Inhomogenous Regularization with Limited and Indirect Data

For an ill-posed inverse problem, particularly with incomplete and limited measurement data, regularization is an essential tool for stabilizing the inverse problem. Among various forms of regularization, the lp penalty term provides a suite of regularization of various characteristics depending on the value of p. When there are no explicit features to determine p, a spatially varying inhomogeneous p can be incorporated to apply different regularization characteristics that change over the domain. This approach has been investigated and used for denoising problems where the first or the second derivatives of the true signal provide information to design the spatially varying exponent p distribution. This study proposes a strategy to design the exponent distribution when the first and second derivatives of the true signal are not available, such as in the case of indirect and limited measurement data. The proposed method extracts statistical and patch-wise information using multiple reconstructions from a single measurement, which assists in classifying each patch to predefined features with corresponding p values. We validate the robustness and effectiveness of the proposed approach through a suite of numerical tests in 1D and 2D, including a sea ice image recovery from partial Fourier measurement data. Numerical tests show that the exponent distribution is insensitive to the choice of multiple reconstructions.