Linear inverse problems are typically ill-posed in the sense of Hadamard and they require regularization strategies in order to compute meaningful approximation of the desired solution.
Traditional regularization methods solve an optimization problem whose objective function consists of a data-fit term, which measures how well an image matches the observations, and one or more penalty terms, referred to as regularization terms, which promote some desirable properties of the sought-for solution. The quality of the computed solution strongly depends on the choice of the regularization parameters weighting the penalty terms.
An advantage of multi-penalty regularization, when compared with one-penalty regularization, is that different features of the solution can be enhanced by using several regularization terms. However, a drawback of multi-penalty regularization is that one has to select the values of several regularization parameters.
In this talk, we present spatially adapted multi-penalty regularization for solving linear inverse problems to promote simultaneously distinct features of the sought-for objects. In the presented approach, the function to be minimized contains one $\ell_1$ penalty term and several spatially adapted $\ell_2$ penalty terms. An iterative procedure is illustrated for the automatic computation of all the regularization parameters. As a case study, we present the application of adaptive multi-penalty regularization to the inversion of two-dimensional NMR data.