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Regularization promotes well-posedness in solving an inverse problem withincomplete measurement data. The regularization term is typically designedbased on a priori characterization of the unknown signal, such as sparsity orsmoothness. The standard inhomogeneous regularization incorporates a spatiallychanging exponent $p$ of the standard $\ell_p$ norm-based regularization torecover a signal whose characteristic varies spatially. This study proposes aweighted inhomogeneous regularization that extends the standard inhomogeneousregularization through new exponent design and weighting using spatiallyvarying weights. The new exponent design avoids misclassification whendifferent characteristics stay close to each other. The weights handle anotherissue when the region of one characteristic is too small to be recoveredeffectively by the $\ell_p$ norm-based regularization even after identifiedcorrectly. A suite of numerical tests shows the efficacy of the proposedweighted inhomogeneous regularization, including synthetic image experimentsand real sea ice recovery from its incomplete wave measurements.

Weighted inhomogeneous regularization for inverse problems with indirect and incomplete measurement data