For nonconvex and nonsmooth restoration models, the lower bound theory reveals their good edge recovery ability, and related analysis can help to design convergent algorithms. Existing such discussions are focused on isotropic regularization models, or only the lower bound theory of anisotropic model with a quadratic fidelity. In this paper, we consider a general image recovery model with a non-Lipschitz anisotropic composite regularization term and an $\ell_q$ norm ($1\leq q<+\infty$) data fidelity term. We establish the lower bound theory for the anisotropic model with an $\ell_1$ fidelity, which applies to impulsive noise removal problems. For the general case with $1\leq q<+\infty$, a support inclusion analysis is provided. To solve this non-Lipschitz composite minimization model, we are then motivated to introduce a support shrinking strategy in the iterative algorithm and relax the support constraint to a thresholding support constraint, which is more computationally practical. The objective function at each iteration is also linearized to construct a strongly convex subproblem. To make the algorithm more implementable, we compute an approximation solution to this subproblem at each iteration, but not exactly solve it. The global convergence result of the proposed inexact iterative thresholding and support shrinking algorithm with proximal linearization is established. The experiments on image restoration and two stage image segmentation demonstrate the effectiveness of the proposed algorithm.
翻译:对于非 convex 和非 smoot 恢复模型, 下约束理论显示它们具有良好的边缘恢复能力, 相关分析可以帮助设计趋同算法。 现有的这类讨论侧重于异端正规化模型, 或仅侧重于具有二次对等性忠实的反异质模型的较低约束理论。 在本文中, 我们考虑一个具有非利普施茨 异质复合整流术语和 $\ell_ q美元标准值的普通图像恢复模型( $\leq q ⁇ infty$ ) 数据忠实化术语。 我们为反异端模型建立较低约束理论, 使用 $\ ell_ 1$ 美元对等值的正弦化算法。 对于使用 $\ leq q ⁇ intyl, 提供一种支持包容分析。 要解决这个非利普施氏性复合复合最小化的复合模型, 我们随后有动力在迭代算式解算法中引入一个支持缩减策略, 将支持约束度降低到一个门槛性支持约束值, 这是更实际的。 在每类正缩缩缩缩缩缩缩缩缩的图像阶段, 将一个更精确的缩缩缩化后, 。