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Abstract
In this paper, we propose a chaotic block-matching and three-dimensional (C-BM3D) filtering algorithm to remove the noise and enhance the security in the turbulent channel of free space optical (FSO) communication. We experimentally demonstrate the performance of C-BM3D by comparing it with chaotic non-local means filtering (C-NLM), chaotic Gaussian filtering and chaotic Median filtering based on Log-normal and Gamma-Gamma turbulence models. The results show that the peak signal-to-noise ratios (PSNRs) of C-BM3D in the weak turbulence under Log-normal and Gamma-Gamma models are up to 96.2956 and 93.2853, respectively. The C-BM3D also achieves superior image similarity in Log-normal turbulent channel, with its structural similarity index measures (SSIMs) nearly equal to 1. Additionally, the signal-to-noise ratio (SNR) of C-BM3D ranks the highest, and its bit error rate (BER) improves by at least 15 dB compared to that of the other three algorithms. The experimental results indicate that the C-BM3D can be a good candidate for the next generation of FSO communication in security and turbulence resistance.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Free space optical (FSO) communication has been widely applied to low earth orbit (LEO) satellite, underwater and unmanned aerial vehicle (UAV) communications because of its license-free, easy deployment, high capacity, etc. For example, C. Li et al. studied a wavelength-division-multiplexing (WDM) four-level pulse amplitude modulation (PAM4) FSO-underwater wireless optical communication (UWOC) integrated system based on 100 Gb/s channel capacity [1]. Since FSO systems do not require license fees, FSO has been viewed as an alternative technology to conventional radiofrequency (RF) systems [2–4] in geostationary satellites [5–7]. To further improve the FSO communication performance, relay-assisted FSO system has been proposed in [8], which outperforms the pure FSO system. FSO links [9] achieve robust and low-latency communications between autonomous underwater vehicle (AUV) and UAV. Authors in [10] also combined UAV with dual-hop RF/FSO relay system to achieve extensive and reliable communications in space-air-ground integrated networks (SAGIN).
However, turbulence is one of the main factors that adversely influences the performance of FSO communication. If a laser transmits through turbulence, intensity fluctuation and optical power attenuation exist, which affects communication performance in turn [11]. There have been many studies about transmitting general information in FSO channels [12–15], which achieved great progress in resisting turbulence. However, there is little research on FSO image transmission. The previous experiments of traditional Gaussian filtering [16] and Median filtering [17] verified that although they have good denoising effects, they are at the expense of communication quality in FSO transmission. The newly proposed non-local means (NLM) filtering [18] can make full use of redundant information and maintain the details of the image, but the noise of similar blocks increases when searching for weights. Based on the above reduction methods, block-matching and three-dimensional (BM3D) filtering is proposed [19] to combine the spatial and frequency domain algorithms, which achieves better visual effects and higher similarity. The latest studies [20–23] show that, compared with traditional filtering algorithms, two-step iterative shrink/threshold segmentation and enhanced Lagrange algorithm, the introduction of the BM3D algorithm not only improves the image reconstruction quality but also enhances the anti-noise capability of the technique.
Due to the exposure of optical signals in FSO communication, the transmitted signals are prone to be eavesdropped and accessed by illegal attackers [24]. The chaotic technique has been widely used in image encryption [25] and security enhancement [26,27] of fiber optic communication thanks to its high sensitivity to initial values, strong unpredictability, etc. For instance, WDM signals can achieve secure transmission on 50 km standard single-mode optical fiber using the private chaotic phase in disarray [28]. Chaotic compressive sensing (CS) successfully enhances the orthogonal frequency division multiplexing-passive optical network (OFDM-PON) transmission security [29]. Chaotic three-dimensional (3D) constellation scrambling could effectively enhance the security and transmission performance of coherent optical orthogonal frequency-division multiplexing (CO-OFDM) systems [30].
In this paper, we combine logistic chaotic mapping with the BM3D filtering algorithm to resist the turbulence and enhance the security of FSO communication. Specifically, the image is encrypted at the transmitter by digital chaos and decrypted at the receiver. At the receiver, the image with turbulence will be filtered by the C-BM3D and other typical filtering algorithms. To clarify the superiority of C-BM3D, we will compare the bit error rate (BER), structural similarity index measure (SSIM) and peak signal-to-noise ratio (PSNR) performance of different filtering algorithms in different turbulent conditions.
2. Principle
2.1 Atmospheric turbulence in FSO channels
Among common statistical models for simulating turbulent conditions, the Log-normal and Gamma-Gamma models are widely used to study turbulence-resistant techniques [31]. The probability distribution function (PDF) of Log-normal is expressed as [32]:
2.2 Logistic chaotic mapping
Logistic mapping is a typical nonlinear iterative equation, as shown in the equation:
where μ is the control parameters, μ∈ (0, 3.5699456], xk is the chaotic sequence, x0∈ (0, 1), and k is the iteration time step.One main characteristic of chaotic systems is the high initial sensitivity. When μ = 0.190903252535798, x1 = 0.699076722656686, x2 = 0.699076722656687, the initial sensitivity of the logistic chaotic mapping is shown in Fig. 1. After certain iterations, the mixed chaotic sequence splits into two sequences with different translations. As the number of iterations increases, the values of the two sequences show bigger difference. It shows that the chaotic system still maintains the initial value sensitivity of logistic chaotic mapping.
2.3 BM3D algorithm
BM3D algorithm [33] can find similar blocks to reference blocks of two-dimensional (2D) images. It stacks 2D-image similar blocks into 3D groups, which can be performed by collaborative filtering. We estimate the recovered image by aggregating the processed results and original image blocks. The overall process of the algorithm is shown in Fig. 2, which can be divided into two steps: Hard-thresholding filtering and Wiener filtering.
We use Zx to represent a square block of N1×N1 on the noisy image z with the top-left coordinate x. x∈X, X is the image domain. Zs is a 3D block, and the top-left coordinates of all 2D blocks are defined by the set S. To represent the reference block in each 3D block, we add the R (reference block) identifier under the corresponding coordinate, namely xR. Similarly, we use Yx to represent a block on a noiseless image y. Because Step1 and Step2 are the same structure, the meanings of some parameters are also the same. In order to distinguish the parameters in the two steps, we add the “ht” and “wie” marks, respectively. For the basic estimation image of Step1, we use ${\hat{y}^{basic}}$ to represent it, and the final estimation image obtained by Step2 is ${\hat{y}^{final}}$.
- 1. Basic estimates: Hard-thresholding filtering
(1) Estimates by block: For each block of the noisy image:
a) Grouping. After finding similar blocks to the reference block, they are altogether stacked into a 3D array. When the noise variance is large or the block is relatively small, it is not accurate to find similar blocks based on the noisy image z. Therefore, the two blocks on the noisy image z are separated by 2D normalized transformation. Then, the coefficients whose amplitude is less than a certain threshold are set to zero, so that the matching error of the two blocks can be expressed as the mean square error of these coefficients.
$$d({Z_{{x_R}}},{Z_x}) = \frac{{||\gamma ^{\prime}(\tau _{2D}^{ht}({Z_{{x_R}}})) - \gamma ^{\prime}(\tau _{2D}^{ht}({Z_x}))||_2^2}}{{{{(N_1^{ht})}^2}}},$$where $\gamma ^{\prime}$ is a hard threshold operator with threshold ${\lambda _{2D}}\sigma $, ||.||22 is the 2-norm, $N_1^{ht}$ is the block size used in basic estimates, $\tau _{2D}^{ht}$ is the 2D normalized transformation.After calculating the matching error between the current reference block and all other blocks according to Eq. (6), we only keep those blocks whose error is less than a certain threshold and get the corresponding coordinate set:
where $\tau _{match}^{ht}$ is the maximum error threshold for determining the similarity of two blocks, which is generally set according to experience. The current reference block will be judged to be a similar block because its matching error is 0, so there is at least one element in the coordinate set. By stacking all the similar blocks together, we get a 3D array ${Z_{S_{xR}^{ht}}}$ of shape $N_1^{ht} \times N_1^{ht} \times |S_{{x_R}}^{ht}|$, where $|S_{{x_R}}^{ht}|$ represents the number of elements in the collection.b) Collaborative hard thresholding. After obtaining the 3D array corresponding to the reference block, we can carry out 3D collaborative transformation and filtering, which can be formally expressed as (8). Firstly, the 3D normalized transformation is applied to the above groups. After obtaining the non-zero component from the weight coefficient by hard threshold, the inverse 3D normalized transformation is performed to get the estimation of all grouped blocks. Finally, they are returned to their original positions.
$$\hat{Y}_{S_{{x_R}}^{ht}}^{ht} = \tau {_{3D}^{ht}{^{ - 1}}}(\gamma (\tau _{3D}^{ht}({Z_{S_{{x_R}}^{ht}}}))),$$where $\gamma$ is a hard threshold operator with threshold ${\lambda _{3D}}\sigma $, ${Z_{S_{{x_R}}^{ht}}}$ is the formed 3D groups, $\tau _{3D}^{ht}$ is the 3D normalized transformation and $\tau {_{3D}^{ht}{^{ - 1}}}$ is the inverse 3D normalized transformation.(2) Aggregation: The superposition weights are calculated by the number of non-zero components. Finally, the estimated image is obtained by dividing the superimposed blocks by the weights of each block.
$${\hat{y}^{basic}}(x) = \frac{{\sum\limits_{{x_R} \in X} {\sum\limits_{{x_m} \in S_{{x_R}}^{ht}} {\omega _{{x_R}}^{ht}\hat{Y}_{{x_m}}^{ht,{x_R}}(x)} } }}{{\sum\limits_{{x_R} \in X} {\sum\limits_{{x_m} \in S_{{x_R}}^{ht}} {\omega _{{x_R}}^{ht}{\chi _{{x_m}}}(x)} } }},\forall x \in X.$$We assume that each block has been filled to the same size as the original image by zeros and ${\chi _{{x_m}}}:X \to \{ 0,1\}$ is used to determine whether a pixel x is on block xm.
- 2. Final estimates: Wiener Filtering
(1) Estimates by block: For each block of the obtained image from basic estimates:
a) Grouping. Within the basic estimates range, block matching (BM) is used to find the location of all blocks that are similar to the reference block, namely R. In this section, we find similar blocks by (10) based on the basic estimation image. Because the current noise is relatively small, it is not necessary to do transformation and hard-thresholding like (6).
$$S_{{x_R}}^{wie} = \left\{ {x \in X:\frac{{||\hat{Y}_{{x_R}}^{basic} - \hat{Y}_x^{basic}||_2^2}}{{{{(N_1^{wie})}^2}}} < \tau_{match}^{wie}} \right\},$$where $N_1^{wie}$ is the block size used in Wiener filtering, $\tau _{match}^{wie}$ is the maximum d-distance for which two blocks are considered similar, $\hat{Y}_{{x_R}}^{basic}$ and $\hat{Y}_x^{basic}$ are the basic estimate blocks located at xR and x, respectively. Now there are two groups formed, one from the noisy image and another one from the basic estimates: ${\hat{Y}^{wie}}_{S_{xR}^{wie}}$ and ${Z_{S_{xR}^{wie}}}$. ${\hat{Y}^{wie}}_{S_{xR}^{wie}}$ is obtained by stacking together the basic estimate blocks $\hat{Y}_{x \in S_{{x_R}}^{wie}}^{basic}$, and ${Z_{S_{xR}^{wie}}}$ is obtained by stacking together the noisy blocks ${Z_{x \in S_{{x_R}}^{wie}}}$.b) Collaborative Wiener filtering. The 3D normalized transformation is applied to the above two groups. The Wiener filtering is performed on the noisy one. Next, the inverse 3D normalized transformation is applied to the filtered Wiener shrinkage coefficients to estimate all blocks. Finally, these block estimates are returned to their original positions. The estimates are shown as:
$${\hat{Y}^{wie}}_{S_{{x_R}}^{wie}} = \tau _{3D}^{wi{e^{ - 1}}}({W_{S_{{x_R}}^{wie}}}\tau _{3D}^{wie}({Z_{S_{{x_R}}^{wie}}})),$$where ${W_{S_{{x_R}}^{wie}}}$ is the Wiener shrinkage coefficient, ${Z_{S_{{x_R}}^{wie}}}$ is the noisy blocks in final estimates.(2) Aggregation: The final estimate ${\hat{y}^{final}}(x)$ of the real image is calculated by aggregating all the obtained local estimates using the weighted average.
3. Simulation process and analysis
In this experiment, we aim to simulate and verify the denoising effect and communication quality of the C-BM3D algorithm by comparing it with the C-NLM algorithm, chaotic Gaussian filtering and chaotic Median filtering under Log-normal and Gamma-Gamma turbulence models. Specifically, the matrix size of the experimental image is 256 × 256. Firstly, the image is encrypted by logistic chaotic mapping. The logistic map is used to generate an array of sequence of random numbers. After obtaining random sequence, the image is cluttered and diffused at the bit plane. Finally, the bitstream of 1*65536 is got by XOR operation on the random image and the random sequence by bit, which is transmitted through the FSO channel. The decryption process is the reverse of encryption. At the receiver, the image recovered from the received bitstream will be filtered by C-BM3D, C-NLM, chaotic Gaussian filtering and chaotic Median filtering. The experimental FSO communication system is shown in Fig. 3, and the simulation parameters are listed in Table 1.
Table 1. The simulation parameters
Additionally, we calculate the SNR and BER to evaluate the communication quality after the four algorithms. And we calculate the PSNR and SSIM of the recovered image and original image to judge the image quality. The formulas are as:
where Ps represents the average power of the signal and Pn represents the average power of the noise. where Ie represents the number of bits accepted in error, and I represents the total number of transmitted bits.3.1 Image quality under turbulence of different intensity
In this section, we aim to compare the PSNR and SSIM of filtering algorithms to estimate the image quality under Log-normal and Gamma-Gamma turbulence models by combining Opti-System with MATLAB. The experiment will be carried out in three different intensities of turbulence in each model: weak, moderate and strong. The values of attenuation (a) are set at 5 dB, 12 dB, 15 dB, respectively, and the values of index refraction structure (c2n) are set at 8 × 10−15m−2/3, 3 × 10−14m−2/3, 1 × 10−13m−2/3, respectively. The results of Log-normal and Gamma-Gamma turbulence models are illustrated in Table 2 and Table 3, respectively. From the data in the two tables below, the PSNRs of C-BM3D in weak turbulence under Log-normal and Gamma-Gamma models are up to 96.2956 and 93.2853, respectively, which are much higher than that of the other three algorithms. Furthermore, C-BM3D and chaotic Gaussian filtering achieve superior image similarity in the Log-normal turbulent channel, with their SSIMs nearly equal to 1. However, it is also apparent that the PSNRs of C-BM3D in strong turbulence under the Gamma-Gamma model present dramatic decrease, even as low as 26.0561. Under strong turbulence, the noise is so dense that the similar blocks may contain some noise. The denoising effect decreases with the estimation error increases. For chaotic Median filtering and C-NLM, their values do not experience obvious change, which have been kept at lower numbers than C-BM3D and chaotic Gaussian filtering. Overall, these results indicate that the C-BM3D has good image denoising and reconstruction performance, with its higher PSNRs and SSIMs in Log-normal and Gamma-Gamma turbulent channels.
Table 2. Comparison in Log-normal Turbulent Channel
Table 3. Comparison in Gamma-Gamma Turbulent Channel
3.2 Communication quality under turbulence of different intensity
In this section, we demonstrate the SNR distribution of filtering algorithms to estimate their communication quality under the Gamma-Gamma turbulence model (Since C-BM3D has already shown superior PSNR performance in the Log-normal turbulent channel from the former section). The experiment will be carried out in four different intensities of turbulence, as shown in Fig. 4. The values of a are set at 5 dB, 10 dB, 12 dB and 15 dB, respectively, and the values of c2n are set at 8 × 10−15m−2/3, 8 × 10−14m−2/3, 8 × 10−13m−2/3 and 8 × 10−12 m−2/3, respectively. It is obvious from Fig. 4 that the SNR of C-BM3D has ranked the highest, even peaking at over 100, while the SNRs of other algorithms have kept at less than 40. Thus, C-BM3D has greater advantages in improving communication quality in FSO communication. However, it can also be seen that C-BM3D shows inferior stability when it comes to strong turbulence under the Gamma-Gamma model.
Fig. 4. The SNR distribution of C-NLM, C-BM3D, chaotic Gaussian filtering, chaotic Median filtering under different Gamma-Gamma turbulent conditions: (a) a = 5 dB/km, $c_n^2 = 8 \times {10^{ - 15}}{m^{ - 2/3}}$; (b) a = 10 dB/km, $c_n^2 = 8 \times {10^{ - 14}}{m^{ - 2/3}}$; (c) a = 12 dB/km, $c_n^2 = 8 \times {10^{ - 13}}{m^{ - 2/3}}$; (d) a = 15 dB/km, $c_n^2 = 8 \times {10^{ - 12}}{m^{ - 2/3}}$.
To further verify the performance of C-BM3D and logistic chaotic mapping, we calculate the BER distribution of the BM3D, NLM, Gaussian filtering, Median filtering and FSO channel with and without chaos, as shown in Fig. 5. Specifically, Fig. 5(a) and Fig. 5(b) demonstrate the results under Log-normal and Gamma-Gamma turbulence models, respectively. It can be seen from Fig. 5 that the BER difference between with and without chaos of every algorithm is little, indicating that logistic chaotic mapping has no adverse effect on the FSO communication quality but also improves communication security. Importantly, the BER values of the BM3D algorithm and FSO channel are almost equal, indicating that the BM3D algorithm improves the image quality without sacrificing the communication quality. From Fig. 5(a), the BM3D algorithm improves at least 24 dB BER performance than the other three algorithms in Log-normal turbulent condition. The superiority of the BM3D algorithm is also verified under the Gamma-Gamma turbulence model. From Fig. 5(b), the BER value of the BM3D algorithm remains unchanged at approximately 0.014 in weak turbulence of the Gamma-Gamma model, improving by at least 15 dB compared with other algorithms. Thus, the BM3D algorithm improves the image quality and communication quality greatly, which can be a good candidate for the next generation of FSO communication in security and turbulence resistance.
Fig. 5. The BER distribution of C-NLM, C-BM3D, chaotic Gaussian filtering, chaotic Median filtering under Log-normal (a) and Gamma-Gamma (b) turbulent conditions.
4. Conclusion
In this paper, we compare the communication performance of C-BM3D, C-NLM, chaotic Gaussian filtering and chaotic Median filtering under Log-normal and Gamma-Gamma turbulence models in the FSO system. The results show that the PSNRs of C-BM3D in the weak turbulence under Log-normal and Gamma-Gamma models are up to 96.2956 and 93.2853, respectively. The C-BM3D also achieves superior image similarity in the Log-normal turbulent channel, with its SSIMs nearly equal to 1. Additionally, the SNR of C-BM3D has ranked the highest, and its BER has remained at the lowest among the four algorithms in FSO communication, improving at least 15 dB BER performance compared with other algorithms. Overall, C-BM3D outperforms other methods since it not only recovers the image with superior quality from the chaotic system of the FSO communication system but also enhances communication quality and security. However, one problem is that the C-BM3D lacks stability in strong turbulence, which needs to be further improved and studied in the future.
Funding
National Natural Science Foundation of China (62025105, 62301097, U21B2005); China Postdoctoral Science Foundation (2023T160775); Natural Science Foundation of Chongqing (CSTB2022NSCQ-MSX0542); Chongqing Municipal Education Commission (KJQN202200610).
Disclosures
The authors declare no conflicts of interest.
Data availability
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.
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