## 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

Encryption or watermarking of 2-D and 3-D data has been extensively studied to provide copy protection and/or copyright protection for digital image, audio, and video [1–14]. Optical encryption techniques can be used to protect data stored in the material or digital data transmitted over the conventional transmission channel [9–14]. However, if one knows the encryption key, his decrypted data should be identical to the original data and no more protection of the data from then. Watermarking techniques can complement the encryption by embedding a secret imperceptible signal (or *watermark*) directly into the original data (or *host*) in such a way that it always remains present against the attacks by a third party [1–8].

Digital watermarking of 2-D data or 3-D object by digital holography or double-random phase encoding was successfully verified by use of optics [3–7]. Several methods of encrypting or watermarking the original data by virtual optics were also reported in which both the encryption and the decryption were performed digitally on a computer [8–10].

In this paper we derive a formula for the optimum weighting factor, with which the watermark is embedded into the digital hologram of a 3-D host object, that minimizes the total mean-square-error (MSE) of the reconstructed 3-D host object and the decoded watermark. 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.

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

Digital hologram of a 3-D host object can be optically obtained from the phase-shifting digital holography. First, Fresnel diffraction of the host object is recorded at CCD(Charge-Coupled Device) by an interference with a plane reference wave. Then the digital hologram of the host object can be calculated from four different interference patterns generated by changing the reference beam phase to 0, -π/2, -*π*, and -3*π*/2, respectively [10,13–15]. The digital hologram of a 3-D host object *g*(*x*
_{o}, *y*
_{o}; z) is given by:

where *FrT*{ }_{z=do} represents Fresnel transformation over a distance *d _{o}*.

The watermark is first encrypted by double-random phase encoding technique before it is embedded into the digital hologram of a host object as:

where *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].

Then the watermarked digital hologram is obtained from Eqs. (1) and (2) as:

where *w* is a weighting factor of the watermark.

The host object can be reconstructed from the watermarked digital hologram by an inverse Fresnel transformation in the conventional method as:

where *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.

The watermark itself can be decoded from the hologram by double-random phase decoding as:

where *n _{g}*(

*x*

_{1},

*y*

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

The watermark must be embedded into the host such that it is not perceptible by a human eye but is still robust against the attacks by a third party. Since in this case the host should not be degraded much, it is needed to find the optimum weighting factor. We define the optimum weighting factor to be the one that minimizes the total mean-square-error *E*(*w*) of the reconstructed host object and the decoded watermark. *E*(*w*) is defined as:

$$\cong \frac{1}{{N}_{x}{N}_{y}}\sum _{x=1}^{{N}_{x}}\sum _{y=1}^{{N}_{y}}\left[{\mid {n}_{f}(x,y)\mid}^{2}{w}^{2}+{\left(1-w\right)}^{2}{f}^{2}(x,y)+{\mid {n}_{g}(x,y)\mid}^{2}\right].$$

Assumed in the derivation is *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.

Then the optimum weighting factor *w _{opt}* is obtained at the minimum of

*E*(

*w*) as:

Note that *w _{opt}* depends on the watermark

*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}.

The digital hologram, which is obtained by the phase-shifting digital holography, can be encrypted for security reasons as is proposed in [10]. The encrypted digital hologram can also be watermarked for the copyright protection. In this case the optimum weighting factor is derived in the same method as above only when the digital hologram *G*(*x*, *y*) is replaced by the encrypted digital hologram *G _{e}*(

*x*,

_{e}*y*):

_{e}where *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

In the experiment a toy car was used as a 3-D host object of a size 3.5cm×3.0cm×7.0cm. Its digital hologram was recorded by the phase-shifting digital holography, with a laser beam of wavelength 514.5nm, at a CCD located at a distance *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.

In the next experiment the watermark was embedded into the digital hologram with three different weighting factors of *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.

Similarly, the watermark was embedded into the encrypted-hologram with three different weighting factors of *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.

Figure 5 shows the total MSE, measured in the experiment, of the reconstructed host object and the decoded watermark as a function of the weighting factor. The minimum of MSE is measured at the weighting factor of 0.59 and 0.75 for the watermarked hologram and for the watermarked encrypted-hologram, respectively. It is noted that the measured optimum values agree well with the theoretical predictions of Eq. (7). It is also noted that the MSE of the reconstructed host object increases as the weighting factor whereas that of the decoded watermark decreases.

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

Next, the robustness of the watermark and the original 3-D host object was investigated against occlusion attacks. Figures 6(a)–6(b) show the real and imaginary parts of the watermarked hologram with an occlusion attack of 25% and Figs. 6(c)–6(d) those with an occlusion attack of 50%. Figures 6(e)–6(h) show the decoded watermarks and the reconstructed host objects in these cases.

The similar experiments were done for the watermarked encrypted-hologram. Figure 7 shows the total MSE measured as a function of the weighting factor in the watermarked hologram and the watermarked encrypted-hologram, respectively, in the presence of occlusion attacks. The minimum of the total MSE was measured at the weighting factor of 0.59 and 0.75 in Figs. 7(a) and 7(b), respectively. As can be noted from Eq. (7) the optimum weighting factor remains the same even in the presence of an occlusion attack because the energies of the decoded watermark and the noise are both reduced by the same factor in the presence of the occlusion attack. The four MSE curves in Fig. 7(b) have the same features as those in Fig. 7(a) except that they are displaced horizontally and vertically. The four curves are located close together in Fig. 7(b) and the occlusion attacks come out less effective in the encrypted-hologram. This results from a stationary whiteness property of the encrypted-hologram [10].

## 5. Conclusion

In summary, we derived a formula for the optimum weighting factor with which the watermark was embedded into the digital hologram of a 3-D host object. It was shown that the optimum weighting factor minimized the total mean-square-error(MSE) of the reconstructed host object and the decoded watermark. It was also shown that the optimum weighting factor could produce the least error in the reconstructed host object and the decoded watermark regardless of the amount of an occlusion attack, which could be the advantage of our formula for the optimum weighting factor. It was also shown that our formula could be applied to watermark the encrypted-digital-hologram of a 3-D host object.

## Acknowledgments

This work was supported by the Advanced Materials and Process Research Center at Sungkyunkwan University (grant R12-2002-057-02002-0).

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