Graph approximation and generalized Tikhonov regularization for signal deblurring

Given a compact linear operator $\mathcal{K}$, the (pseudo) inverse ${\mathcal{K}^{\dagger} }$ is usually substituted by a family of regularizing operators ${\mathcal{R}_\alpha }$ which depends on $\mathcal{K}$ itself. Naturally, in the actual computation we are forced to approximate the true continuous operator $\mathcal{K}$ with a discrete operator ${\mathcal{K}^{(n)}}$ characterized by a finesses discretization parameter n, and obtaining then a discretized family of regularizing operators $\mathcal{R}_\alpha ^{(n)}$. In general, the numerical scheme applied to discretize $\mathcal{K}$ does not preserve, asymptotically, the full spectrum of $\mathcal{K}$. 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 $\mathcal{R}_\alpha ^{(n)}$. This approach is combined with a graph based regularization technique with respect to the penalty term.