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：正则化通过约束方法使反问题从不完全测量数据中求解具有适定性。正则化项通常基于对未知信号的先验特征描述（如稀疏性或光滑性）进行设计。标准非齐次正则化通过引入随空间变化的指数$p$扩展了基于$\ell_p$范数的标准正则化方法，以恢复空间特征变化的信号。本研究提出一种加权非齐次正则化方法，通过新型指数设计和空间变权重加权扩展了标准非齐次正则化。新型指数设计可避免不同特征区域邻近时产生的误分类问题，而权重机制则能处理另一类问题：即当某一特征区域过小时，即使正确识别该区域，基于$\ell_p$范数的正则化仍难以有效恢复该区域特征。通过合成图像实验及基于不完全波场测量数据的真实海冰恢复数值测试，证实了所提出的加权非齐次正则化方法的有效性。