Regularization promotes well-posedness in solving an inverse problem with incomplete measurement data. The regularization term is typically designed based on a priori characterization of the unknown signal, such as sparsity or smoothness. The standard inhomogeneous regularization incorporates a spatially changing exponent $p$ of the standard $\ell_p$ norm-based regularization to recover a signal whose characteristic varies spatially. This study proposes a weighted inhomogeneous regularization that extends the standard inhomogeneous regularization through new exponent design and weighting using spatially varying weights. The new exponent design avoids misclassification when different characteristics stay close to each other. The weights handle another issue when the region of one characteristic is too small to be recovered effectively by the $\ell_p$ norm-based regularization even after identified correctly. A suite of numerical tests shows the efficacy of the proposed weighted inhomogeneous regularization, including synthetic image experiments and real sea ice recovery from its incomplete wave measurements.

翻译：正则化可促进不完整测量数据反问题求解的适定性。正则化项通常基于未知信号的先验特征（如稀疏性或光滑性）设计。标准非齐次正则化通过引入空间变化的指数$p$（基于标准$\ell_p$范数正则化）来恢复具有空间变化特征的信号。本研究提出一种加权非齐次正则化方法，通过新的指数设计和使用空间变化权重的加权策略扩展了标准非齐次正则化。新的指数设计避免了在不同特征空间位置接近时出现的误分类问题。权重机制则解决了另一个问题：即使正确识别出某一特征区域，当该区域过小时，基于$\ell_p$范数的正则化难以有效恢复其信号。通过一系列数值实验验证了所提加权非齐次正则化的有效性，包括合成图像实验以及从不完整波浪测量数据中恢复真实海冰的实例。