基于TV与gamma范数的非凸秩近似在矩阵补全中的应用

现实生活中，由于运动模糊、光学模糊等因素的影响，获取的图像往往是模糊、不完整的，即图像内容质量下降、细节特征被掩盖，从而影响图像的视觉效果及应用.矩阵补全(Matrix Completion， MC)的目的是将获得的模糊、不完整图像以最大的保真度恢复出完整清晰的图像.该文采用gamma范数代替传统的核范数作为秩函数的非凸近似，与核范数比较，gamma范数大大减弱了大奇异值的贡献，使得较小奇异值的贡献接近零；同时引入全变差(Total Variation， TV)正则项保留图像本身真实的边缘信息和细节信息，从而避免恢复出的图像过度光滑；接着，应用增广拉格朗日乘子法求解模型；最后，通过数值实验验证文章提出的算法较现有求解矩阵补全的算法更高效.

The application of non-convex rank approximation based on TV and gamma norm in matrix completion

In real life， due to the influence of motion blur， optical blur and other factors， the observed image is often fuzzy and incomplete， that is， the image quality may be degraded and details are covered， which affects the visual effect and further applications. Matrix completion aims to recover a clean and complete image from the fuzzy and incomplete image with the greatest fidelity. In this paper， the gamma norm is used instead of the traditional nuclear norm as the non-convex approximation of the rank function. Compared with the nuclear norm， the gamma norm greatly weakens the contribution of the large singular value， and makes the contribution of the smaller singular value to close to zero. Meanwhile， the Total Variation (TV) regularization is exploited to preserve the edge structure and detail information of the image， so as to avoid the over smoothing of the recovered image. Then we use the augmented Lagrangian multiplier method to solve our proposed model. Finally， numerical experiments validate the proposed algorithm is more efficient than the existing algorithms for matrix completion.