In recent years, the development and utilization of nuclear energy have become an important field at the frontier of the world's scientific and technological competitions. Inertial confinement fusion (ICF), as controlled fusion, injects a large amount of energy into the target pellet containing fusion fuel in a very short period, and the fuel inside the pellet undergoes compressional implosion under the action of extremely high temperature, pressure, and density to cause thermonuclear fusion reactions. Throughout the fusion process, the fusion fails due to the uneven symmetry of compressional implosion caused by various factors, which also limits the development of ICF research. Measuring the velocity distribution of shock waves can predict the compression state reached by the target pellet and provide reliable reference data for further optimization of ICF compressional implosion processes. CUP-VISAR is a significant diagnostic instrument for shock wave velocity measurement in the late stage of ICF implosion by recording the interferometric fringes formed by Doppler shift. The CUP-VISAR system provides a new way of thinking about the research on ultra-high temporal resolution 2D imaging of ICF. Currently, the two-step iterative shrinkage thresholding (TWIST) algorithm is mainly employed to solve the optimization problem, which has a large amount of matrix operation during the iterative solution process and thus leads to the defect of long reconstruction time. In this study, a fast and better-quality data reconstruction algorithm is adopted for CUP-VISAR measurement systems.

The traditional reconstruction algorithm of total variational regular constraint compression sampling is based on the total variational model to associate the sparse sampling matrix with its gradient domain to recover the edge and detail information of the sampled data during eliminating noise and artifacts. The traditional total variational regular constraint is mainly utilized to reconstruct two-dimensional data based on one-dimensional sampled data, but the adopted compression diagnosis process of CUP-VISAR systems is to collect three-dimensional data into two-dimensional data. The traditional total variational regularization algorithm is leveraged to recover the 3D diagnostic data of CUP-VISAR systems. It is necessary to expand the 3D linear observation matrix operators into a diagonal matrix for each frame and arrange them successively to form a two-dimensional matrix. However, the sampled data of the fringe distribution of the shock wave velocity field need to be arranged as a one-dimensional matrix, which is reconstructed into three-dimensional data after employing the full variational regularization algorithm. The diagonal matrix expanded by the linear observation matrix operator contains a large number of zero elements that do not contain effective information and expand the matrix computation amount. Thus, the requirements including memory and CPU are relatively high, and this traditional total variation regularization reconstruction algorithm needs generally long reconstruction time. For diagnostic data with more than 40 frame, the partitioning method should be adopted for the reconstruction. However, the partitioning method may exert an obvious block effect on the data at the partitioning edge, which significantly reduces the reconstruction quality. Therefore, as the total variation regularization algorithm cannot directly reconstruct the two-dimensional data, it needs to be further extended and optimized to reconstruct the two-dimensional data collected by CUP-VISAR into three-dimensional data and improve the algorithm reconstruction speed spontaneously. We propose an improved total variation regularization reconstruction algorithm TVAL3H by combining the enhanced Lagrange function method and the alternate minimization method. The convex optimization problem to be solved is divided into two sub-problems which are solved by the iterative threshold shrinkage method and the Barzilai-Borwein gradient method respectively. In the reconstruction process, the TVAL3 algorithm is directly extended to 3D reconstruction by the Hadamard product method, which can effectively reconstruct the sampled data of the shock wave velocity field, significantly improve the reconstruction speed of the algorithm, and guarantee the reconstruction quality.

Simulation reconstruction analysis results of the bending stripes show that the proposed TVAL3H algorithm improves peak signal to noise ratio (PSNR) by 6.86 dB (25 frame)-1.20 dB (150 frame) (Fig. 6) and structure similarity (SSIM) by 26.67% (25 frame)-14.10% (150 frame) (Fig. 6), and reduces time consumption by 92.15% (25 frame)-78.30% (150 frame) (Fig. 6) compared with the conventional TVAL3 algorithm. The time consumption is reduced by 57.79% (100 frame GAP)-77.20% (25 frame ADMM) while the PSNR and SSIM differences are smaller compared with the GAP and ADMM algorithms (Fig. 6). At the same reconstruction time scale, the PSNR of the proposed reconstruction algorithm improves by 1.92 dB (25 frame)-0.84 dB (150 frame) and 1.85 dB (25 frame)-0.80 dB (150 frame) compared with GAP and ADMM algorithms respectively in different frame conditions (Fig. 7). SSIM improves 9.23% (25 frame)-4.48% (150 frame) and 8.85% (25 frame)-4.46% (150 frame) (Fig. 7) compared with GAP and ADMM algorithms respectively.

We propose and implement a three-dimensional extended reconstruction algorithm TVAL3H with total variation regularization for CUP-VISAR compressed sampling of diagnostic data, which simulates the ultrafast compression of shock wave fringe images in the CUP-VISAR diagnostic process. The reconstruction experiments on simulation data of the shock wave velocity diagnostic interference fringe based on the CUP-VISAR system are completed by simulating the compressed ultrafast photography process in the CUP-VISAR diagnostic process and combining the aperture problem during the actual compression coding in different coding aperture conditions of 1×1 and 7×7. The proposed TVAL3H algorithm has advantages in reconstruction speed compared with TWIST and TWIST-DCT algorithms. For the 350×780 dimensional diagnostic data at 25 and 50 frame, the spent reconstruction time is within 400 s. TVAL3H algorithm significantly improves the reconstruction PSNR and SSIM at 25, 50, 100, and 150 frame compared with the conventional TVAL3 algorithm, with significantly reduced construction time. Compared with the GAP and ADMM algorithms, the reconstruction speed is significantly improved with little difference in PSNR and SSIM. By unifying the recovery time of the TVAL3H algorithm with GAP and ADMM to the same time scale, the PSNR and SSIM of TVAL3H recovery fringes are better than those of GAP and ADMM at different frames. The reconstruction results are better than those of GAP and ADMM.