Abstract

In this paper, we analyze the influence of two kinds of deviation errors on sectional image reconstruction for an optical scanning holography system using a random-phase pupil. The first deviation occurs in the lateral pixel position while the second occurs in the pixel value of the decoding function. Theoretical analysis and numerical simulation show that these two deviations may lead to noise in the reconstructed image. Additional discussions include the signal-to-noise ratio of the reconstructed image.

© 2012 Optical Society of America

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References

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  1. T.-C. Poon, Optical Scanning Holography with MATLAB (Springer, 2007).
  2. E. Y. Lam, X. Zhang, H. Vo, T.-C. Poon, and G. Indebetouw, “Three-dimensional microscopy and sectional image reconstruction using optical scanning holography,” Appl. Opt. 48, H113–H119 (2009).
    [CrossRef]
  3. G. Indebetouw, “Properties of a scanning holographic microscopy: improved resolution, extended depth-of-focus, and/or optical sectioning,” J. Mod. Opt. 49, 1479–1500 (2002).
    [CrossRef]
  4. J. Ke, T.-C. Poon, and E. Y. Lam, “Depth resolution enhancement in optical scanning holography with a dual-wavelength laser source,” Appl. Opt. 50, H285–H296 (2011).
    [CrossRef]
  5. T. Kim, “Optical sectioning by optical scanning holography and a Wiener filter,” Appl. Opt. 45, 872–879 (2006).
    [CrossRef]
  6. H. Kim, S.-W. Min, B. Lee, and T.-C. Poon, “Optical sectioning for optical scanning holography using phase-space filtering with Wigner distribution functions,” Appl. Opt. 47, D164–D175 (2008).
    [CrossRef]
  7. X. Zhang, E. Y. Lam, and T.-C. Poon, “Reconstruction of sectional images in holography using inverse imaging,” Opt. Express 16, 17215–17226 (2008).
    [CrossRef]
  8. X. Zhang and E. Y. Lam, “Edge-preserving sectional image reconstruction in optical scanning holography,” J. Opt. Soc. Am. A 27, 1630–1637 (2010).
    [CrossRef]
  9. F.-J. Zhao, X.-C. Qu, X. Zhang, T.-C. Poon, T. Kim, Y. S. Kim, and J.-M. Liang, “Solving inverse problems for optical scanning holography using an adaptively iterative shrinkage-thresholding algorithm,” Opt. Express 20, 5942–5954 (2012).
    [CrossRef]
  10. T. Kim and T.-C. Poon, “Autofocusing in optical scanning holography,” Appl. Opt. 48, H153–H159 (2009).
    [CrossRef]
  11. X. Zhang, E. Y. Lam, T. Kim, Y. S. Kim, and T.-C. Poon, “Blind sectional image reconstruction for optical scanning holography,” Opt. Lett. 34, 3098–3100 (2009).
    [CrossRef]
  12. P. W. M. Tsang, K. W. K. Cheung, T. Kim, Y. S. Kim, and T.-C. Poon, “Fast reconstruction of sectional images in digital holography,” Opt. Lett. 36, 2650–2652 (2011).
    [CrossRef]
  13. X. Zhou, K. Dobson, Y. Shinoda, and T.-C. Poon, “Sectional image reconstruction in optical scanning holography using a random-phase pupil,” Opt. Lett. 35, 2934–2936 (2010).
    [CrossRef]
  14. T.-C. Poon and G. Indebetouw, “Three-dimensional point spread functions of an optical heterodyne scanning image processor,” Appl. Opt. 42, 1485–1492 (2003).
    [CrossRef]
  15. T.-C. Poon and T. Kim, “Optical image recognition of three-dimensional objects,” Appl. Opt. 38, 370–381 (1999).
    [CrossRef]
  16. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef]
  17. R. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).
  18. B. Javidi, A. Sergent, G. Zhang, and L. Guibert, “Fault tolerance properties of a double random phase encoding encryption technique,” Opt. Eng. 36, 992–998 (1997).
    [CrossRef]
  19. F. Goudail, F. Bollaro, B. Javidi, and P. Refregier, “Influence of a perturbation in a double phase-encoding system,” J. Opt. Soc. Am. A 15, 2629–2638 (1998).
    [CrossRef]
  20. B. Wang, C.-C. Sun, W.-C. Su, and A. E. T. Chiou, “Shift-tolerance property of an optical double-random phase-encoding encryption system,” Appl. Opt. 39, 4788–4793 (2000).
    [CrossRef]
  21. D. Lai, X. Zhou, D.-F. Zhou, and D.-H. Li, “Study on the influence of key errors on the deciphered image in the double random phase encryption system by applying affine cryptography,” J. Mod. Opt. 55, 167–176 (2008).
    [CrossRef]
  22. X. Zhou, S. Yuan, S. W. Wang, and J. Xie, “Affine cryptosystem of double-random-phase encryption based on the fractional Fourier transform,” Appl. Opt. 45, 8434–8439 (2006).
    [CrossRef]
  23. T. Nomura and B. Javidi, “Optical encryption system with a binary key code,” Appl. Opt. 39, 4783–4787 (2000).
    [CrossRef]

2012 (1)

2011 (2)

2010 (2)

2009 (3)

2008 (3)

2006 (2)

2003 (1)

2002 (1)

G. Indebetouw, “Properties of a scanning holographic microscopy: improved resolution, extended depth-of-focus, and/or optical sectioning,” J. Mod. Opt. 49, 1479–1500 (2002).
[CrossRef]

2000 (2)

1999 (1)

1998 (1)

1997 (1)

B. Javidi, A. Sergent, G. Zhang, and L. Guibert, “Fault tolerance properties of a double random phase encoding encryption technique,” Opt. Eng. 36, 992–998 (1997).
[CrossRef]

1982 (1)

1972 (1)

R. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Bollaro, F.

Cheung, K. W. K.

Chiou, A. E. T.

Dobson, K.

Fienup, J. R.

Gerchberg, R.

R. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Goudail, F.

Guibert, L.

B. Javidi, A. Sergent, G. Zhang, and L. Guibert, “Fault tolerance properties of a double random phase encoding encryption technique,” Opt. Eng. 36, 992–998 (1997).
[CrossRef]

Indebetouw, G.

Javidi, B.

Ke, J.

Kim, H.

Kim, T.

Kim, Y. S.

Lai, D.

D. Lai, X. Zhou, D.-F. Zhou, and D.-H. Li, “Study on the influence of key errors on the deciphered image in the double random phase encryption system by applying affine cryptography,” J. Mod. Opt. 55, 167–176 (2008).
[CrossRef]

Lam, E. Y.

Lee, B.

Li, D.-H.

D. Lai, X. Zhou, D.-F. Zhou, and D.-H. Li, “Study on the influence of key errors on the deciphered image in the double random phase encryption system by applying affine cryptography,” J. Mod. Opt. 55, 167–176 (2008).
[CrossRef]

Liang, J.-M.

Min, S.-W.

Nomura, T.

Poon, T.-C.

F.-J. Zhao, X.-C. Qu, X. Zhang, T.-C. Poon, T. Kim, Y. S. Kim, and J.-M. Liang, “Solving inverse problems for optical scanning holography using an adaptively iterative shrinkage-thresholding algorithm,” Opt. Express 20, 5942–5954 (2012).
[CrossRef]

P. W. M. Tsang, K. W. K. Cheung, T. Kim, Y. S. Kim, and T.-C. Poon, “Fast reconstruction of sectional images in digital holography,” Opt. Lett. 36, 2650–2652 (2011).
[CrossRef]

J. Ke, T.-C. Poon, and E. Y. Lam, “Depth resolution enhancement in optical scanning holography with a dual-wavelength laser source,” Appl. Opt. 50, H285–H296 (2011).
[CrossRef]

X. Zhou, K. Dobson, Y. Shinoda, and T.-C. Poon, “Sectional image reconstruction in optical scanning holography using a random-phase pupil,” Opt. Lett. 35, 2934–2936 (2010).
[CrossRef]

T. Kim and T.-C. Poon, “Autofocusing in optical scanning holography,” Appl. Opt. 48, H153–H159 (2009).
[CrossRef]

E. Y. Lam, X. Zhang, H. Vo, T.-C. Poon, and G. Indebetouw, “Three-dimensional microscopy and sectional image reconstruction using optical scanning holography,” Appl. Opt. 48, H113–H119 (2009).
[CrossRef]

X. Zhang, E. Y. Lam, T. Kim, Y. S. Kim, and T.-C. Poon, “Blind sectional image reconstruction for optical scanning holography,” Opt. Lett. 34, 3098–3100 (2009).
[CrossRef]

H. Kim, S.-W. Min, B. Lee, and T.-C. Poon, “Optical sectioning for optical scanning holography using phase-space filtering with Wigner distribution functions,” Appl. Opt. 47, D164–D175 (2008).
[CrossRef]

X. Zhang, E. Y. Lam, and T.-C. Poon, “Reconstruction of sectional images in holography using inverse imaging,” Opt. Express 16, 17215–17226 (2008).
[CrossRef]

T.-C. Poon and G. Indebetouw, “Three-dimensional point spread functions of an optical heterodyne scanning image processor,” Appl. Opt. 42, 1485–1492 (2003).
[CrossRef]

T.-C. Poon and T. Kim, “Optical image recognition of three-dimensional objects,” Appl. Opt. 38, 370–381 (1999).
[CrossRef]

T.-C. Poon, Optical Scanning Holography with MATLAB (Springer, 2007).

Qu, X.-C.

Refregier, P.

Saxton, W. O.

R. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Sergent, A.

B. Javidi, A. Sergent, G. Zhang, and L. Guibert, “Fault tolerance properties of a double random phase encoding encryption technique,” Opt. Eng. 36, 992–998 (1997).
[CrossRef]

Shinoda, Y.

Su, W.-C.

Sun, C.-C.

Tsang, P. W. M.

Vo, H.

Wang, B.

Wang, S. W.

Xie, J.

Yuan, S.

Zhang, G.

B. Javidi, A. Sergent, G. Zhang, and L. Guibert, “Fault tolerance properties of a double random phase encoding encryption technique,” Opt. Eng. 36, 992–998 (1997).
[CrossRef]

Zhang, X.

Zhao, F.-J.

Zhou, D.-F.

D. Lai, X. Zhou, D.-F. Zhou, and D.-H. Li, “Study on the influence of key errors on the deciphered image in the double random phase encryption system by applying affine cryptography,” J. Mod. Opt. 55, 167–176 (2008).
[CrossRef]

Zhou, X.

Appl. Opt. (11)

J. Ke, T.-C. Poon, and E. Y. Lam, “Depth resolution enhancement in optical scanning holography with a dual-wavelength laser source,” Appl. Opt. 50, H285–H296 (2011).
[CrossRef]

T. Kim, “Optical sectioning by optical scanning holography and a Wiener filter,” Appl. Opt. 45, 872–879 (2006).
[CrossRef]

H. Kim, S.-W. Min, B. Lee, and T.-C. Poon, “Optical sectioning for optical scanning holography using phase-space filtering with Wigner distribution functions,” Appl. Opt. 47, D164–D175 (2008).
[CrossRef]

E. Y. Lam, X. Zhang, H. Vo, T.-C. Poon, and G. Indebetouw, “Three-dimensional microscopy and sectional image reconstruction using optical scanning holography,” Appl. Opt. 48, H113–H119 (2009).
[CrossRef]

T. Kim and T.-C. Poon, “Autofocusing in optical scanning holography,” Appl. Opt. 48, H153–H159 (2009).
[CrossRef]

T.-C. Poon and G. Indebetouw, “Three-dimensional point spread functions of an optical heterodyne scanning image processor,” Appl. Opt. 42, 1485–1492 (2003).
[CrossRef]

T.-C. Poon and T. Kim, “Optical image recognition of three-dimensional objects,” Appl. Opt. 38, 370–381 (1999).
[CrossRef]

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef]

B. Wang, C.-C. Sun, W.-C. Su, and A. E. T. Chiou, “Shift-tolerance property of an optical double-random phase-encoding encryption system,” Appl. Opt. 39, 4788–4793 (2000).
[CrossRef]

X. Zhou, S. Yuan, S. W. Wang, and J. Xie, “Affine cryptosystem of double-random-phase encryption based on the fractional Fourier transform,” Appl. Opt. 45, 8434–8439 (2006).
[CrossRef]

T. Nomura and B. Javidi, “Optical encryption system with a binary key code,” Appl. Opt. 39, 4783–4787 (2000).
[CrossRef]

J. Mod. Opt. (2)

D. Lai, X. Zhou, D.-F. Zhou, and D.-H. Li, “Study on the influence of key errors on the deciphered image in the double random phase encryption system by applying affine cryptography,” J. Mod. Opt. 55, 167–176 (2008).
[CrossRef]

G. Indebetouw, “Properties of a scanning holographic microscopy: improved resolution, extended depth-of-focus, and/or optical sectioning,” J. Mod. Opt. 49, 1479–1500 (2002).
[CrossRef]

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

Opt. Eng. (1)

B. Javidi, A. Sergent, G. Zhang, and L. Guibert, “Fault tolerance properties of a double random phase encoding encryption technique,” Opt. Eng. 36, 992–998 (1997).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Optik (1)

R. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Other (1)

T.-C. Poon, Optical Scanning Holography with MATLAB (Springer, 2007).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic of OSH system. p1(x,y) and p2(x,y) are pupil functions; I(x,y;z) is the input transmittance intensity function; f is the focal length of lens l1; z is the distance of the object from the focal plane of lens l1; is an electronic multiplier; LPF is a low-pass filter; PD is a photo detector.

Fig. 2.
Fig. 2.

Schematic of OSH system when a lateral deviation occurs. p1d(x) and p2d(x) are decoding pupil functions; p2d(xx0) is p2d(x) with a lateral shift x0. Note that a delta function is to be scanned to obtain the point spread function of the system as discussed in the text.

Fig. 3.
Fig. 3.

Schematic of the random-phase pupil with pixel value error.

Fig. 4.
Fig. 4.

Sectional image reconstruction with the decryption pupil function lateral deviation. (a) The original image; (b) x0=0mm, y0=0mm; (c) x0=2mm, y0=0mm; (d) x0=10mm, y0=0mm; (e) x0=0mm, y0=2mm; (f) x0=0mm, y0=10mm. The focus of lens l1 is f=7.5cm; the reconstruct longitudinal is zd=8cm.

Fig. 5.
Fig. 5.

Relationship curves between the normalized decoding pupil lateral shift and SNR of the reconstruction sectional image.

Fig. 6.
Fig. 6.

Reconstructed images with error pixels of decoding pupil. (a) 1/196, (b) 1/16, and (c) 1/4.

Fig. 7.
Fig. 7.

Relationship curve between the ratios of the error pixels and the image SNR.

Equations (18)

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i(x,y)=F1{F{I(x,y;z=zc)}·OTFΩ(kx,ky;z=zc)},
OTFΩ(kx,ky;zc)=exp[jzc2k0(kx2+ky2)]×p1c*(x,y)p2c(x+fk0kx,y+fk0ky)exp[jzcf(xkx+yky)]dxdy,
ϕout(x,y)=F1{F{I(x,y;zc)}·OTFΩ(kx,ky;zc)·OTFΩ(kx,ky;zd)}.
ϕout(x,y)=F1{F{I(x,y;zc)}exp[jzdzc2k0(kx2+ky2)]}.
p1c(x,y)·p2d(x,y)=1,
P1c(zifkx,zifky)·P2d(zdfkx,zdfky)=1,
ϕout(x,y)=ziF1{F{I(x,y;zi)}×P1c(zifkx,zifky)·P2d(zdfkx,zdfky)·exp[jzdzi2k0(kx2+ky2)]}.
p2dΔ(x)=p2d(xx0)·1D·rect(xx0/2Dx0),
P2dΔ(zdfkx)=[P2d(zdfkx)·exp(j2πzdfkxx0)][(Dx0D)·exp(jπx0zdfkx)sinc((Dx0)zdfkx)].
ϕout(x,y)=ziF1{F{I(x;zi)}P1c*(zifkx)P2dΔ(zdfkx)exp(jzdzi2k0kx2)}=ziF1{F{I(x;zi)}P1c*(zifkx){[P2d(zdfkx)exp(j2πzdfkxx0)](Dx0D)exp(jπx0zdfkx)sinc((Dx0)zdfkx)}exp(jzdzi2k0kx2)}=ziF1{F{I(x;zi)}P1c*(zifkx){ξP2d(zdfkxξ)exp[j2π(zdfkxξ)x0]×(Dx0D)exp(jπx0ξ)sinc[(Dx0)ξ]}exp(jzdzi2k0kx2)}.
ϕout(x)=ziF1{F{I(x;zi)}P1c*(zifkx)P2d(zdfkx)×exp(j2πzdfkxx0)(Dx0D)exp(jzdzi2k0kx2)+R(kx)},
ϕout(x)=F1{F{I(x;z=zd)}P1c*(zdfkx)P2d(zdfkx)×exp(j2πzdfkxx0)(Dx0D)+R(kx)}=(Dx0D)I(xzdfx0;zd)+r(x).
|ϕout(x)|2=(Dx0D)2|I(xzdfx0;zd)|2+r(x),
SNR=10log10{(Dx0)2D2(Dx0)2}.
p2dΔ(x)=1D[p2d(x)rect(xx/2Dx)+p3d(x)rect(x(xD)/2x)],
P2dΔ(zdfkx)=P2d(zdfkx)(DxD)exp(jπxzdfkx)sinc[(Dx)zdfkx]+P3d(zdfkx)xDexp[jπ(xD)zd2fkx]sinc(xzdfkx).
ϕout(x)=ziF1{F{I(x;zi)}P1c*(zifkx)P2d(zdfkx)(DxD)exp(jzdzi2k0kx2)+R(kx)}.
ϕout(x)=F1{F{I(x;z=zd)}P1c*(zdfkx)P2d(zdfkx)(DxD)+R(kx)}=(DxD)I(x;zd)+r(x).

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