Abstract

Imaging technologies such as dynamic viewpoint generation are engineered for incoherent radiation using the traditional light field, and for coherent radiation using electromagnetic field theory. We present a model of coherent image formation that strikes a balance between the utility of the light field and the comprehensive predictive power of Maxwell’s equations. We synthesize research in optics and signal processing to formulate, capture, and form images from quasi light fields, which extend the light field from incoherent to coherent radiation. Our coherent cameras generalize the classic beamforming algorithm in sensor array processing and invite further research on alternative notions of image formation.

© 2009 Optical Society of America

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References

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  1. M. Levoy and P. Hanrahan, “Light field rendering,” in Proceedings of ACM SIGGRAPH 96 (ACM, 1996), pp. 31-42.
    [CrossRef]
  2. R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, Stanford University, Calif. (2005).
  3. W. Chun and O. S. Cossairt, “Data processing for three-dimensional displays,” United States Patent 7,525,541 (April 28, 2009).
  4. R. Ziegler, S. Bucheli, L. Ahrenberg, M. Magnor, and M. Gross, “A bidirectional light field-hologram transform,” Comput. Graph. Forum 26, 435-446 (2007).
    [CrossRef]
  5. T. L. Szabo, Diagnostic Ultrasound Imaging: Inside Out (Elsevier, 2004).
  6. M. J. Bastiaans, “Application of the Wigner distribution function in optics,” in The Wigner Distribution--Theory and Applications in Signal Processing, W.Mecklenbräuker and F.Hlawatsch, eds. (Elsevier Science B.V., 1997), pp. 375-426.
  7. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 58, 1256-1259 (1968).
    [CrossRef]
  8. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
    [CrossRef]
  9. J. Ville, “Théorie et applications de la notion de signal analytique,” Cables Transm. 2A, 61-74 (1948).
  10. K. D. Stephan, “Radiometry before World War II: Measuring infrared and millimeter-wave radiation 1800-1925,” IEEE Antennas Propag. Mag. 47, 28-37 (2005).
    [CrossRef]
  11. S. Chandrasekhar, Radiative Transfer (Dover, 1960).
  12. A. T. Friberg, G. S. Agarwal, J. T. Foley, and E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386-1393 (1992).
    [CrossRef]
  13. P. Moon and G. Timoshenko, “The light field,” J. Math. Phys. 18, 51-151 (1939). [Translation of A. Gershun, The Light Field (Moscow, 1936)].
  14. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).
  15. A. Walther, “Radiometry and coherence,” J. Opt. Soc. Am. 63, 1622-1623 (1973).
    [CrossRef]
  16. E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6-17 (1978).
    [CrossRef]
  17. G. S. Agarwal, J. T. Foley, and E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67-72 (1987).
    [CrossRef]
  18. E. H. Adelson and J. R. Bergen, “The plenoptic function and the elements of early vision,” in Computational Models of Visual Processing, M.S.Landy and J.A.Movshon, eds. (MIT Press, 1991), pp. 3-20.
  19. S. J. Gortler, R. Grzeszczuk, R. Szeliski, and M. F. Cohen, “The lumigraph,” in Proceedings of ACM SIGGRAPH 96 (ACM, 1996), pp. 43-54.
    [CrossRef]
  20. B.Boashash, ed. Time Frequency Signal Analysis and Processing (Elsevier, 2003).
  21. R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983).
  22. A. Adams and M. Levoy, “General linear cameras with finite aperture,” in Proc. Eurographics Symposium on Rendering (Eurographics, 2007).
  23. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  24. A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary states of coherence,” J. Opt. Soc. Am. 69, 192-198 (1979).
    [CrossRef]
  25. P. Flandrin, Time-Frequency/Time-Scale Analysis (Academic, 1999).
  26. D. J. Griffiths, Introduction to Quantum Mechanics (Pearson Education, 2005).
  27. G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. Mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161-2186 (1970).
    [CrossRef]
  28. J. T. Foley and E. Wolf, “Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236-241 (1985).
    [CrossRef]
  29. J. R. Guerci, “Theory and application of covariance matrix tapers for robust adaptive beamforming,” IEEE Trans. Signal Process. 47, 977-985 (1999).
    [CrossRef]
  30. Z. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in Proceedings of ICCP 09 (IEEE, 2009).
  31. J. G. Kirkwood, “Quantum statistics of almost classical assemblies,” Phys. Rev. 44, 31-37 (1933).
    [CrossRef]
  32. A. Rihaczek, “Signal energy distribution in time and frequency,” IEEE Trans. Inf. Theory 14, 369-374 (1968).
    [CrossRef]
  33. ZEMAX Development Corporation, Bellevue, Wash., Optical Design Program User's Guide (2006).
  34. M. A. Alonso, “Radiometry and wide-angle wave fields. I. Coherent fields in two dimensions,” J. Opt. Soc. Am. A 18, 902-909 (2001).
    [CrossRef]
  35. R. G. Littlejohn and R. Winston, “Corrections to classical radiometry,” J. Opt. Soc. Am. A 10, 2024-2037 (1993).
    [CrossRef]
  36. J. R. Shewell and E. Wolf, “Inverse diffraction and a new reciprocity theorem,” J. Opt. Soc. Am. 58, 1596-1603 (1968).
    [CrossRef]
  37. H. L. Van Trees, Optimum Array Processing (Wiley, 2002).
    [CrossRef]
  38. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge Univ. Press, 1995).
  39. A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett. 95, 043904 (2005).
    [CrossRef] [PubMed]
  40. R. W. Schoonover, A. M. Zysk, P. S. Carney, J. C. Schotland, and E. Wolf, “Geometrical optics limit of stochastic electromagnetic fields,” Phys. Rev. A 77, 043831 (2008).
    [CrossRef]

2008 (1)

R. W. Schoonover, A. M. Zysk, P. S. Carney, J. C. Schotland, and E. Wolf, “Geometrical optics limit of stochastic electromagnetic fields,” Phys. Rev. A 77, 043831 (2008).
[CrossRef]

2007 (1)

R. Ziegler, S. Bucheli, L. Ahrenberg, M. Magnor, and M. Gross, “A bidirectional light field-hologram transform,” Comput. Graph. Forum 26, 435-446 (2007).
[CrossRef]

2005 (2)

K. D. Stephan, “Radiometry before World War II: Measuring infrared and millimeter-wave radiation 1800-1925,” IEEE Antennas Propag. Mag. 47, 28-37 (2005).
[CrossRef]

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

2001 (1)

1999 (1)

J. R. Guerci, “Theory and application of covariance matrix tapers for robust adaptive beamforming,” IEEE Trans. Signal Process. 47, 977-985 (1999).
[CrossRef]

1993 (1)

1992 (1)

1987 (1)

G. S. Agarwal, J. T. Foley, and E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67-72 (1987).
[CrossRef]

1985 (1)

J. T. Foley and E. Wolf, “Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236-241 (1985).
[CrossRef]

1979 (1)

1978 (1)

1973 (1)

1970 (1)

G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. Mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161-2186 (1970).
[CrossRef]

1968 (3)

1948 (1)

J. Ville, “Théorie et applications de la notion de signal analytique,” Cables Transm. 2A, 61-74 (1948).

1939 (1)

P. Moon and G. Timoshenko, “The light field,” J. Math. Phys. 18, 51-151 (1939). [Translation of A. Gershun, The Light Field (Moscow, 1936)].

1933 (1)

J. G. Kirkwood, “Quantum statistics of almost classical assemblies,” Phys. Rev. 44, 31-37 (1933).
[CrossRef]

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Adams, A.

A. Adams and M. Levoy, “General linear cameras with finite aperture,” in Proc. Eurographics Symposium on Rendering (Eurographics, 2007).

Adelson, E. H.

E. H. Adelson and J. R. Bergen, “The plenoptic function and the elements of early vision,” in Computational Models of Visual Processing, M.S.Landy and J.A.Movshon, eds. (MIT Press, 1991), pp. 3-20.

Agarwal, G. S.

A. T. Friberg, G. S. Agarwal, J. T. Foley, and E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386-1393 (1992).
[CrossRef]

G. S. Agarwal, J. T. Foley, and E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67-72 (1987).
[CrossRef]

G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. Mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161-2186 (1970).
[CrossRef]

Ahrenberg, L.

R. Ziegler, S. Bucheli, L. Ahrenberg, M. Magnor, and M. Gross, “A bidirectional light field-hologram transform,” Comput. Graph. Forum 26, 435-446 (2007).
[CrossRef]

Alonso, M. A.

Bastiaans, M. J.

M. J. Bastiaans, “Application of the Wigner distribution function in optics,” in The Wigner Distribution--Theory and Applications in Signal Processing, W.Mecklenbräuker and F.Hlawatsch, eds. (Elsevier Science B.V., 1997), pp. 375-426.

Bergen, J. R.

E. H. Adelson and J. R. Bergen, “The plenoptic function and the elements of early vision,” in Computational Models of Visual Processing, M.S.Landy and J.A.Movshon, eds. (MIT Press, 1991), pp. 3-20.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).

Boyd, R. W.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983).

Brédif, M.

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, Stanford University, Calif. (2005).

Bucheli, S.

R. Ziegler, S. Bucheli, L. Ahrenberg, M. Magnor, and M. Gross, “A bidirectional light field-hologram transform,” Comput. Graph. Forum 26, 435-446 (2007).
[CrossRef]

Carney, P. S.

R. W. Schoonover, A. M. Zysk, P. S. Carney, J. C. Schotland, and E. Wolf, “Geometrical optics limit of stochastic electromagnetic fields,” Phys. Rev. A 77, 043831 (2008).
[CrossRef]

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

Chun, W.

W. Chun and O. S. Cossairt, “Data processing for three-dimensional displays,” United States Patent 7,525,541 (April 28, 2009).

Cohen, M. F.

S. J. Gortler, R. Grzeszczuk, R. Szeliski, and M. F. Cohen, “The lumigraph,” in Proceedings of ACM SIGGRAPH 96 (ACM, 1996), pp. 43-54.
[CrossRef]

Cossairt, O. S.

W. Chun and O. S. Cossairt, “Data processing for three-dimensional displays,” United States Patent 7,525,541 (April 28, 2009).

Duval, G.

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, Stanford University, Calif. (2005).

Flandrin, P.

P. Flandrin, Time-Frequency/Time-Scale Analysis (Academic, 1999).

Foley, J. T.

A. T. Friberg, G. S. Agarwal, J. T. Foley, and E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386-1393 (1992).
[CrossRef]

G. S. Agarwal, J. T. Foley, and E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67-72 (1987).
[CrossRef]

J. T. Foley and E. Wolf, “Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236-241 (1985).
[CrossRef]

Friberg, A. T.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Gortler, S. J.

S. J. Gortler, R. Grzeszczuk, R. Szeliski, and M. F. Cohen, “The lumigraph,” in Proceedings of ACM SIGGRAPH 96 (ACM, 1996), pp. 43-54.
[CrossRef]

Griffiths, D. J.

D. J. Griffiths, Introduction to Quantum Mechanics (Pearson Education, 2005).

Gross, M.

R. Ziegler, S. Bucheli, L. Ahrenberg, M. Magnor, and M. Gross, “A bidirectional light field-hologram transform,” Comput. Graph. Forum 26, 435-446 (2007).
[CrossRef]

Grzeszczuk, R.

S. J. Gortler, R. Grzeszczuk, R. Szeliski, and M. F. Cohen, “The lumigraph,” in Proceedings of ACM SIGGRAPH 96 (ACM, 1996), pp. 43-54.
[CrossRef]

Guerci, J. R.

J. R. Guerci, “Theory and application of covariance matrix tapers for robust adaptive beamforming,” IEEE Trans. Signal Process. 47, 977-985 (1999).
[CrossRef]

Hanrahan, P.

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, Stanford University, Calif. (2005).

M. Levoy and P. Hanrahan, “Light field rendering,” in Proceedings of ACM SIGGRAPH 96 (ACM, 1996), pp. 31-42.
[CrossRef]

Horowitz, M.

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, Stanford University, Calif. (2005).

Kirkwood, J. G.

J. G. Kirkwood, “Quantum statistics of almost classical assemblies,” Phys. Rev. 44, 31-37 (1933).
[CrossRef]

Levoy, M.

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, Stanford University, Calif. (2005).

M. Levoy and P. Hanrahan, “Light field rendering,” in Proceedings of ACM SIGGRAPH 96 (ACM, 1996), pp. 31-42.
[CrossRef]

A. Adams and M. Levoy, “General linear cameras with finite aperture,” in Proc. Eurographics Symposium on Rendering (Eurographics, 2007).

Z. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in Proceedings of ICCP 09 (IEEE, 2009).

Littlejohn, R. G.

Magnor, M.

R. Ziegler, S. Bucheli, L. Ahrenberg, M. Magnor, and M. Gross, “A bidirectional light field-hologram transform,” Comput. Graph. Forum 26, 435-446 (2007).
[CrossRef]

Mandel, L.

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

Moon, P.

P. Moon and G. Timoshenko, “The light field,” J. Math. Phys. 18, 51-151 (1939). [Translation of A. Gershun, The Light Field (Moscow, 1936)].

Ng, R.

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, Stanford University, Calif. (2005).

Rihaczek, A.

A. Rihaczek, “Signal energy distribution in time and frequency,” IEEE Trans. Inf. Theory 14, 369-374 (1968).
[CrossRef]

Schoonover, R. W.

R. W. Schoonover, A. M. Zysk, P. S. Carney, J. C. Schotland, and E. Wolf, “Geometrical optics limit of stochastic electromagnetic fields,” Phys. Rev. A 77, 043831 (2008).
[CrossRef]

Schotland, J. C.

R. W. Schoonover, A. M. Zysk, P. S. Carney, J. C. Schotland, and E. Wolf, “Geometrical optics limit of stochastic electromagnetic fields,” Phys. Rev. A 77, 043831 (2008).
[CrossRef]

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

Shewell, J. R.

Stephan, K. D.

K. D. Stephan, “Radiometry before World War II: Measuring infrared and millimeter-wave radiation 1800-1925,” IEEE Antennas Propag. Mag. 47, 28-37 (2005).
[CrossRef]

Szabo, T. L.

T. L. Szabo, Diagnostic Ultrasound Imaging: Inside Out (Elsevier, 2004).

Szeliski, R.

S. J. Gortler, R. Grzeszczuk, R. Szeliski, and M. F. Cohen, “The lumigraph,” in Proceedings of ACM SIGGRAPH 96 (ACM, 1996), pp. 43-54.
[CrossRef]

Timoshenko, G.

P. Moon and G. Timoshenko, “The light field,” J. Math. Phys. 18, 51-151 (1939). [Translation of A. Gershun, The Light Field (Moscow, 1936)].

Van Trees, H. L.

H. L. Van Trees, Optimum Array Processing (Wiley, 2002).
[CrossRef]

Ville, J.

J. Ville, “Théorie et applications de la notion de signal analytique,” Cables Transm. 2A, 61-74 (1948).

Walther, A.

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
[CrossRef]

Winston, R.

Wolf, E.

R. W. Schoonover, A. M. Zysk, P. S. Carney, J. C. Schotland, and E. Wolf, “Geometrical optics limit of stochastic electromagnetic fields,” Phys. Rev. A 77, 043831 (2008).
[CrossRef]

A. T. Friberg, G. S. Agarwal, J. T. Foley, and E. Wolf, “Statistical wave-theoretical derivation of the free-space transport equation of radiometry,” J. Opt. Soc. Am. B 9, 1386-1393 (1992).
[CrossRef]

G. S. Agarwal, J. T. Foley, and E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67-72 (1987).
[CrossRef]

J. T. Foley and E. Wolf, “Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236-241 (1985).
[CrossRef]

E. Wolf, “Coherence and radiometry,” J. Opt. Soc. Am. 68, 6-17 (1978).
[CrossRef]

G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. Mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161-2186 (1970).
[CrossRef]

J. R. Shewell and E. Wolf, “Inverse diffraction and a new reciprocity theorem,” J. Opt. Soc. Am. 58, 1596-1603 (1968).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).

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

Zhang, Z.

Z. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in Proceedings of ICCP 09 (IEEE, 2009).

Ziegler, R.

R. Ziegler, S. Bucheli, L. Ahrenberg, M. Magnor, and M. Gross, “A bidirectional light field-hologram transform,” Comput. Graph. Forum 26, 435-446 (2007).
[CrossRef]

Zysk, A. M.

R. W. Schoonover, A. M. Zysk, P. S. Carney, J. C. Schotland, and E. Wolf, “Geometrical optics limit of stochastic electromagnetic fields,” Phys. Rev. A 77, 043831 (2008).
[CrossRef]

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

Cables Transm. (1)

J. Ville, “Théorie et applications de la notion de signal analytique,” Cables Transm. 2A, 61-74 (1948).

Comput. Graph. Forum (1)

R. Ziegler, S. Bucheli, L. Ahrenberg, M. Magnor, and M. Gross, “A bidirectional light field-hologram transform,” Comput. Graph. Forum 26, 435-446 (2007).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

K. D. Stephan, “Radiometry before World War II: Measuring infrared and millimeter-wave radiation 1800-1925,” IEEE Antennas Propag. Mag. 47, 28-37 (2005).
[CrossRef]

IEEE Trans. Inf. Theory (1)

A. Rihaczek, “Signal energy distribution in time and frequency,” IEEE Trans. Inf. Theory 14, 369-374 (1968).
[CrossRef]

IEEE Trans. Signal Process. (1)

J. R. Guerci, “Theory and application of covariance matrix tapers for robust adaptive beamforming,” IEEE Trans. Signal Process. 47, 977-985 (1999).
[CrossRef]

J. Math. Phys. (1)

P. Moon and G. Timoshenko, “The light field,” J. Math. Phys. 18, 51-151 (1939). [Translation of A. Gershun, The Light Field (Moscow, 1936)].

J. Opt. Soc. Am. (5)

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

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

Opt. Commun. (2)

J. T. Foley and E. Wolf, “Radiometry as a short-wavelength limit of statistical wave theory with globally incoherent sources,” Opt. Commun. 55, 236-241 (1985).
[CrossRef]

G. S. Agarwal, J. T. Foley, and E. Wolf, “The radiance and phase-space representations of the cross-spectral density operator,” Opt. Commun. 62, 67-72 (1987).
[CrossRef]

Phys. Rev. (2)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
[CrossRef]

J. G. Kirkwood, “Quantum statistics of almost classical assemblies,” Phys. Rev. 44, 31-37 (1933).
[CrossRef]

Phys. Rev. A (1)

R. W. Schoonover, A. M. Zysk, P. S. Carney, J. C. Schotland, and E. Wolf, “Geometrical optics limit of stochastic electromagnetic fields,” Phys. Rev. A 77, 043831 (2008).
[CrossRef]

Phys. Rev. D (1)

G. S. Agarwal and E. Wolf, “Calculus for functions of noncommuting operators and general phase-space methods in quantum mechanics. I. Mapping theorems and ordering of functions of noncommuting operators,” Phys. Rev. D 2, 2161-2186 (1970).
[CrossRef]

Phys. Rev. Lett. (1)

A. M. Zysk, P. S. Carney, and J. C. Schotland, “Eikonal method for calculation of coherence functions,” Phys. Rev. Lett. 95, 043904 (2005).
[CrossRef] [PubMed]

Other (19)

ZEMAX Development Corporation, Bellevue, Wash., Optical Design Program User's Guide (2006).

Z. Zhang and M. Levoy, “Wigner distributions and how they relate to the light field,” in Proceedings of ICCP 09 (IEEE, 2009).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).

H. L. Van Trees, Optimum Array Processing (Wiley, 2002).
[CrossRef]

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

S. Chandrasekhar, Radiative Transfer (Dover, 1960).

T. L. Szabo, Diagnostic Ultrasound Imaging: Inside Out (Elsevier, 2004).

M. J. Bastiaans, “Application of the Wigner distribution function in optics,” in The Wigner Distribution--Theory and Applications in Signal Processing, W.Mecklenbräuker and F.Hlawatsch, eds. (Elsevier Science B.V., 1997), pp. 375-426.

M. Levoy and P. Hanrahan, “Light field rendering,” in Proceedings of ACM SIGGRAPH 96 (ACM, 1996), pp. 31-42.
[CrossRef]

R. Ng, M. Levoy, M. Brédif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Tech. Rep. CTSR 2005-02, Stanford University, Calif. (2005).

W. Chun and O. S. Cossairt, “Data processing for three-dimensional displays,” United States Patent 7,525,541 (April 28, 2009).

E. H. Adelson and J. R. Bergen, “The plenoptic function and the elements of early vision,” in Computational Models of Visual Processing, M.S.Landy and J.A.Movshon, eds. (MIT Press, 1991), pp. 3-20.

S. J. Gortler, R. Grzeszczuk, R. Szeliski, and M. F. Cohen, “The lumigraph,” in Proceedings of ACM SIGGRAPH 96 (ACM, 1996), pp. 43-54.
[CrossRef]

B.Boashash, ed. Time Frequency Signal Analysis and Processing (Elsevier, 2003).

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, 1983).

A. Adams and M. Levoy, “General linear cameras with finite aperture,” in Proc. Eurographics Symposium on Rendering (Eurographics, 2007).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

P. Flandrin, Time-Frequency/Time-Scale Analysis (Academic, 1999).

D. J. Griffiths, Introduction to Quantum Mechanics (Pearson Education, 2005).

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Figures (6)

Fig. 1
Fig. 1

We can compute the value of each pixel in an image produced by an arbitrary virtual camera, defined as the power emitted from a scene surface patch toward a virtual aperture, by integrating an appropriate bundle of light field rays that have been previously captured with remote hardware.

Fig. 2
Fig. 2

We capture a discrete quasi light field l by sampling the scalar field at regularly spaced sensors and processing the resulting measurements. We may optionally apply an aperture stop T to mimic traditional light field capture, but this restricts us to capturing quasi light fields with poor localization properties.

Fig. 3
Fig. 3

The spectrogram does not resolve a plane wave propagating past the edge of an opaque screen as well as other quasi light fields, such as the Wigner and Rihaczek. We capture all three quasi light fields by sampling the scalar field with sensors and processing the measurements according to Eqs. (25, 26, 28). The ringing and blurring in the light field plots indicate the diffraction fringes and energy localization limitations.

Fig. 4
Fig. 4

To ensure that integrating bundles of remote light field rays in the near zone results in an accurate image, we derive a light field L ρ R ( r , s ) in the measurement plane from the infinitesimal flux d Φ at the point r P where the ray originates from the scene surface patch. We thereby avoid making the assumptions that the measurement plane is far from the scene and that the light field is constant along rays.

Fig. 5
Fig. 5

Near-zone light fields result in cameras that align spherical wavefronts diverging from the point of focus r P , in accordance with the Huygens–Fresnel principle of diffraction, while quasi light fields result in cameras that align plane wavefront approximations in accordance with Fraunhofer diffraction. Quasi light fields are therefore accurate only in the far zone. (a) We derive both cameras by approximating the integral over a bundle of rays by the summation of discrete light field rays, and (b) we interpret the operation of each camera by how they align sensor measurements along wavefronts from r P .

Fig. 6
Fig. 6

Images of nearby objects formed from pure quasi light fields are blurry. In the scene, a small backlit pinhole moves across the field of view of a sensor array that implements three cameras, each computing one pixel value for each pinhole position, corresponding to a fixed surface patch. As the pinhole crosses the fixed scene surface patch, the near-zone camera resolves the pinhole down to its actual size of 1 mm , while the far-zone camera records a blur 66 cm wide.

Equations (47)

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P = σ Ω r L ( r , s ) cos ψ d 2 s d 2 r ,
U ( ρ s ) = 2 π i k s z exp ( i k ρ ) ρ a ( s ) ,
a ( s ) = ( k 2 π ) 2 U ( r ) exp ( i k s r ) d 2 r
F ( ρ s ) = ( 2 π k ) 2 a ( s ) a * ( s ) s z 2 ρ 2 s ,
d Φ = ( 2 π k ) 2 s z 2 a ( s ) a * ( s ) d Ω .
I ( s ) = d Φ d Ω = ( 2 π k ) 2 s z 2 a ( s ) a * ( s ) .
I ( s ) = s z [ ( k 2 π ) 2 s z U ( r + 1 2 r ) U * ( r 1 2 r ) exp ( i k s r ) d 2 r ] d 2 r .
L W ( r , s ) = ( k 2 π ) 2 s z W ( r , s λ ) ,
W ( r , s ) = U ( r + 1 2 r ) U * ( r 1 2 r ) exp ( i 2 π s r ) d 2 r
[ x ̂ , s ̂ x ] = i λ 2 π , [ y ̂ , s ̂ y ] = i λ 2 π ,
r R | L ̂ | r C = U ( r R ) U * ( r C ) ,
L Ω ( r , s ) = ( k 2 π ) 2 s z L ̃ Ω ( r , s )
L Ω ( r , s ) = k 2 ( 2 π ) 4 s z Ω ̃ ( u , k r ) exp [ i u ( r r ) ] exp ( i k s r ) U ( r + 1 2 r ) U * ( r 1 2 r ) d 2 u d 2 r d 2 r ,
Ω ̃ ( u , v ) = Π ( a , b ) exp [ i ( a u + b v ) ] d 2 a d 2 b
L Ω ( r , s ) = ( k 2 π ) 2 s z Π ( r r , s s ) W ( r , s λ ) d 2 r d 2 s = ( k 2 π ) 2 s z Π ( r , s ) W ( r , s λ ) .
1 Ω ̃ be an entire analytic function with no zeros on the real component axes .
Ω ̃ ( 0 , v ) = 1 for all v ,
Π ( a , b ) = K ( a + λ 2 v , a λ 2 v ) exp ( i 2 π b v ) d 2 v .
r R = r + 1 2 r , r C = r 1 2 r
L ( r , s ) = ( k 2 π ) 2 s z U ( r R ) { K ( r R r , r C r ) × exp [ i k s ( r R r C ) ] } U * ( r C ) d 2 r R d 2 r C .
U ( r ) U ( r r 0 ) exp ( i k s 0 r )
L ( r , s ) L ( r r 0 , s s 0 ) .
L ( r , s ) = ( k 2 π ) 2 s z { U ( r R ) exp [ i k s ( r r R ) ] } K ( r R r , r C r ) × { U ( r C ) exp [ i k s ( r r C ) ] } * d 2 r R d 2 r C .
U ( r R ) K ( r R r , r C r ) U * ( r C ) .
l S ( y , s y ) = | n T ( n d ) U ( y + n d ) exp ( i k n d s y ) | 2 .
l W ( y , s y ) = n U ( y + n d 2 ) U * ( y n d 2 ) exp ( i k n d s y ) .
L R ( r , s ) = s z U * ( r ) exp ( i k s r ) a ( s ) .
l R ( y , s y ) = U * ( y ) exp ( i k y s y ) n U ( n d ) exp ( i k n d s y ) .
U ( r P ) = i k 2 π U ( r M ) z P | r P r M | exp ( i k | r P r M | ) | r P r M | d 2 r M .
F ( r P ) = 1 4 π k ν [ U * t U + U t U * ] .
| r P r M | | r P |
| r P | | r P r M | 1 ,
F ( ρ s ) = ( 2 π k ) 2 a ̃ ( ρ s ) a ̃ * ( ρ s ) s z 2 ρ 2 s ,
a ̃ ( ρ s ) = ( k 2 π ) 2 U ( r M ) exp ( i k | ρ s r M | ) d 2 r M .
d Φ = ( 2 π k ) 2 s z 2 a ̃ ( ρ s ) a ̃ * ( ρ s ) d Ω .
L ρ R ( r , s ) = s z U * ( r ) exp ( i k ρ ) a ̃ ( r ρ s ) = ( k 2 π ) 2 s z U * ( r ) exp ( i k ρ ) U ( r M ) exp ( i k | r ρ s r M | ) d 2 r M ,
P R = | n d | < A 2 { [ U ( n d ) exp ( i k n d s y n ) ] * m U ( m d ) exp ( i k m d s y n ) } ,
P ρ R = [ | n d | < A 2 U ( n d ) exp ( i k Δ n ) ] * [ m U ( m d ) exp ( i k Δ m ) ] .
g = m T ( m d ) U ( m d ) exp ( i k Δ m ) ,
Δ m Δ 0 m d s y m m d s y 0 ,
g = m T ( m d ) U ( m d ) exp ( i k m d s y 0 ) .
P ρ R = g 1 * g 2 ,
g 1 = | n d | < A 2 U ( n d ) exp ( i k Δ n )
g 2 = m U ( m d ) exp ( i k Δ m ) ,
P ρ B = | g 1 | 2 .
P R ( g 1 ) * g 2 ,
| P ρ R | = | g 1 | 2 | g 2 | 2 .

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