  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.
