Encryption of 3 D Point Cloud Object with Deformed Fringe

A 3D point cloud object encryption method was proposed with this study. With the method, a mapping relationship between 3D coordinates was formulated andZ coordinate was transformed to deformed fringe by a phase coding method.The deformed fringe and gray image were used for encryption and decryption with simulated off-axis digital Fresnel hologram. Results indicated that the proposed method is able to accurately decrypt the coordinates and gray image of the 3D object. The method is also robust against occlusion attacks.


Introduction
Image encryption has been extensively studied in the past.It includes encryption of 2D and 3D objects through computational methods.For example, discrete fractional Fourier transform, Fibonacci transformation, and discrete cosine transform have been used to for image or 3D object encryption [1][2][3].Alternatively, holographic encryption has also been applied for encrypting and decrypting information.Holographic encryption has advantages in providing flexibility in selection of various encryption parameters including optical wavelength, diffraction distance, and polarization.The encryption of 2D image uses optical methods such as wavelength multiplexing method [4,5], two-step phaseshifting interferometry [6], computer-generated hologram [7], joint transform correlator [8], and modulation of plane wave [9] as reference wave.The significance of 3D information encryption is also obvious and many methods for the encryption of 3D objects such as using digital holography [10][11][12] and computer-generated [13] have been proposed.However, for most of 3D object encryption, object information has yet been fully reconstructed in terms of accuracy and robustness, although the 3D information is very important for practical utilization [14,15].It is mostly due to the complexity of the computation and sensitivity of the parameters inherited in the methods.
More robust methods were created to reduce the sensitivity of the encryption parameters hence to increase the validity of the encryption under different environment.Among them, the integral imaging method has been popularized due to its use of structured pinholes to enhance the data redundancy and encryption robustness, yet the method involves use of complex experimental setup and simulation [1,16].Advanced 3D digital holography that has been also applied for 3D image encryption, though robust, lacks accuracy and is insufficiently strong in encryption process due to the direct use of object coordinate information in the encryption process [17].
In this paper, we proposed robust and accurate, yet less computation intensive 3D point cloud object encryption.At the center of the method, a mapping relationship between , , and  was formulated. coordinate of the 3D object was phase coded and subsequently transformed to deformed fringe.With deformed fringe and gray image of the 3D object, a conventional off-axis digital Fresnel holography method was utilized to encrypt and record their information.Following the decryption and reconstruction, both deformed fringe and gray image were reconstructed.3D object was hence reconstructed with , , , and gray image information.

Methods
The angular orthogonal projection geometry [18] is used to convert the 3D point objects of one specific perspective 2 Advances in Optical Technologies into - plane and  coordinate.Figure 1 is the angular orthogonal projection and the projection image coordinate (  ,   ) and the object point (  ,   ,   , gray  ) from a gray model are related by   =   cos () −   sin () ,   =   cos () −   sin () cos () −   sin () sin () . ( For a specific situation when  = /2 and  = /2, A little difference from the traditional angular orthogonal projection is that two arrays gray and  are used to save the gray value and  coordinates.The calculation of  and  coordinates can be extracted from (3), which can be expressed as Steps of proposed method for encryption and decryption of the 3D object are shown in Figure 3.
The detailed description of the two steps is presented in below.
Step 1 (transformation of  coordinate to deformed fringe).The method starts from decomposing the 3D object into - plane and  coordinate for a given view port.It is followed by transforming  coordinate of the 3D object into phase (, ) shown below: where   = max{},   = min{}, and  is a constant between 0 and 1. max{} is the calculation of maximum value of  and min{} is the calculation of minimum value of .With a proper selection of , the range of  could be within [−, ].  = 1.7893,   = 0, and  = 0.9 was used.
The phase is further coded to deformed fringe with phase coding method inspired by Fourier transform profilometry [19,20], which can be expressed as where  and  are some constants and   carrier frequency is 0.11. coordinate and deformed fringe are shown in Figure 4. Figure 4(a) is the normalized  coordinate of 3D object.Figure 4(b) is the deformed fringe coded from phase calculated from the mapping formula between height and phase.Figure 4(c) is the gray image of 3D object.Step 2 (encryption and decryption).Multiplexing methods [4,5] are often used in information encryption with Fresnel hologram.The space multiplexing method, the combination of deformed fringe and gray image as object, is used to simulate off-axis Fresnel holography shown in Figure 5.In the simulation, 4 optical system constructed from lenses  1 and  2 was used.Laser with  = 632.8m was used in the simulation.Two random phase masks, which are also used as encryption keys, PR 1 and PR 2 , were used to encrypt the object and reference waves.The object wave on the hologram plane is described: where Fresnel{•,   } is the Fresnel diffraction in (8) of • with a diffraction distance   .  = 600 mm.obj(, ) is the distribution of object and  PR 1 (, ) is the phase due to random phase mask PR 1 .Consider The Fresnel hologram is written as where ( ℎ ,  ℎ ) = exp[ PR 1 (, )] and asterisk denotes the conjugate wave of ( ℎ ,  ℎ ) and ( ℎ ,  ℎ ).It consists of DCs (first two components), first order, and conjugate terms (third and fourth items).The resolution of hologram is 1400 × 1400 and the sampling interval in hologram plane is 10 m in both  and  directions.The angle between the reference and object waves is 2.4 degrees.The hologram reconstruction and decryption process is written as where  PR 2 (, ) is the phase due to random phase mask PR 2 .
Following the hologram reconstruction, gray image and deformed fringe are obtained.For deformed fringe, with a specific choice of carrier frequency   shown in ( 6) and filtering in frequency domain, the deformed fringe can be simplified to [16,17]  The phase (, ) can be calculated based on simplified deformed fringe: real{  } is the real part of   and imag{  } is the imaginary part of   .The phase was truncated within the range of [−, ] avoiding the phase unwrap problem [16,17].With phase (, ),  coordinate can be reconstructed based on (5).Using reconstructed gray image and  coordinate and sampling intervals Δ and Δ in  and  directions, the  and  coordinates of 3D object can also be reconstructed using (4).

Simulated Occlusion Attacks. The occlusion attacks of the
Fresnel hologram simulation were conducted.It was done through removal of partial Fresnel hologram data.The occlusion ratio is defined as the ratio of the removed Fresnel hologram data dividing the total Fresnel hologram data.The mean square error (MSE) between real  coordinates and reconstructed  coordinates was calculated and correlation coefficient (CC) of reconstructed gray image comparing with original gray image was also tested when occlusion ratio was different.

Direct 𝑍 Coordinate versus Deformed Fringe Encryption.
In order to comparably quantify the robustness of using deformed fringe for  coordinate encryption, comparison of performance of direct (just using)  coordinate versus deformed fringe encryption was conducted.It was used to illustrate the relationship between occlusion ratio and mean square error of  coordinate reconstruction by just using  coordinate versus using deformed fringe for Fresnel hologram encryption.

Results
The encrypted Fresnel hologram and reconstructed deformed fringe and gray image of 3D object are shown in Figure 6.Correlation coefficient between the reconstructed gray image and original gray image is 0.9824.With wrong phase masks, both deformed fringe and gray image could not be decrypted as illustrated in Figure 8.

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Results of deformed fringe and gray image showed the reduced phase and gray image quality with increased occlusion attack ratio shown in Figure 9.
The mean square error (MSE) and correlation coefficients (CC) used to evaluate the decrypted quality between the reconstructed images without occlusion attacks and the reconstructed images under different occlusion attacks are displayed in Figures 10(a  Advances in Optical Technologies without occlusion and with different occlusion attacks when  coordinate is directly used for encryption.And the red line presents the MSE when the phase coding method is used.The phase coding method can reduce the reconstructed  coordinate error compared with the directly encrypted  coordinate under 50% occlusion attacks.Above 50% attacks, the comparison is meaningless because  coordinate cannot be calculated from both methods.

Conclusion
In this research, a 3D point cloud object encryption method was proposed.The method uses off-axis digital Fresnel hologram and phase masks for encrypting and decrypting 3D information of the object, in contrast to conventional digital hologram encryption method, which has been restricted to encryption and decryption of 2D or 3D objects with Fresnel hologram.The mapping relationship between , , and  coordinates was formulated and, for  coordinate, a transformation was conducted to translate it to deformed fringe.With deformed fringe, Fresnel hologram encryption and decryption were performed.
Results showed that the proposed method is able to encrypt and decrypt the deformed fringe and gray image of the 3D object.The proposed method is also sufficiently accurate just yielding small errors between reconstructed  coordinate and real  coordinate as shown in Figure 7(d).The method is also robust for different occlusion attacks as shown in Figure 9. Additionally, in comparison to direct  coordinate encryption, deformed fringe is associated with smaller MSE for the occlusion attack ratio under 50% as illustrated in Figure 10.It indicates that the deformed fringe is more robust than directly using  coordinate for encryption.This might be because deformed fringe is less sensitive to increased noise-signal ratio relative to simply using  coordinate.
One possible contribution of this research is that, by transforming the  coordinate of 3D object to deformed fringe, it increases the encryption strength because without knowing the mapping relationship between deformed fringe and  coordinate,  coordinate decryption and reconstruction would be challenging.
One major limitation of the research should be noted.This study has only investigated  coordinate deformed fringe encryption and decryption.It would be ideal if all , , , and gray image of the 3D object can be transformed, encoded, and encrypted, which can further increase the encryption strength.This however could be also an interesting topic to study in the future.
object Phase coding for Z array x, y, z coordinates x, y, z coordinates

Figure 3 :
Figure 3: Steps of encryption and decryption for 3D point cloud object.

Figure 7 :
Figure 7: Reconstructed phase (a) and  coordinate (b), comparison between original and reconstructed phase (c) and  (d).

Figure 8 :
Figure 8: Reconstructed deformed fringe (a) and gray image (b) with wrong phase masks.

Figure 10 :
Figure 10: Mean square error (MSE) values of reconstructed  coordinate for different occlusion attacks (a).Correlation coefficients (CC) values for reconstructed gray image under different occlusion attacks (b).
The sampling intervals in the projection plane are Δ and Δ, respectively, in  and  coordinate and the resolution of projection plane is  by .34.87 × 17.89 mm 3 .The th point of the 3D object can be represented by   (  ,   ,   , gray  ), where  is within the range of 1 to .  is the number of object points, where Δ and Δ are both 0.02 mm, which gives the resolution of 300 × 300 corresponding to  rows multiplied by  columns in - plane for the point cloud object as is shown in Figure2.