## Single shot high resolution digital holography |

Optics Express, Vol. 21, Issue 3, pp. 2581-2591 (2013)

http://dx.doi.org/10.1364/OE.21.002581

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### 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

## 1. Introduction

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]

*et al.*[5

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]

*et al.*[7

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]

*et al.*[9

9. G. L. Chen, C. Y. Lin, M. K. Kuo, and C. C. Chang, “Numerical suppression of zero-order image in digital holography,” Opt. Express **15**(14), 8851–8856 (2007). [CrossRef] [PubMed]

*et al.*[10

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]

*et al.*[11

11. M. Liebling, T. Blu, and M. A. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A **21**(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. SPIE **5207**, 553–559 (2003). [CrossRef]

*et al.*[13,14] made use of 1D &2D Gabor wavelets for the recovery of the complex object fields from an off axis digital hologram. We would like to remark that most of the dc and twin image removal techniques in above references depend either on the assumption that the dc and cross terms in the hologram are well separated in the Fourier domain or require multiple hologram frames as in phase shifting methods. As already stated above, this requirement of separation of the dc and the cross terms is often considered as a key factor governing the resolution achievable in single frame DH imaging. In this paper, we describe a new computational approach for image recovery that gives high quality image recovery from a single hologram frame even when the dc and the cross terms have substantial overlap with each other. We treat the problem of recovery of the complex object field

15. E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory **52**(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]

## 2. Image recovery in DH as a constrained optimization problem

*q*denotes some neighborhood

*p*. The weight function

*p*and

*q*. These types of penalty functions are often used in medical imaging literature for iterative image reconstruction [18

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

^{th}iteration by

*t*is a positive constant representing the step size which may be selected by standard line-search methods [20]. Operationally, we have implemented the iterative method in Eq. (4) by alternating between the least square term and the smoothness term. The L2-error is reduced by using Eq. (4) with ‘

*O*may certainly be used [18

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

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]

*G*may thus be selected to suppress the carrier-frequency component and its second harmonic that may get introduced into the solution. We observe that using an averaging filter with a spatial extent of approximately equal to half the carrier-fringe period (corresponding to the second harmonic of the carrier fringe) is sufficient for this purpose as we illustrate in the following section.

## 3. Experiments on single shot image recovery in Digital Holography

### 3.1 Image recovery by simulating the physical hologram replay process

### 3.2 Image recovery by spatial filtering approach

### 3.3 Image recovery by solving minimum L2-norm problem

*t*was selected using line search during each iteration and was found to be in the range 0.01 to 0.001. We observe that the iteration is fully in the hologram plane domain and requires purely algebraic operations and only a few seconds are required for completing 20 iterations for a hologram frame size of 1944x2592. The computation was performed using MATLAB (our code is not necessarily optimized in any way) on a PC with 3.3 GHz Xeon processor. We observe that the recovered field is modulated by the carrier fringes and their harmonic components. When this field is Fresnel back-propagated to the image plane, the recovery as shown in Fig. 5(c), 5(d) has artifacts due to this modulation. Figure 5(d) is a magnified version of a part of Fig. 5(c). In particular multiple overlapping orders are seen in the recovery as a result of the modulation. It is however important to note that the contribution of the dc terms is seen to be removed and some of the high frequency features in the center of the resolution chart appear better resolved as compared to those in Fig. 4(d).

### 3.4 Image recovery by solving minimum L2-norm problem with smoothness constraint

*t*was selected using line search during each iteration and was again found to be in the range 0.01 to 0.001. The computation time required was approximately 10 seconds due to the additional step of convolution with the averaging filter. Once again we would like to point out that the use of specialized hardware or optimization of code could result in much better timing performance although this is not the main focus of the present work. Figures 6(a) and 6(b) show amplitude and phase of the recovered field

### 3.5 Complex object wave and image recovery for a synthetic hologram

## 4. Conclusion

## References and links

1. | U. Schnars and W. P. Jüptner, |

2. | L. Yaroslavsky, |

3. | E. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am. |

4. | T. M. Kreis and W. P. Juptner, “Suppression of the dc term in digital holography,” Opt. Eng. |

5. | Y. Takaki, H. Kawai, and H. Ohzu, “Hybrid holographic microscopy free of conjugate and zero-order images,” Appl. Opt. |

6. | I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. |

7. | E. Cuche, P. Marquet, and C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. |

8. | K. Khare and N. George, “Direct coarse sampling of electronic holograms,” Opt. Lett. |

9. | G. L. Chen, C. Y. Lin, M. K. Kuo, and C. C. Chang, “Numerical suppression of zero-order image in digital holography,” Opt. Express |

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. |

11. | M. Liebling, T. Blu, and M. A. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A |

12. | M. Liebling, T. Blu, and M. A. Unser, “Non-linear Fresnelet approximation for interference term suppression in digital holography,” Proc. SPIE |

13. | J. Zhong, J. Weng, and C. Hu, “Reconstruction of digital hologram by use of the wavelet transform,” in |

14. | J. Zhong and J. Weng, “Reconstruction of digital hologram by use of the wavelet transform,” in |

15. | E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory |

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. |

17. | M. Bertero and P. Boccacchi, |

18. | J. A. Fessler, “Penalized weighted least square image reconstruction for positron emission tomography,” IEEE Trans. Med. Image. |

19. | D. H. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Proc. H Microwaves Opt. Antennas |

20. | S. Boyd and L. Vandenberghe, |

21. | S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. |

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. |

23. | J. W. Goodman, |

**OCIS Codes**

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(090.0090) Holography : Holography

(110.0110) Imaging systems : Imaging systems

**ToC Category:**

Holography

**History**

Original Manuscript: November 7, 2012

Revised Manuscript: December 21, 2012

Manuscript Accepted: December 25, 2012

Published: January 28, 2013

**Citation**

Kedar Khare, P. T. Samsheer Ali, and Joby Joseph, "Single shot high resolution digital holography," Opt. Express **21**, 2581-2591 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-2581

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### References

- 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).
- E. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am.52(10), 1123–1128 (1962). [CrossRef]
- T. M. Kreis and W. P. Juptner, “Suppression of the dc term in digital holography,” Opt. Eng.36(8), 2357–2360 (1997). [CrossRef]
- 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]
- I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett.22(16), 1268–1270 (1997). [CrossRef] [PubMed]
- 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]
- K. Khare and N. George, “Direct coarse sampling of electronic holograms,” Opt. Lett.28(12), 1004–1006 (2003). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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)
- 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]
- 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]
- M. Bertero and P. Boccacchi, Introduction to Inverse Problems in Imaging (IOP, 1998).
- J. A. Fessler, “Penalized weighted least square image reconstruction for positron emission tomography,” IEEE Trans. Med. Image.13(2), 290–300 (1994). [CrossRef]
- D. H. Brandwood, “A complex gradient operator and its application in adaptive array theory,” IEE Proc. H Microwaves Opt. Antennas 130, 11–16 (1983).
- S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004).
- 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]
- 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]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York 1996).

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