Regularization is a critical technique for ensuring well-posedness in solving inverse problems with incomplete measurement data. Traditionally, the regularization term is designed based on prior knowledge of the unknown signal's characteristics, such as sparsity or smoothness. Inhomogeneous regularization, which incorporates a spatially varying exponent $p$ in the standard $\ell_p$-norm-based framework, has been used to recover signals with spatially varying features. This study introduces weighted inhomogeneous regularization, an extension of the standard approach incorporating a novel exponent design and spatially varying weights. The proposed exponent design mitigates misclassification when distinct characteristics are spatially close, while the weights address challenges in recovering regions with small-scale features that are inadequately captured by traditional $\ell_p$-norm regularization. Numerical experiments, including synthetic image reconstruction and the recovery of sea ice data from incomplete wave measurements, demonstrate the effectiveness of the proposed method.

翻译：正则化是确保不完备测量数据反问题求解适定性的关键技术。传统上，正则化项的设计基于对未知信号特性（如稀疏性或平滑性）的先验知识。非齐次正则化通过在标准基于$\ell_p$范数的框架中引入空间变化的指数$p$，已被用于恢复具有空间变化特征的信号。本研究提出加权非齐次正则化方法，该方法是标准框架的扩展，引入了新颖的指数设计与空间变化权重。所提出的指数设计缓解了当不同特征在空间上接近时的误分类问题，而权重则解决了传统$\ell_p$范数正则化在恢复小尺度特征区域时捕获能力不足的挑战。数值实验（包括合成图像重建和从不完备波浪测量数据中恢复海冰数据）验证了所提方法的有效性。