Abstract

We present a new passive depth detection method for quasi-monochromatic and spatially incoherent objects. We utilize the wavefront discrimination properties of a volume holographic pupil combined with a measurement of the degree of coherence of the diffracted field. Depth detection is posed as the Bayesian hypothesis testing on the outcome of the coherence measurement. We present the analysis of our proposed optical system and experimental results confirming binary depth discrimination with high confidence for featureless objects.

© 2008 Optical Society of America

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References

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  1. B. Curless, “Overview of active vision techniques: SIGGRAPH 99 Course on 3D Photography” (1999), http://www.cs.cmu.edu/~seitz/course/SIGG00/slides/curless-active.pdf.
  2. A. P. Cracknell and L. Hayes, Introduction to Remote Sensing (Taylor &Francis, 2007).
  3. C. J. R. Sheppard and D. M. Shotton, Confocal Laser Scanning Microscopy (Springer, 1997).
  4. W. Sun and G. Barbastathis, “Rainbow volume holographic imaging,” Opt. Lett. 30, 976-978 (2005).
    [CrossRef] [PubMed]
  5. B. K. P. Horn, Robot Vision (MIT Press, 1986).
  6. J. Y. Bouguet and P. Perona, “3D photography using shadows in dual-space geometry,” Int. J. Comput. Vis. 35, 129-149(1999).
    [CrossRef]
  7. D. Marr and T. Poggio, “A theory of human stereo vision,” Artificial Intelligence Lab Publications, AI Memos AIM-451 (MIT, 1977), http://dspace.mit.edu/handle/1721.1/6093.
  8. N. Joshi, W. Matusik, and S. Avidan, “Natural video matting using camera arrays,” ACM Trans. Graphics 25, 779-786(2006).
    [CrossRef]
  9. J. Ens and P. Lawrence, “An investigation of methods for determining depth from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 97-108 (1993).
    [CrossRef]
  10. S. Chaudhuri and A. Rajagopalan, Depth from Defocus: a Real Aperture Imaging Approach (Springer, 1999).
    [CrossRef]
  11. E. R. Dowski, Jr., and W. T. Cathey, “Single-lens single-image incoherent passive-ranging systems,” Appl. Opt. 33, 6762-6773 (1994).
    [CrossRef] [PubMed]
  12. G. Barbastathis and D. J. Brady, “Multidimensional tomographic imaging using volume holography,” Proc. IEEE 87, 2098-2120 (1999).
    [CrossRef]
  13. G. Barbastathis, M. Balberg, and D. J. Brady, “Confocal microscopy with a volume holographic filter,” Opt. Lett. 24, 811-813 (1999).
    [CrossRef]
  14. W. Liu, G. Barbastathis, and D. Psaltis, “Volume holographic hyperspectral imaging,” Appl. Opt. 43, 3581-3599 (2004).
    [CrossRef] [PubMed]
  15. W. Liu, D. Psaltis, and G. Barbastathis, “Real-time spectral imaging in three spatial dimensions,” Opt. Lett. 27, 854-856 (2002).
    [CrossRef]
  16. A. Sinha, W. Sun, T. Shih, and G. Barbastathis, “Volume holographic imaging in transmission geometry,” Appl. Opt. 43, 1533-1551 (2004).
    [CrossRef] [PubMed]
  17. W. Sun, K. Tian, and G. Barbastathis, “Hyper-spectral imaging with volume holographic lenses,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Applications Systems Technologies, Technical Digest (CD) (Optical Society of America, 2005), paper CFP2.
    [PubMed]
  18. P. Wissmann, “Simulation and optimization of active and passive holographic imaging systems for shape and surface metrology,” Diploma thesis (RWTH Aachen University2007).
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    [CrossRef]
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    [CrossRef]
  25. F. Zernike, “The concept of degree of coherence and its applications to optical problems,” Physica 5, 785-795 (1938).
    [CrossRef]
  26. D. L. Marks, “Four-dimensional coherence sensing,” Ph.D. thesis (University of Illinois at Urbana--Champaign, 2001).
  27. A. R. Smith and J. F. Blinn, “Blue screen matting,” in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1996), pp. 259-268.
  28. P. Vlahos, “Comprehensive electronic compositing system,” U.S. Patent 4,344,085 (10 August 1982 ).
  29. P. Vlahos, “Encoded signal color image compositing,” U.S. Patent 4,409,611 (11 October 1983).
  30. P. E. Vlahos, P. Vlahos, and D. F. Fellinger, “Automated encoded signal color image compositing,” U.S. Patent 4,589,013 (13 May 1986).
  31. P. Vlahos, “Comprehensive electronic compositing system,” U.S. Patent 4,625,231 (25 November 2 1986 ).
  32. J. W. Goodman, Statistical Optics (Wiley, 2000).
  33. G. Barbastathis, “The transfer function of volume holographic optical systems,” in Photorefractive Materials and Their Applications, Springer Series in Optical Science, J. -P. Huignard and P. Günter, eds. (Springer-Verlag, 2006), pp. 51-76.
  34. J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).
  35. B. J. Thompson and R. Sudol, “Finite-aperture effects in the measurement of the degree of coherence,” J. Opt. Soc. Am. A 1, 598-604 (1984).
    [CrossRef]
  36. A. S. Marathay and D. B. Pollock, “Young's interferencce fringes with finite-sized sampling apertures,” J. Opt. Soc. Am. A 1, 1057-1059 (1984).
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2006 (1)

N. Joshi, W. Matusik, and S. Avidan, “Natural video matting using camera arrays,” ACM Trans. Graphics 25, 779-786(2006).
[CrossRef]

2005 (1)

2004 (2)

2002 (1)

1999 (5)

1996 (3)

1994 (1)

1993 (1)

J. Ens and P. Lawrence, “An investigation of methods for determining depth from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 97-108 (1993).
[CrossRef]

1984 (2)

1938 (1)

F. Zernike, “The concept of degree of coherence and its applications to optical problems,” Physica 5, 785-795 (1938).
[CrossRef]

1934 (1)

P. H. V. Cittert, “Die wahrscheinliche Schwingungsverteilung in einer von einer Lichtquelle direkt oder mittels einer Linse beleuchteten Ebene,” Physica 1, 201-210 (1934).
[CrossRef]

Arimoto, H.

Avidan, S.

N. Joshi, W. Matusik, and S. Avidan, “Natural video matting using camera arrays,” ACM Trans. Graphics 25, 779-786(2006).
[CrossRef]

Balberg, M.

Barbastathis, G.

W. Sun and G. Barbastathis, “Rainbow volume holographic imaging,” Opt. Lett. 30, 976-978 (2005).
[CrossRef] [PubMed]

W. Liu, G. Barbastathis, and D. Psaltis, “Volume holographic hyperspectral imaging,” Appl. Opt. 43, 3581-3599 (2004).
[CrossRef] [PubMed]

A. Sinha, W. Sun, T. Shih, and G. Barbastathis, “Volume holographic imaging in transmission geometry,” Appl. Opt. 43, 1533-1551 (2004).
[CrossRef] [PubMed]

W. Liu, D. Psaltis, and G. Barbastathis, “Real-time spectral imaging in three spatial dimensions,” Opt. Lett. 27, 854-856 (2002).
[CrossRef]

G. Barbastathis, M. Balberg, and D. J. Brady, “Confocal microscopy with a volume holographic filter,” Opt. Lett. 24, 811-813 (1999).
[CrossRef]

G. Barbastathis and D. J. Brady, “Multidimensional tomographic imaging using volume holography,” Proc. IEEE 87, 2098-2120 (1999).
[CrossRef]

W. Sun, K. Tian, and G. Barbastathis, “Hyper-spectral imaging with volume holographic lenses,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Applications Systems Technologies, Technical Digest (CD) (Optical Society of America, 2005), paper CFP2.
[PubMed]

G. Barbastathis, “The transfer function of volume holographic optical systems,” in Photorefractive Materials and Their Applications, Springer Series in Optical Science, J. -P. Huignard and P. Günter, eds. (Springer-Verlag, 2006), pp. 51-76.

Berger, J. O.

J. O. Berger, Statistical Decision Theory and Bayesian Analysis, Springer Series in Statistics (Springer-Verlag, 1985).

Blinn, J. F.

A. R. Smith and J. F. Blinn, “Blue screen matting,” in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1996), pp. 259-268.

Bouguet, J. Y.

J. Y. Bouguet and P. Perona, “3D photography using shadows in dual-space geometry,” Int. J. Comput. Vis. 35, 129-149(1999).
[CrossRef]

Brady, D. J.

Cathey, W. T.

Chaudhuri, S.

S. Chaudhuri and A. Rajagopalan, Depth from Defocus: a Real Aperture Imaging Approach (Springer, 1999).
[CrossRef]

Cittert, P. H. V.

P. H. V. Cittert, “Die wahrscheinliche Schwingungsverteilung in einer von einer Lichtquelle direkt oder mittels einer Linse beleuchteten Ebene,” Physica 1, 201-210 (1934).
[CrossRef]

Cracknell, A. P.

A. P. Cracknell and L. Hayes, Introduction to Remote Sensing (Taylor &Francis, 2007).

Curless, B.

B. Curless, “Overview of active vision techniques: SIGGRAPH 99 Course on 3D Photography” (1999), http://www.cs.cmu.edu/~seitz/course/SIGG00/slides/curless-active.pdf.

Dowski, E. R.

Ens, J.

J. Ens and P. Lawrence, “An investigation of methods for determining depth from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 97-108 (1993).
[CrossRef]

Fellinger, D. F.

P. E. Vlahos, P. Vlahos, and D. F. Fellinger, “Automated encoded signal color image compositing,” U.S. Patent 4,589,013 (13 May 1986).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

J. W. Goodman, Statistical Optics (Wiley, 2000).

Hayes, L.

A. P. Cracknell and L. Hayes, Introduction to Remote Sensing (Taylor &Francis, 2007).

Horn, B. K. P.

B. K. P. Horn, Robot Vision (MIT Press, 1986).

Itoh, K.

Joshi, N.

N. Joshi, W. Matusik, and S. Avidan, “Natural video matting using camera arrays,” ACM Trans. Graphics 25, 779-786(2006).
[CrossRef]

Lawrence, P.

J. Ens and P. Lawrence, “An investigation of methods for determining depth from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 97-108 (1993).
[CrossRef]

Liu, W.

Marathay, A. S.

Marks, D. L.

D. L. Marks, R. A. Stack, and D. J. Brady, “Three-dimensional coherence imaging in the Fresnel domain,” Appl. Opt. 38, 1332-1342 (1999).
[CrossRef]

D. L. Marks, “Four-dimensional coherence sensing,” Ph.D. thesis (University of Illinois at Urbana--Champaign, 2001).

Marr, D.

D. Marr and T. Poggio, “A theory of human stereo vision,” Artificial Intelligence Lab Publications, AI Memos AIM-451 (MIT, 1977), http://dspace.mit.edu/handle/1721.1/6093.

Matusik, W.

N. Joshi, W. Matusik, and S. Avidan, “Natural video matting using camera arrays,” ACM Trans. Graphics 25, 779-786(2006).
[CrossRef]

Perona, P.

J. Y. Bouguet and P. Perona, “3D photography using shadows in dual-space geometry,” Int. J. Comput. Vis. 35, 129-149(1999).
[CrossRef]

Poggio, T.

D. Marr and T. Poggio, “A theory of human stereo vision,” Artificial Intelligence Lab Publications, AI Memos AIM-451 (MIT, 1977), http://dspace.mit.edu/handle/1721.1/6093.

Pollock, D. B.

Psaltis, D.

Rajagopalan, A.

S. Chaudhuri and A. Rajagopalan, Depth from Defocus: a Real Aperture Imaging Approach (Springer, 1999).
[CrossRef]

Rosen, J.

Sheppard, C. J. R.

C. J. R. Sheppard and D. M. Shotton, Confocal Laser Scanning Microscopy (Springer, 1997).

Shih, T.

Shotton, D. M.

C. J. R. Sheppard and D. M. Shotton, Confocal Laser Scanning Microscopy (Springer, 1997).

Sinha, A.

Smith, A. R.

A. R. Smith and J. F. Blinn, “Blue screen matting,” in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1996), pp. 259-268.

Stack, R. A.

Sudol, R.

Sun, W.

W. Sun and G. Barbastathis, “Rainbow volume holographic imaging,” Opt. Lett. 30, 976-978 (2005).
[CrossRef] [PubMed]

A. Sinha, W. Sun, T. Shih, and G. Barbastathis, “Volume holographic imaging in transmission geometry,” Appl. Opt. 43, 1533-1551 (2004).
[CrossRef] [PubMed]

W. Sun, K. Tian, and G. Barbastathis, “Hyper-spectral imaging with volume holographic lenses,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Applications Systems Technologies, Technical Digest (CD) (Optical Society of America, 2005), paper CFP2.
[PubMed]

Thompson, B. J.

Tian, K.

W. Sun, K. Tian, and G. Barbastathis, “Hyper-spectral imaging with volume holographic lenses,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Applications Systems Technologies, Technical Digest (CD) (Optical Society of America, 2005), paper CFP2.
[PubMed]

Vlahos, P.

P. Vlahos, “Comprehensive electronic compositing system,” U.S. Patent 4,625,231 (25 November 2 1986 ).

P. Vlahos, “Comprehensive electronic compositing system,” U.S. Patent 4,344,085 (10 August 1982 ).

P. Vlahos, “Encoded signal color image compositing,” U.S. Patent 4,409,611 (11 October 1983).

P. E. Vlahos, P. Vlahos, and D. F. Fellinger, “Automated encoded signal color image compositing,” U.S. Patent 4,589,013 (13 May 1986).

P. E. Vlahos, P. Vlahos, and D. F. Fellinger, “Automated encoded signal color image compositing,” U.S. Patent 4,589,013 (13 May 1986).

Wissmann, P.

P. Wissmann, “Simulation and optimization of active and passive holographic imaging systems for shape and surface metrology,” Diploma thesis (RWTH Aachen University2007).

Yariv, A.

Yoshimori, K.

Zernike, F.

F. Zernike, “The concept of degree of coherence and its applications to optical problems,” Physica 5, 785-795 (1938).
[CrossRef]

ACM Trans. Graphics (1)

N. Joshi, W. Matusik, and S. Avidan, “Natural video matting using camera arrays,” ACM Trans. Graphics 25, 779-786(2006).
[CrossRef]

Appl. Opt. (4)

IEEE Trans. Pattern Anal. Mach. Intell. (1)

J. Ens and P. Lawrence, “An investigation of methods for determining depth from focus,” IEEE Trans. Pattern Anal. Mach. Intell. 15, 97-108 (1993).
[CrossRef]

Int. J. Comput. Vis. (1)

J. Y. Bouguet and P. Perona, “3D photography using shadows in dual-space geometry,” Int. J. Comput. Vis. 35, 129-149(1999).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Lett. (5)

Physica (2)

P. H. V. Cittert, “Die wahrscheinliche Schwingungsverteilung in einer von einer Lichtquelle direkt oder mittels einer Linse beleuchteten Ebene,” Physica 1, 201-210 (1934).
[CrossRef]

F. Zernike, “The concept of degree of coherence and its applications to optical problems,” Physica 5, 785-795 (1938).
[CrossRef]

Proc. IEEE (1)

G. Barbastathis and D. J. Brady, “Multidimensional tomographic imaging using volume holography,” Proc. IEEE 87, 2098-2120 (1999).
[CrossRef]

Other (18)

W. Sun, K. Tian, and G. Barbastathis, “Hyper-spectral imaging with volume holographic lenses,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science and Photonic Applications Systems Technologies, Technical Digest (CD) (Optical Society of America, 2005), paper CFP2.
[PubMed]

P. Wissmann, “Simulation and optimization of active and passive holographic imaging systems for shape and surface metrology,” Diploma thesis (RWTH Aachen University2007).

B. K. P. Horn, Robot Vision (MIT Press, 1986).

B. Curless, “Overview of active vision techniques: SIGGRAPH 99 Course on 3D Photography” (1999), http://www.cs.cmu.edu/~seitz/course/SIGG00/slides/curless-active.pdf.

A. P. Cracknell and L. Hayes, Introduction to Remote Sensing (Taylor &Francis, 2007).

C. J. R. Sheppard and D. M. Shotton, Confocal Laser Scanning Microscopy (Springer, 1997).

D. Marr and T. Poggio, “A theory of human stereo vision,” Artificial Intelligence Lab Publications, AI Memos AIM-451 (MIT, 1977), http://dspace.mit.edu/handle/1721.1/6093.

S. Chaudhuri and A. Rajagopalan, Depth from Defocus: a Real Aperture Imaging Approach (Springer, 1999).
[CrossRef]

D. L. Marks, “Four-dimensional coherence sensing,” Ph.D. thesis (University of Illinois at Urbana--Champaign, 2001).

A. R. Smith and J. F. Blinn, “Blue screen matting,” in Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques (ACM, 1996), pp. 259-268.

P. Vlahos, “Comprehensive electronic compositing system,” U.S. Patent 4,344,085 (10 August 1982 ).

P. Vlahos, “Encoded signal color image compositing,” U.S. Patent 4,409,611 (11 October 1983).

P. E. Vlahos, P. Vlahos, and D. F. Fellinger, “Automated encoded signal color image compositing,” U.S. Patent 4,589,013 (13 May 1986).

P. Vlahos, “Comprehensive electronic compositing system,” U.S. Patent 4,625,231 (25 November 2 1986 ).

J. W. Goodman, Statistical Optics (Wiley, 2000).

G. Barbastathis, “The transfer function of volume holographic optical systems,” in Photorefractive Materials and Their Applications, Springer Series in Optical Science, J. -P. Huignard and P. Günter, eds. (Springer-Verlag, 2006), pp. 51-76.

J. W. Goodman, Introduction to Fourier Optics (Roberts, 2005).

J. O. Berger, Statistical Decision Theory and Bayesian Analysis, Springer Series in Statistics (Springer-Verlag, 1985).

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Figures (8)

Fig. 1
Fig. 1

Schematic of the experimental setup. IR, variable iris; VH, volume hologram; L1, L2, lenses; M, mirror; RP, roof prism; PZT, piezo actuator for dithering.

Fig. 2
Fig. 2

Geometry of 4 - f system with a volume holographic pupil.

Fig. 3
Fig. 3

Thirty cross sections of the interference fringes along the x axis obtained from thirty corresponding longitudinal positions of the PZT actuator, where f 1 = f 2 = 150 mm , L = 1 mm , and θ s = 30 ° . Since the center part of the fringe is more spatially coherent, the interference fringes oscillate near the center, but the oscillations die out away from the center where the field becomes incoherent.

Fig. 4
Fig. 4

Simulation results of visibility and aperture size for the FG object (blue solid curve) and the BG object (green dashed curve), where δ z = 180 mm , λ = 532 nm , the CCD pixel size is 5 × 5 μm , L = 1.5 mm , θ s = 30 ° , and f 1 = f 2 = 200 mm .

Fig. 5
Fig. 5

Enlarged portion of Fig. 3. Either too small or larger pixel size loses discrimination ability.

Fig. 6
Fig. 6

Experimental results of visibility and aperture size for the FG (blue solid curve) and BG object (green dashed curve), where δ z = 150 mm , λ = 532 nm , the CCD pixel size is 5 × 5 μm , f 1 = f 2 = 200 mm , L = 1 mm , and θ s = 30 ° . The red dotted line is the result with the laser only, without using the rotational diffuser. Other parameters are identical to the simulation in Fig. 4. Error bars indicate the standard deviation of the measurements.

Fig. 7
Fig. 7

Histogram of the experimental result at 25 mm aperture size of Fig. 6. The FG object (blue curve with solid circles) and the BG object (green curve with squares) are plotted together. The red dashed line indicates the best visibility threshold computed by the Bayesian estimation.

Fig. 8
Fig. 8

Experiment with FG object and BG object simultaneously present, where f 1 = f 2 = 200 mm , L = 1 mm , δ z = 200 mm , and θ s = 30 ° . (a) Raw image on the CCD camera (left) and enlargement of the slitlike visible region (right). The red arrow indicates the location where fringe visibility is maximum. (b) Intensity profile (dashed blue, left-hand axis) and visibility (solid green, right-hand axis) of the two objects.

Equations (15)

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ϵ ( x , y , z ) = ϵ 0 exp { j 2 π λ f ( θ s x θ s 2 2 z ) } rect ( x a ) rect ( y b ) rect ( z L ) ,
P ( x , y , t ) = Re { p ( x , y ) e j ω p t } ,
Γ P ( x 1 , y 1 , x 2 , y 2 , τ ) = P ( x 1 , y 1 , t + τ ) P * ( x 2 , y 2 , t ) ,
Γ Q ( x 1 , y 1 , x 2 , y 2 , τ ) = Q ( x 1 , y 1 , t + τ ) Q * ( x 2 , y 2 , t ) ,
J p ( x 1 , y 1 , x 2 , y 2 ) = p ( x 1 , y 1 ) p * ( x 2 , y 2 ) ,
J q ( x 1 , y 1 , x 2 , y 2 ) = q ( x 1 , y 1 ) q * ( x 2 , y 2 ) .
q ( x , y ) = ϵ 0 a b L d x d y p ( x , y ) sinc { a ( x λ p f 1 + x λ p f 2 θ s λ f ) } sinc { b λ p ( y f 1 + y f 2 ) } sinc { L 2 ( x 2 + y 2 λ p f 1 2 x 2 + y 2 λ p f 2 2 + θ s 2 λ f ) } .
Γ Q ( x 1 , y 1 , x 2 , y 2 , τ ) = ϵ 0 2 ( a b L ) 2 d x 1 d y 1 d x 2 d y 2 Γ P ( x 1 , y 1 , x 2 , y 2 , τ ) sinc { a ( x 1 λ p f 1 + x 1 λ p f 2 θ s λ f ) } sinc { a ( x 2 λ p f 1 + x 2 λ p f 2 θ s λ f ) } sinc { b ( y 1 λ p f 1 + y 1 λ p f 2 ) } sinc { b ( y 2 λ p f 1 + y 2 λ p f 2 ) } sinc { L 2 ( x 1 2 + y 1 2 λ p f 1 2 x 1 2 + y 1 2 λ p f 2 2 + θ s 2 λ f ) } sinc { L 2 ( x 2 2 + y 2 2 λ p f 1 2 x 2 2 + y 2 2 λ p f 2 2 + θ s 2 λ f ) } .
J q ( x 1 , y 1 , x 2 , y 2 ) = ϵ 0 2 ( a b L ) 2 d x 1 d y 1 d x 2 d y 2 J p ( x 1 , y 1 , x 2 , y 2 ) sinc { a ( x 1 λ p f 1 + x 1 λ p f 2 θ s λ f ) } sinc { a ( x 2 λ p f 1 + x 2 λ p f 2 θ s λ f ) } sinc { b ( y 1 λ p f 1 + y 1 λ p f 2 ) } sinc { b ( y 2 λ p f 1 + y 2 λ p f 2 ) } sinc { L 2 ( x 1 2 + y 1 2 λ p f 1 2 x 1 2 + y 1 2 λ p f 2 2 + θ s 2 λ f ) } sinc { L 2 ( x 2 2 + y 2 2 λ p f 1 2 x 2 2 + y 2 2 λ p f 2 2 + θ s 2 λ f ) } .
I q ( x , y ) = ϵ 0 2 ( a b L ) 2 d x 1 d y 1 d x 2 d y 2 J p ( x 1 , y 1 , x 2 , y 2 ) sinc { a ( x 1 λ p f 1 + x λ p f 2 θ s λ f ) } sinc { a ( x 2 λ p f 1 + x λ p f 2 θ s λ f ) } sinc { b ( y 1 λ p f 1 + y λ p f 2 ) } sinc { b ( y 2 λ p f 1 + y λ p f 2 ) } sinc { L 2 ( x 1 2 + y 1 2 λ p f 1 2 x 2 + y 2 λ p f 2 2 + θ s 2 λ f ) } sinc { L 2 ( x 2 2 + y 2 2 λ p f 1 2 x 2 + y 2 λ p f 2 2 + θ s 2 λ f ) } .
J p , FG ( x 1 , y 1 , x 2 , y 2 ) = I 0 δ ( Δ x , Δ y ) ,
J p , BG ( x 1 , y 1 ; x 2 , y 2 ) = 1 ( λ p δ z ) 2 exp { j 2 π λ p δ z ( x 2 2 + y 2 2 x 1 2 y 1 2 ) } I s ( ξ , η ) exp { j 2 π λ p δ z ( Δ x ξ + Δ y η ) } d ξ d η ,
I ( x , y ) = | q ( x , y ) | 2 + | q ( x , y ) | 2 + 2 Re { q * ( x , y ) q ( x , y ) } = | q ( x , y ) | 2 + | q ( x , y ) | 2 + 2 | J q ( x , y , x , y ) | cos ( 2 π λ p d + ϕ ) ,
V ( x , y ) = I max ( x , y ) I min ( x , y ) I max ( x , y ) + I min ( x , y ) ,
μ ( x , y ; x , y ) = J q ( x , y , x , y ) | q ( x , y ) | | q ( x , y ) | .

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