## Optimal watermarking of digital hologram of 3-D object

Optics Express, Vol. 13, Issue 8, pp. 2881-2886 (2005)

http://dx.doi.org/10.1364/OPEX.13.002881

Acrobat PDF (1709 KB)

### Abstract

The optimum condition for watermarking the digital hologram of a 3-D host object is analyzed. It is shown in the experiment that the digital hologram watermarked with the optimum weighting factor produces the least errors in the reconstructed 3-D host object and the decoded watermark even in the presence of an occlusion attack.

© 2005 Optical Society of America

## 1. Introduction

1. W. Bender, D. Gruhl, N. Morimoto, and A. Lu, “Techniques for data hiding,” IBM Syst. J. **35**, 313–336 (1996). [CrossRef]

14. E. Tajahuerce and B. Javidi, “Encrypting three-dimensional information with digital holography,” Appl. Opt. **39**, 6595–6601 (2000). [CrossRef]

9. L. Yu, X. Peng, and L. Cai, “Parameterized multi-dimensional data encryption by digital optics,” Opt. Commun. **203**, 67–77 (2002). [CrossRef]

14. E. Tajahuerce and B. Javidi, “Encrypting three-dimensional information with digital holography,” Appl. Opt. **39**, 6595–6601 (2000). [CrossRef]

*watermark*) directly into the original data (or

*host*) in such a way that it always remains present against the attacks by a third party [1

1. W. Bender, D. Gruhl, N. Morimoto, and A. Lu, “Techniques for data hiding,” IBM Syst. J. **35**, 313–336 (1996). [CrossRef]

8. X. Peng, L. Yu, and L. Cai, “Digital watermarking in three-dimensional space with a virtual-optics imaging modality,” Opt. Commun. **226**, 155–165 (2003). [CrossRef]

3. N. Takai and Y. Mifune, “Digital watermarking by a holographic technique,” Appl. Opt. **41**, 865–873 (2002). [CrossRef] [PubMed]

7. L. Cai, M. He, Q. Liu, and X. Yang, “Digital image encryption and watermarking by phase-shifting interferometry,” Appl. Opt. **43**, 3078–3084 (2004). [CrossRef] [PubMed]

8. X. Peng, L. Yu, and L. Cai, “Digital watermarking in three-dimensional space with a virtual-optics imaging modality,” Opt. Commun. **226**, 155–165 (2003). [CrossRef]

10. H. Kim, D.-H. Kim, and Yeon H. Lee, “Encryption of digital hologram of 3-D object by virtual optics,” Opt. Express12, 4912–4921 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4912. [CrossRef] [PubMed]

## 2. Watermarking of digital hologram of 3-D object

*π*, and -3

*π*/2, respectively [10

10. H. Kim, D.-H. Kim, and Yeon H. Lee, “Encryption of digital hologram of 3-D object by virtual optics,” Opt. Express12, 4912–4921 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4912. [CrossRef] [PubMed]

13. E. Tajahuerce, O. Matoba, S. C. Verrall, and B. Javidi, “Optoelectronic information encryption with phase-shifting interferometry,” Appl. Opt. **39**, 2313–2320 (2000). [CrossRef]

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

*g*(

*x*

_{o},

*y*

_{o}; z) is given by:

*FrT*{ }

_{z=do}represents Fresnel transformation over a distance

*d*.

_{o}*f*(

*x*

_{1},

*y*

_{1}) is a real and positive function representing the watermark, and

*p*(

*x*

_{1},

*y*

_{1}) and

*b*(

*u*,

*v*) are statistically independent random phase masks uniformly distributed in [0, 2

*π*]. Here

*FT*{ } and

*IFT*{ } represent Fourier and inverse Fourier transformations, respectively. Figure 1 shows the conventional watermarking of the digital hologram of a host object [5

5. S. Kishk and B. Javidi, “Watermarking of three-dimensional objects by digital holography,” Opt. Lett. **28**, 167–169 (2003). [CrossRef] [PubMed]

*w*is a weighting factor of the watermark.

*IFrT*{ } represents inverse Fresnel transformation and

*n*(

_{f}*x*

_{o},

*y*

_{o})=

*IFrT*{

*F*(

*x*,

*y*)}

_{z=do}represents an additive noise resulting from the watermark.

*n*(

_{g}*x*

_{1},

*y*

_{1}) represents an additive noise resulting from the digital hologram of the host object.

*E*(

*w*) of the reconstructed host object and the decoded watermark.

*E*(

*w*) is defined as:

*f*

^{2}(

*x*,

*y*)≫|

*n*(

_{g}*x*,

*y*)|

^{2}for pixels of a nonzero value and therefore ∑∑

*f*(

*x*,

*y*)|

*f*(

_{d}*x*,

*y*)|≅

*w*∑∑

*f*

^{2}(

*x*,

*y*). Also note that ∑∑

*f*(

*x*,

*y*)Re{

*n*(

_{g}*x*,

*y*)}≅0 because

*n*(

_{g}*x*,

*y*) becomes a complex Gaussian white noise with a zero mean, where Re{ } represents the real part.

*N*and

_{x}*N*are the numbers of CCD pixels in

_{y}*x*and

*y*-coordinates. Note that subscripts o and 1 are omitted in the final equation for simplicity.

*w*is obtained at the minimum of

_{opt}*E*(

*w*) as:

*w*depends on the watermark

_{opt}*f*(

*x*,

*y*) itself and the watermark noise in the reconstructed host object, which is given by

*n*(

_{f}*x*,

*y*)=

*IFrT*{

*F*(

*x*,

*y*)}

_{z=do}.

10. H. Kim, D.-H. Kim, and Yeon H. Lee, “Encryption of digital hologram of 3-D object by virtual optics,” Opt. Express12, 4912–4921 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4912. [CrossRef] [PubMed]

*G*(

*x*,

*y*) is replaced by the encrypted digital hologram

*G*(

_{e}*x*,

_{e}*y*):

_{e}*ϕ*(

_{e}*x*,

*y*) is a computer-generated random phase mask attached to the digital hologram

*G*(

*x*,

*y*) and

*FrT*{ }

_{z=de}is Fresnel transformation over a distance

*d*. It can be easily shown that the optimum weighting factor is obtained of the same form of Eq. (7) except for

_{e}*n*(

_{f}*x*,

*y*), which is given by

*n*(

_{f}*x*

_{o},

*y*

_{o})=

*IFrT*{

*IFrT*{

*F*(

*x*,

_{e}*y*)}

_{e}_{z=de}exp[-

*jϕ*(

_{e}*x*,

*y*)]}

_{z=do}in this case.

## 3. Experiments

### 3.1 Optimal watermarking of 3-D object

*d*

_{o}=250cm. The CCD had pixels of 640×480 at a pitch of 8.4µm×9.8µm. Figures 2(a) and 2(b) show the real and imaginary parts of the recorded digital hologram. Figure 2(c) shows the reconstructed host object from the digital hologram. Next, the watermark of Fig. 2(f) was encrypted by the double-random phase encoding and embedded into the digital hologram of Figs. 2(a) and 2(b) with the optimum weighting factor

*w*=0.59 obtained from Eq. (7). Figures 2(d) and 2(e) show the real and imaginary parts of the watermarked digital hologram. Before watermarking, the magnitudes of the digital hologram and the encrypted watermark, at each CCD pixel, were normalized by their maximums so that the two signals might have the similar strengths.

*w*=0.2, 0.59, and 1.0. Then the watermark was decoded following the procedure of Eq. (5) and the 3-D host object was reconstructed following the procedure of Eq. (4) as shown in Fig. 3. Note that the quality of the decoded watermark is improved as the weighting factor increases whereas that of the reconstructed host object is degraded.

*w*=0.2, 0.75, and 1.0. Figure 4 shows the decoded watermarks and the reconstructed 3-D host objects. In high quality prints of Figs. 3 and 4 more noises appear in the reconstructed host objects when the weighting factor is larger than the optimum.

### 3.2 Robustness of the watermark and the host object against occlusion attacks

## 5. Conclusion

## Acknowledgments

## References and links

1. | W. Bender, D. Gruhl, N. Morimoto, and A. Lu, “Techniques for data hiding,” IBM Syst. J. |

2. | G. C. Langelaar and R. L. Lagendijk, “Optimal differential energy watermarking of DCT encoded images and video,” IEEE T. Image Process. |

3. | N. Takai and Y. Mifune, “Digital watermarking by a holographic technique,” Appl. Opt. |

4. | S. Kishk and B. Javidi, “Information hiding technique with double phase encoding,” Appl. Opt. |

5. | S. Kishk and B. Javidi, “Watermarking of three-dimensional objects by digital holography,” Opt. Lett. |

6. | S. Kishk and B. Javidi, “3D Watermarking by a 3D hidden object,” Opt. Express11, 874–888 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-874. [CrossRef] [PubMed] |

7. | L. Cai, M. He, Q. Liu, and X. Yang, “Digital image encryption and watermarking by phase-shifting interferometry,” Appl. Opt. |

8. | X. Peng, L. Yu, and L. Cai, “Digital watermarking in three-dimensional space with a virtual-optics imaging modality,” Opt. Commun. |

9. | L. Yu, X. Peng, and L. Cai, “Parameterized multi-dimensional data encryption by digital optics,” Opt. Commun. |

10. | H. Kim, D.-H. Kim, and Yeon H. Lee, “Encryption of digital hologram of 3-D object by virtual optics,” Opt. Express12, 4912–4921 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4912. [CrossRef] [PubMed] |

11. | P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. |

12. | F. Goudail, F. Bollaro, B. Javidi, and P. Refregier, “Influence of a perturbation in a double-phase encoding system,” J. Opt. Soc. Am. A |

13. | E. Tajahuerce, O. Matoba, S. C. Verrall, and B. Javidi, “Optoelectronic information encryption with phase-shifting interferometry,” Appl. Opt. |

14. | E. Tajahuerce and B. Javidi, “Encrypting three-dimensional information with digital holography,” Appl. Opt. |

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

**OCIS Codes**

(090.2880) Holography : Holographic interferometry

(100.2000) Image processing : Digital image processing

(100.6890) Image processing : Three-dimensional image processing

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 16, 2005

Revised Manuscript: March 28, 2005

Published: April 18, 2005

**Citation**

Hyun Kim and Yeon Lee, "Optimal watermarking of digital hologram of 3-D object," Opt. Express **13**, 2881-2886 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-8-2881

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### References

- W. Bender, D. Gruhl, N. Morimoto, and A. Lu, �??Techniques for data hiding,�?? IBM Syst. J. 35, 313-336 (1996). [CrossRef]
- G. C. Langelaar and R. L. Lagendijk, �??Optimal differential energy watermarking of DCT encoded images and video,�?? IEEE T. Image Process. 6, 148-158 (2001). [CrossRef]
- N. Takai and Y. Mifune, �??Digital watermarking by a holographic technique,�?? Appl. Opt. 41, 865-873 (2002). [CrossRef] [PubMed]
- S. Kishk and B. Javidi, �??Information hiding technique with double phase encoding,�?? Appl. Opt. 41, 5462-5470 (2002). [CrossRef] [PubMed]
- S. Kishk and B. Javidi, �??Watermarking of three-dimensional objects by digital holography,�?? Opt. Lett. 28, 167-169 (2003). [CrossRef] [PubMed]
- S. Kishk and B. Javidi, �??3D Watermarking by a 3D hidden object,�?? Opt. Express 11, 874-888 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-874.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-874.</a> [CrossRef] [PubMed]
- L. Cai, M. He, Q. Liu, and X. Yang, �??Digital image encryption and watermarking by phase-shifting interferometry,�?? Appl. Opt. 43, 3078-3084 (2004). [CrossRef] [PubMed]
- X. Peng, L. Yu, and L. Cai, �??Digital watermarking in three-dimensional space with a virtual-optics imaging modality,�?? Opt. Commun. 226, 155-165 (2003). [CrossRef]
- L. Yu, X. Peng, and L. Cai, �??Parameterized multi-dimensional data encryption by digital optics,�?? Opt. Commun. 203, 67-77 (2002). [CrossRef]
- H. Kim, D.-H. Kim, and Yeon H. Lee, �??Encryption of digital hologram of 3-D object by virtual optics,�?? Opt. Express 12, 4912-4921 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4912.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4912.</a> [CrossRef] [PubMed]
- P. Refregier and B. Javidi, �??Optical image encryption based on input plane and Fourier plane random encoding,�?? Opt. Lett. 20, 767-769 (1995). [CrossRef] [PubMed]
- 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]
- E. Tajahuerce, O. Matoba, S. C. Verrall, and B. Javidi, �??Optoelectronic information encryption with phase-shifting interferometry,�?? Appl. Opt. 39, 2313-2320 (2000). [CrossRef]
- E. Tajahuerce and B. Javidi, �??Encrypting three-dimensional information with digital holography,�?? Appl. Opt. 39, 6595-6601 (2000). [CrossRef]
- I. Yamaguchi and T. Zhang, �??Phase-shifting digital holography,�?? Opt. Lett. 22, 1268-1270 (1997). [CrossRef] [PubMed]

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