Abstract

The Hilbert transform as been investigated abundantly in coherent imaging. To the best of our knowledge, it is for the first time investigated in the context of incoherent imaging. We present a two-pupil optical heterodyne scanning system and analyze mathematically the design of its two pupils such that the optical system can perform the Hilbert transform on incoherent objects. Computer simulations of the idea clarify the theoretical results.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  6. T.-C. Poon, "Scanning holography and two-dimensional image processing by acousto-optic two-pupil synthesis," J. Opt. Soc. Am. 2, 521-527 (1985).
    [CrossRef]
  7. T.-C. Poon and T. Kim, Engineering Optics with MATLAB, (World Scientific, 2006).
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    [CrossRef]

2003 (1)

1999 (1)

1997 (1)

1985 (1)

T.-C. Poon, "Scanning holography and two-dimensional image processing by acousto-optic two-pupil synthesis," J. Opt. Soc. Am. 2, 521-527 (1985).
[CrossRef]

1979 (1)

1975 (1)

1967 (1)

S. Lowenthal and Y. Belvaux, "Observation of phase objects by optically processed Hilbert transform," Appl. Phys. Lett. 11, 49-51 (1967).
[CrossRef]

Belvaux, Y.

S. Lowenthal and Y. Belvaux, "Observation of phase objects by optically processed Hilbert transform," Appl. Phys. Lett. 11, 49-51 (1967).
[CrossRef]

Buczynshi, R.

Gale Wilson, R.

Kim, T.

Korpel, A.

Kowalczyk, M.

Lohmann, A. W.

Lowenthal, S.

S. Lowenthal and Y. Belvaux, "Observation of phase objects by optically processed Hilbert transform," Appl. Phys. Lett. 11, 49-51 (1967).
[CrossRef]

Nowicki, S.

Poon, T.-C.

Ramirez, J. G.

Sagan, A.

Szoplik, T.

Tepichin, E.

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Figures (2)

Fig. 1.
Fig. 1.

Typical two-pupil optical heterodyne image processor. [Figure 3.11 of T.-C. Poon, Optical scanning holography with MATLAB (Springer, 2007). With kind permission of Springer Science and Business Media.]

Fig. 2.
Fig. 2.

(a). Hologram Hre (x, y) and (b). hologram Him (x, y), (c). real part of and (d). imaginary part of reconstruction from HC (x, y) = Hre (x, y) + jHim (x, y).

Equations (24)

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F { g ( x ) } = G ( k x ) = g ( x ) exp ( j k x x ) d x ,
g a + ( x ) = 1 π 0 G ( k x ) exp ( j k x x ) d k x
g a + ( x ) = 2 1 2 π G ( k x ) U ( k x ) exp ( j k x x ) d k x = 2 F 1 { G ( k x ) U ( k x ) } ,
U ( k x ) = 1 for k x > 0 and U ( k x ) = 0 for k x < 0 .
g a + ( x ) = g ( x ) j g ̂ ( x )
g ̂ ( x ) = H { g ( x ) } = g ( x ) π ( x x ) d x .
g ( x ) = Re { g a + ( x ) } ; g ̂ ( x ) = Im { g a + ( x ) } .
g ( x ) = 2 Re [ F 1 { G ( k x ) U ( k x ) } ] ; g ̂ ( x ) = 2 Im [ F 1 { G ( k x ) U ( k x ) } ] .
g a ( x ) = 1 π 0 G ( k x ) exp ( j k x x ) d k x = 2 F 1 { G ( k x ) U ( k x ) }
= g ( x ) + j g ̂ ( x ) ,
g ( x ) = 2 Re [ F 1 { G ( k x ) U ( k x ) } ] ; g ̂ ( x ) = 2 Im [ F 1 { G ( k x ) U ( k x ) } ] .
i c x y = Re [ F 1 { F { Γ 0 ( x , y ; z ) 2 } OTF Ω } ] ;
i s x y = Im [ F 1 { F { Γ 0 ( x , y ; z ) 2 } OTF Ω } ] ,
OTF Ω = exp [ j z 2 k 0 ( k x 2 + k y 2 ) ]
× p 1 * x y ) p 2 x + f k 0 k x y + f k 0 k y exp [ j z f ( x k x + y k y ) ] d x d y
OTF Ω = exp [ j z 2 k 0 ( k x 2 + k y 2 ) ] U ( k x ) U ( k y ) .
H r e x y = Re [ F 1 { F { Γ 0 2 } exp [ j z 2 k 0 ( k x 2 + k y 2 ) ] U ( k x ) U ( k y ) } ] ;
H i m x y = Im [ F 1 { F { Γ 0 2 } exp [ j z 2 k 0 ( k x 2 + k y 2 ) ] U ( k x ) U ( k y ) } ] .
H r e x y = Γ 0 ( x , y ; z ) 2 * k 0 2 π z sin [ k 0 2 z ( x 2 + y 2 ) ]
+ H { Γ 0 ( x , y ; z ) 2 } * k 0 2 π z cos [ k 0 2 z ( x 2 + y 2 ) ] ;
H i m x y = H { Γ 0 ( x , y ; z ) 2 } * k 0 2 π z sin [ k 0 2 z ( x 2 + y 2 ) ]
Γ 0 ( x , y ; z ) 2 * k 0 2 π z cos [ k 0 2 z ( x 2 + y 2 ) ] ,
H C x y = [ j Γ 0 ( x , y ; z ) 2 + H { Γ 0 ( x , y ; z ) 2 } ] * exp [ j k 0 2 z ( x 2 + y 2 ) ] .
H C ( x , y ) * h ( x , y ; z ) ,

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