In order to obtain higher imaging resolution, it is usually necessary to increase the aperture of the optical system. However, a single large aperture optical system has problems such as difficult processing and heavy weight. Synthetic aperture technology uses multiple sub-apertures to obtain the resolution equivalent to a single large aperture in space according to a certain arrangement. However, the following problems are imaging blurring, mainly due to the reduction of light transmission area and system common phase error. The reduction of the light transmission area will cause the change of the point spread function of the system, which will lead to the attenuation of the optical transfer function in the middle and low frequency parts.The common phase error is caused by the synthetic aperture system in the process of light path construction and instrument assembly. The common phase error makes the beams of each sub-aperture no longer have the same phase, and the beams passing through each sub-aperture cannot form interference after reaching the image plane, which leads to the blurring of the final image, even in severe cases, it's can not be imaged. All of the above results in the degradation of the imaging quality of the synthetic aperture system. Some large errors can be solved by improving or debugging the optical system, while the inevitable errors can only be restored through the angle of the image. To solve this problem, a sparse prior image restoration method based on norm is proposed. Firstly, the degradation process of the synthetic aperture system is modeled, the influence of different filling factors on the modulation transfer function is analyzed, and the PSF with piston error is calculated. Further, through statistics, it is found that most of the dark channel values of the image degraded by the synthetic aperture system become non-minimum values, which is found to be caused by the convolution process through mathematical verification. Therefore, this paper proposes an image restoration algorithm model based on the dark channel theory. In the objective function, the dark channel and gradient prior are constrained in the form of L₀ norm as the regularization term, and the point spread function calculated under the optical system array structure is used as the initial fuzzy kernel. Finally, to solve the L₀ norm and nonlinear problems, a semi-quadratic splitting method is proposed to calculate alternately, and obtain the estimated clear image. In the experiment part, the simulation experiment and the actual scene experiment are carried out respectively. Peak Signal-to-Noise Ratio(PSNR), Structure Similarity (SSIM) and Gray Mean Gradient (GMG) are used as evaluation indicators, and are compared with the Wiener Filter method and Richardson-Lucy method. In the simulation experiment, five groups of different piston error parameters are set first, and the image is degraded according to these parameters. The average PSNR, SSIM and GMG of the method in this paper reach 23.13 dB, 0.77 and 20.31. Evaluation and comparison show that the method in this paper is applicable to the degraded image with piston error. After that, seven groups of different scene images are set, and the average PSNR, SSIM and GMG of the method in this paper reach 23.79 dB, 0.80 and 30.28 with small variance, which verifies the stability of the method in different scenes restoration. Finally, in the experiment of the actual scene, a special synthetic aperture camera is used to take remote sensing images and real scene images respectively, and the synthetic aperture array structure parameters are set to be consistent with the simulation. In the remote sensing image experiment, the average PSNR, SSIM and GMG of the method in this paper reach 23.04 dB, 0.65 and 24.58. In the real scene experiment, the average PSNR, SSIM and GMG of the method in this paper reach 24.11 dB, 0.90 and 15.27, which shows the effectiveness of the method in the actual scene.