Directional convergence of spectral regularization method associated to families of closed operators (Q2392973)

The regularized solutions of a linear inverse ill-posed problem are constructed using the generalized Tikhonov functionals with penalizers given by linear combinations of seminorms induced by closed operators. Convergence is proved. The efficiency is demonstrated on the image restoration problem formulated in terms of regularized solutions to the following 2D Fredholm equation using generalized Tikhonov methods with convex combinations of seminorms: \[ {\mathcal{K}} I \equiv \int \int\limits_{\Omega} K(x,y, x^{\prime}, y^{\prime}) I(x^{\prime}, y^{\prime})\, d x^{\prime} \, d y^{\prime} = \tilde{I}(x, y). \] Here, \(\Omega \subset {\mathbb {R}}^2\) is bounded, \(I \in L^2(\Omega) \) is the desired image, and the kernel \(K\) is some point spread function (PSF), e.g., in this paper it is assumed that \[ K(x-x^{\prime}, y-y^{\prime})= \frac{1}{2\pi \omega \tilde{\omega}} \exp \left(-\frac{1}{2\omega^2}(x-x^{\prime})^2 - \frac{1}{2\tilde{\omega}^2}(y-y^{\prime})^2 \right). \] Such an inverse problem of image restoration is well known and is ill-posed in the sense of Hadamard.