In the present work, we investigate the inverse problem of identifying simultaneously the denoised image and the weighting parameter that controls the balance between two diffusion operators for an evolutionary partial differential equation (PDE). The problem is formulated as a non-smooth PDE-constrained optimization model. This PDE is constructed by second- and fourth-order diffusive tensors that combines the benefits from the diffusion model of Perona–Malik in the homogeneous regions, the Weickert model near sharp edges and the fourth-order term in reducing staircasing. The existence and uniqueness of solutions for the proposed PDE-constrained optimization system are provided in a suitable Sobolev space. Also, an optimization problem for the determination of the weighting parameter is introduced based on the Primal–Dual algorithm. Finally, simulation results show that the obtained parameter usually coincides with the better choice related to the best restoration quality of the image.

在目前的工作中，我们研究了同时识别去噪图像和控制进化偏微分方程 (PDE) 的两个扩散算子之间平衡的加权参数的逆问题。该问题被表述为非光滑 PDE 约束优化模型。该 PDE 由二阶和四阶扩散张量构建，结合了均匀区域中 Perona-Malik 扩散模型、锐边附近的 Weickert 模型和减少阶梯的四阶项的优点。在合适的 Sobolev 空间中提供了所提出的 PDE 约束优化系统的解的存在性和唯一性。此外，基于Primal-Dual算法引入了用于确定权重参数的优化问题。最后，