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摘要:
利用压缩感知技术对大气湍流波前探测数据进行压缩，可使测量数据量大幅度减少，能有效降低数据的传输与存储压力，有利于湍流波前的实时测量；但压缩条件要求波前信号是稀疏的或在某个变换域内能够稀疏表示。本文对大气湍流波前斜率信号的稀疏性进行了初步研究，基于大气湍流的统计特性，在频域内对湍流功率谱作黄金分割采样(GS)，建立符合大气湍流斜率物理特征的稀疏基，明确了湍流波前斜率的稀疏性。利用该GS稀疏基对波前斜率进行稀疏分解，并通过仿真实验对比了不同稀疏基对波前斜率的稀疏分解效果。在此基础上，以GS基作为训练基的初始化字典，进行K奇异值分解字典训练(KSVD)，得到训练基(KSVD-GS)，分析了该训练基对波前斜率信号的稀疏表示性能。本文验证了波前斜率能够稀疏分解，建立了一个较好的稀疏基，为压缩感知的应用提供了前提基础。
Abstract:Using compressive sensing technology in atmospheric turbulent wavefront detected data compression can greatly reduce the amount of measured data, can effectively reduce the pressure of data transmission and storage, which is good for real-time measurement of turbulent wavefront. However, the wavefront signal is required to be sparse or can be sparsely represented in one transform domain. In this paper, a preliminary study of the sparsity of the atmospheric turbulent wavefront gradient signal is carried out. Based on the statistical characteristics of atmospheric turbulence, the golden section (GS) is used to make the turbulent power spectrum in the frequency domain, and the sparse basis is established to meet the physical characteristics of the turbulent gradient, then the sparsity of the gradient of the turbulent wavefront is clarified. The sparse decomposition of the wavefront gradient is simulated by using the GS sparse base, and the sparse decomposition effect of different sparsity bases on the wavefront gradient is compared. On this basis, using the GS basis as the initialization training dictionary, K singular value decomposition (KSVD) dictionary training is carried out to get the training base (KSVD-GS), and then the sparse representation performance of this training base to the wavefront gradient signal is analyzed. This paper verifies that the wavefront gradient can be sparsely decomposed and build a better sparse basis, and provides the precondition for the application of compressive sensing.
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Overview: In astronomical imaging observation, the image quality of observation object decreases because of the influence of atmospheric turbulence and noise. Adaptive optics technique is an effective method to correct atmospheric turbulence disturbance. Wavefront sensor, as the eye of the adaptive system, can detect the distorted wavefront which is affected by atmospheric turbulence in real time. As the aperture of the telescope expands, the resolution is improved and the compressive sensing technique is used to measure the atmospheric turbulent wavefront gradient. Compressive sensing can greatly reduce the amount of measured data, and effectively reduce the pressure of data transmission and storage, which is good for real-time measurement of the turbulent wavefront. But this requires the measurement signal is sparse or can be sparsely represented in one transform domain. In this paper, the sparsity of atmospheric turbulence wavefront gradient signal is studied. Based on the statistical characteristics of atmospheric turbulence, the turbulent power spectrum is sampled by golden section (GS) in the frequency domain, to establish a sparse basis that conforms to the physical characteristics of the atmospheric turbulence, and this basis verifies the sparsity of turbulent wavefront gradient. The sparse decomposition of the wavefront gradient is simulated by using the GS sparse base, and the sparse decomposition effect on the wavefront gradient is compared under different bases such as discrete Fourier transform(DFT), over complete discrete Fourier transform (ODFT), and Zernike. Changing the sparse coefficient value K, the sparse representation performances of different sparse basis were discussed. Simulation results show that the sparse decomposition performance of sparse basis GS established in this paper is better than that other sparse bases, the PSNR of sparse basis is improved 2 dB~5 dB, and the MAER of sparse basis is 0~0.04 decreased. Then the gradients of 60 phase screens are selected for sparse decomposition, which fully verifies that GS basis effect is better than other sparse bases. On the GS basis, the training base (KSVD-GS) is obtained through K-singular value decomposition (KSVD) method, the sparse representation performance of the training basis of the wavefront gradient signal is analyzed, the PSNR is increased 2 dB, and the MAER is decreased 0.01. Finally, by increasing the noise and comparing the robustness of each sparse base, the robustness of the GS base is better than that of other sparse bases. In this paper, we mainly study the sparse decomposition of the wavefront gradient and provide the precondition for the application of compressive sensing.
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图 1 大气湍流相位屏及相位结构函数。(a)湍流相位屏; (b)相位结构函数对数图
Figure 1. Atmospheric turbulence phase screen and structure function of logarithmic. (a) Atmospheric turbulence; (b) Structure function logarithmic
图 2 湍流波前X、Y方向的斜率。(a)波前X方向斜率Gx; (b)波前Y方向斜率Gy
Figure 2. The turbulent wavefront gradient in X, Y direction. (a) Wavefront X-direction gradient Gx; (b) Wavefront Y-direction gradient Gy
图 4 在稀疏系数是20×256下各稀疏基复原的湍流及误差图。(a) DFT复原湍流，PSNR为34.45；(b) DFT湍流误差，MAER为0.038; (c) ODFT复原湍流，PSNR为35.56；(d) ODFT湍流误差，MAER为0.034; (e) Zernike复原湍流，PSNR为32.82; (f) Zernike湍流误差，MAER为0.047; (g) GS复原湍流，PSNR为40.29；(h) GS湍流误差，MAER为0.019
Figure 4. When the sparse coefficient is 20 x 256 the restored turbulence and the error graph in each sparse basis. (a) DFT restored turbulence, PSNR is 34.45; (b) DFT error of turbulence, MAER is 0.038;(c) ODFT restored turbulence, PSNR is 35.56; (d) ODFT error of turbulence, MAER is 0.034; (e) Zernike restored turbulence, PSNR is 32.82; (f) Zernike error of turbulence, MAER is 0.047; (g) GS restored turbulence, PSNR is 40.29; (h) GS error of turbulence, MAER is 0.019
图 5 各稀疏基在不同稀疏系数下的比较。(a)不同稀疏系数下的PSNR；(b)不同稀疏系数下的MAER
Figure 5. The comparison of each sparse basis under different sparse coefficient. (a) PSNR in different sparse coefficient; (b) MAER in different sparse coefficient
图 6 各个稀疏基在不同稀疏系数下的比较。(a)不同稀疏系数下的PSNR；(b)不同稀疏系数下的MAER
Figure 6. The comparison of each sparse basis under different coefficient. (a) PSNR in different sparse coefficient; (b) MAER in different sparse coefficient
图 7 各个稀疏基在不同噪声下的比较。(a)不同稀疏系数下的PSNR；(b)不同稀疏系数下的MAER
Figure 7. The comparison of different sparse basis under different SNR. (a) PSNR of different sparse coefficient; (b) MAER of different sparse coefficient
表 1 K=20 时不同稀疏基的稀疏表示性能及运行时间
Table 1. The sparse decomposition performance and running time in each sparse basis when K=20
Basis PSNR/dB MAER Time/s DFT 36.9066 0.0289 1.735 ODFT 37.252 0.0280 2.430 Zernike 34.837 0.0393 0.8436 GS 39.1987 0.0236 0.8666 KSVD-GS 38.7901 0.0251 0.8369 
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