Abstract

We present a technique for phase-shifting digital in-line holography which compensates for lateral object motion. By collecting two frames of interference between object and reference fields with identical reference phase, one can estimate the lateral motion that occurred between frames using the cross-correlation. We also describe a very general linear framework for phase-shifting holographic reconstruction which minimizes additive white Gaussian noise (AWGN) for an arbitrary set of reference field amplitudes and phases. We analyze the technique’s sensitivity to noise (AWGN, quantization, and shot), errors in the reference fields, errors in motion estimation, resolution, and depth of field. We also present experimental motion-compensated images achieving the expected resolution.

© 2006 Optical Society of America

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References

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  1. T. Kreis, Handbook of Holographic Interferometry (Wiley-VCH, Weinheim, 2005).
  2. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, Greenwood Village, 2004).
  3. I. Yamaguchi and T. Zhang, "Phase-shifting digital holography," Opt. Lett. 22, 1268-1270 (1997).
    [CrossRef] [PubMed]
  4. S. Lai, B. King, and M. A. Neifeld, "Wave front reconstruction by means of phase-shifting digital in-line holography," Opt. Commun. 173, 155-160 (2000).
    [CrossRef]
  5. K. Larkin, "A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns," Opt. Express 9, 236-253 (2001).
    [CrossRef] [PubMed]
  6. R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
    [CrossRef]

2001

2000

S. Lai, B. King, and M. A. Neifeld, "Wave front reconstruction by means of phase-shifting digital in-line holography," Opt. Commun. 173, 155-160 (2000).
[CrossRef]

1998

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
[CrossRef]

1997

Armstrong, E. E.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
[CrossRef]

Barnard, K. J.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
[CrossRef]

Bognar, J. G.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
[CrossRef]

Hardie, R. C.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
[CrossRef]

King, B.

S. Lai, B. King, and M. A. Neifeld, "Wave front reconstruction by means of phase-shifting digital in-line holography," Opt. Commun. 173, 155-160 (2000).
[CrossRef]

Lai, S.

S. Lai, B. King, and M. A. Neifeld, "Wave front reconstruction by means of phase-shifting digital in-line holography," Opt. Commun. 173, 155-160 (2000).
[CrossRef]

Larkin, K.

Neifeld, M. A.

S. Lai, B. King, and M. A. Neifeld, "Wave front reconstruction by means of phase-shifting digital in-line holography," Opt. Commun. 173, 155-160 (2000).
[CrossRef]

Watson, E. A.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
[CrossRef]

Yamaguchi, I.

Zhang, T.

Opt. Commun.

S. Lai, B. King, and M. A. Neifeld, "Wave front reconstruction by means of phase-shifting digital in-line holography," Opt. Commun. 173, 155-160 (2000).
[CrossRef]

Opt. Eng.

R. C. Hardie, K. J. Barnard, J. G. Bognar, E. E. Armstrong, and E. A. Watson, "High-resolution image reconstruction from a sequence of rotated and translated frames and its application to an infrared imaging system," Opt. Eng. 37, 247-260 (1998).
[CrossRef]

Opt. Express

Opt. Lett.

Other

T. Kreis, Handbook of Holographic Interferometry (Wiley-VCH, Weinheim, 2005).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, Greenwood Village, 2004).

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Figures (11)

Fig. 1.
Fig. 1.

Experimental setup for motion-compensated digital in-line holography. The object is transmissive, three-dimensional, and movable with a translation stage. The setup is effectively a Mach-Zehnder interferometer with an attenuator and phase shifter in one arm, and the object in the other arm.

Fig. 2.
Fig. 2.

Several images of an USAF-1951 resolution target. (a) shows a traditional image of the resolution target. (b) shows a holographic reconstruction of the intensity at the plane of the horizon ally-moving target without motion compensation. (c) shows the same reconstruction with motion compensation. (d) shows a single raw frame of the holographic data.

Fig. 3.
Fig. 3.

Two reconstructions at different depths from a single holographic data set. (a) shows a reconstruction at z=107 mm. The top target is “in focus” and the bottom target is not. (b) Using the same data but reconstructing at z=244 mm, the reverse is true.

Fig. 4.
Fig. 4.

An illustration of how diffraction limits resolution for holographic reconstruction. So that the localized grating can be reconstructed, at least two diffraction orders (including m=0) must be recorded by the sensor. This limits the minimum grating size that can be resolved.

Fig. 5.
Fig. 5.

Portion of experimental setup when using a relay lens. The holographic sensor observes the image, allowing control of the object resolution by adjusting the magnification.

Fig. 6.
Fig. 6.

Simulated SNR E as a function of sensor SNR for linear (solid) and nonlinear (dotted) reconstruction. The upper (high SNR E ) lines are for M=3 and the lower (low SNR E ) lines are for M=6. The region to the left is dominated by AWGN and the region to the right by shot noise.

Fig. 7.
Fig. 7.

An illustration of the intensities that may fall on the sensor given an object field of maximum amplitude f max and maximum reference field amplitude R max. The central region between the dashed lines of height 4f max R max represents the part of the sensor’s full dynamic range that is actually used.

Fig. 8.
Fig. 8.

Amplitude transfer function (ATF) as a function of spatial frequency kx for several values of z. In each case, the image is 500×500 9µm pixels. The vertical lines show the spatial frequency kx =1/(2rd ) corresponding to the predicted resolution at that value of z.

Fig. 9.
Fig. 9.

Several simulated images corresponding to the ATF plots in Fig. 8. The top row of images shows the entire 500×500 for each value of z, whereas the lower images are 120×120 pixels, showing the central region. Both the noise and the object field are smoothed at increasing z.

Fig. 10.
Fig. 10.

Illustration of how the resolving power with a defocus of Δz is the same for holographic reconstruction and traditional imaging with a lens of focal length F=z/2 and diameter D=Np .

Fig. 11.
Fig. 11.

MSE of a sample image as a function of defocus (solid) and axial shifts (dashed). The shifts were performed for M=3,6,9, with increasing error for larger M because the overall shift is larger. For defocus, the object was placed at z=z 0z. For the shifted images, each frame was acquired with the object at zj =z 0+[j-(M+1)/2]Δ z . In all cases, z 0=65 mm and the image size 500×500 9µm pixels.

Equations (32)

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X 1 , M + 1 ( x , y ) = I 1 I M + 1 = 1 [ ( I 1 ) * ( I M + 1 ) ]
I ¯ j ( x , y ) = I j ( x + j 1 M Δ x , y + j 1 M Δ y ) ,
I ¯ j = R j 2 + f 2 + 2 ( R j f + R j f ) ,
D j = I ¯ j R j 2 = f 2 + 2 ( R j f + R j f ) .
D = ( 2 R + 1 f ) f ,
D = [ D 1 D 2 ] , R = [ R 1 R 1 R 2 R 2 ] , 1 = [ 1 1 ] , f = [ f f ] .
P R p = T ,
where R p = [ R 1 R 1 1 R 2 R 2 1 ] and T = [ 1 0 0 0 1 0 ] .
g ̂ ( x , y ) = f ( x , y ) * K ( x , y ) = 1 [ ( f ( x , y ) ) ( K ( x , y ) ) ] ,
K ( x , y ) = e ikz i λ z e ik 2 z ( x 2 + y 2 ) .
r d = Δ 2 λ d N p .
r d 1 M i λ d N p .
σ fg 2 = σ fg 2 + σ fg 2
= 1 4 j = 1 M [ ( P j σ Ig ) 2 + ( P j σ Ig ) 2 ]
= 1 4 σ Ig 2 P F 2 ,
σ Iq 2 = ( I max ( 2 3 ) 2 b ) 2 ,
σ fq 2 = 1 4 σ Iq 2 P F 2 .
σ fs 2 = α 4 j = 1 M ( P j 2 + P j 2 ) [ R j 2 + f 2 + 2 ( R j f + R j f ) ] ,
R j = R e i ( j 1 ) 2 π M ,
σ f 2 = σ fg 2 + σ fq 2 + σ fs 2 = σ Ig 2 MR 2 + σ Iq 2 MR 2 + α M ( 1 + f 2 R 2 ) .
SNR f = f σ f = [ MR 2 f 2 α ( R 2 + f 2 ) + σ fg 2 + σ fq 2 ] 1 2 .
σ E 2 = σ Ig 2 E R + σ Iq 2 E R + α M ( 1 + E f E R )
SNR E = f σ f = [ E f E R α ( E f + E R ) + M σ fg 2 + M σ fq 2 ] 1 2 .
f ̂ = [ I + ( P ̂ P ) R ] f
= ( I + E ) f ,
rms [ E ] = [ 1 0 0 1 ] 2 M σ ε ( real ε j ) ,
= [ 0 1 1 0 ] 2 M σ ε ( imaginary ε j ) ,
= [ 1 1 1 1 ] 1 M σ ε ( complex ε j ) ,
DOF = 2 z R = π 2 ( r d λ ) 2 λ = π 2 λ ( d N P ) 2 .
f ̂ ( x , y ) = f ( x , y ) * ( x , y ) and
g ̂ ( x , y ) = [ f ( x , y ) * B ( x , y ) ] * K ( x , y )
= B ( x , y ) * [ f ( x , y ) * K ( x , y ) ] .

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