Inverse Weighted Sparse Regularization and Its Application in Radon Transform

In the reconstruction problem of compressed sensing, to address the challenge of adapting common sparse constraints to diverse data, we propose a data-driven inverse-weighted regularization for adaptive data matching to enhance the ability of sparse constraints. Specifically, we formulate a weighted regularization term based on the data transform domain, positing that higher values in the sparsity-promoting transform domain correspond to a greater probability of effective signals. Therefore, when solving sparse optimization problems, we inversely weight this portion based on the inverse relationship with the coefficient magnitude, thereby reducing its impact and mitigating damage to effective signals. However, recognizing that noise and other irrelevant signals are sparse and approximately uniformly distributed in the transform domain, we can increase the weight of this portion to boost the sparsity constraint in the transform domain, thereby enhancing noise suppression. Consequently, we presented the corresponding solution algorithm and convergence proof for inverse-weighted sparse regularization, along with an application example in the context of the Radon transform. Experimental data tests indicate that inverse-weighted sparse regularization enhances the capability of sparse constraints, protects effective signals, suppresses noise, and improves the recovery accuracy of compressive sensing algorithms, as demonstrated in natural image enhancement and seismic multiple suppression.