CrossRef]

Other (10)

F. T. Yu, Entropy and Information Optics (Marcel Dekker, 2000).

F. T. Yu, Optics and Information Theory (Wiley, 1976).

D. Gabor, “Light and information,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1961), Vol. I, pp. 109–153.

F. Gori, Advanced Topics in Shannon Sampling and Interpolation Theory (Springer-Verlag, 1993), pp. 37–83.

H. L. Van Trees, Detection, Estimation and Modulation Theory, Part I (Wiley, 2001).

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

K. B. Wolf, Integral Transforms in Science and Engineering (Plenum, 1979).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

R. A. Horn and C. R. Johnson, Matrix Analysis (Cambridge University, 1990).

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

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Figures (8)

Fig. 1.
Fig. 1.

Error versus cost budget CB, β=1/8 (varying SNR).

Fig. 2.
Fig. 2.

Error versus cost budget CB, β=1 (varying SNR).

Fig. 3.
Fig. 3.

Number of samples and optimum sampling interval versus cost budget, β=1/8, SNR=.

Fig. 4.
Fig. 4.

Number of samples and optimum sampling interval versus cost budget, β=1/8, SNR=1.

Fig. 5.
Fig. 5.

Number of samples and optimum sampling interval versus cost budget, β=1, SNR=.

Fig. 6.
Fig. 6.

Number of samples and optimum sampling interval versus cost budget, β=1, SNR=1.

Fig. 7.
Fig. 7.

Error versus cost budget CB, β=1/8 (varying SNR). The dotted lines are for optimal sampling strategies and the corresponding dashed and solid lines are for sampling theorem-based strategies.

Fig. 8.
Fig. 8.

Error versus cost budget CB, β=1 (varying SNR). The dotted lines are for optimal sampling strategies and the corresponding dashed and solid lines are for sampling theorem-based strategies.

Equations (28)

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

g(x)=L{f(x)}+n(x),
si=g(ξi)+mi,
ε(CB)=minΔx,x0,ME[D|f(x)f^(x|s)|2dx],
Kfs(x)=h(x)Ks,
E[|f(x)f^D(x)|2dx]=E[xD|f(x)f^D(x)|2dx]+E[xD|f(x)f^D(x)|2dx]
=E[xD|f(x)f^D(x)|2dx]+E[xD|f(x)|2dx]
=ε(CB)+xDKf(x,x)dx.
Kf(x1,x2)=Afexp(x12+x224σI2)exp((x1x2)22σν2)exp(jk2R(x12x22)).
β=σνσI.
KσI,β,R(x1,x2)=Afexp(x12+x224σI2)exp((x1x2)22(βσI)2)exp(jk2R(x12x22)).
KσI,β,(x1,x2)=KσI,β,(x1σIσI,x2σIσI),
E[f¯(x)f¯*(x)]=KσI,β,R(x1,x2)
=KσI,β,(x1,x2)exp(jk2R(x12x22))
=E[f(x)f*(x)]exp(jk2R(x12x22)).
Ks¯=Kg¯+Km
=TKgT+TKmT
=TKsT,
d¯(x)=exp((jk/2R)x2))d(x)T.
E[|f¯(x)f¯^(x|s¯)|2]=KσI,β,R(x,x)d¯(x)Ks¯s¯1d¯(x)
=KσI,β,(x,x)d(x)Kss1d(x)
=E[|f(x)f^(x|s)|2].
Ms=2rσI1/(2rσI,F)=2r2π(1β2+14)0.5.
Kf(x1,x2)=k=0λkϕk(x1)ϕk*(x2),
f(x)=k=1zkϕk(x),xD,
E[D|f(x)k=1Nzkϕk(x)|2dx]=E[D|k=1zkϕk(x)k=1Nzkϕk(x)|2dx]
=E[D|k=N+1zkϕk(x)|2dx]
=k=N+1E[|zk|2]
=k=N+1λk.

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