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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 8 — Apr. 21, 2003
  • pp: 874–888
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3D object watermarking by a 3D hidden object

Sherif Kishk and Bahram Javidi  »View Author Affiliations


Optics Express, Vol. 11, Issue 8, pp. 874-888 (2003)
http://dx.doi.org/10.1364/OE.11.000874


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Abstract

In this paper we present a method to watermark a 3D object with another hidden 3D object using digital holography. The watermark or the hidden information is a 3D object that is embedded in the digital hologram of a 3D host object. The digital holograms are obtained optically by phase shift interferometery. The hologram of the hidden 3D object is double phase encoded before embedding it to the host 3D object hologram. Then, the watermarked hologram is double phase encoded again using different set of codes. The resultant watermarked hologram is very secure because of the multi-key nature of the watermarking process. We discuss the effect of distortion caused by hologram quantization and occlusion of some of the hologram pixels. We present tests to illustrate the effect of using a window of the hologram to reconstruct the hidden 3D object and the host 3D object. Both mathematical analysis and simulations are presented to illustrate the system performance. To the best of our knowledge, this is the first report of embedding a 3D objects within another 3D object.

© 2003 Optical Society of America

1. Introduction

Watermarking [1

1. N. F. Johnson, Z. Duric, and S. Jajodia, Information Hiding : Steganography and Watermarking - Attacks and Countermeasures (Advances in Information Security, Volume 1,Kluwer Academic2001).

3

3. G. C. Langelaar, I. Setyawan, and R. L. Lagendijk, “Watermarking Digital Image and Video Data. A state of the Art overview,” IEEE Signal Processing Magazine 17, 20–46 (2000). [CrossRef]

] can be considered as a method to authenticate a source by hiding a piece of information within another piece of information. A good watermarking algorithm should meet a number of conditions [4

4. S. Kishk and B. Javidi, “Information Hiding Technique using Double Phase Encoding,” Appl. Opt. 41, 5470–5482 (2002). [CrossRef]

]. For example, the host data quality should not be affected in a significant way by the hidden data. If the watermark will be used to authenticate the source, the watermark should be robust to removal or modification trials. Another important issue with watermarking is how secure it is, in other words how hard it is to decode the hidden information by an un-authorized user even if the watermarking technique is known. Several techniques were proposed to watermark and hide one-dimensional and two-dimensional information [4

4. S. Kishk and B. Javidi, “Information Hiding Technique using Double Phase Encoding,” Appl. Opt. 41, 5470–5482 (2002). [CrossRef]

8

8. Po-Chyi Su, C.-C. Jay Kuo, and Houng-jyh Wang, “Wavelet-Based Digital Image Watermarking,” Opt. Express , 3, 491–496 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-12-491 [CrossRef] [PubMed]

]. Some techniques have presented watermarking of 3D objects [9

9. S. Kishk and B. Javidi “Watermarking of a 3D Object Using Digital Holography,” Opt. Let. 28, 167–169 (2003). [CrossRef]

,10

10. R. Ohbuchi, A. ukaiyama, and S. Takahashi “A frequency-domain approach to watermarking 3D shapes,” Computer Graphics Forum 21, 373–382 Sp. Iss. SI(2002). [CrossRef]

].

Double phase encoding [11

11. P. Refregier and B. Javidi, “Optical image encryption using input plane and Fourier plane random encoding,” Opt. Let. 20, 767–769 (1995). [CrossRef]

13

13. F. Goudail, F. Bollaro, B. Javidi, and P. Refregier, “Influence of perturbation in a double phase-encoding system,” J. Opt. Soc. Am. A 15, 2629–2638 (1998). [CrossRef]

] is a technique that transforms the input information to a white Gaussian noise. As a esult, it is difficult to decode the original information without knowing the codes used in the encoding process.

Digital holography [14

14. H. J. Caulfield, Handbook of Optical Holography, (Academic press, London, 1979).

,16

16. U. Schnars and W. P. O. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994). [CrossRef] [PubMed]

] can be considered as a technique to sense and visualize 3D objects. A digital hologram contains information about different views of a 3D object so a hologram can replace a set of 2D images taken from different perspectives. In general, a digital hologram stores the complex Fresnel diffraction pattern generated by a 3D object. Onaxis digital holography is easy to implement and more precise than off-axis generated digital holography. The holograms presented in this paper are generated optically using phase shift interferometery [17

17. I Yamaguchi and T Zhang, “Phase-Shifting Digital Holography,” Opt. Lett. 22, 1268–1270 (1997). [CrossRef] [PubMed]

19

19. J. Schwider, “Advanced Evaluation Techniques in Interferometry,” in Progress in Optics, E. Wolf, ed (North Holland, Amesterdam) XXVIII, 271–359 (1990)

] and stored in a digital computer. The 3D object reconstruction from a digital hologram is performed using digital computer by approximating the Fresnel integral digitally.

We propose a technique to use digital holography to hide a 3D object stored in the form of a digital hologram within another 3D object. The proposed technique generates the digital hologram of both the hidden 3D object and the host 3D object optically using phase shift interferometery. Then, the hologram of the hidden 3D object is encoded using double phase encoding. The encoded hologram of the hidden 3D object is embedded within the hologram of the host 3D object. The watermarked hologram is double phase encoded again using a different set of random phase codes. The second double phase encoding process provides a higher level of security and will guarantee the whiteness of the transmitted hologram. The watermarked hologram is robust to pixels occlusion in terms of reconstructing different poses of the 3D object due to the whiteness of the final transmitted data. To the best of our knowledge this is the first presentation for watermarking a 3D object by another 3D object.

The performance of the proposed technique is tested against several types of distortions and removal trials for the hidden 3D object. The effect of distortion caused by quantization of the transmitted hologram is investigated. Also, the effect of using a subset window of the transmitted hologram to reconstruct the 3D object is tested.

Mathematical analysis and simulations are presented to demonstrate the system performance. The analysis and simulations of the proposed algorithm illustrate the system performance in terms of the quality of decoded 3D host and 3D hidden object, and the robustness against removal trials and distortions.

The paper is organized as follows: Section 2 presents background on digital holography and double phase encoding. Section 3 discusses a description of the proposed technique and mathematical analysis. In Section 4, we illustrate analytically the system performance when distortions are present. Section 5 presents simulations and discussions, and Section 6 is the conclusion.

2. Background

2.1 Digital holography:

As described above, a digital hologram can be generated either using on-axis holography or off-axis holography. We use on-axis phase shifting interferometery because of its suitability for use with charged coupled device (CCD) cameras. Four steps phase shifting interferometer is used to generate the digital hologram. In phase shifting digital interferometery the interference pattern between the diffraction pattern of the 3D object of interest and a reference beam is generated using CCD camera. The method relies on storing four interference patterns between the diffraction pattern of the object and the reference beam. The phase of the reference beam is different at each stored interference pattern. Phase shifting of the reference beam is achieved using two phase retarders that give four phase shifts of 0, -π/2, -π, -3π/2. Fig. 1 shows a phase shifting interferometer.

Fig. 1. Phase shifting interferometer

Let the diffraction pattern of the 3D object, D(x,y), at the CCD be given by:

D(x,y)=A(x,y)exp(jϕ(x,y))
(1)

where A(x,y) is the amplitude of the diffraction of the 3D object and ϕ(x,y) is the phase of the diffraction of a 3D object. The reference beam at the CCD is given by:

R(x,y;α)=ARexpj(ϕR+α)
(2)

where AR and ϕ R are the amplitude and phase of the diffraction of the reference beam and α is a phase controlled using the phase retarder plates. The intensity pattern recorded by the CCD camera is:

I(x,y;α)=A2(x,y)+AR2+2A(x,y)ARcos[ϕ(x,y)ϕRα].
(3)

Let I1(x,y)=I(x,y;0), I2(x,y)=I(x,y;-π/2), I3(x,y)=I(x,y;-π), and I4(x,y)=I(x,y;-3π/2).

Using these four measurements for the interference intensities, it is straightforward to show that the 3D object hologram phase is:

ϕO(x,y)=arctan[I4(x,y)I2(x,y)I1(x,y)I3(x,y)],
(4)

and the amplitude is:

AO(x,y)=14{[I1(x,y)I3(x,y)]2+[I2(x,y)I4(x,y)]2}12
(5)

The complex hologram of the 3D object, Ho(x,y) is given by

HO(x,y)=AO(x,y)exp[jϕO(x,y)]
(6)

2.2 Double phase encoding:

Double phase encoding redistributes the energy of the input image such that the encoding data is white Gaussian noise-like. Let the input to the double phase encoder be a complex signal H(x,y). b1(x,y) and b2(x,y) are two statistically independent uniformly distributed random variables from 0 to 1. (x,y) are the spatial coordinates. Let ψ 1(x,y)=exp[j2πb1(x,y)] and π2(ξ,γ)=exp[j2πb1(ξ,γ)] be two phase functions. The double phase encoding signal, Hd(x,y) is given by:

Hd(x,y)={H(x,y)ψ1(x,y)}IFT[ψ2(ξ,γ)]
(7)

where (ξ,γ) are the coordinates in the Fourier plane and IFT stands for inverse Fourier transform. And the symbol ⊗ stands for convolution. The decoding process is straight forward [11

11. P. Refregier and B. Javidi, “Optical image encryption using input plane and Fourier plane random encoding,” Opt. Let. 20, 767–769 (1995). [CrossRef]

].

Using an analysis similar to [6

6. Joseph Rosen and Bahram Javidi, “Hiding images in Halftone Pictures,” Appl. Opt. 40, 3346 (2001). [CrossRef]

] it can be shown that the encoded double phase encoded complex signal is a white Gaussian noise having a variance given by:

σ2=1N.My=0M1x=0N1H(x,y)H*(x,y)
(8)

where N,M are the size of the input signal in the spatial coordinates, and * is the complex conjugate.

3. Hiding a 3D object in another 3D object

Figure 2 shows a block diagram of the proposed technique illustrating the transmitter and receiver. As illustrated in the diagram we optically construct the digital hologram for both the host 3D object and the hidden 3D object using phase shifting interferemetery. Then, the hologram of the hidden 3D object is double phase encoded, scaled, and embedded to the hologram of the host 3D object. The watermarked hologram is double phase encoded again using a different set of random phase codes. At the receiver end, the watermarked hologram is double phase decoded to recover the watermarked hologram. Then, the watermarked hologram is double phase encoded again to recover the hologram of the hidden 3D object. The watermarked hologram and the double phase decoded holograms are used to reconstruct the host 3D object and the hidden 3D object, respectively.

Let us denote the host object hologram Hhost(x,y), and the hidden object hologram Hhidden(x,y). The transmitted watermarked hologram, Hw(x,y), is given by:

Hw(x,y)={[Hhost(x,y)+α[Hhidden(x,y)ψ1(x,y)ψ2(x,y)]]ψ1(x,y)}ψ2(x,y)
(9)

where α is an arbitrary constant and ψ1, and ψ2 and ψ’1 and ψ’2 are random phase codes as defined before. It can be shown that because of the added hidden hologram, the amplitude of the host object hologram is distorted by a Raleigh distributed noise with variance, σw2 :

σw2=12.N.My=0M1x=0N1Hhidden(x,y)Hhidden*(x,y)
(10)

The phase of the host 3D object hologram is distorted by a uniformly distributed noise.

Fig. 2. Block diagram of the proposed system (a) Transmitter (b) Receiver

The receiver consists of two components: one component recovers the host 3D object and the other component recovers the 3D hidden object as shown in Fig. 2. In detecting the hidden 3D object the first double phase decoding process is a loss less operation unless the transmitted signal is distorted. The decoded 3D hidden hologram after the second double phase decoder is given by:

H˜hidden(x,y)=αHhidden(x,y)+IFT{Ĥhost(ξ,γ)ψ*2(ξ,γ)}ψ*1(x,y)
(11)

Equation (11) indicates that the decoded hidden hologram is distorted by the second term of Eq. (11). It can be shown that the second term of Eq. (11) is an additive white complex Gaussian noise with zero mean and a variance, σh2 :

σw2=1N.My=0M1x=0N1Hhost(x,y)Hhost*(x,y)
(12)

The reconstructed hidden 3D object, Õ d (u, v; d 1), at plane (u, v)at a distance d 1 from the output plane (x,y) can be reconstructed using the following inverse Fresnel discrete integral

O˜d(u,v;d1)=exp[jπλd1(Δu2u2+Δv2v2)]x=0N1y=0M1{H˜hidden(x,y)
exp[jπλd1(Δx2x+Δy2y)]exp[j2π(xuN+yvM)]}
(13)

where (Δxy), and (Δuv) are the spatial resolution at the CCD plane, and the 3D object reconstruction plane respectively. The resolution at the 3D object reconstruction plane is given by:

Δu=λd1NΔx,and
Δv=λd1MΔy
(14)

Using Eq. (11) and Eq. (13), the reconstructed 3D hidden object is given by:

O˜d(u,v;d1)=αOd(u,v;d1)+exp[jπλd1(Δu2u2+Δv2v2)].
x=0N1y=0M1{(IFT{H˜host(ξ,γ)ψ*2(ξ,γ)}ψ1*(x,y))
exp[jπλd1(Δx2x+Δy2y)]exp[j2π(xuN+yvM)]}
(15)

Od(u,v;d1) is the original hidden 3D object reconstructed at a distance d1 from the output plane. The second term of Eq. (15) is a distortion added to the recovered hidden 3D object. It can be proven that this added distortion can be represented by additive Gaussian white noise having zero mean and a variance given by Eq. (12) (Appendix A).

On the other hand, the recovered 3D host object, Õ h (u, v; d 2), is given by:

O˜h(u,v;d2)=Oh(u,v;d2)+exp[jπλd2(Δu2u2+Δv2v2)]
x=0N1=01M1{α[Hhidden(x,y)ψ*1(x,y)ψ*2(x,y)]
exp[jπλd2(Δx2x+Δy2y)]exp[j2π(xuN+yvM)]}
(16)

σhost2=α2N.My=0M1x=0N1Hhidden(x,y)Hhidden*(x,y)
(17)

4. Proposed system response to noise and distortion

In this section we investigate the effect of different types of noise and distortion on both the recovered 3D hidden object and the reconstructed 3D host object.

One problem that arises from the use of digital holography is the large file size of the digital hologram. One way to overcome this problem is to quantize the digital hologram20. In this paper we used both a uniform and optimum quantizer.

As before, we denote the double phase encoded watermarked hologram as Hw (x, y), and the quantized double phase encoded watermarked hologram as w (x, y). The number of quantization bits is B. We can represent the transmitted quantized hologram as:

H˜w(x,y)=Hw(x,y)+ΔHw(x,y)
(18)

where ΔHw (x, y), is the error introduced by the quantizer. We use a uniform quantizer. Taking into consideration the statistical nature of a transmitted double phase encoded digital hologram, it is straightforward to show that the distortion caused to the watermarked hologram due to quantization is a uniformly distributed white process with variance equal to Δ2/12 where Δ is the quantization step size The reconstructed 3D hidden object using quantized watermarked hologram is given by:

O˜d(u,v;d1)=αOd(u,v;d1)+exp[λd1(Δu2u2+Δv2v2)].
{x=0N1y=0M1(IFT{H˜host(ξ,γ)ψ2*(ξ,γ)}ψ1*(x,y)).
exp[jπλd1(Δx2x+Δy2y)]exp[j2π(xuN+yvM)]+
x=0N1y=0M1ΔHw(x,y)exp[jπλd(Δx2x+Δy2y)]exp[j2π(xuN+yvM)]}.
(19)

The reconstructed 3D host object is given by:

O˜d(u,v;d2)=Od(u,v;d2)+exp[jπλd2(Δu2u2+Δv2v2)].
{u=0N1v=0M1α[Hhidden(x,y)ψ1(x,y)ψ2(x,y)].
exp[jπλd2(Δx2x+Δy2y)]exp[j2π(xuN+yvM)]+
x=0N1y=0M1ΔHw(x,y)exp[λd2(Δx2x+Δy2y)]exp[j2π(xuN+yvM)]}
(20)

Double phase encoding is a unitary transform and the process of reconstructing the 3D object is also a unitary process. From Eqs. (19) and (20) it is easy to show that the quantization noise is un-correlated with other distortion caused by either the hidden 3D object or the host 3D object. The variance of the noise added to the reconstructed 3D host and 3D hidden objects are given by:

σhost2=α2N.My=0M1x=0N1Hhidden(x,y)Hhidden*(x,y)+Δ212
(21)
σw2=1N.My=0M1x=0N1Hhost(x,y)Hhost*(x,y)+Δ212
(22)

Since the watermarked hologram is Gaussian distributed, using companding technique or optimum quantizer will improve the performance of the quantizer.

Occlusion of some areas of the transmitted watermarked hologram can occur as an attacking method to remove the hidden hologram. The watermarked hologram is white noiselike. As a result, occluding any area of the watermarked hologram, either in the spatial domain or in the frequency domain, will have the same effect. Let the transmitted occluded hologram, Hwo , is given by:

Hwo(x,y)=Hw(x,y)(1W(x,y))=Hw(x,y)Hw(x,y)W(x,y)
(23)

where W is a window function that has a value of one in the occluded areas and zero otherwise. As this equation indicates the effect of occluding some of the hologram pixels will be additive noise and reduction in the energy [4

4. S. Kishk and B. Javidi, “Information Hiding Technique using Double Phase Encoding,” Appl. Opt. 41, 5470–5482 (2002). [CrossRef]

] of the reconstructed 3D objects.

5. Experimental results and simulations

We perform a set of experiments to evaluate the proposed 3D object watermarking by another 3D object under several conditions. We used the minimum mean square error metric to measure the quality of the recovered 3D objects. To take the z-axis depth information into consideration when calculating the mean square error we use the mean of the mean square error of the 3D objects reconstructed at different planes. The 3D object complex amplitude distribution can be reconstructed at any distance d from the output plane using:

Od(u,v;d)=exp[j2πλ(zd)]λ(zd)exp[jπλ(zd)(u2+v2)]·
O(x,y,z)exp[jπλ(zd)(x2+y2)]
exp[j2πλ(zd)(xu+yv)]dxdydz
(24)

Only the 3D object reconstructed at a distance d with d equal z will be focused in the reconstruction plane. Figure 3 shows 3D object reconstructed at different planes within the depth of the object. Equation (25) illustrates the equation used to find the mean of the mean square error along different construction planes:

error=1Pp=1P1N×My=0M1x=0N1[Od(u,v;d(p))O(u,v;d(p))]2O(u,v;d(p))
(25)

where d(p) is a vector that contains the reconstruction distances of length P, O(u,v;d(p)) is the original reconstructed object at a distance d(p) without watermarking, and Od(u,v;d(p)) is the reconstructed object at a distance d(p) with watermarking.

Fig. 3. 3D object reconstructed at different distances from the output plane
Fig. 4. The original host and hidden 3D objects.

Figure 5 shows the correlation for the watermarked hologram with and without the second double phase encoding process. It is clear from Fig. 5 that the watermarked hologram without the second double phase encoding process is not white. Therefore, means occluding some areas of the watermarked hologram might prevent us from reconstructing certain poses of the 3D host object. Figure 6 shows the reconstructed 3D host and 3D hidden objects. The resultant 3D host and 3D hidden objects have an error, as defined in equation 25 using P=5, of 7.7972e-004 and 0.15 respectively. The arbitrary constant α is chosen to preserve the quality of the host 3D object since it is the object of interest, and the hidden 3D object should have only the minimal energy that give us the ability to detect it in the presence of distortions.

Fig. 5. (a) The autocorrelation for the watermarked hologram without using the second double phase encoding process. (b) The autocorrelation for the watermarked hologram when using the second double phase encoding process.

Table 1. error when using a uniform quantizer

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Fig. 6. The Reconstructed 3D objects from the double phase encoded watermarked hologram without distortions (a) the 3D host object(b) The 3D hidden object
Fig. 7. The reconstructed 3D host object using uniform quantization (a) 8 bits (b) 2 bits
Fig. 8. The reconstructed 3D hidden object using uniform quantization (a) 8 bits (b) 2 bits

Fig. 9. The reconstructed 3D objects using different portions of the digital holograms using quantized watermarked hologram. (a,b) 1/16 of the hologram area and 8 bits quantization (c,d) 1/4 of the hologram area and 4 bits quantization (e,f) 1/2 of the hologram area and 2 bits quantization

Table 2. Error when using only 25% of the hologram and a uniform quantizer

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Fig. 10. (a) The transmitted hologram having 50% occlusion and 4 bits quantization (b) The transmitted hologram having 75% occlusion and 4 bits quantization (c) The reconstructed 3D host object using the hologram in a (d) The reconstructed 3D hidden object using the hologram in a (e) The reconstructed 3D host object using the hologram in b (f) The reconstructed 3D hidden object using the hologram in b.

Fig. 11. The effect of number of quantization levels and occluding parts of the transmitted double phase encoded watermarked hologram when using a uniform quantizer

In Fig. 13 we compare the error when using a uniform quantizer with error when using an optimum quantizer. We present the following experiment to illustrate the watermarking system ability against blind detection. Figure 14(a) shows the recovered hidden object assuming we know three out of four of the codes, and the inverse Fresnel transform reconstruction distance. Only the hidden image spatial domain code is unknown. Figure 14(b) shows the decoded hidden object when all the codes are known except for the hidden object Fourier domain phase code.

Fig. 12. The reconstructed 3D objects using an optimum quantizer with 4 bits quantization and 25% of the holograms (a) host object (b) hidden object
Fig. 13. The error when using an optimum quantizer compared to the error when using a uniform quantizer, 25% of the hologram is used.
Fig. 14. Effect of blind decoding on the hidden object (a) only the hidden object spatial domain phase code is unknown. (b) only the hidden object Fourier domain phase code is unknown.

6. Conclusion

In this paper, we have presented a technique for hiding a 3D object within another 3D host object. Phase shifting interfermetery is used to generate the digital hologram, optically, for both the hidden object and the host object. The hologram of the hidden object is double phase encoded, scaled, and then embedded in the host object hologram. The resultant watermarked hologram is double phase encoded for a second time to generate the final transmitted signal. The second double phase encoding process adds more security and robustness against attacks. Mathematical analysis and numerical simulation are presented to illustrate the system performance. System performance is investigated when the system encounters several types of distortions, for example occlusion, quantization, and using a fraction of the hologram to reconstruct different poses of the3D objects. Mathematical analysis and digital experiments show that the proposed system has a reasonable performance. To the best of our knowledge this is the first report for hiding a 3D object inside another 3D object.

Appendix A

For simplicity we use one-dimensional notation. Let us define the one-dimensional version of the second term of Eq. (15) as:

X(u)exp[jπλd(Δu2u2)]·x=0N1(IFT{Ĥhost(ξ)ψ2(ξ)}ψ1(x))exp[jπλd(Δx2x)]exp[j2π(xuN)].
(A-1)

The autocorrelation, Rx(u,u’), of X(u) is defined as:

Rx(u,u)=EX*(u)X(u)
(A-2)

Equation (A-2) can be written as:

Rx(u,u)=E[exp[jπλd(Δu2u2)]x=0N1ξ=0N11NĤhost*(ξ)ψ2*(ξ)ψ1*(x)exp[jπλd(Δx2x)]exp[j2π(xuN)].exp[jπλd(Δu2u2)]x=0N1ξ=0N11NĤhost(ξ)ψ2(ξ)ψ1(x)exp[jπλd(Δx2x)]exp[j2π(xuN)]]
(A-3)

Note that ψ′2(ξ)=exp[jn 2(ξ)], and ψ′1(ξ)=exp[jn 1(x)] where n 2(ξ), and n 1(x) are two independent uniformly distributed random variables. Therefore,

Rx(u,u)=1N2E[exp[jπλdΔu2(u2u2)]x=0N1x=0N1ξ=0N1ξN1H*(ζ)H(ξ)ψ2(ξ)ψ2(ξ')ψ1(x)ψ1(x)exp[jπλdΔx2(xx)]exp[j2π(xuNxuN)]]
(A-4)

Equation (A-4) can be written as:

Rx(u,u)=1N2exp[jπλdΔu2(u2u2)]x=0N1x=0N1ξ=0N1ξN1H*(ζ)H(ξ)En2[ψ2(ξ)ψ2(ξ)]En1[ψ1(x)ψ12(x)]
exp[jπλdΔx2(xx)]exp[j2π(xuNxuN)]
(A-5)

where En2 [.] is the ensemble average over the random variable n2 and En1[.] is the ensemble average over the random variable n1 . It is straight forward to prove that: En2[ψ2(ξ)ψ2(ξ)]=δ(ξξ')andEn1[ψ1(x)ψ12(x)]=δ(xx), and

R(u,u)=1N2exp[jπλdΔu2(u2u2)]x=0N1ζ=0N1H(ξ)2exp[j2πx(uNuN)].
(A-6)

Using the definition of a discrete delta function : δ(uu)=1Nx=0N1exp[j2πx(uu)] , we can write:

R(u,u)=1Nexp[jπλdΔu2(u2u2)ξ=0N1H(ξ)2δ(uu)]
(A-7)

Therefore,

R(u,u)=1Nξ=0N1H(ξ)2δ(uu)
(A-8)

References and links

1.

N. F. Johnson, Z. Duric, and S. Jajodia, Information Hiding : Steganography and Watermarking - Attacks and Countermeasures (Advances in Information Security, Volume 1,Kluwer Academic2001).

2.

W. Bender, D. Gruhl, N. Morimoto, and L. Lu, “Techniques for data hiding,” IBM Systems Journal 35, 313–336 (1996). [CrossRef]

3.

G. C. Langelaar, I. Setyawan, and R. L. Lagendijk, “Watermarking Digital Image and Video Data. A state of the Art overview,” IEEE Signal Processing Magazine 17, 20–46 (2000). [CrossRef]

4.

S. Kishk and B. Javidi, “Information Hiding Technique using Double Phase Encoding,” Appl. Opt. 41, 5470–5482 (2002). [CrossRef]

5.

Chuan-Fu Wu and Wen-Shyong Hsieh, “Digital watermarking using zerotree of DCT,” Consumer Electronics, IEEE Transactions on , 46, 87–94 (2000). [CrossRef]

6.

Joseph Rosen and Bahram Javidi, “Hiding images in Halftone Pictures,” Appl. Opt. 40, 3346 (2001). [CrossRef]

7.

C. Hosinger and M. Rabbani, “Data Embedding using Phase Dispersion,” The International Conference on Information Technology: Coding and Computing (ITCC 2000), Las Vegas, Nev. 27–29 March (2000).

8.

Po-Chyi Su, C.-C. Jay Kuo, and Houng-jyh Wang, “Wavelet-Based Digital Image Watermarking,” Opt. Express , 3, 491–496 (1998), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-12-491 [CrossRef] [PubMed]

9.

S. Kishk and B. Javidi “Watermarking of a 3D Object Using Digital Holography,” Opt. Let. 28, 167–169 (2003). [CrossRef]

10.

R. Ohbuchi, A. ukaiyama, and S. Takahashi “A frequency-domain approach to watermarking 3D shapes,” Computer Graphics Forum 21, 373–382 Sp. Iss. SI(2002). [CrossRef]

11.

P. Refregier and B. Javidi, “Optical image encryption using input plane and Fourier plane random encoding,” Opt. Let. 20, 767–769 (1995). [CrossRef]

12.

R. K. Wang, I. A. Watson, and C. Chatwin, “Random phase encoding for optical secutity,” Opt. Eng. 35, 2464–2460 (1996). [CrossRef]

13.

F. Goudail, F. Bollaro, B. Javidi, and P. Refregier, “Influence of perturbation in a double phase-encoding system,” J. Opt. Soc. Am. A 15, 2629–2638 (1998). [CrossRef]

14.

H. J. Caulfield, Handbook of Optical Holography, (Academic press, London, 1979).

15.

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

16.

U. Schnars and W. P. O. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994). [CrossRef] [PubMed]

17.

I Yamaguchi and T Zhang, “Phase-Shifting Digital Holography,” Opt. Lett. 22, 1268–1270 (1997). [CrossRef] [PubMed]

18.

J. H. Bruninig, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital Wavefront measuring interferometer for Testing Optical Surfaces and Lenses,” Appl. Opt. 13, 2693 (1974). [CrossRef]

19.

J. Schwider, “Advanced Evaluation Techniques in Interferometry,” in Progress in Optics, E. Wolf, ed (North Holland, Amesterdam) XXVIII, 271–359 (1990)

20.

T. Naughton, Y. Frauel, B. Javidi, and E. Tajahuerce, “Compression of digital holograms for threedimensional object reconstruction and recognition,” Appl. Opt. 11, 4124–4132 (2002). [CrossRef]

OCIS Codes
(070.4560) Fourier optics and signal processing : Data processing by optical means
(090.0090) Holography : Holography
(100.0100) Image processing : Image processing
(100.6890) Image processing : Three-dimensional image processing

ToC Category:
Research Papers

History
Original Manuscript: March 13, 2003
Revised Manuscript: April 2, 2003
Published: April 21, 2003

Citation
Sherif Kishk and Bahram Javidi, "3D object watermarking by a 3D hidden object," Opt. Express 11, 874-888 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-8-874


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References

  1. N. F. Johnson, Z. Duric, and S. Jajodia, Information Hiding : Steganography and Watermarking �?? Attacks and Countermeasures (Advances in Information Security, Volume 1, Kluwer Academic 2001).
  2. W. Bender, D. Gruhl, N. Morimoto and L. Lu, �??Techniques for data hiding,�?? IBM Systems J. 35, 313-336 (1996). [CrossRef]
  3. G. C. Langelaar, I. Setyawan and, R. L. Lagendijk, "Watermarking Digital Image and Video Data. A state of the Art overview,�?? IEEE Signal Processing Magazine 17, 20-46 (2000). [CrossRef]
  4. S. Kishk, and B. Javidi, "Information Hiding Technique using Double Phase Encoding,�?? Appl. Opt. 41, 5470-5482 (2002). [CrossRef]
  5. Chuan-Fu Wu; Wen-Shyong Hsieh, "Digital watermarking using zerotree of DCT,�?? IEEE Trans. Consumer Electron. 46, 87-94 (2000) [CrossRef]
  6. Joseph Rosen, Bahram Javidi, �??Hiding images in Halftone Pictures,�?? Appl. Opt. 40, 3346 (2001) [CrossRef]
  7. C. Hosinger and M. Rabbani, �??Data Embedding using Phase Dispersion, "The International Conference on Information Technology: Coding and Computing (ITCC 2000), Las Vegas, Nev. 27-29 March (2000).
  8. Po-Chyi Su,C.-C. Jay Kuo,Houng-jyh Wang, "Wavelet-Based Digital Image Watermarking,�?? Opt. Express 3, 491-496 (1998), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-12-491">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-12-491</a> [CrossRef] [PubMed]
  9. S. Kishk and B. Javidi �??Watermarking of a 3D Object Using Digital Holography,�?? Opt. Lett. 28, 167-169 (2003). [CrossRef]
  10. R. Ohbuchi,A. ukaiyama,S. Takahashi �??A frequency-domain approach to watermarking 3D shapes,�?? Computer Graphics Forum 21, 373-382 Sp. Iss. SI(2002). [CrossRef]
  11. P. Refregier and B. Javidi, �??Optical image encryption using input plane and Fourier plane random encoding,�?? Opt. Lett. 20, 767-769 (1995). [CrossRef]
  12. R. K. Wang, I. A. Watson, and C. Chatwin, �??Random phase encoding for optical secutity,�?? Opt. Eng. 35, 2464-2460 (1996). [CrossRef]
  13. F. Goudail, F. Bollaro, B. Javidi, and P. Refregier, �??Influence of perturbation in a double phase-encoding system,�?? J. Opt. Soc. Am. A 15, 2629-2638 (1998). [CrossRef]
  14. H. J. Caulfield, Handbook of Optical Holography, (Academic press, London, 1979).
  15. J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, NY, 1996).
  16. U. Schnars, and W. P. O. Jüptner, �??Direct recording of holograms by a CCD target and numerical reconstruction,�?? Appl. Opt. 33, 179-181 (1994). [CrossRef] [PubMed]
  17. I Yamaguchi, and T Zhang, �??Phase-Shifting Digital Holography,�?? Opt. Lett. 22, 1268-1270 (1997) [CrossRef] [PubMed]
  18. J. H. Bruninig, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, �??Digital Wavefront measuring interferometer for Testing Optical Surfaces and Lenses,�?? Appl. Opt. 13, 2693 (1974). [CrossRef]
  19. J. Schwider, �??Advanced Evaluation Techniques in Interferometry,�?? in Progress in Optics, E. Wolf, ed (North Holland, Amesterdam) XXVIII, 271-359 (1990)
  20. T. Naughton, Y. Frauel, B. Javidi, and E. Tajahuerce, "Compression of digital holograms for three dimensional object reconstruction and recognition,�?? Appl. Opt. 11, 4124-4132 (2002). [CrossRef]

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