Abstract

We demonstrate a novel computational method for high resolution image recovery from a single digital hologram frame. The complex object field is obtained from the recorded hologram by solving a constrained optimization problem. This approach which is unlike the physical hologram replay process is shown to provide high quality image recovery even when the dc and the cross terms in the hologram overlap in the Fourier domain. Experimental results are shown for a Fresnel zone hologram of a resolution chart, intentionally recorded with a small off-axis reference beam angle. Excellent image recovery is observed without the presence of dc or twin image terms and with minimal speckle noise.

© 2013 OSA

Full Article  |  PDF Article

Corrections

Kedar Khare, P. T. Samsheer Ali, and Joby Joseph, "Single shot high resolution digital holography: Erratum," Opt. Express 21, 5634-5634 (2013)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-21-5-5634

References

  • View by:
  • |
  • |
  • |

  1. U. Schnars and W. P. Jüptner, Digital Holography–Digital Hologram Recording, Numerical Reconstruction and Related Techniques (Springer-Verlag, Berlin, 2005).
  2. L. Yaroslavsky, Digital Holography and Digital Image Processing (Kluwer Academic, 2010).
  3. E. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am.52(10), 1123–1128 (1962).
    [CrossRef]
  4. T. M. Kreis and W. P. Juptner, “Suppression of the dc term in digital holography,” Opt. Eng.36(8), 2357–2360 (1997).
    [CrossRef]
  5. Y. Takaki, H. Kawai, and H. Ohzu, “Hybrid holographic microscopy free of conjugate and zero-order images,” Appl. Opt.38(23), 4990–4996 (1999).
    [CrossRef] [PubMed]
  6. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett.22(16), 1268–1270 (1997).
    [CrossRef] [PubMed]
  7. E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt.39(23), 4070–4075 (2000).
    [CrossRef] [PubMed]
  8. K. Khare and N. George, “Direct coarse sampling of electronic holograms,” Opt. Lett.28(12), 1004–1006 (2003).
    [CrossRef] [PubMed]
  9. G. L. Chen, C. Y. Lin, M. K. Kuo, and C. C. Chang, “Numerical suppression of zero-order image in digital holography,” Opt. Express15(14), 8851–8856 (2007).
    [CrossRef] [PubMed]
  10. L. Ma, H. Wang, Y. Li, and H. Zhang, “Elimination of zero-order diffraction and conjugate image in off-axis digital holography,” J. Mod. Opt.56(21), 2377–2383 (2009).
    [CrossRef]
  11. M. Liebling, T. Blu, and M. A. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A21(3), 367–377 (2004).
    [CrossRef] [PubMed]
  12. M. Liebling, T. Blu, and M. A. Unser, “Non-linear Fresnelet approximation for interference term suppression in digital holography,” Proc. SPIE5207, 553–559 (2003).
    [CrossRef]
  13. J. Zhong, J. Weng, and C. Hu, “Reconstruction of digital hologram by use of the wavelet transform,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2009), paper DWB16.
  14. J. Zhong and J. Weng, “Reconstruction of digital hologram by use of the wavelet transform,” in Holography, Research and Technologies, J. Rosen, ed.(InTech, 2011)
  15. E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52(2), 489–509 (2006).
    [CrossRef]
  16. K. Khare, C. J. Hardy, K. F. King, P. A. Turski, and L. Marinelli, “Accelerated MR imaging using compressive sensing with no free parameters,” Magn. Reson. Med.68(5), 1450–1457 (2012).
    [CrossRef] [PubMed]
  17. M. Bertero and P. Boccacchi, Introduction to Inverse Problems in Imaging (IOP, 1998).
  18. J. A. Fessler, “Penalized weighted least square image reconstruction for positron emission tomography,” IEEE Trans. Med. Image.13(2), 290–300 (1994).
    [CrossRef]
  19. D. H. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Proc. H Microwaves Opt. Antennas 130, 11–16 (1983).
  20. S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004).
  21. S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell.PAMI-6(6), 721–741 (1984).
    [CrossRef] [PubMed]
  22. E. Y. Sidky, Y. Duchin, X. Pan, and C. Ullberg, “A constrained, total-variation minimization algorithm for low-intensity x-ray CT,” Med. Phys.38(S1Suppl 1), S117–S125 (2011).
    [CrossRef] [PubMed]
  23. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York 1996).

2012 (1)

K. Khare, C. J. Hardy, K. F. King, P. A. Turski, and L. Marinelli, “Accelerated MR imaging using compressive sensing with no free parameters,” Magn. Reson. Med.68(5), 1450–1457 (2012).
[CrossRef] [PubMed]

2011 (1)

E. Y. Sidky, Y. Duchin, X. Pan, and C. Ullberg, “A constrained, total-variation minimization algorithm for low-intensity x-ray CT,” Med. Phys.38(S1Suppl 1), S117–S125 (2011).
[CrossRef] [PubMed]

2009 (1)

L. Ma, H. Wang, Y. Li, and H. Zhang, “Elimination of zero-order diffraction and conjugate image in off-axis digital holography,” J. Mod. Opt.56(21), 2377–2383 (2009).
[CrossRef]

2007 (1)

2006 (1)

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52(2), 489–509 (2006).
[CrossRef]

2004 (1)

2003 (2)

K. Khare and N. George, “Direct coarse sampling of electronic holograms,” Opt. Lett.28(12), 1004–1006 (2003).
[CrossRef] [PubMed]

M. Liebling, T. Blu, and M. A. Unser, “Non-linear Fresnelet approximation for interference term suppression in digital holography,” Proc. SPIE5207, 553–559 (2003).
[CrossRef]

2000 (1)

1999 (1)

1997 (2)

T. M. Kreis and W. P. Juptner, “Suppression of the dc term in digital holography,” Opt. Eng.36(8), 2357–2360 (1997).
[CrossRef]

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett.22(16), 1268–1270 (1997).
[CrossRef] [PubMed]

1994 (1)

J. A. Fessler, “Penalized weighted least square image reconstruction for positron emission tomography,” IEEE Trans. Med. Image.13(2), 290–300 (1994).
[CrossRef]

1984 (1)

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell.PAMI-6(6), 721–741 (1984).
[CrossRef] [PubMed]

1983 (1)

D. H. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Proc. H Microwaves Opt. Antennas 130, 11–16 (1983).

1962 (1)

Blu, T.

M. Liebling, T. Blu, and M. A. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A21(3), 367–377 (2004).
[CrossRef] [PubMed]

M. Liebling, T. Blu, and M. A. Unser, “Non-linear Fresnelet approximation for interference term suppression in digital holography,” Proc. SPIE5207, 553–559 (2003).
[CrossRef]

Brandwood, D. H.

D. H. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Proc. H Microwaves Opt. Antennas 130, 11–16 (1983).

Candes, E.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52(2), 489–509 (2006).
[CrossRef]

Chang, C. C.

Chen, G. L.

Cuche, E.

Depeursinge, C.

Duchin, Y.

E. Y. Sidky, Y. Duchin, X. Pan, and C. Ullberg, “A constrained, total-variation minimization algorithm for low-intensity x-ray CT,” Med. Phys.38(S1Suppl 1), S117–S125 (2011).
[CrossRef] [PubMed]

Fessler, J. A.

J. A. Fessler, “Penalized weighted least square image reconstruction for positron emission tomography,” IEEE Trans. Med. Image.13(2), 290–300 (1994).
[CrossRef]

Geman, D.

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell.PAMI-6(6), 721–741 (1984).
[CrossRef] [PubMed]

Geman, S.

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell.PAMI-6(6), 721–741 (1984).
[CrossRef] [PubMed]

George, N.

Hardy, C. J.

K. Khare, C. J. Hardy, K. F. King, P. A. Turski, and L. Marinelli, “Accelerated MR imaging using compressive sensing with no free parameters,” Magn. Reson. Med.68(5), 1450–1457 (2012).
[CrossRef] [PubMed]

Juptner, W. P.

T. M. Kreis and W. P. Juptner, “Suppression of the dc term in digital holography,” Opt. Eng.36(8), 2357–2360 (1997).
[CrossRef]

Kawai, H.

Khare, K.

K. Khare, C. J. Hardy, K. F. King, P. A. Turski, and L. Marinelli, “Accelerated MR imaging using compressive sensing with no free parameters,” Magn. Reson. Med.68(5), 1450–1457 (2012).
[CrossRef] [PubMed]

K. Khare and N. George, “Direct coarse sampling of electronic holograms,” Opt. Lett.28(12), 1004–1006 (2003).
[CrossRef] [PubMed]

King, K. F.

K. Khare, C. J. Hardy, K. F. King, P. A. Turski, and L. Marinelli, “Accelerated MR imaging using compressive sensing with no free parameters,” Magn. Reson. Med.68(5), 1450–1457 (2012).
[CrossRef] [PubMed]

Kreis, T. M.

T. M. Kreis and W. P. Juptner, “Suppression of the dc term in digital holography,” Opt. Eng.36(8), 2357–2360 (1997).
[CrossRef]

Kuo, M. K.

Leith, E.

Li, Y.

L. Ma, H. Wang, Y. Li, and H. Zhang, “Elimination of zero-order diffraction and conjugate image in off-axis digital holography,” J. Mod. Opt.56(21), 2377–2383 (2009).
[CrossRef]

Liebling, M.

M. Liebling, T. Blu, and M. A. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A21(3), 367–377 (2004).
[CrossRef] [PubMed]

M. Liebling, T. Blu, and M. A. Unser, “Non-linear Fresnelet approximation for interference term suppression in digital holography,” Proc. SPIE5207, 553–559 (2003).
[CrossRef]

Lin, C. Y.

Ma, L.

L. Ma, H. Wang, Y. Li, and H. Zhang, “Elimination of zero-order diffraction and conjugate image in off-axis digital holography,” J. Mod. Opt.56(21), 2377–2383 (2009).
[CrossRef]

Marinelli, L.

K. Khare, C. J. Hardy, K. F. King, P. A. Turski, and L. Marinelli, “Accelerated MR imaging using compressive sensing with no free parameters,” Magn. Reson. Med.68(5), 1450–1457 (2012).
[CrossRef] [PubMed]

Marquet, P.

Ohzu, H.

Pan, X.

E. Y. Sidky, Y. Duchin, X. Pan, and C. Ullberg, “A constrained, total-variation minimization algorithm for low-intensity x-ray CT,” Med. Phys.38(S1Suppl 1), S117–S125 (2011).
[CrossRef] [PubMed]

Romberg, J.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52(2), 489–509 (2006).
[CrossRef]

Sidky, E. Y.

E. Y. Sidky, Y. Duchin, X. Pan, and C. Ullberg, “A constrained, total-variation minimization algorithm for low-intensity x-ray CT,” Med. Phys.38(S1Suppl 1), S117–S125 (2011).
[CrossRef] [PubMed]

Takaki, Y.

Tao, T.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52(2), 489–509 (2006).
[CrossRef]

Turski, P. A.

K. Khare, C. J. Hardy, K. F. King, P. A. Turski, and L. Marinelli, “Accelerated MR imaging using compressive sensing with no free parameters,” Magn. Reson. Med.68(5), 1450–1457 (2012).
[CrossRef] [PubMed]

Ullberg, C.

E. Y. Sidky, Y. Duchin, X. Pan, and C. Ullberg, “A constrained, total-variation minimization algorithm for low-intensity x-ray CT,” Med. Phys.38(S1Suppl 1), S117–S125 (2011).
[CrossRef] [PubMed]

Unser, M. A.

M. Liebling, T. Blu, and M. A. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A21(3), 367–377 (2004).
[CrossRef] [PubMed]

M. Liebling, T. Blu, and M. A. Unser, “Non-linear Fresnelet approximation for interference term suppression in digital holography,” Proc. SPIE5207, 553–559 (2003).
[CrossRef]

Upatnieks, J.

Wang, H.

L. Ma, H. Wang, Y. Li, and H. Zhang, “Elimination of zero-order diffraction and conjugate image in off-axis digital holography,” J. Mod. Opt.56(21), 2377–2383 (2009).
[CrossRef]

Yamaguchi, I.

Zhang, H.

L. Ma, H. Wang, Y. Li, and H. Zhang, “Elimination of zero-order diffraction and conjugate image in off-axis digital holography,” J. Mod. Opt.56(21), 2377–2383 (2009).
[CrossRef]

Zhang, T.

Appl. Opt. (2)

IEE Proc. H Microwaves Opt. Antennas (1)

D. H. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Proc. H Microwaves Opt. Antennas 130, 11–16 (1983).

IEEE Trans. Inf. Theory (1)

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52(2), 489–509 (2006).
[CrossRef]

IEEE Trans. Med. Image. (1)

J. A. Fessler, “Penalized weighted least square image reconstruction for positron emission tomography,” IEEE Trans. Med. Image.13(2), 290–300 (1994).
[CrossRef]

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

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell.PAMI-6(6), 721–741 (1984).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

L. Ma, H. Wang, Y. Li, and H. Zhang, “Elimination of zero-order diffraction and conjugate image in off-axis digital holography,” J. Mod. Opt.56(21), 2377–2383 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Magn. Reson. Med. (1)

K. Khare, C. J. Hardy, K. F. King, P. A. Turski, and L. Marinelli, “Accelerated MR imaging using compressive sensing with no free parameters,” Magn. Reson. Med.68(5), 1450–1457 (2012).
[CrossRef] [PubMed]

Med. Phys. (1)

E. Y. Sidky, Y. Duchin, X. Pan, and C. Ullberg, “A constrained, total-variation minimization algorithm for low-intensity x-ray CT,” Med. Phys.38(S1Suppl 1), S117–S125 (2011).
[CrossRef] [PubMed]

Opt. Eng. (1)

T. M. Kreis and W. P. Juptner, “Suppression of the dc term in digital holography,” Opt. Eng.36(8), 2357–2360 (1997).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Proc. SPIE (1)

M. Liebling, T. Blu, and M. A. Unser, “Non-linear Fresnelet approximation for interference term suppression in digital holography,” Proc. SPIE5207, 553–559 (2003).
[CrossRef]

Other (7)

J. Zhong, J. Weng, and C. Hu, “Reconstruction of digital hologram by use of the wavelet transform,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (CD) (Optical Society of America, 2009), paper DWB16.

J. Zhong and J. Weng, “Reconstruction of digital hologram by use of the wavelet transform,” in Holography, Research and Technologies, J. Rosen, ed.(InTech, 2011)

M. Bertero and P. Boccacchi, Introduction to Inverse Problems in Imaging (IOP, 1998).

U. Schnars and W. P. Jüptner, Digital Holography–Digital Hologram Recording, Numerical Reconstruction and Related Techniques (Springer-Verlag, Berlin, 2005).

L. Yaroslavsky, Digital Holography and Digital Image Processing (Kluwer Academic, 2010).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York 1996).

S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Mach-Zehnder interferometer setup; M1, M2: Mirrors, Tilting of mirror M2 can introduce small off-axis angle. The Fresnel zone hologram of USAF resolution chart is recorded with distance between the object and sensor plane z = 19 cm and an off-axis angle of ~0.6 degrees.

Fig. 2
Fig. 2

(a) Fresnel hologram of USAF resolution chart; (b) Amplitude of the Fourier transform of the recorded hologram pattern shown on logarithmic scale. The dc and the cross terms are seen to overlap in the Fourier transform domain making it hard to separate them by spatial filtering approaches.

Fig. 3
Fig. 3

Image recovery by simulating the physical hologram replay process. The small off-axis angle causes the recoveries from the dc and the cross terms to overlap producing a poor quality image.

Fig. 4
Fig. 4

Image recovery using Fourier filtering of hologram in Fig. 2(a). (a),(c) show the square shaped filter regions used for recovering the object field in the hologram plane. For the filter in (a), the recovery (b) shows poor resolution. For a larger sized filter as in (c) there is dc term contribution causing speckle noise. For even larger filter as in (e), the dc contribution completely obscures the recovery (f).

Fig. 5
Fig. 5

Complex object field and image recovery by solution of minimum L2-norm problem as in Eq. (7). (a) Amplitude and (b) phase of the complex object field in the hologram plane, (c) image recovery (amplitude) by Fresnel back-propagation of the field corresponding to (a), (b). (d) is a magnified version of a part of image (c). The recovered field is modulated by the carrier and its harmonics leading to multiple orders and noise in the recovered image. The contribution of the dc terms is seen to be removed and some of the high frequency features are better resolved compared to those in Fig. 4(d).

Fig. 6
Fig. 6

Complex object field and image recovery by solution of constrained optimization problem as in Eq. (5). (a) Amplitude and (b) phase of the complex object field in the hologram plane, (c) image recovery (amplitude) by Fresnel back-propagation of the field corresponding to (a), (b). (d) is a magnified version of a part of image (c).The contribution of the dc and the twin image is seen to be removed and some of the high frequency features appear better resolved compared to those in Figs. 4(d), 5(d). The speckle noise is observed to be minimal.

Fig. 7
Fig. 7

Numerical constrained optimization procedure applied to a synthetic hologram. (a) Object, Fresnel diffraction computation showing (b) amplitude and (c) phase of the diffracted field; (d) synthetically computed off-axis hologram; Recovery of (e) amplitude and (f) phase of the complex object field in the hologram plane; (g) back Fresnel propagation of the recovered field to image plane; (h) absolute difference between original object (a) and the recovered image (g).

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

H=|O | 2 +|R | 2 +OR*+O*R.
C(O,O*)= 1 2 ||H(|O | 2 +|R | 2 +OR*+O*R)| | 2 +αψ(O,O*).
ψ(O,O*)= p q N p w pq | O p O q | 2 .
O* C(O,O*)=[H(|O | 2 +|R | 2 +OR*+O*R)].(O+R)+α O* ψ(O,O*).
O (n+1) = O (n) t [ O* C] O= O (n) .
O (n+1) =G{ O (n) +t[H(|O | 2 +|R | 2 +OR*+O*R)].(O+R)}.
h(x,y,z)= i λz exp[ iπ λz ( x 2 + y 2 )].
O (n+1) = O (n) +t[H(|O | 2 +|R | 2 +OR*+O*R)].(O+R).

Metrics