3D object watermarking by a 3D hidden object

Sherif Kishk; Bahram Javidi

doi:10.1364/OE.11.000874

1. Introduction

Watermarking [1–3] 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]. 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 [4–8]. Some techniques have presented watermarking of 3D objects [9,10].

Double phase encoding [11–13] 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 [14,16] 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 [17–19] 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

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

D (x, y) = A (x, y) \exp (j ϕ (x, y))

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) = A_{R} \exp ⌊ j (ϕ_{R} + α) ⌋

where A_R 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) = A^{2} (x, y) + A_{R}^{2} + 2 A (x, y) A_{R} \cos [ϕ (x, y) - ϕ_{R} - α] .

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:

ϕ_{O} (x, y) = \arctan [\frac{I_{4} (x, y) - I_{2} (x, y)}{I_{1} (x, y) - I_{3} (x, y)}],

and the amplitude is:

A_{O} (x, y) = \frac{1}{4} {{[I_{1} (x, y) - I_{3} (x, y)]}^{2} + {[I_{2} (x, y) - I_{4} (x, y)]}^{2}}^{\frac{1}{2}}

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

H_{O} (x, y) = A_{O} (x, y) \exp [j ϕ_{O} (x, y)]

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, H_d(x,y) is given by:

H_{d} (x, y) = {H (x, y) ψ_{1} (x, y)} \otimes IFT [ψ_{2} (ξ, γ)]

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].

Using an analysis similar to [6] it can be shown that the encoded double phase encoded complex signal is a white Gaussian noise having a variance given by:

σ^{2} = \frac{1}{N . M} \sum_{y = 0}^{M - 1} \sum_{x = 0}^{N - 1} H (x, y) H^{*} (x, y)

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 H_host(x,y), and the hidden object hologram H_hidden(x,y). The transmitted watermarked hologram, H_w(x,y), is given by:

H_{w} (x, y) = {[H_{host} (x, y) + α [H_{hidden} (x, y) ψ_{1} (x, y) \otimes ψ_{2} (x, y)]] {ψ'}_{1} (x, y)} \otimes {ψ'}_{2} (x, y)

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, $σ_{w}^{2}$ :

σ_{w}^{2} = \frac{1}{2 . N . M} \sum_{y = 0}^{M - 1} \sum_{x = 0}^{N - 1} H_{hidden} (x, y) H_{hidden}^{*} (x, y)

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

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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:

{\tilde{H}}_{hidden} (x, y) = α H_{hidden} (x, y) + IFT {{\hat{H}}_{host} (ξ, γ) ψ^{*} 2 (ξ, γ)} ψ^{*} 1 (x, y)

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, $σ_{h}^{2}$ :

σ_{w}^{2} = \frac{1}{N . M} \sum_{y = 0}^{M - 1} \sum_{x = 0}^{N - 1} H_{host} (x, y) H_{host}^{*} (x, y)

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

{\tilde{O}}_{d} (u, v; d_{1}) = \exp [- \frac{j π}{λ d_{1}} (Δ u^{2} u^{2} + Δ v^{2} v^{2})] \sum_{x = 0}^{N - 1} \sum_{y = 0}^{M - 1} {{\tilde{H}}_{hidden} (x, y)

where (Δx,Δy), and (Δu,Δv) 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:

Δ u = \frac{λ d_{1}}{N Δ x}, and

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

{\tilde{O}}_{d} (u, v; d_{1}) = α O_{d} (u, v; d_{1}) + \exp [- \frac{j π}{λ d_{1}} (Δ u^{2} u^{2} + Δ v^{2} v^{2})] .

O_d(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:

{\tilde{O}}_{h} (u, v; d_{2}) = O_{h} (u, v; d_{2}) + \exp [- \frac{j π}{λ d_{2}} (Δ u^{2} u^{2} + Δ v^{2} v^{2})]

O_h(u,v;d₂) is the original 3D host object reconstructed at a distance d ₂ from the output plane. According to the previous equation the reconstructed 3D host object is distorted by the second term of equation 16. It can be shown that the distortion added to the reconstructed 3D host object is an additive Gaussian white process with zero mean and variance, $σ_{host}^{2}$ , equal:

σ_{host}^{2} = \frac{α^{2}}{N . M} \sum_{y = 0}^{M - 1} \sum_{x = 0}^{N - 1} H_{hidden} (x, y) H_{hidden}^{*} (x, y)

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 H_w (x, y), and the quantized double phase encoded watermarked hologram as H͂_w (x, y). The number of quantization bits is B. We can represent the transmitted quantized hologram as:

{\tilde{H}}_{w} (x, y) = H_{w} (x, y) + Δ H_{w} (x, y)

where ΔH_w (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:

{\tilde{O}}_{d} (u, v; d_{1}) = α O_{d} (u, v; d_{1}) + \exp [- \frac{jπ}{λ d_{1}} (Δ u^{2} u^{2} + Δ v^{2} v^{2})] .

The reconstructed 3D host object is given by:

{\tilde{O}}_{d} (u, v; d_{2}) = O_{d} (u, v; d_{2}) + \exp [- \frac{j π}{λ d_{2}} (Δ u^{2} u^{2} + Δ v^{2} v^{2})] .

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:

σ_{host}^{2} = \frac{α^{2}}{N . M} \sum_{y = 0}^{M - 1} \sum_{x = 0}^{N - 1} H_{hidden} (x, y) H_{hidden}^{*} (x, y) + \frac{Δ^{2}}{12}

σ_{w}^{2} = \frac{1}{N . M} \sum_{y = 0}^{M - 1} \sum_{x = 0}^{N - 1} H_{host} (x, y) H_{host}^{*} (x, y) + \frac{Δ^{2}}{12}

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, H_wo , is given by:

H_{w o} (x, y) = H_{w} (x, y) (1 - W (x, y)) = H_{w} (x, y) - H_{w} (x, y) W (x, y)

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] 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:

O_{d} (u, v; d) = \int_{- \infty}^{\infty} \frac{- \exp [j \frac{2 π}{λ} (z - d)]}{λ (z - d)} \exp [j \frac{π}{λ (z - d)} (u^{2} + v^{2})] \cdot

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 = \frac{1}{P} \sum_{p = 1}^{P} \frac{1}{N \times M} \sum_{y = 0}^{M - 1} \sum_{x = 0}^{N - 1} \frac{{[O_{d} (u, v; d (p)) - O (u, v; d (p))]}^{2}}{O (u, v; d (p))}

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 O_d(u,v;d(p)) is the reconstructed object at a distance d(p) with watermarking.

Figure 4 shows the original reconstructed 3D object without watermarking for both the host object and the hidden object. In the next experiment we illustrate the effect of the second double phase encoding process on the autocorrelation of the watermarked hologram.

Fig. 3. 3D object reconstructed at different distances from the output plane

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Fig. 4. The original host and hidden 3D objects.

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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.

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The next experiment illustrates the effect of quantizing the transmitted double phase encoded hologram. In the first set of experiments we use a uniform quantizer. Figure 7 shows the reconstructed 3D host when using 8, and 2 bits uniform quantizer. Figure 8 shows the reconstructed 3D hidden object when using 8, and 2 bits uniform quantizer. Table 1 presents the effect of the number of quantization bits on the error in the reconstructed objects.

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

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Fig. 7. The reconstructed 3D host object using uniform quantization (a) 8 bits (b) 2 bits

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Fig. 8. The reconstructed 3D hidden object using uniform quantization (a) 8 bits (b) 2 bits

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As in regular holograms we can reconstruct different views of the 3D object using different areas of the hologram. In the next experiment we use 6.25%, 25% and 50% of the hologram area to reconstruct both the 3D host object and the 3D hidden object. Figure 9 shows the reconstructed objects using different number of quantization bits and different portions of the transmitted hologram. Table 2 illustrates the resulting error for the host 3D object.

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

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

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The next experiment illustrates the effect of occluding parts of the transmitted double phase encoded watermarked hologram. This can be done as a trial to remove the hidden 3D object. Figure 10 shows the occluded transmitted holograms and the reconstructed 3D host and 3D object when the occlusion level is set to 50% and 75% percent of the total hologram area. In this experiment the transmitted hologram is quantized using 4 bits quantizer. The reconstructed 3D host error is .0224 and .075 respectively.

The effect of the second double phase encoding step is clear in this experiment. Without this second double phase encoding, occluding a portion of the watermarked hologram will prevent us from reconstructing a certain pose for the 3D object. And, the location of the occluded areas does not impact the quality of the reconstructed objects. For example if we repeat the previous experiment with 50% occlusion and 4 bits quantizer but without the second double phase encoding we get an error equal .093. In Fig. 11 we demonstrate the effect of combining occluding a percentage of the transmitted double phase encoding watermarked hologram and quantization on the host 3D object error.

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.

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In the next experiment we use an optimum quantizer to quantize the transmitted double phase encoded watermarked hologram. We use the fact that the double phase encoded watermarked hologram is white Gaussian to find the optimum quantization level. To ensure a fixed quantization level that is independent on either the host or the hidden 3D object we normalize the energy of the transmitted double phase encoded watermarked hologram. In this experiment we transmit only 25% of the holograms. Figure 12 shows the reconstructed 3D host object having an error equal to 0.0019 and the reconstructed 3D hidden object.

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

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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

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Fig. 13. The error when using an optimum quantizer compared to the error when using a uniform quantizer, 25% of the hologram is used.

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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.

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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) \equiv \exp [- \frac{j π}{λ d} (Δ u^{2} u^{2})] \cdot \sum_{x = 0}^{N - 1} (IFT {{\hat{H}}_{host} (ξ) {ψ'}_{2} (ξ)} {ψ'}_{1} (x)) \exp [- \frac{j π}{λ d} (Δ x^{2} x)] \exp [j 2 π (\frac{x u}{N})] .

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

R_{x} (u, u') = E ⌊ X^{*} (u) X (u') ⌋

Equation (A-2) can be written as:

R_{x} (u, u') = E [\begin{matrix} \exp [\frac{j π}{λ d} (Δ u^{2} u^{2})] \sum_{x = 0}^{N - 1} \sum_{ξ = 0}^{N - 1} \frac{1}{N} {\hat{H}}_{host}^{*} (ξ) {ψ'}_{2}^{*} (ξ) {ψ'}_{1}^{*} (x) \exp [\frac{j π}{λ d} (Δ x^{2} x)] \exp [- j 2 π (\frac{x u}{N})] . \\ \exp [- \frac{j π}{λ d} (Δ u^{2} {u'}^{2})] \sum_{x' = 0}^{N - 1} \sum_{ξ' = 0}^{N - 1} \frac{1}{N} {\hat{H}}_{host} (ξ') {ψ'}_{2} (ξ') {ψ'}_{1} (x') \exp [- \frac{j π}{λ d} (Δ x^{2} x')] \exp [j 2 π (\frac{x' u'}{N})] \end{matrix}]

Note that ψ′₂(ξ)=exp[j2πn ₂(ξ)], and ψ′₁(ξ)=exp[j2πn ₁(x)] where n ₂(ξ), and n ₁(x) are two independent uniformly distributed random variables. Therefore,

R_{x} (u, u') = \frac{1}{N^{2}} E [\begin{matrix} \exp [\frac{j π}{λ d} Δ u^{2} (u^{2} - {u'}^{2})] \sum_{x = 0}^{N - 1} \sum_{x' = 0}^{N - 1} \sum_{ξ = 0}^{N - 1} \sum_{ξ'}^{N - 1} H^{*} (ζ) H (ξ') {ψ'}_{2} (ξ) {ψ'}_{2} (ξ') {ψ'}_{1} (x) {ψ'}_{1} (x') \\ \exp [\frac{j π}{λ d} Δ x^{2} (x - x')] \exp [- j 2 π (\frac{x u}{N} - \frac{x' u'}{N})] \end{matrix}]

Equation (A-4) can be written as:

R_{x} (u, u') = \frac{1}{N^{2}} \exp [\frac{j π}{λ d} Δ u^{2} (u^{2} - {u'}^{2})] \sum_{x = 0}^{N - 1} \sum_{x' = 0}^{N - 1} \sum_{ξ = 0}^{N - 1} \sum_{ξ'}^{N - 1} H^{*} (ζ) H (ξ') E_{n_{2}} [{ψ'}_{2} (ξ') {ψ'}_{2} (ξ')] E_{n 1} [{ψ'}_{1} (x) {ψ'}_{12} (x')]

where E_n2 [.] is the ensemble average over the random variable n₂ and E_n1[.] is the ensemble average over the random variable n₁ . It is straight forward to prove that: $E_{n_{2}} [{ψ'}_{2} (ξ) {ψ'}_{2} (ξ')] = δ (ξ - ξ') and E_{n 1} [{ψ'}_{1} (x) {ψ'}_{12} (x')] = δ (x - x'),$ and

R (u, u') = \frac{1}{N^{2}} \exp [\frac{j π}{λ d} Δ u^{2} (u^{2} - {u'}^{2})] \sum_{x = 0}^{N - 1} \sum_{ζ = 0}^{N - 1} {∣ H (ξ) ∣}^{2} \exp [- j 2 π x (\frac{u}{N} - \frac{u'}{N})] .

Using the definition of a discrete delta function : $δ (u - u') = \frac{1}{N} \sum_{x = 0}^{N - 1} \exp [- j 2 π x (u - u')]$ , we can write:

R (u, u') = \frac{1}{N} \exp [\frac{j π}{λ d} Δ u^{2} (u^{2} - {u'}^{2}) \sum_{ξ = 0}^{N - 1} {∣ H (ξ) ∣}^{2} δ (u - u')]

Therefore,

R (u, u') = \frac{1}{N} \sum_{ξ = 0}^{N - 1} {∣ H (ξ) ∣}^{2} δ (u - u')

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	8 bits	4 bits	2 bits
Host Object	.0015	0.0019	0.0086
Hidden Object	0.1463	0.1558	0.2078

	8 bits	4 bits	2 bits
Host Object	.0015	0.0019	0.0086
Hidden Object	0.1463	0.1558	0.2078

3D object watermarking by a 3D hidden object

Abstract

1. Introduction

2. Background

2.1 Digital holography:

2.2 Double phase encoding:

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

4. Proposed system response to noise and distortion

5. Experimental results and simulations

6. Conclusion

Appendix A

References and links

Cited By

Figures (14)

Tables (2)

Equations (48)

Optics Express