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Abstract
In allusion to the privacy and security problems in 3D point cloud classification, a novel privacy protection method for 3D point cloud classification based on optical chaotic encryption scheme is proposed and implemented in this paper for the first time. The mutually coupled spin-polarized vertical-cavity surface-emitting lasers (MC-SPVCSELs) subject to double optical feedback (DOF) are studied to generate optical chaos for permutation and diffusion encryption process of 3D point cloud. The nonlinear dynamics and complexity results demonstrate that the MC-SPVCSELs with DOF have high chaotic complexity and can provide tremendously large key space. All the test-sets of ModelNet40 dataset containing 40 object categories are encrypted and decrypted by the proposed scheme, and then the classification results of 40 object categories for original, encrypted, and decrypted 3D point cloud are entirely enumerated through the PointNet++. Intriguingly, the class accuracies of the encrypted point cloud are nearly all equal to 0.0000% except for the plant class with 100.0000%, indicating the encrypted point cloud cannot be classified and identified. The decryption class accuracies are very close to the original class accuracies. Therefore, the classification results verify that the proposed privacy protection scheme is practically feasible and remarkably effective. Additionally, the encryption and decryption results show that the encrypted point cloud images are ambiguous and unrecognizable, while the decrypted point cloud images are identical to original images. Moreover, this paper improves the security analysis via analyzing 3D point cloud geometric features. Eventually, various security analysis results validate that the proposed privacy protection scheme has high security level and good privacy protection effect for 3D point cloud classification.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the ceaseless advancement of 3D acquisition sensors in the direction of miniaturization, affordability and commercialization, their application fields have gradually expanded from military and industry to the public consumption [1,2]. These 3D data can be more abundant than 2D image data in depicting the geometry, shape and proportion of the surrounding environment, thus its applications are in the fields of unmanned driving, remote sensing mapping, quality testing, robots, face devices, security monitoring, medical care, logistics, etc [3–6]. The common 3D data formats include wireframe models, point cloud (PC) models, meshes, volumetric grids, and texture models, where point cloud model is the preferred choice for modeling many application scenarios due to it retains the raw geometric information in 3D space without any discretization [3,7–9]. Against such a backdrop, the classification of 3D point cloud has promptly become a hotspot research field and has been applied to many practical tasks, such as establishing urban environment [10], autonomous vehicle navigation [11,12], and road marking extraction [13,14]. However, while the point cloud classification has vital impact on these practical tasks, its security issue is also confronted with formidable challenges. Therefore, the privacy protection is an imminent problem to be studied and solved in point cloud classification.
To remedy the privacy protection problem of point cloud data aforementioned, several point cloud encryption techniques based on chaos theory have been proposed successively, owing to chaotic system has the characteristics of ergodicity, aperiodic, non-convergence, unpredictability, and sensitivity rely on initial conditions and system parameters [15–20]. The Chebyshev chaotic map is utilized in key scheduling algorithm for two round point cloud encryption process with random permutations and geometric rotations, which preserves the dimensional and spatial stability of the original point cloud [21]. The Logistic chaotic mapping is exploited for 3D point cloud encryption with two individual schemes, one scheme applies logistic chaotic mapping to generate three random vectors (RV), each random vector is sorted to randomly confuse each coordinate of the 3D point clouds; the other scheme uses logistic chaotic mapping to produce a $4\times 4$ random transformation matrix (RTM) consisting of a $3\times 3$ rotation matrix and a $3\times 1$ translate matrix, then each 3D point is confused to another random place in the homogeneous coordinate [22]. The off-axis digital Fresnel hologram and phase masks approaches are applied for encrypting and decrypting 3D point cloud object, in which the Z coordinate is phase coded and converted to deformed fringe. It is regrettable that this scheme demands the grayscale image array of 3D point cloud object, which prevents the encryption approach from being directly used for 3D point cloud encryption [23]. In the recent years, the chaotic cat mapping is also adopted for 3D point cloud encryption [24]. Resemble to literature [22], literature [24] implements two encryption algorithms for 3D point cloud, one encryption algorithm employs three random permutation matrices generated by 2D cat mapping to scramble each coordinate of the 3D point clouds, while the other algorithm uses the invertible random transformation matrix generated by the Logistics mapping and 3D cat mapping to encrypt the relevant 3D point clouds. A two-level encryption scheme based on a novel chaotic map is proposed for point cloud data [25], in which the ascending sort (AS) and descending sort (DS) of six different random sequences are used for first and second encryption level respectively.
Compared with the traditional chaos above-mentioned, optical chaos has demonstrated some distinctive and eminent superiorities in numerous associated aspects concerning bandwidth, speed, security, cost, attenuation, complexity, etc [20,26,27]. Due to these unique intrinsic features of optical chaos, the optical chaos is an ideal candidate method for 3D point cloud encryption. Nevertheless, to the best of our knowledge, there are no relevant researches on 3D point cloud encryption based on optical chaos. In addition, the above-mentioned 3D point cloud encryption schemes are still lacking in the following aspects: (i) Almost all the above existing 3D point cloud encryption schemes only implement scrambling encryption algorithm, but lack of diffusion algorithm, i.e only change the corresponding position of 3D point cloud data without changing its value. (ii) The above-mentioned existing 3D point cloud encryption schemes lack the comparison measurement of 3D point cloud features before and after encryption. (iii) All of the above literatures lack the encryption effect test for the encrypted 3D point cloud in practical application tasks, such as the classification task of 3D point cloud.
Combining the advantages of optical chaos in encryption and improving the defects in the above 3D point cloud encryption, a novel privacy protection scheme for 3D point cloud classification based on optical chaos is proposed in this paper. First and foremost, the nonlinear dynamic characteristics of mutually coupled (MC) spin-polarized vertical-cavity surface-emitting lasers (SPVCSELs) subject to double optical feedback (DOF) is studied via bifurcation diagram, and the complexity comparison of SPVCSELs under MC and MC plus DOF is also studied by spectral entropy (SE) and $C_0$ complexity, subsequently its optical chaotic state is demonstrated in detail through time series, power spectra, spectrum, phase diagram, attractor, and 0-1 test. The generated optical chaotic data is exploited for permutation and diffusion encryption process of 3D point cloud, in which the permutation process is shuffled according to the ascending order of optical chaos data, while the diffusion process is completed by XOR operation of point cloud data and optical chaos data. Furthermore, we have encrypted and decrypted all the data in the test dataset (ModelNet40) of 3D point cloud classifier (PointNet++), then individually tested the classification accuracy of the encrypted and decrypted point cloud data, and given the comparison results of the original, encrypted and decrypted 3D point cloud classification accuracy. The encryption and decryption results of 3D point cloud test dataset are clearly presented as well. Eventually, we make a detailed security analysis of 3D point cloud encryption, especially a comparative analysis of point cloud features before and after encryption.
Therefore, compared with the existing literatures, the main contributions and improvements of our work can be summarized as follows:
(1) To the best of our knowledge, this is the first research paper to exploit optical chaotic encryption scheme for privacy protection of 3D point cloud classification.
(2) This paper explores the performance of encrypted and decrypted 3D point cloud images in the 3D point cloud classifier, and entirely gives the classification accuracy results of 40 object categories of encrypted and decrypted 3D point cloud images. This result directly reflects the effectiveness of privacy protection in the 3D point cloud classification.
(3) The mutually coupled spin-polarized vertical-cavity surface-emitting lasers subject to double optical feedback used in this paper can provide an extremely large key space, which can exceed most of the existing key space of 3D point cloud encryption scheme.
(4) The 3D point cloud encryption scheme based on optical chaos in this paper includes both permutation encryption and diffusion encryption, unlike most existing point cloud encryption schemes that only use permutation encryption.
(5) This paper improves the security analysis of 3D point cloud encryption by analyzing eight geometric features of 3D point clouds before and after encryption.
The rest of this paper is structured as follows. Section 2 introduces the system model and encryption scheme. Section 3 presents the accuracy results of point cloud encryption and decryption in 3D point cloud classifier, and discusses the relevant encryption and decryption results. Section 4 shows the security analysis results of 3D point cloud encryption in detail. Finally, Section 5 summarizes the whole paper.
2. System model and methods
This section introduces the dataset and implementation platform of 3D point cloud, then presents the theoretical model of mutually coupled spin-polarized vertical-cavity surface-emitting lasers subject to double optical feedback, and finally displays the encryption scheme of 3D point cloud based on the above foundation.
2.1 Dataset and implementation platform
For this paper, we implement with the famous ModelNet40 dataset for 3D point cloud classification, which has been a prevalent benchmark for point cloud object classification [28–31]. The ModelNet40 dataset has a total of 12,311 CAD models containing 40 object categories, where there are 9,843 objects for training and 2,468 objects for testing [31]. Since the ModelNet40 dataset is synthetic, it has the the characteristics of complete, well-segmented, and noiseless [30]. In this paper, we exploit the uniformly resampled dense point cloud variant as pretreated in literature [31], which can be downloaded from https://shapenet.cs.stanford.edu/media/modelnet40_normal_resampled.zip. In this dataset, each point cloud object is composed of 10,000 points that are normalized to the range of [-1,1] with 6 significant digits after the decimal point, where each point has six point cloud attributes, the first three are the coordinates of the point cloud (x, y, z), and the latter three are the normal vectors ($\textbf {n}_x$, $\textbf {n}_y$, $\textbf {n}_z$) [30]. We evaluate the ModelNet40 classification benchmark on PointNet++ [32] with one Intel XEON E5-2630 CPU and one NVIDIA TITAN XP GPU, following the default train/test split and utilizing the default parameters as the original implementation method.
2.2 Theoretical model for MC-SPVCSELs with DOF
The optical chaos applied to the encryption and decryption of 3D point cloud in this paper is generated by the mutually coupled spin-polarized vertical-cavity surface-emitting lasers subject to double optical feedback, which are driven by the circularly polarized light as the optical pumping. The system schematic of MC-SPVCSELs with DOF are demonstrated in Fig. 1. Two SPVCSELs (entitled as SPVCSEL1 and SPVCSEL2) are mutually coupled with each other via the corresponding variable optical delay line (VODL) and variable attenuator (VA), where the VODLs are used to control the coupling delay time and the VAs are utilized to manipulate the coupling strength. In the meantime, the outputs of two SPVCSELs are individually reflected back to SPVCSEL1 and SPVCSEL2 through the relevant mirror (M). Under the above system schematic, in consideration of the spin-flip model (SFM) [33], mutually coupled VCSELs [34], optical feedback VCSELs [35], optically pumped SPVCSELs [36–39], the rate equations for MC-SPVCSELs with DOF can be depicted by the following formulas:
Ultimately, $F$ characterizes the effect of spontaneous emission noise with a zero mean complex Gaussian noise source, which can be written as follows:
where $\beta _{sp}$ signifies the spontaneous emission rate, $\xi _{1}$ and $\xi _{2}$ represent the complex Gaussian white noises with zero mean.
Fig. 1. The system schematic of MC-SPVCSELs with DOF. SPVCSEL: spin-polarized vertical-cavity surface-emitting lasers; VODL: variable optical delay line; VA: variable attenuator; M: mirror.
2.3 3D point cloud encryption and decryption scheme
The privacy protection for 3D point cloud classification is implemented via 3D point cloud encryption scheme based on optical chaos. Figure 2 exhibits the 3D point cloud encryption and decryption scheme based on optical chaos, where the dashed line box in the upper part of the figure represents the 3D point cloud encryption process, and the dashed line box in the lower part represents the corresponding 3D point cloud decryption process. Moreover, the yellow boxes in the figure represent the corresponding 3D point cloud data (PCD), the green boxes represent the secret key, the blue boxes represent the encryption and decryption operations, and the gray boxes represent the 3D point cloud classification process. Firstly, the MC-SPVCSELs with DOF generate one million optical chaos (OC) data, named OC1 in Fig. 2. Ten thousand optical chaotic data are randomly selected from OC1 without repetition and named as OCPx, where Px represents the permutation process for x. Afterwards, remove the selected OCPx from OC1 to get OC2. Repeat the above operations to successively obtain the secret keys required in the permutation and diffusion process for 3D point cloud data ($x, y, z, \textbf {n}_x, \textbf {n}_y, \textbf {n}_z$), as shown in the green boxes in the first and second lines of Fig. 2. These operation processes can be described as follows:
where $i$ represents the serial number of different optical chaos data ($i = 1, 2, \dots, 12$), the function numel(A) is used to calculate the number of elements in the array A, the randperm(A,K) is a function that returns a row vector containing K unique integers selected randomly from 1 to A. The OCXX represents the optical chaotic data used in the encryption process for 3D point cloud data, where the former X denotes the encryption process, i.e. P denotes the permutation process, D denotes the diffusion process, and the latter X represents the 3D PCD target ($x, y, z, \textbf {n}_x, \textbf {n}_y, \textbf {n}_z$) that needs to be encrypted. Additionally, the symbol $\sim$ denotes the logical NOT operation, the ismember(A,B) is a function for arrays A and B that returns an array of the same size as A containing true (logical value 1) where the elements of A are in B and false (logical value 0) otherwise.Fig. 2. 3D point cloud encryption and decryption scheme based on optical chaos. $OC_i$: the different optical chaos data ($i = 1, 2, \dots, 12$); OCXX: the randomly selected data from $OC_i$ without repetition, where the former X denotes the encryption process, i.e. P denotes the permutation process, D denotes the diffusion process, and the latter X represents the 3D PCD target ($x, y, z, \textbf {n}_x, \textbf {n}_y, \textbf {n}_z$); AS: sort the data with ascending order; TI6: transform to signed integers with 6 significant digits; XOR: bitwise XOR operation; NORM: normalization process; I-NORM: the inverse process of NORM; TD6: transform to signed decimals with 6 significant digits; I-AS: the inverse process of AS.
The original 3D point cloud data are permuted with OCPX by the following expression:
where the $SortIndex(PCD,OCPX,'Ascending')$ is a function that returns the sorted P_PCD sequence of PCD by sorting the elements of OCPX in ascending (AS) order. Before the exclusive OR (XOR) operation, the P_PCD and OCDX data are reserved for 6 digits after the decimal point and then transformed to signed integers with 6 significant digits (TI6), which can be described as follows: where the $roundn(A,num)$ is a function that rounds each element of A to the nearest multiple of $10^{num}$, num has positive and negative sign, with positive number to the left of the decimal point and negative number to the right of the decimal point. Thereafter, the bitwise XOR operation for diffusion process of PCD can be executed by the following formula: where the symbol $\otimes$ represents the bitwise XOR operation. In order to successfully pass the test of 3D point cloud classification, the diffused data needs to be normalized to the range of [-1, 1] with 6 significant digits after the decimal point, just like the original 3D point cloud data. The process of normalization (NORM) is as follows:3. Results and discussion
This section displays the nonlinear dynamic behavior of MC-SPVCSELs with DOF via the bifurcation diagram, especially the optical chaotic state is detailedly investigated. Significantly, the classification results of the whole forty object categories for original PCD, encrypted PCD, and decrypted PCD are precisely demonstrated with intensive discussion. Besides, five vivid 3D point cloud images in different object categories of test set are chosen for exhibiting the encryption and decryption results visually.
3.1 Optical chaotic states of MC-SPVCSELs with DOF
The rate Eqs. (1)–(5) for MC-SPVCSELs with DOF can be numerically solved using fourth-order Runge-Kutta algorithm. In the calculation process, the adopted parameters of MC-SPVCSELs with DOF are shown as below: $\alpha = 3$, $\kappa = 150$ ns$^{-1}$, $\gamma = 1$ ns$^{-1}$, $\gamma _p = 50$ ns$^{-1}$, $\gamma _a = 0.1$ ns$^{-1}$, $\omega _0 = 2.2176 \times 10^{15}$ rad/s (corresponding optical wavelength is 850 nm), $\triangle \omega = 0$ GHz. To preserve the symmetry of the optical system, these parameters are independently set to be same: $\gamma _s^1 =\gamma _s^2 = 30$ ns$^{-1}$, $P^1 =P^2 = -0.1$, $\omega _1 = \omega _2 = 2.2176 \times 10^{15}$ rad/s, ${k_{inj}}_{1} = {k_{inj}}_{2} = 3$ ns$^{-1}$, ${\tau _{inj}}_{1} = {\tau _{inj}}_{2} = 2$ ns, ${k_{fb}}_{1} = {k_{fb}}_{2} = 3$ ns$^{-1}$, ${\tau _{fb}}_{1} = {\tau _{fb}}_{2} = 2$ ns.
Figure 3 shows the bifurcation diagrams of the RCP intensity extrema as a function of $\eta$ for MC-SPVCSELs with DOF and MC-SPVCSELs without DOF, together with the relevant complexity comparisons, where the red parts denote the MC-SPVCSELs without DOF, and the blue parts denote the MC-SPVCSELs with DOF. Here, the $\eta$ represents both $\eta ^1$ and $\eta ^2$, and the relationship between them is $\eta ^1 =\eta ^2$. As can be seen from Fig. 3(a) and Fig. 3(b), the MC-SPVCSELs with DOF has larger chaotic regions than the MC-SPVCSELs without DOF, especially the region of $\eta <10$. In order to better evaluate the complexity of optical chaotic system, the spectral entropy (SE) and $C_0$ complexity are introduced to calculate the disorder complexity of chaotic time series in the frequency domain. The calculation methods of SE and $C_0$ complexity have been expounded detailedly in literatures [40–44], thus not tired in words here. The Fig. 3(c) and Fig. 3(d) demonstrate the results of SE and $C_0$ complexity for MC-SPVCSELs with DOF and MC-SPVCSELs without DOF. In Fig. 3(c) and Fig. 3(d), SE and $C_0$ complexity of MC-SPVCSELs with DOF are greater than MC-SPVCSELs without DOF in most ranges of $\eta$, particularly in the region of $\eta <10$, which are consistent with Fig. 3(a) and Fig. 3(b). For more intuitively observing the difference between MC+DOF and MC over the entire $\eta$ range, their respective average values are calculated. The average values of SE and $C_0$ complexity are shown by the dashed lines in Fig. 3(c) and Fig. 3(d), where can find that the average SE and $C_0$ complexity of MC-SPVCSELs with DOF are both higher than MC-SPVCSELs without DOF. Hence, compared with the previous SPVCSELs, the MC-SPVCSELs with DOF has the following two advantages in the privacy protection of 3D point cloud classification: i) it has higher chaotic complexity and can improve the encryption effect of 3D point cloud; ii) it has more external disturbances and can availably expand the key space. In addition, the overall trend of SE and $C_0$ complexity increase with the raise of $\eta$ in Fig. 3(c) and Fig. 3(d), thereby we observe the values of SE and $C_0$ complexity as $\eta =20$: the SE of MC and MC+DOF are 0.7626 and 0.7629, while $C_0$ complexity of MC and MC+DOF are 0.2891 and 0.2946. On this basis, we focus on the optical chaotic state as $\eta =20$, the specific situation is shown in Fig. 4.
Fig. 3. Bifurcation diagram of the RCP intensity extrema as a function of $\eta$ for MC-SPVCSELs with DOF and MC-SPVCSELs without DOF, together with the relevant complexity comparison, where the red parts denote the MC-SPVCSELs without DOF, and the blue parts denote the MC-SPVCSELs with DOF. (a) Bifurcation diagram of $\eta$ for MC-SPVCSELs without DOF; (b) Bifurcation diagram of $\eta$ for MC-SPVCSELs with DOF; (c) SE of $\eta$ for MC-SPVCSELs with DOF and MC-SPVCSELs without DOF; (d) $C_0$ of $\eta$ for MC-SPVCSELs with DOF and MC-SPVCSELs without DOF.
Fig. 4. Optical chaotic state of MC-SPVCSELs with DOF for $\eta =20$. (a) Time series; (b) Power spectra; (c) Optical spectra; (d) Phase portrait; (e) Attractor; (f)-(h) 0-1 test.
Figure 4 demonstrates the optical chaotic state of MC-SPVCSELs with DOF for $\eta =20$. In Fig. 4(a), the time series present haphazard intensity pulses and similar to the noisy volatility. In Fig. 4(b) and Fig. 4(c), the power spectrum and optical spectra have broadened pedestals and wide peak continuous spectrum. In Fig. 4(d), the phase portrait has scattered points randomly distributed in a large range. In Fig. 4(e), the attractor is amorphous, aperiodic, dense and convoluted trajectory. Moreover, Fig. 4(f)-(h) show the results of 0-1 test, which can distinguish regular and chaotic dynamics in deterministic dynamical systems without phase space reconstruction [45–47]. The relevant theory principle and computational mechanism of the 0-1 test have been elaborated thoroughly in the literatures [44–49], thereupon not reiterated it herein. In Fig. 4(f)-(h), $K(c)$ is the asymptotic growth rate ($c$ is the randomly chosen constant in $(0, \pi )$), $M(t)$ is the mean square displacement ($t = 1, 2, \dots, T$), $p$ and $q$ are the translation variables of time series. On the basis of 0-1 test above-mentioned, it can be observed that the values of $K(c)$ closely approximates to 1 in Fig. 4(f), the values of $M(t)$ increase linearly with $t$ in Fig. 4(g), and the diagram of $p$-$q$ is resemble to the Brownian motion in Fig. 4(h). Consequently, all the results presented in Fig. 4(a)-(h) are the typical features of the chaos. In virtue of the chaotic states of Fig. 4(a)-(h), the optical chaotic output under this laser parameter is used to generate the OC1 for 3D point cloud encryption in Fig. 3.
3.2 Classification results for 3D point cloud
In this work, all of the 2,468 objects in the test set of ModelNet40 dataset containing 40 object categories are encrypted and decrypted by our proposed 3D point cloud encryption and decryption scheme. The classification results of ModelNet40 dataset for original PCD (OPCD), encrypted PCD (EPCD), and decrypted PCD (DPCD) are listed in Table 1.
Table 1. Classification results of ModelNet40 dataset for OPCD, EPCD, and DPCD.
As exhibited in Table 1, the class accuracies of 40 object categories are totally enumerated, where the second column, third column, and fourth column denote the classification results of original PCD, encrypted PCD, and decrypted PCD, respectively. The classification results of the original PCD illustrate that our experimental platform can utilize PointNet++ to perform the classification task of the ModelNet40 dataset and has good classification results, owing to the mean class accuracy and instance accuracy of original PCD are $90.4166\%$ and $92.8964\%$. In the third column of Table 1, the class accuracies of the encrypted PCD are almost all equal to $0.0000\%$ except for the plant class with $100.0000\%$ class accuracy, revealing that the encrypted 3D point cloud cannot be classified and identified. In other words, all the encrypted 3D point clouds are classified as plants, which is meaningless to 3D point cloud classification. Here, the mean class accuracy and instance accuracy of encrypted PCD are $2.5000\%$ and $4.0453\%$, respectively. Therefore, our proposed 3D point cloud encryption scheme based on optical chaos can effectively protect the privacy of 3D PCD in 3D point cloud classification. Furthermore, the class accuracies of the decrypted PCD are nearly identical to the original PCD, with the exception of several categories that are slightly different. In the meantime, the mean class accuracy and instance accuracy of decrypted PCD are $90.1562\%$ and $92.7751\%$, which are very close to the classification results of original PCD. In conclusion, the classification results in Table 1 verify that the proposed privacy protection scheme for 3D point cloud classification is practically feasible and remarkably effective.
3.3 Encryption and decryption results for 3D point cloud
For purpose of directly demonstrating the specific situation of 3D point cloud encryption and decryption in Table 1, we take five 3D point cloud images belonging to different categories in test set of ModelNet40 as examples in the following article to show the corresponding encryption and decryption results, as shown in Fig. 5. The Fig. 5(a)-(e) are the results of airplane, bed, car, chair, and cup, respectively. Besides, the first column of Fig. 5 shows the original 3D point cloud images, while the second column and third column show the encrypted results and decrypted results of the relevant 3D point cloud images. It can be seen from the second column of Fig. 5, the encrypted 3D point cloud images are blurred and unrecognizable, which are uniformly and dispersively distributed in the whole range of [-1,1] for each x, y, z coordinates. In addition, it can be observed from the third column of Fig. 5 that the decrypted 3D point cloud images are exactly the same as the original images without any difference after making difference comparison between them. Therefore, the results in Fig. 5 illustrate that the privacy of 3D point cloud data can be well protected, and the decrypted point cloud image recovered from the encrypted point cloud image does not affect the 3D point cloud classification at all, which is consistent with the results in Table 1 and can validate the classification results in Table 1 from another perspective.
Fig. 5. Encryption and decryption results for 3D point cloud, where the first column shows the original 3D point cloud images, the second column and third column show the encrypted results and decrypted results of the relevant 3D point cloud images. (a) Airplane; (b) Bed; (c) Car; (d) Chair; (e) Cup.
4. Security analysis
An excellent 3D point cloud encryption scheme should own the capacity to counteract various malicious attacks. In an effort to corroborate the security performance of our proposed privacy protection scheme for 3D point cloud classification, this section analyzes the security performance from the aspects of key space, key sensitivity, statistical attack, encryption and decryption speed, 3D point cloud features, and noise attack in especial.
4.1 Key space analysis
In order to afford sufficient security for 3D point cloud encryption, the key space should be large enough to prevent brute-force attack. In our proposed 3D point cloud encryption scheme, the secret key includes the following parameters: $\eta ^1$, $\eta ^2$, $P^1$, $P^2$, $\gamma _s^1$, $\gamma _s^2$, $\omega _1$, $\omega _2$, ${k_{inj}}_{1}$, ${k_{inj}}_{2}$, ${\tau _{inj}}_{1}$, ${\tau _{inj}}_{2}$, ${k_{fb}}_{1}$, ${k_{fb}}_{2}$, ${\tau _{fb}}_{1}$, ${\tau _{fb}}_{2}$. According to the precision of 64-bit double data is $10^{-15}$ resulting in a $10^{15}$ range, the key space of our proposed 3D point cloud encryption scheme is approximately $(10^{15})^{16} = 10^{240} \approx 2^{798}$, which is far larger than $2^{100}$ and even much greater than $2^{256}$ of the Advanced Encryption Standard (AES) algorithm [22,24,25,50]. Furthermore, Table 2 presents the comparison results of our key space with some other 3D point cloud encryption schemes, where $J$ is the size of point cloud in literatures [22] and [24], and $L = floor(sqrt(J)+1)$. The comparison results in Table 2 demonstrate that the key space of our proposed scheme is much larger than Ref. [21] and Ref. [25], meanwhile larger than Ref. [22] as $J \le 10$ together with Ref. [25] as $J < 3$. Therefore, the key space of our proposed scheme is sufficiently large to resist the brute-force attack.
Table 2. Comparison results of key space.
4.2 Key sensitivity analysis
A good 3D point cloud encryption scheme is exceedingly sensitive to the key, specifically, totally different encryption results are caused by diminutive key modifications. It should be noted in advance that the spontaneous emission noise F of MC-SPVCSELs with DOF (see Eq. (5)) and the selection pretreatment of the optical chaotic key (see Eq. 6) can bring some randomness to the encryption system. For eliminating the interference brought by the above two aspects to the key sensitivity analysis, we remove the noise term and fixed the selected optical chaotic data in this section.
Figure 6 shows the key sensitivity analysis results of 3D point cloud encryption scheme, where the tiny change is set to be $\Delta = 10^{-14}$ and $\eta ^1 = 20$. Figure 6(a) is the original 3D point cloud image, while Fig. 6(b) and Fig. 6(c) are the encrypted 3D point cloud image from Fig. 6(a) with $\eta ^1$ and $\eta ^1+\Delta$, respectively. To compare the distinction between Fig. 6(b) and Fig. 6(c), we make a difference between them and the corresponding result is displayed in Fig. 6(d). It can be seen from Fig. 6(d), the difference diagram presents a spherical distribution within the broader range [-2,2] of the x, y, z coordinates, which indicates a 100% distinction between Fig. 6(b) and Fig. 6(c). Moreover, the encrypted 3D point cloud image of Fig. 6(b) is decrypted by $\eta ^1$ in Fig. 6(e), which is identical to the original 3D point cloud image in Fig. 6(a). However, the decrypted 3D point cloud image from Fig. 6(b) by $\eta ^1+\Delta$ is unidentifiable in Fig. 6(f). Therefore, the results of the above key sensitivity analysis fully demonstrate that the proposed 3D point cloud encryption scheme is awfully sensitive to the key. It is worth noting that the results in Fig. 6 are obtained by removing the noise and fixing the optical chaotic data, whereas the key sensitivity can be much more sensitive in the actual encryption scheme than the results in Fig. 6.
Fig. 6. Key sensitivity analysis results of 3D point cloud encryption scheme, where the tiny change is set to be $\Delta = 10^{-14}$ and $\eta ^1 = 20$. (a) Original 3D point cloud image; (b) Encrypted 3D point cloud image with $\eta ^1$; (c) Encrypted 3D point cloud image with $\eta ^1+\Delta$; (d) Difference between (b) and (c); (e) Decrypted 3D point cloud image from (b) with $\eta ^1$; (f) Decrypted 3D point cloud image from (b) with $\eta ^1+\Delta$.
4.3 Statistical attack analysis
A qualified 3D point cloud encryption scheme should be able to effectively conceal the statistical characteristics of the 3D point cloud image and thus resist statistical attack, so as to prevent the eavesdropper from reconstructing the 3D point cloud image according to the leaked statistical characteristics. The viewpoint feature histogram (VFH) is a new characteristic representation regarding the global descriptor for the whole point cloud, applying to the point cloud clustering recognition and six degree of freedom (DOF) pose estimation [22,24]. The default VFH implementation totally includes 308 binning subdivisions, using 45$\times$3 binning subdivisions for three directional angle information between the centroid point of the point cloud and its individual points, plus another 45 binning subdivisions for the distances between each point and the centroid, and 128 binning subdivisions for the viewpoint component of the viewpoint directional angle information between the viewpoint and the normal of each point, which results in a 308-byte array of float values. Therefore, the horizontal axis of VFH represents the number of histogram bins (NHB), and the vertical axis of VFH represents the percentage of points falling in each bin (PPFEB).
Figure 7 presents the VFH results of the 3D point cloud images before and after encryption, where the first column and second column demonstrate the original 3D point cloud images and the relevant VFH results, besides the third column and fourth column show the encrypted 3D point cloud images and the relevant VFH results. The Fig. 7(a)-(e) represent the VFH results of airplane, bed, car, chair, and cup, respectively. In the second column of Fig. 7, the VFHs of original 3D point cloud images expose multitudinous characteristic peaks with large fluctuations, offering some information of the original 3D point cloud image for the eavesdropper. Nevertheless, in the fourth column of Fig. 7, the VFHs of encrypted 3D point cloud images are smoother fluctuations and significantly different from the VFHs of original 3D point cloud images, revealing the statistical characteristics of 3D point cloud images are concealed. For more specific explanation, we take the VFHs of original and encrypted airplane in Fig. 7(a) as an example to illustrate the differences between them. In the VFH of original airplane, there are three small feature peaks in the 2nd, 22nd, and 44th bins. However, in the VFH of encrypted airplane, there are no obvious feature peaks in these bins. In the 68th bin, the VFHs of both original and encrypted airplane show the most obvious peaks, but their values are different: 68.3199 in the original VFH and 98.4264 in the encrypted VFH. In the 115th bin, the VFH of original airplane show the third highest feature peak with value of 23.0120, but it is different in the VFH of encrypted airplane, where the second highest feature peak appears in the 119th bin with the value of 20.1419. Ultimately, In the 245th bin, the VFH of original airplane show the second highest feature peak with the value of 43.7287, but the VFH of encrypted airplane show the third highest feature peak with the value of 11.7102, besides the width of these two peak are different. In summary, the VFH distribution of original and encrypted airplane is different with respect to each bin. Even though there are feature peaks at the same bin, their values are different, which reflects that the point before and after encryption is different from the directional angle and distance between the centroid point and its individual point, as well as the viewpoint directional angle, i.e. the point after encryption is greatly changed from the point before encryption and then located in different positions in space, thus the point cloud outline formed by all these points is completely different. As a consequence, the VFH results of 3D point cloud image explicate that the proposed 3D point cloud encryption scheme can make the statistical attack impracticable.
Fig. 7. VFH results of the 3D point cloud image before and after encryption, where the first column and second column demonstrate the original 3D point cloud images and the relevant VFH results, besides the third column and fourth column show the encrypted 3D point cloud images and the relevant VFH results. (a) Airplane; (b) Bed; (c) Car; (d) Chair; (e) Cup.
4.4 Encryption and decryption speed analysis
A pragmatic 3D point cloud encryption scheme is expected to be fast enough to meet the requirements of real-time encryption. In this work, the proposed 3D point cloud encryption scheme is implemented via Matlab 2019 software on personal computer with Intel Core i7-8700 CPU 3.20 GHz, and 8 GB RAM under Windows 10 operation system. In addition, the proposed encryption scheme mainly contains the permutation and diffusion encryption process for 3D point cloud data $x, y, z, \textbf {n}_x, \textbf {n}_y, \textbf {n}_z$, hence the time complexity of the proposed 3D point cloud encryption scheme is $O(12N_{PC})$, where the $N_{PC}$ denotes the number of points in a 3D point cloud object. Table 3 shows the encryption and decryption times of the proposed 3D point cloud encryption scheme for five example 3D point cloud images. As can be seen from Table 3, the encryption and decryption times are about 33 ms and 34 ms. As a result, the proposed 3D point cloud encryption scheme is fast enough to support the real-time encryption.
4.5 3D Point cloud feature analysis
Diverse 3D features can be acquired via considering the fundamental nature of the local 3D neighborhood and the local 3D shape features generated by the spatial disposition of 3D points in the neighborhood [51]. Assuming that each point $U$ has a spherical neighborhood centered on this point U with a radius less than $r$, afterwards the famous principal component analysis (PCA) algorithm is conducted on such point set composed of the above spherical neighborhood to extract the eigenvalues ($\lambda _1$, $\lambda _2$, $\lambda _3$) and eigenvectors ($\textbf {e}_1$, $\textbf {e}_2$, $\textbf {e}_3$) of the covariance matrix, where the three eigenvalues satisfy the condition of $\lambda _1 \ge \lambda _2 \ge \lambda _3 \ge 0$[52–55]. These three eigenvalues can be exploited to calculate the following eight geometric features of 3D point cloud: eigenentropy, change of curvature, omnivariance, anisotropy, linearity, planarity, sphericity, and sum of eigenvalues.
Table 3. Encryption and decryption time.
The eigenentropy denotes a measurement depicting the order or disorder in each eigenvector direction of 3D points within its local 3D neighborhood. Thereupon, the measurement of eigenentropy ($E_\lambda$) can be written by the Shannon entropy as following formula [51–53,55]:
The change of curvature is also called the surface variation, displaying the changes in the shape of local surfaces and calculating the deviation of the points from the tangent plane. Hence, the change of curvature ($C_\lambda$) is the ratio of the smallest eigenvalue to the sum and can be defined as following formula [51–53,55]:
The omnivariance ($O_\lambda$) can describe the 3D volumetric distribution of individual points in its neighborhood, which can be computed by the following formula [52,53]:
The anisotropy is a feature of the 3D point cloud surface adaptation to its topology, and is applied to distinguish object-oriented and non-object-oriented. The anisotropy ($A_\lambda$) can be acquired via proportioning the difference between the smallest eigenvectors and largest eigenvectors [52,53]:
Furthermore, the eigenvalues can be utilized for calculating the dimensionality features of 3D point cloud via linearity, planarity and sphericity, which denote the 1D, 2D, and 3D features, receptively. Specific instructions, the linearity feature can be exploited to extract the linear objects, the planarity can be used for ascertaining the smoothness property, and the sphericity can be adopted to describe the scattering of high curvature points. The linearity ($L_\lambda$), planarity ($P_\lambda$) and sphericity ($S_\lambda$) can be expressed as follows [51–53]:
Ultimately, the sum of eigenvalues ($Sum_\lambda$) delineates the total variance, which can be written in the following formula [52,53]:
Thus far, the above eight feature indicators can measure 3D point cloud features in multiple aspects. Figure 8 shows the contrast results of the eight feature indicators for the example 3D point cloud image "airplane" before and after encryption, where the red dots and blue dots represent the associated feature indicators of original airplane and encrypted airplane, the $N_{PC}$ denotes the number of points in a 3D point cloud object. Under the calculation process of eight feature indicators, the specifying number of nearest neighbors $k$ in K-Nearest Neighbors (KNN) algorithm is set to be $k = 10$. It can be seen from Fig. 8(a), the eigenentropies with blue dots of encrypted airplane are nearly all larger than the original airplane with red dots, indicating that the encrypted airplane is more likely approach to the disorder with good encryption outcome. In Fig. 8(b), the change of curvatures with blue dots of encrypted airplane are almost all greater than the original airplane with red dots, revealing that the encrypted airplane have higher deviations between the points and the surface. In Fig. 8(c), the omnivariances with blue dots of encrypted airplane are almost all bigger than the original airplane with red dots, implying the 3D volumetric distribution of encrypted airplane are boarder than the original airplane. In Fig. 8(d), the anisotropies of encrypted airplane are lower than the original airplane, explicating the encrypted airplane is more tend to isotropy and non-oriented objects. In Fig. 8(e)-(g), the linearities and planarities of encrypted airplane are little smaller than the original airplane, while the sphericities of encrypted airplane are larger than the original airplane, proving its one-dimensional and two-dimensional features are weakened and its three-dimensional distribution is more dispersed after encryption. In Fig. 8(h), the sum of eigenvalues of encrypted airplane are greater than the original airplane, which demonstrates the total variance are increased. To summarize the contrast results of above eight feature indicators in Fig. 8(a)-(h), the encrypted point cloud image tends to be more disorderly arrangement and has a more dispersed distribution in the three-dimensional space, which verifies the excellent encryption effect and effectiveness of the proposed 3D point cloud encryption scheme.
Fig. 8. Contrast results of the eight feature indicators for the example 3D point cloud image "airplane" before and after encryption, where the red dots and blue dots represent the associated feature indicators of original airplane and encrypted airplane, the $N_{PC}$ denotes the number of points in a 3D point cloud object, and the $k$ value is set to be $k = 10$. (a) Eigenentropy; (b) Change of curvature; (c) Omnivariance; (d) Anisotropy; (e) Linearity; (f) Planarity; (g) Sphericity; (h) Sum of eigenvalues.
In an effort to more concisely and intuitively measure the eight feature indicators in Fig. 8, we quantified the feature values of all points in the point cloud by calculating their average values, which are shown in the second row and third row in Table 4. In addition, the eight feature indicators corresponding to the other four 3D point cloud examples are also illustrated in Table 4. It can be observed from Table 4, the eigenentropy, change of curvature, omnivariance, sphericity, and sum of eigenvalues of the encrypted point cloud example images have been increased by several times and even dozens of times, conversely, the anisotropy, linearity, and planarity of the encrypted point cloud example images have been reduced less than doubled. The eigenentropy, change of curvature, omnivariance, anisotropy, linearity, planarity, sphericity, and sum of eigenvalues of encrypted point cloud example images are about 0.046, 0.13, 0.0022, 0.74, 0.43, 0.3, 0.25, 0.008 respectively, regardless of the original values of the eight feature indicators. Therefore, the results in Table 4 verify the results in Fig. 8 and once again illustrate the effectiveness of the proposed 3D point cloud encryption scheme.
Table 4. Contrast results of the average values for eight feature indicators before and after encryption.
Moreover, the number of nearest neighbors $k$ value is closely related to the values of eight feature indicators. Therefore, we explore the impact of $k$ value on the eight feature indicators for the example 3D point cloud image "airplane" before and after encryption, and the corresponding results are shown in Fig. 9. Meanwhile, the red pentagrams and blue pentagrams represent the related feature indicators of original airplane and encrypted airplane, the orange circles denote the ratios of original airplane feature values and encrypted airplane feature values. As can be seen in Fig. 9(a), the eigenentropy of both original and encrypted point clouds increases with the raise of $k$ value, yet its relative ratio decreases continuously, while the similar variation trend also appears in Fig. 9(c) and Fig. 9(h). As observed in Fig. 9(b), the change of curvature of both original and encrypted point clouds elevates with the increase of $k$ value, however its relative ratio decreases first and then increases to a maximum peak at $k = 3$, hereupon gradually decreases in the range of $3 < k < 16$, and finally increases slowly in the region of $k$ greater than 16. Besides, the orange curve of relative ratio gradually flattens in the range of $k > 25$, and its relative rate of change is relatively stable, while the resemble variation tendency also arises in Fig. 9(g). On the contrary, the anisotropy, linearity, and their relative ratio of original and encrypted point clouds are all decreases with the increase of $k$ value. Different from previous results, it can be found in Fig. 9(f), the planarity of original point cloud increases with the ascent of $k$, nevertheless the planarity of encrypted point cloud first increases with the raise of $k$ and then decreases after $k > 7$. Anomalously, the planarity of encrypted point cloud is greater than the planarity of original point cloud as $k < 5$, thus the $k < 5$ is an undesirable range. In summary, the advisable range of $k$ value is recommended to be [5,25].
Fig. 9. The impact of $k$ value on eight feature indicators for the example 3D point cloud image "airplane" before and after encryption, where the red pentagrams and blue pentagrams represent the related feature indicators of original airplane and encrypted airplane, the orange circles denote the ratios of original airplane feature values and encrypted airplane feature values. (a) Eigenentropy; (b) Change of curvature; (c) Omnivariance; (d) Anisotropy; (e) Linearity; (f) Planarity; (g) Sphericity; (h) Sum of eigenvalues.
4.6 Noise attack analysis
The encrypted 3D point cloud unavoidable be polluted via diverse noises in the actual storage and transmission environment, which brings great difficulties to the recovery from encrypted 3D point cloud. Therefore, anti-noise performance is also an important aspect of evaluating the performance of encryption scheme. The mean square error (MSE) and peak signal to noise ratio (PSNR) are used to measure the quality of decrypting 3D point cloud after noise attacks, which can be written as follows:
Figure 10 presents the noise attack results by virtue of polluting the encrypted 3D point cloud airplane utilizing Gaussian noise with zero mean value and different standard deviation (SD). Figure 10(a)-(f) show the decrypted 3D point cloud airplanes under SDs are 0.0001, 0.0003, 0.0005, 0.0007, 0.0009, and 0.0011, and all the decrypted 3D point clouds can be visually recognized as the airplane similar to the original point cloud.
Fig. 10. The noise attack results under different SDs. (a) SD = 0.0001; (b) SD = 0.0003; (c) SD = 0.0005; (d) SD = 0.0007; (e) SD = 0.0009; (f) SD= 0.0011.
To measure the performance of the anti-noise attack more specifically, Table 5 lists the MSEs and PSNRs under different SDs. As shown in Table 2, the MSE increases with the increase of SD, while the PSNR decreases with the increase of SD, and importantly all the PSNRs are greater than 31 dB. Therefore, the results of Fig. 30 and Table 5 demonstrate that our 3D point cloud encryption scheme has good anti-noise performance and can resist noise attack.
Table 5. The noise attack results of MSEs and PSNRs under different SDs.
5. Conclusion
In this paper, a novel privacy protection scheme for 3D point cloud classification based on optical chaos is proposed and implemented for the first time as farthest knowledge as we know. The MC-SPVCSELs with DOF is constructed to support the optical chaos for 3D point cloud encryption and decryption process. The complexity comparison is presented for MC-SPVCSELs with DOF and MC-SPVCSELs without DOF through the bifurcation diagram, SE, and $C_0$ complexity, while the MC-SPVCSELs with DOF show higher chaotic complexity. The optical chaotic state of MC-SPVCSELs with DOF is comprehensively displayed via time series, power spectra, optical spectra, phase portrait, attractor, and 0-1 test. In this work, all of the 2,468 objects in the test set of ModelNet40 dataset containing 40 object categories are encrypted and decrypted by our proposed 3D point cloud encryption and decryption scheme. The mean class accuracy and instance accuracy of original PCD and decrypted PCD are indeed very close to each other, due to 90.4166% and 92.8964% for original PCD, together with 90.1562% and 92.7751% for decrypted PCD. Interestingly, the class accuracies of the encrypted PCD are nearly all equal to $0.0000\%$ apart from the plant class with $100.0000\%$ class accuracy, leading to the mean class accuracy and instance accuracy of encrypted PCD are $2.5000\%$ and $4.0453\%$. The classification results of original PCD, encrypted PCD, and decrypted PCD demonstrate our proposed privacy protection scheme is effective for 3D point cloud classification. Moreover, the encryption and decryption results for airplane, bed, car, chair, and cup are respectively exhibited, the encrypted 3D point cloud images are ambiguous and illegible, while the decrypted 3D point cloud images are identical to the original images without difference. In addition, the diversified security analyses are illustrated, containing the key space analysis, key sensitivity analysis, statistical attack analysis, encryption and decryption speed analysis, and particularly the 3D point cloud geometric feature analysis. These security analysis results demonstrate that the proposed privacy protection scheme has a sufficient large key space to counteract brute-force attack, high sensitivity to the key, VFHs with concealing characteristic peaks making statistical attack infeasible, fast enough encryption and decryption speed for real-time encryption, 3D point cloud geometric features representing a higher degree of disorder and a more dispersed 3D space, and good anti-noise performance to resist noise attack, which show high security level and good privacy protection effect for 3D point cloud classification. In the future, more privacy protection schemes based on optical chaos suitable for 3D point clouds can be explored to perform different methods of 3D object classification, object part segmentation, and semantic scene segmentation on diverse point cloud datasets.
Funding
Chongqing Talent Plan (cstc2022ycjh-bgzxm0165); Innovation Research 2035 Pilot Plan of Southwest University and Fundamental Research Funds for the Central Universities (SWU-XDPY22013); Special funds for Postdoctoral research of Chongqing (2010010004713415); Chongqing Normal University Ph.D. Startup Fund (21XLB035).
Acknowledgments
We thank the Chongqing Talent Plan under Grant cstc2022ycjh-bgzxm0165, the Innovation Research 2035 Pilot Plan of Southwest University and Fundamental Research Funds for the Central Universities under Grant SWU-XDPY22013, the Special funds for Postdoctoral research of Chongqing under Grant 2010010004713415, and the Chongqing Normal University Ph.D. Startup Fund under Grant 21XLB035.
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time and can be obtained from the authors upon reasonable request.
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