The total variation method has been widely used because it can preserve important sharp edges and target boundaries in the image, but one of its shortcomings is that the \(L_{1}\) norm would cause excessive punishment. To overcome excessive penalization, suppress step effects, and maintain a clean contour, this article introduces the \(L_{q}\) non-convex function to design a new regularization term and proposes a new non-convex variational model. For non-convex variational optimization problems, the algorithm employs the alternating direction method of multipliers to obtain optimal approximate solutions, and rigorous convergence proofs are provided based on rich mathematical theory. By comparing with advanced fractional-order and other total variation recovery algorithms, it can be observed that this algorithm achieves better visual effects and quantitative analysis results on images with complex texture structures.

全变分方法因其能够保留图像中重要的锐利边缘和目标边界而得到广泛应用，但其缺点之一是\(L_{1}\)范数会造成过度的惩罚。为了克服过度惩罚、抑制阶跃效应并保持清晰的轮廓，本文引入\(L_{q}\)非凸函数来设计新的正则化项，并提出一种新的非凸变分模型。针对非凸变分优化问题，算法采用乘子交替方向法获得最优近似解，并基于丰富的数学理论提供了严格的收敛性证明。通过与先进的分数阶等全变分恢复算法的比较，可以看出该算法对具有复杂纹理结构的图像取得了更好的视觉效果和定量分析结果。