Abstract

We present a proof of the conjecture reported in Stein and Barbastathis [“Axial imaging necessitates loss of lateral shift invariance,” Appl. Opt. 41, 6055–6061 (2002)] that axial imaging necessitates loss of lateral shift invariance. We apply the Wigner distribution function (WDF) to the axial imaging process and compare laterally shift variant and invariant imaging systems. Two conditions for axial imaging are established: (i) objects with spatially variant local spatial frequency content to create sufficient change in the WDF with defocus and (ii) properly designed shift variant imaging kernels to estimate the slope of the sheared WDF. The lateral shift variance is a necessary condition for axial imaging. We use two examples from Stein and Barbastathis to show how axial imaging is interpreted in Wigner space.

© 2009 Optical Society of America

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References

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  1. A. Stein and G. Barbastathis, “Axial imaging necessitates loss of lateral shift invariance,” Appl. Opt. 41, 6055-6061 (2002).
    [CrossRef] [PubMed]
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    [CrossRef]
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  4. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749-759 (1932).
    [CrossRef]
  5. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent-light,” J. Opt. Soc. Am. A 3, 1227-1238 (1986).
    [CrossRef]
  6. W. Pan, “Double Wigner distribution function of a first-order optical system with a hard-edge aperture,” Appl. Opt. 47, 45-51 (2008).
    [CrossRef]
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    [CrossRef]
  8. E. H. Adelson and J. Y. A. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 99-106 (1992).
    [CrossRef]
  9. A. Levin, W. T. Freeman, and F. Durand, “Understanding camera trade-offs through a Bayesian analysis of light field projections,” Tech. Rep. MIT-CSAIL-TR-2008-049 (Massachusetts Institute of Technology, 2008).
  10. C. J. R. Sheppard and D. M. Shotton, Confocal Laser Scanning Microscopy (Springer, 1997).
  11. K. J. Gåsvik, Optical Metrology, 3rd ed. (Wiley, 2002).
    [CrossRef]
  12. W. Sun and G. Barbastathis, “Rainbow volume holographic imaging,” Opt. Lett. 30, 976-978 (2005).
    [CrossRef] [PubMed]
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2008

2005

2002

1992

E. H. Adelson and J. Y. A. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 99-106 (1992).
[CrossRef]

1986

1984

K. H. Brenner and J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213-223(1984).
[CrossRef]

1973

1970

1965

1932

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

Adelson, E. H.

E. H. Adelson and J. Y. A. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 99-106 (1992).
[CrossRef]

Allen, J. B.

Barbastathis, G.

Bastiaans, M. J.

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent-light,” J. Opt. Soc. Am. A 3, 1227-1238 (1986).
[CrossRef]

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, 1997), pp. 375-426.

Brenner, K. H.

K. H. Brenner and J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213-223(1984).
[CrossRef]

Durand, F.

A. Levin, W. T. Freeman, and F. Durand, “Understanding camera trade-offs through a Bayesian analysis of light field projections,” Tech. Rep. MIT-CSAIL-TR-2008-049 (Massachusetts Institute of Technology, 2008).

Freeman, W. T.

A. Levin, W. T. Freeman, and F. Durand, “Understanding camera trade-offs through a Bayesian analysis of light field projections,” Tech. Rep. MIT-CSAIL-TR-2008-049 (Massachusetts Institute of Technology, 2008).

Gåsvik, K. J.

K. J. Gåsvik, Optical Metrology, 3rd ed. (Wiley, 2002).
[CrossRef]

Johnson, W. O.

Levin, A.

A. Levin, W. T. Freeman, and F. Durand, “Understanding camera trade-offs through a Bayesian analysis of light field projections,” Tech. Rep. MIT-CSAIL-TR-2008-049 (Massachusetts Institute of Technology, 2008).

Lohmann, A. W.

Meadows, D. M.

Ojeda-Castañeda, J.

K. H. Brenner and J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213-223(1984).
[CrossRef]

Pan, W.

Paris, D. P.

Sheppard, C. J. R.

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

Shotton, D. M.

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

Stein, A.

Sun, W.

Takasaki, H.

Wang, J. Y. A.

E. H. Adelson and J. Y. A. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 99-106 (1992).
[CrossRef]

Wigner, E.

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

Appl. Opt.

IEEE Trans. Pattern Anal. Mach. Intell.

E. H. Adelson and J. Y. A. Wang, “Single lens stereo with a plenoptic camera,” IEEE Trans. Pattern Anal. Mach. Intell. 14, 99-106 (1992).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

K. H. Brenner and J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213-223(1984).
[CrossRef]

Opt. Lett.

Phys. Rev.

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

Other

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, 1997), pp. 375-426.

A. Levin, W. T. Freeman, and F. Durand, “Understanding camera trade-offs through a Bayesian analysis of light field projections,” Tech. Rep. MIT-CSAIL-TR-2008-049 (Massachusetts Institute of Technology, 2008).

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

K. J. Gåsvik, Optical Metrology, 3rd ed. (Wiley, 2002).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Wigner representation of incoherent light and Fresnel propagation: (a) since incoherent light has all spatial frequency values, its WDF appears to be a vertical strip with intensity variations along x. (b) the empty rectangle is before and the solid rectangle is after propagation of δ z .

Fig. 2
Fig. 2

Axial imaging scenario in one-dimensional geometry measures the distance between the focused and defocused objects, in which the defocus corresponds to the x shear of the WDF in Wigner space.

Fig. 3
Fig. 3

Forward/Backward imaging in Wigner space. The intensity integration kernels at the output plane are back projected onto the input plane.

Fig. 4
Fig. 4

Comparison of backprojected integration kernels in shift invariant and variant systems: (a) Shift invariant systems in which the integration kernels are all identical. (b) Shift variant systems in which the kernels could be different in shape, value, orientation, and vertical extent.

Fig. 5
Fig. 5

Backprojected integration kernels of various camera systems.

Fig. 6
Fig. 6

Local spatial frequency of sinusoidal and chirp gratings. For simpler visualization, the spatial frequency in intensity is denoted by u I . (a) The local spatial frequency of a sinusoidal amplitude grating is nonzero but constant along the x direction. There is ambiguity in finding correspondence. (b) The local spatial frequency of a chirp grating varies along the x direction, which satisfies the first condition.

Fig. 7
Fig. 7

Principle of a confocal microscope in Wigner space.

Fig. 8
Fig. 8

Binocular systems in Wigner space. Vertical lines of WDF are mapped in different pixels, where the slope of the sheared WDF can be estimated.

Equations (25)

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W g ( x , u ) = g ( x + x 2 ) g * ( x - x 2 ) e - i 2 π x u d x ,
I ( x ) = W ( x , u ) d u .
W ( x , u ) = I ( x ) .
W 2 ( x , u ) = W 1 ( x - λ δ z u , u ) ,
E 2 ( x 2 ) = h ( x 2 ; x 1 ) E 1 ( x 1 ) d x 1 ,
E 2 ( x 2 ) = h ( x 1 - x 2 ) E 1 ( x 1 ) d x 1 .
W 2 ( x 2 , u 2 ) = K si ( x 2 - x 1 , u 2 ) W 1 ( x 1 , u 2 ) d x 1 ,
K si ( x , u ) = h ( x + x 2 ) h * ( x - x 2 ) e - i 2 π u x d x .
W 2 ( x 2 , u 2 ) = K sv ( x 2 , u 2 ; x 1 , u 1 ) W 1 ( x 1 , u 1 ) d x 1 d u 1 ,
K sv ( x 2 , u 2 ; x 1 , u 1 ) = h ( x 2 + ξ 2 ; x 1 + η 2 ) h * ( x 2 - ξ 2 ; x 1 - η 2 ) e - i 2 π ( u 2 ξ - u 1 η ) d ξ d η .
E 2 ( x 2 ) = E 1 ( x 1 ) h ( x 2 - x 1 ) d x 1 .
W 2 ( x 2 , u 2 ) = E 2 ( x 2 + ξ 2 ) E 2 * ( x 2 ξ 2 ) e j 2 π u 2 ξ d ξ = [ E 1 ( x 1 ) h ( x 2 + ξ 2 x 1 ) d x 1 ] [ E 1 * ( x 1 ) h * ( x 2 ξ 2 x 1 ) d x 1 ] e i 2 π u 2 ξ d ξ = [ h ( x 2 + ξ 2 x 1 ) h * ( x 2 ξ 2 x 1 ) e i 2 π u 2 ξ d ξ ] E 1 ( x 1 ) E 1 * ( x 1 ) d x 1 d x 1 ,
x 1 = p + q 2 ,
x 1 = p - q 2 ,
W 2 ( x 2 , u 2 ) = [ h ( x 2 + ξ 2 p q 2 ) h * ( x 2 ξ 2 p + q 2 ) e i 2 π u 2 ξ d ξ ] × E 1 ( p + q 2 ) E 1 * ( p q 2 ) d p d q = [ h ( x 2 p + ξ + q 2 ) h * ( x 2 p ξ q 2 ) e i 2 π u 2 ξ d ξ ] E 1 ( p + q 2 ) E 1 * ( p q 2 ) d p d q = [ h ( x 2 p + ξ + q 2 ) h * ( x 2 p ξ q 2 ) e i 2 π u 2 ( ξ q ) d ξ ] E 1 ( p + q 2 ) E 1 * ( p q 2 ) e i 2 π u 2 q d p d q .
W 2 ( x 2 , u 2 ) = E 1 ( p + q 2 ) E 1 * ( p q 2 ) e i 2 π u 2 q d p d q [ h ( x 2 p + t 2 ) h * ( x 2 p t 2 ) e i 2 π u 2 t d t ] = K si ( x 2 p ) E 1 ( p + q 2 ) E 1 * ( p q 2 ) e i 2 π u 2 q d p d q ,
K si ( x 2 - p , u 2 ) = h ( x 2 - p + t 2 ) h * ( x 2 - p - t 2 ) e - i 2 π u 2 t d t .
W 2 ( x 2 , u 2 ) = K si ( x 2 p ) [ E 1 ( p + q 2 ) E 1 * ( p q 2 ) e i 2 π u 2 q d q ] d p = K si ( x 2 p , u 2 ) W 1 ( p , u 2 ) d p .
W 2 ( x 2 , u 2 ) = K si ( x 2 - x 1 , u 2 ) W 1 ( x 1 , u 2 ) d x 1 .
E 2 ( x 2 ) = E 1 ( x 1 ) h ( x 2 ; x 1 ) d x 1 .
W 2 ( x 2 , u 2 ) = E 2 ( x 2 + ξ 2 ) E 2 * ( x 2 ξ 2 ) e i 2 π u 2 ξ d ξ = [ E 1 ( x 1 ) h ( x 2 + ξ 2 ; x 1 ) d x 1 ] [ E 1 ( x 1 ) h ( x 2 ξ 2 ; x 1 ) d x 1 ] * e i 2 π u 2 ξ d ξ = [ h ( x 2 + ξ 2 ; p + q 2 ) h * ( x 2 ξ 2 ; p q 2 ) e i 2 π u 2 ξ d ξ ] E 1 ( p + q 2 ) E 1 * ( p q 2 ) d p d q .
K sv ( x 2 , u 2 ; x 1 , u 1 ) = h ( x 2 + ξ 2 ; x 1 + η 2 ) h * ( x 2 - ξ 2 ; x 1 - η 2 ) e - i 2 π ( u 2 ξ - u 1 η ) d ξ d η ,
h ( x 2 + ξ 2 ; p + q 2 ) h * ( x 2 - ξ 2 ; p - q 2 ) e i 2 π u 2 ξ d ξ = K sv ( x 2 , u 2 ; p , u 1 ) e i 2 π u 1 q d u 1 .
W 2 ( x 2 , u 2 ) = [ K sv ( x 2 , u 2 ; p , u 1 ) e i 2 π u 1 q d u 1 ] E 1 ( p + q 2 ) E 1 * ( p q 2 ) d p d q = d u 1 dp K sv ( x 2 , u 2 ; p , u 1 ) [ E 1 ( p + q 2 ) E 1 * ( p q 2 ) e i 2 π u 1 q d q ] = K sv ( x 2 , u 2 ; p , u 1 ) W 1 ( p , u 1 ) d p d u 1 .
W 2 ( x 2 , u 2 ) = K sv ( x 2 , u 2 ; x 1 , u 1 ) W 1 ( x 1 , u 1 ) d x 1 d u 1 .

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