Graph approximation and generalized Tikhonov regularization for signal deblurring

Given a compact linear operator K, the (pseudo) inverse K^† is usually substituted by a family of regularizing operators Rα which depends on K itself. Naturally, in the actual computation we are forced to approximate the true continuous operator K with a discrete operator K^(n) characterized by a finesses discretization parameter n, and obtaining then a discretized family of regularizing operators R_α ^(n). In general, the numerical scheme applied to discretize K does not preserve, asymptotically, the full spectrum of K. In the context of a generalized Tikhonov-type regularization, we show that a graph-based discretization scheme that guarantees, asymptotically, a zero maximum relative spectral error can significantly improve the approximated solutions given by R_α ^(n). This approach is combined with a graph based regularization technique with respect to the penalty term.